Phd Thesis Ana Girao

CHARACTERIZATION OF THE DUCTILITY OF

BOLTED END PLATE BEAM-TO-COLUMN STEEL CONNECTIONS

Ana Margarida Girão Coelho

Thesis presented in fulfilment of the requirements for the de-gree of Doctor of Philosophy in Civil Engineering under the scientific advising of Prof. Dr. Luís Simões da Silva and Prof. Ir. Frans S. K. Bijlaard.

Tese apresentada para obtenção do grau de doutor em Engenharia Civil sob orientação científica do Prof. Dr. Luís Simões da Silva e do Prof. Frans S. K. Bijlaard.

Universidade de Coimbra July 2004

O trabalho apresentado nesta tese de doutoramento foi financiado pelo Ministério da Ciência e Ensino Superior, ao abrigo do programa PRODEP (Concurso Público 4/5.3/PRODEP/2000) e com apoio da Fundação para a Ciência e Tecnologia (Bolsa de Doutoramento SFRH/BD/5125/2001). Coimbra, 2004.

To Miguel, my son and Encarnação and Hermínio, my parents.

ACKNOWLEDGEMENTS

The author would like to express her sincere gratitude to Prof. Dr. Luís A. P. Simões da Silva (University of Coimbra) and Prof. Ir. Frans S. K. Bijlaard (Delft University of Technology). Prof. Simões da Silva and Prof. Bijlaard are a model for their technical expertise, professionalism, scientific knowledge and ethics. Financial support from the Portuguese Ministry of Science and Higher Educa-tion (Ministério da Ciência e Ensino Superior) under contract grants from PRODEP (Concurso Público 4/5.3/PRODEP/2000) and Fundação para a Ciência e Tecnologia (Grant SFRH/BD/5125/2001) is gratefully acknowledged. The assistance provided by Mr. Nol Gresnigt, Mr. Henk Kolstein and Mr. Edwin Scharp from the Department of Steel and Timber Structures of the Delft University of Technology is most appreciated. To Corrie van der Wouden and Jan Willem van de Kuilen, thank you for your friendship. This research project was also made possible by the assistance of several people at the Department of Civil Engineering of the Faculty of Science and Technology of the University of Coim-bra. Thank you Aldina Santiago, Luciano Lima, Luís Borges, Luís Neves, Pedro Simão, Rui Simões and Sandra Jordão. The friendship and support of my sister Rita and all my friends is also very much appreciated. Thank you all. To Carina, a special word of appreciation for the works with the cover of this thesis. To Cláudio, thank you for your patience, love and understanding.

TABLE OF CONTENTS

ABSTRACT

RESUMO (Portuguese abstract)

NOTATION

PART I STATE-OF-THE-ART AND LITERATURE REVIEW

1

1 MODELLING OF THE MOMENT-ROTATION CHARACTERISTICS OF BOLTED JOINTS: BACKGROUND REVIEW

3

1.1 General introduction 3 1.1.1 Literature review 4 1.1.2 Scope of the work, objectives and research approach 7 1.1.3 Outline of the dissertation 9 1.2 Definitions 10 1.3 Methods for modelling the rotational behaviour of beam-to-

column joints 12

1.3.1 Generality 12 1.3.2 The component method 12 1.4 Characterization of basic components of bolted joints in terms

of plastic resistance and initial stiffness 14

1.4.1 T-stub model for characterization of the tension zone of bolted joints

15

1.4.1.1 Plastic resistance of single T-stub connec-tions

15

1.4.1.2 Initial stiffness of single T-stub connec-tions

19

1.4.2 Characterization of the several joint components 24 1.5 Characterization of the post-limit behaviour of basic compo-

nents of bolted joints 29

1.5.1 Column web in shear (component with high duc-tility)

30

1.5.2 Column flange in bending, end plate in bending and bolts in tension (T-stub idealization)

32

1.5.3 Column web in compression (component with limited ductility)

32

1.5.4 Column web in tension (component with limited ductility)

34

1.6 Evaluation of the moment-rotation response of bolted joints by means of component models

34

1.6.1 Eurocode 3 component model 37 1.6.1.1 Model for stiffness evaluation 37 1.6.1.2 Model for resistance evaluation 38 1.6.1.3 Idealization of the moment-rotation curve 39 1.6.2 Guidelines for evaluation of the ductility of bolted

joints 39

1.7 References 44 Appendix A: Design provisions for characterization of resistance

and stiffness of T-stubs 50

A.1 Basic formulations for prediction of plastic resistance of bolted T-stubs

50

A.1.1 Type-1 mechanism 50 A.1.2 Type-2 mechanism 50 A.1.3 Type-3 mechanism 51 A.1.4 Supplementary mechanism 51 A.2 Influence of the moment-shear interaction on resistance formu-

lations 51

A.2.1 Type-1 mechanism 52 A.2.2 Type-2 mechanism 53 A.3 Influence of the bolt dimensions on resistance formulations 54 A.4 Formulations for prediction of elastic stiffness of bolted T-

stubs 56

A.4.1 Elastic theory for the evaluation of the elastic stiff-ness of a bolted T-stub

56

A.4.2 Simplification of the stiffness coefficients for in-clusion in design codes

57

PART II FURTHER DEVELOPMENTS ON THE T-STUB MODEL

59

2 IMPROVEMENTS ON THE T-STUB MODEL: INTRODUCTION 61 2.1 Introduction 61 2.2 Failure modes 62 2.3 References 65 3 EXPERIMENTAL ASSESSMENT OF THE BEHAVIOUR OF T-STUB CON-

NECTIONS 67

3.1 Introduction 67 3.2 Description of the experimental programme 67 3.2.1 Geometrical properties of the specimens 67 3.2.2 Mechanical properties of the specimens 69

3.2.2.1 Tension tests on the bolts 69 3.2.2.2 Tension tests on the structural steel 73 3.2.3 Testing procedure 75 3.2.4 Aspects related to the welding procedure 78 3.3 Experimental results 82 3.3.1 Reference test series WT1 82 3.3.2 Failure modes and general characteristics of the

overall behaviour of the test specimens 87

3.4 Concluding remarks 90 3.5 References 92 4 NUMERICAL ASSESSMENT OF THE BEHAVIOUR OF T-STUB CONNEC-

TIONS 93

4.1 Introduction 93 4.2 Previous research 94 4.3 Description of the model 96 4.4 Calibration of the finite element model 99 4.4.1 Geometry 100 4.4.2 Boundary and load conditions 101 4.4.3 Mechanical properties of steel components 102 4.4.4 Specimen discretization 102 4.4.5 Contact analysis 104 4.5 Failure criteria 104 4.6 Numerical results for HR T-stub T1 106 4.7 Numerical results for WP T-stub WT1 110 4.8 Considerations on the numerical modelling of the heat af-

fected zone in WP T-stubs 113

4.9 Concluding remarks 115 4.10 References 116 Appendix B: Preliminary study for calibration of the finite element

model (e.g. HR-T-stub T1) 119

B.1 Mesh convergence study 119 B.2 Influence of the definition of the constitutive law and ele-

ment formulation on the overall behaviour 121

B.3 Calibration of the joint element stiffness 121 Appendix C: Stress and strain numerical results for HR-T-stub T1 123 C.1 Load steps for stress and strain contours 123 C.2 Von Mises equivalent stresses, σeq 123 C.3 Stresses σxx and strains εxx 124 C.4 Stresses σyy 126 C.5 Stresses σzz 128 C.6 Principal stresses and strains, σ11 and ε11 128 C.7 Displacement results in xy cross-section 132

5 PARAMETRIC STUDY 135 5.1 Description of the specimens 135 5.2 Influence of the assembly type and the weld throat thickness 135 5.3 Influence of geometric parameters 147 5.3.1 Gauge of the bolts 149 5.3.2 Pitch of the bolts and end distance 149 5.3.3 Edge distance and flange thickness 151 5.4 Influence of the bolt and flange steel grade 158 5.5 Experimental results for the stiffened test specimens and the

rotated configurations 169

5.5.1 Influence of a transverse stiffener 169 5.5.2 Influence of the T-stub orientation 174 5.6 Summary of the parametric study and concluding remarks 175 5.7 References 178 6 SIMPLIFIED METHODOLOGIES FOR ASSESSMENT OF THE BEHAVIOUR

OF SINGLE T-STUB CONNECTIONS 179

6.1 Introduction 179 6.2 Previous research 179 6.2.1 Jaspart proposal (1991) 180 6.2.2 Faella and co-workers model (2000) 181 6.2.3 Swanson model (1999) 182 6.2.4 Beg and co-workers proposals for evaluation of

the deformation capacity (2002) 185

6.2.5 Examples 186 6.2.5.1 Evaluation of initial stiffness 186 6.2.5.2 Evaluation of plastic resistance 187 6.2.5.3 Piecewise multilinear approximation of the

overall response and evaluation of the deformation capacity and ultimate resis-tance

187

6.2.5.4 Summary 193 6.3 Proposal and validation of a beam model for characteriza-

tion of the force-deformation response of T-stubs 194

6.3.1 Description of the model 194 6.3.1.1 Fracture conditions 196 6.3.1.2 Bolt deformation behaviour 196 6.3.1.3 Flange constitutive law 197 6.3.2 Analysis of the model in the elastic range 199 6.3.3 Analysis of the model in the elastoplastic range 204 6.3.4 Sophistication of the proposed method: modelling

of the bolt action as a distributed load 214

6.3.5 Influence of the distance m for the WP T-stubs 215 6.4 Concluding remarks 216 6.5 References 218

Appendix D: Detailed results obtained from application of the sim-plified methods for assessment of the force-deformation response of single T-stub connections

219

D.1 Geometrical and mechanical characteristics of the specimens 219 D.2 Previous research: exemplification 219 D.2.1 Evaluation of initial stiffness 219 D.2.2 Piecewise multilinear approximation of the overall

response and evaluation of the deformation capac-ity and ultimate resistance

219

D.3 Application of the proposed model: results for HR-T-stub T1 235 D.4 Application of the proposed model: results for WP-T-stub

WT1 239

D.5 Prediction of the nonlinear response of the above connections using the nominal stress-strain characteristics

241

D.6 Comparative graphs: simple beam model and sophisticated beam model accounting for the bolt action

255

D.7 Comparative graphs: influence of the distance m for the WP-T-stubs

264

PART III MONOTONIC BEHAVIOUR OF BEAM-TO-COLUMN BOLTED END PLATE CONNECTIONS

273

7 EXPERIMENTAL TESTS ON BOLTED END PLATE CONNECTIONS 275 7.1 Introduction 275 7.2 Description of the test programme 275 7.2.1 Test details 275 7.2.2 Geometrical properties 277 7.2.3 Mechanical properties 277 7.2.3.1 Tension tests on the bolts 277 7.2.3.2 Tension tests of the structural steel 278 7.2.4 Test arrangement and instrumentation 280 7.2.5 Testing procedure 284 7.3 Test results 284 7.3.1 Moment-rotation curves 288 7.3.2 Behaviour of the tension zone 294 7.3.2.1 End plate deformation behaviour 294 7.3.2.2 Yield line patterns 299 7.3.2.3 Bolt elongation behaviour 299 7.3.2.4 Strain behaviour 300 7.4 Discussion of test results 302 7.4.1 Plastic flexural resistance 303 7.4.2 Initial rotational stiffness 304 7.4.3 Rotation capacity 304 7.5 Concluding remarks 305

7.6 References 306 8 DUCTILITY OF BOLTED END PLATE CONNECTIONS 307 8.1 Introduction 307 8.2 Modelling of bolt row behaviour through equivalent T-stubs 310 8.3 Application to bolted extended end plate connections 310 8.3.1 Component characterization 310 8.3.2 Evaluation of the nonlinear moment-rotation re-

sponse 318

8.3.3 Evaluation of the rotation capacity according to other authors

326

8.3.4 Characterization of the joint ductility 328 8.4 Discussion 330 8.5 References 332 9 CONCLUSIONS AND RECOMMENDATIONS 333 9.1 Conclusions 333 9.2 Future research 336 9.3 References 338 LIST OF REFERENCES 339

ABSTRACT

The analysis of steel-framed building structures with full strength beam-to-column joints is quite standard nowadays. Buildings utilizing such framing systems are widely used in design practice. However, there is growing recogni-tion of significant benefits in designing joints as partial strength, semi-rigid. The design of joints within this partial strength/semi-rigid approach is becom-ing more and more popular. It requires however the knowledge of the full nonlinear moment-rotation behaviour of the joint, which is also a design pa-rameter. Additionally, the joint failure must be ductile, i.e. the joint must have sufficient rotation capacity as the first plastic hinges occur in the joints rather than in the connected members. The research work reported in this thesis deals with this issue and gives particular attention to the characterization of the joint ductility, which is particularly important in the partial strength/semi-rigid joint scenario. The experimental and numerical results of sixty one individual T-stub tests and eight full-scale bolted end plate connection tests are presented and as-sessed based on their resistance, stiffness and ductility characteristics. The re-sults are used to compare existing resistance and stiffness models and to de-velop a simple methodology for evaluation of ductility properties. The T-stub model has been used for many years to model the tension zone of bolted joints. Previous research was mainly concentrated on rolled profiles as T-stub elements. In the case of end plate connections, the T-stub on the end plate side comprises welded plates as T-stub elements. This research also pro-vides insight into the behaviour of this different type of assembly, in terms of resistance, stiffness, deformation capacity and failure modes, in particular. It also explores the main features of the individual T-stub as a standalone con-figuration and evaluates quantitatively and qualitatively the influence of the main geometrical and mechanical parameters on the overall behaviour. A simplified two-dimensional beam model for the assessment of the de-formation response of individual T-stubs is developed based on the experimen-tal observations and the results of the finite element investigation. The model is based on the Eurocode 3 approach and includes the deformations from tension bolt elongation and bending of the T-stub flange. It is able to predict the de-formation capacity of a T-stub with a satisfactory degree of accuracy. This study on individual T-stubs is part of the investigation of end plate behaviour. The outcomes are used to validate a methodology based on the so-called component model to determine the rotational behaviour of bolted end plate connections. Since most of the joint rotation in thin end plates comes

from the end plate deformation, the characterization of bolt row behaviour through equivalent T-stubs is of the utmost importance. A spring model that includes the T-stub idealization of the tension zone is used to derive the nonlinear moment-rotation of the joint. Special attention is given to the charac-terization of the joint ductility. Comparisons of the joint ductility and the cor-responding equivalent T-stub for the end plate side are drawn. Finally, some recommendations for the required ductility expressed in terms of a ductility index are given.

RESUMO

O projecto de estruturas metálicas para edifícios porticados com ligações viga-pilar de resistência total é relativamente comum. No entanto, tem-se vindo a reconhecer os benefícios que decorrem da modelação semi-rígida e de resistência parcial das ligações. Esta abordagem tem-se generalizado no dimensionamento das ligações metálicas. Para o efeito, é necessário avaliar o comportamento momento-rotação real das ligações. Adicionalmente, a rotura das ligações tem de ser dúctil, isto é, as ligações têm de exibir capacidade de rotação suficiente, uma vez que as primeiras rótulas plásticas se formam no nó de ligação e não nos elementos (viga ou pilar). O trabalho de investigação apresentado nesta tese foca este aspecto e dá ênfase à caracterização da ductilidade das ligações, que é particularmente relevante na modelação semi-rígida/resistência parcial. Descrevem-se e discutem-se os resultados experimentais e numéricos de sessenta e um testes em ligações em duplo T (T-stubs) individuais e oito ligações viga-pilar aparafusadas com placa de extremidade. A análise destes resultados inclui a caracterização das propriedades de resistência, rigidez e ductilidade das ligações e a sua confrontação com modelos correntes de ava-liação de resistência e rigidez. Em termos de ductilidade, é proposta uma me-todologia simplificada para a caracterização desta propriedade das ligações. O modelo do T-stub é utilizado na idealização da zona traccionada de ligações aparafusadas. Os trabalhos de investigação anteriores centraram a análise desta ligação mais simples em elementos que utilizam perfis lamina-dos a quente. No caso de ligações com placa de extremidade, os T-stubs equi-valentes na zona da placa englobam elementos soldados. Neste trabalho procura-se descrever o comportamento deste tipo de T-stub, focando os mo-dos de rotura, a resistência, a rigidez e a ductilidade, em particular. Explo-ram-se também as principais características do T-stub isolado e avalia-se qualitativa e quantitativamente a influência dos principais parâmetros geo-métricos e mecânicos no comportamento global. Com base nos resultados experimentais e numéricos (elementos finitos) propõe-se um modelo de viga bidimensional simplificado para caracterização do comportamento força-deformação de T-stubs. O modelo baseia-se na abordagem do Eurocódigo 3 e inclui a deformação do parafuso traccionado e do banzo do T-stub em flexão e permite prever a capacidade de deformação com um grau de precisão satisfatório. Este estudo em T-stubs isolados constitui uma parte do trabalho de inves-tigação do comportamento da placa de extremidade. As conclusões deste es-

tudo são utilizadas na validação de uma metodologia baseada no método das componentes para avaliação do comportamento rotacional de ligações apa-rafusadas com placa de extremidade. Uma vez que a rotação da ligação pro-vém essencialmente da deformação da placa de extremidade, no caso de pla-cas finas, a idealização do seu comportamento por meio de T-stubs equiva-lentes é particularmente relevante. Um modelo mecânico de molas e bielas rí-gidas que inclui a idealização da zona traccionada por intermédio de T-stubs é utilizado para a caracterização da resposta momento-rotação da ligação, com particular ênfase na avaliação da sua ductilidade. Estabelecem-se com-parações entre a ductilidade da ligação e os correspondentes T-stubs equiva-lentes na zona da placa de extremidade. Finalmente, são propostas algumas recomendações para a ductilidade mínima da ligação, expressa em termos de um índice de ductilidade.

NOTATION

Lower cases a’ Effective edge distance according to the Kulak’s prying model aep Throat thickness of a fillet weld at the end plate side aw Throat thickness of a fillet weld a Total displacements b Width; actual width of a T-stub tributary to a bolt row b’ Distance between the inside edge of the bolt shank to 50% dis-

tance into profile root beff Effective width; effective width of a T-stub tributary to a bolt row

for resistance calculations effb′ Effective width of a T-stub tributary to a bolt-row for stiffness

calculations d Length between the bolt axis and the face of the T-stub web dc Clear depth of the column web dw Bolt head, nut or washer diameter, as appropriate d0 Bolt hole clearance e Edge distance ecomp End plate distance e1 End distance (from the centre of the bolt hole to the adjacent edge) fu Ultimate or tensile stress fy Yield stress h Depth hmrn Height of the resultant tension force above the neutral axis at

maximum strain hr Distance of bolt row r to the centre of compression hyfn Height of the flush bolt row above the neutral axis at yield k Empirical factor ke Initial axial stiffness of a spring-component keff.r Effective stiffness coefficient for bolt row r ki (i=1→4)

Auxiliary length values for definition of the bolt conventional length, according to Aggerskov

ki (i=1→3)

Joint element stiffness modulus (1: normal direction; 2,3: tangen-tial direction)

kp-l Post-limit axial stiffness of a spring-component lHAZ Width of the heat affected zone m Distance from bolt centre to 20% distance into profile root or weld mf Average distance from each bolt to the adjacent web and flange

welds below the tension flange

mpl Plastic moment of a plate per unit length n Effective edge distance; number of bolt rows in tension; ratio be-

tween the axial force in the column and the corresponding plastic level

nth Number of threads per unit length of the bolt p Pitch of the bolts pi-j Distance between bolt rows i and j q Parameter qb Uniformly distributed bolt action, statically equivalent to B qee Initial prying gradient qij,k Prying gradient r Fillet radius of the flange-to-web connection s Length sp Length obtained by dispersion at 45º through the end plate sx Ratio transverse stress/yield stress in the column web t Thickness u Degree-of-freedom v Degree-of-freedom w Horizontal distance between bolt axis centrelines (gauge); degree-

of-freedom x Cartesian axis; distance xi Distance of the joint row to the tip of the flanges y Cartesian axis z Lever arm; cartesian axis zi Distance between the ith bolt row to the centre of compression z1 Distance in [mm] between the first bolt row from the tension

flange and the centre of compression Upper cases Ab Nominal area of the bolt shank As Bolt tensile stress area Avc Shear area of a column profile B Bolt force E Young modulus Eh Strain hardening modulus Eu Modulus of the stress-strain curve before collapse F Force; resistance; load; applied load per bolt row in a T-stub FQi Contact force associated to a joint row FRd Full “plastic” (design) resistance Fti Potential resistance of bolt row i in the tension zone Fu Ultimate resistance Fv Vertical forces

1. .0RdF Ratio between the design resistance of mechanism type-1 accounting for shear and that corresponding to the basic formulation

G Tangential modulus of elasticity Hc.low Height of the column below the end plate

Hc.up Height of the column above the end plate I Moment of inertia K Spring axial stiffness (generic) Kb Bolt elastic stiffness according to the Swanson’s bolt model Kcws.h Residual stiffness (Krawinkler et al. model for characterization of

the behaviour of the “column web in shear”) K(flex) Stiffness for the flexible beam approach K(rig) Stiffness for the rigid beam approach L Length; cantilever length Lb Bolt conventional length

*bL Clamping length of the bolts

Lbeam Length of the beam Lcomp Length of the end plate below the compression beam flange Lg Grip length Linfluence.i Influence length of a joint row Lload Distance between the load application point and the face of the

end plate Ls Bolt shank length Ltg Bolt threaded length included in the grip length M Bending moment Mj.Ed Bending moment (lower than Mj.Rd) acting in the joint Mj.Rd Joint flexural plastic (design) resistance N Axial force P Concentrated force Q Prying force R Norm of external forces Sj.ini Initial rotational stiffness of a joint S0 Bolt preload V Shear force Zf Parameter Greek letters α Coefficient obtained from an abacus provided in Eurocode 3;

parameter that represents a ductility limit αf Parameter β Transformation parameter; ratio flexural resistance of flanges /axial

resistance of the bolts; parameter that represents a ductility limit βa; βb Coefficients that account for the shear deformations βu.lim Limit value for the β-ratio to have a collapse failure mode gov-

erned by cracking of the flange material χ Curvature δ Relative displacement; elongation δ(flex) Displacement of a flexible beam at mid-span δ(rig) Displacement of a rigid beam at mid-span δu Deformation capacity of half T-stub

δ Non-dimensional displacement aδ Norm of the iterative displacements

∆ Axial deformation; elongation a∆ Total displacements for a certain increment

RdF∆ Deformation corresponding to the component plastic resistance

∆u Deformation capacity ε Strain; engineering strain; parameter εe Elastic deformation (strain) εh Strain at the strain hardening point εhs Engineering strain at which the maximum engineering stress is

reached εn Natural or logarithmic strain εp Plastic deformation (strain) εu Ultimate strain εuni Uniform strain ε0 Ultimate transverse strain acting in the column web in the case

that the axial force in the column is absent φ Connection rotational deformation; bolt diameter Φ Joint rotation φCd Rotation capacity of a connection ΦCd Rotation capacity of a joint

*CdΦ Joint rotation at which the moment deteriorates back to Mj.Rd after

reaching a moment above Mj.Rd through deformation beyond ΦXd maxMφ Rotation of the connection at maximum load

maxMΦ Rotation of the joint at maximum load

φXd Connection rotation value at which the moment resistance first reaches Mj.Rd

ΦXd Joint rotation value at which the moment resistance first reaches Mj.Rd

γ Shear deformation of the column web panel γd Euclidean displacement norm γdt Euclidean iterative displacement norm γi (i=1→3)

Coefficients

γM Partial safety coefficients (γM0, γM1, γM2) γw Work norm γψ Euclidean residual norm η Stiffness modification factor ϕi Component ductility index ϑj Joint ductility index λ Ratio between n and m

pλ Plate slenderness

κN Parameter that reflects the influence of the level of axial force in the column

κwc Reduction factor to account for the effect of axial force in the col-

umn µ Friction coefficient; ratio between characteristic strain values;

stiffness ratio θ Rotation ρ Reduction factor for plate buckling ρy Yield ratio

*yρ Alternative definition of the yield ratio

σ Stress; nominal or conventional stress σeq Von Mises equivalent stress σn True stress σx Transverse stress acting in the column web τ Shear stress Γ Parameter τy Yield shear stress υ Poisson’s ratio ω Reduction factor to allow for possible effects of interaction with

shear in the column web panel (ω 1, ω2: parameters for computation of ω)

ξ Coefficient ψ Norm of residuals ψi Component ductility index ζ Coefficient taken as 0.8 in Eurocode 3 Subscripts av average b Beam; bolt bfc Beam web and flange in compression bot Bottom T-stub bt Bolts in tension bwt Beam web in tension c Column; compression cfb Column flange in bending cp Circular yield line patterns cwc Column web in compression cws Column web in shear cwt Column web in tension e/el Elastic ep End plate epb End plate in bending f Flange fract Fracture h Bolt head; strain hardening j Joint l Lower T-stub element m Strain hardening range and before collapse max Maximum

min Minimum n Bolt nut nc Non-circular yield line patterns p/pl Plastic p-l Post-limit red Reduced ri Bolt row i Rd Pure plastic conditions; design conditions s Stiffener t Tension top Top T-stub T T-stub element u Ultimate conditions; upper T-stub element w Web; weld wp Web panel wsh Washer X Extension of the end plate above the tension beam flange y Yield 0 T-stub component 1 Type-1 plastic failure mechanism of a T-stub; bolt row 1 11 Principal direction 1 for a stress state 2 Type-2 “plastic” failure mechanism of a T-stub; bolt row 2 3 Type-3 “plastic” failure mechanism of a T-stub; bolt row 3 * Supplementary plastic failure mechanism of a T-stub Abbreviations B Back (from eye position) BF Basic formulation of resistance BM Base metal DTi Reference to a LVDT i F Front (from eye position) FBA Resistance formulation accounting for the bolt action FE Finite element FT Full-threaded bolt HAZ Heat affected zone HR Hot-rolled profile L Left (from eye position) LVDT Linear variable displacement transducer K-R Knee-range of a deformability curve R Right (from eye position) SG Strain gauge ST Short-threaded bolt WM Weld metal WP Welded plates HP Reference to a specific LVDT

1

PART I: STATE-OF-THE-ART AND LITERATURE REVIEW

3

1 MODELLING OF THE MOMENT-ROTATION CHARACTER-

ISTICS OF BOLTED JOINTS: BACKGROUND REVIEW 1.1 GENERAL INTRODUCTION Structural joints, particularly bolted and welded connections found in common steel constructions, exhibit a distinctively nonlinear behaviour. This nonlinear-ity arises because a joint is an assemblage of several components that interact differently at distinct levels of applied loads. The interaction between the ele-mental parts includes elastoplastic deformations, contact, slip and separation phenomena. The analysis of this complex behaviour is usually approximate in nature with drastic simplifications. Tests (both experimental and numerical) are frequently carried out to obtain the actual response, which is then modelled ap-proximately by mathematical expressions that relate the main structural joint properties. Beam-to-column joints in steel-framed building structures have to transfer the beam and floor loads to the columns. Generally, the forces transmitted through the joints can be axial and shear forces, bending and torsion moments. The bending deformations are predominant in most cases, when compared to axial and shear deformations that are hence neglected. The effect of torsion is also negligible in planar frames. Typical beam-to-column moment-resisting joints in steel-framed structures include bolted end plate connections, bolted connections with (flange and/or web) angle cleats and welded connections. Their behaviour is represented by a moment vs. rotation curve (M-Φ) that de-scribes the relationship between the applied bending moment, M and the corre-sponding rotation between the members, Φ. This curve defines three main structural properties: (i) moment resistance, (ii) rotational stiffness and (iii) ro-tation capacity. Historically, moment-resisting joints have been designed for strength and stiffness with little regard to rotational capacity. There is growing recognition that in many situations this practice is questionable and so guid-ance is urged to help designers. Joints can be grouped according to their structural properties. The European code of practice for the design of structural steel joints in buildings, Eurocode 3 [1.1], classifies joints by strength (full strength, partial strength or nominally pinned) and stiffness (rigid, semi-rigid or nominally pinned). A full strength joint exhibits a moment resistance at least equal to that of the connected mem-bers whilst partial strength joints have lower strength than the members. Nomi-nally pinned joints are sufficiently flexible to be regarded as a pin for analysis purposes, i.e. they are not moment resisting and have no rotational stiffness. A rigid joint is stiff enough for the effect of its deformation on the distribution of

State-of-the-art and literature review

4

internal forces and bending moments in the structure to be neglected. A semi-rigid joint does not meet the criteria for a rigid joint or a pin. Naturally, nomi-nally pinned joints have to be ductile, i.e. they have to rotate plastically at some stage of the loading cycle without failure. The semi-rigid/partial strength de-sign philosophy of joints usually leads to more economic and simple solutions. The use of this joint category in steel frames, however, is only feasible if they develop sufficient rotation capacity in order that the intended failure mecha-nism of the whole structure can be formed prior to fracture of the joint. End plate bolted connections that are widely used in steel-frames as mo-ment-resistant connections between steel members usually fall in the semi-rigid/partial strength category. The simplicity and economy associated to their fabrication and erection made this joint typology quite popular in steel-framed structures. In Europe, steel bolted partial strength extended end plate connec-tions are typical for low-rise buildings erected using welding at the shop and bolting on site. Rules for prediction of strength and stiffness of this joint con-figuration have been included in modern design codes as the Eurocode 3. Yet, no quantitative guidance for characterization of the ductility is available. The main topics of this research work are moment-resisting bolted (major axis) connections joining I-sections in steel-framed structures and the charac-terization of their rotational behaviour. Special emphasis is given on extended end plate connections similar to that shown in Fig. 1.1. The main source of de-formability of this connection type is often the tension zone that can be mod-elled with the T-stub approach [1.1-1.5]. The evaluation of the deformation be-haviour of single T-stubs is therefore very important and is also focused on in this work.

(a) Three-dimensional view. (b) Section. (c) Elevation. Fig. 1.1 Unstiffened bolted end plate connection. 1.1.1 Literature review Bolted end plate beam-to-column steel connections have been widely studied over the years. The emphasis in most of the previous research on this subject

Modelling of the M-Φ characteristics of bolted joints: background review

5

was mainly placed on full strength end plate connections and therefore only the resistance and stiffness properties were fully characterized. Thoroughly con-ducted experimental tests were carried out for prediction of the M-Φ curve. However, the information extracted from those experiments was limited to the joint typology that had been tested and could not be extrapolated to other joint configurations. Analytical methodologies based on finite element (FE) analyses can be regarded as an alternative tool for investigation and understanding of joint behaviour, provided that the requirements for a reliable simulation are to-tally fulfilled. Many researchers used both approaches in conjunction. Douty and McGuire [1.6] conducted monotonic experimental tests on end plate connections to study their performance, design and use in plastically de-signed structures. They identified the effect of prying action in increasing the tension bolt force and recognized the importance of material strain hardening. The effect of prying action was further investigated by Aggerskov [1.4,1.7] who carried out a series of fifteen tests on extended end plates. Zoetemeijer [1.2,1.8-1.10] reported on detailed series of tests performed at the Delft University of Technology to propose and validate yield line models for the strength design of the tension region. This zone of end plate connections includes the following basic elemental parts: column flange, end plate and the bolts in tension. Zoetemeijer also proposed some criteria and simple empirical expressions for the estimation of a joint deformation capacity based on a series of experiments described in [1.10]. Packer and Morris [1.3] and Mann and Morris [1.11] focused on this subject too. Similar to Zoetemeijer, they also ide-alized the tension region as a T-stub. Fig. 1.2 identifies the T-stub which ac-counts for the deformation of the column flange and the end plate in bending in the particular case of an extended end plate bolted connection. In this particular case, since the column flange is unstiffened, the T-stub on the column side is orientated at right angles to the end plate T-stub [1.5]. Different investigators also carried out various studies focusing on mechanisms in T-stubs rather than whole plates, particularly to assess the resistance properties of this simple con-nection [1.5,1.12-1.15]. Jenkins et al. [1.16] contributed to a better understanding of end plate be-haviour and proposed standardized end plate connection types to permit a gen-eralization of joint characteristics obtained from numerical modelling. They performed FE analysis to determine the complete M-Φ curve of some joints that was compared with experimental results. This experimental programme included eighteen tests. The principal objective of the programme was to obtain M-Φ relationships but they also directed attention at other features as the evaluation of the axial forces in the bolts. The characterization of the initial rotational stiffness of beam-to-column joints was the main research topic of Davison et al. [1.17] who did various tests on end plate connections with different thickness and identical beam and col-umn sizes. The researchers also investigated the effect of lack of fit [1.18] and concluded that it was negligible.


6

EquivalentT-stub

M

T-stub T-stub

(a) Unstiffened extended end plate connection: T-stub identification and orien-tation.

External load

External load

External load

External load

Stiffener

Weld toe

(b) Model for the column flange side. (c) Model for the end plate side. Fig. 1.2 T-stub identification and representation. Janss and co-workers [1.19] completed a series of tests that were later used by Jaspart [1.20] to propose a methodology for evaluation of plastic resistance and initial rotational stiffness of moment joints. Aggarwal [1.21] and Bose et al. [1.22] carried out comparative tests on end plate connections for which they characterized the moment carrying behaviour. In particular, Bose et al. [1.22] described the observed failure modes that in-volved end plate failure, bolt fracture, bolt stripping, weld fracture and column web buckling. They used these test results to validate finite element models for the analysis of this joint type. In their tests, most of the specimens were full strength joints but they also tested partial strength joints. More recently, Adegoke and Kemp [1.23] reported on three tests on thin end plate partial strength joints that use a similar column/beam set and different plate thickness. These tests provide insight into the joint resistance and ductil-ity properties. The observed failure modes included failure of the end plate and bolt, development of cracks in the end plate along the weld to the beam web in the tension zone that led to fracture of the end plate in the thinner plates [1.23]. Fracture of the bolt in tension below the tension flange determined collapse for


7

the thicker plate [1.23]. The test results were compared with a bilinear M-Φ re-lationship proposed by the authors. They also identified the influence of the plate thickness on the membrane effect and material strain hardening. Amongst those researchers focusing exclusively on the end plate behaviour, Zandonini and Zanon [1.24] performed five static tests on extended end plate connections with four bolts in tension and different plate thickness. In order to isolate the end plate behaviour, the specimens were connected to a counter beam with minor deformability. The moment capacity of the connection in all tests was greater than the plastic moment of the beam. Bursi [1.25] used these test results to evaluate the plastic failure moment capacity of the tested connec-tions by means of numerical modelling. He compared the yield line paths and failure mechanism models defined numerically with experimental evidence and found a good agreement between both. The numerical simulation of bolted connections also represents a significant part of the research work devoted to end plate behaviour. Krishnamurthy and co-workers [1.26-1.28] carried out a comprehensive research programme to in-vestigate the rotational response of this joint type by means of FE analyses. The objective of their research was the development of rotational design crite-ria applicable to end plate connections. They performed three-dimensional FE analyses on bolted connections and correlated the results to previous two-dimensional analyses to enable the prediction of the more accurate three-dimensional values from the less expensive two-dimensional results. Having validated the computer analyses, they proposed equations to predict the general rotational behaviour. However, they overlooked some important phenomena as the flexibility of the column flange, bolt head and nut, or plasticity of the mate-rial. Kukreti and collaborators [1.29-1.31] focused on a similar topic. They de-veloped an analytical methodology based on FE results to characterize the M-Φ behaviour of this joint type. Experimental tests were also carried out to verify the methodology. Bahaari and Sherbourne [1.32-1.36] did a series of FE analy-ses to propose analytical expressions for the design of end plate connections. In their models they considered all major influences on the overall response, in-cluding column, beam, bolt components, material plasticity, strain hardening and contact phenomena. Bursi and Jaspart [1.37-1.38] gave some recommenda-tions on FE modelling of end plate behaviour. Choi and Chung [1.39] devel-oped a refined three-dimensional FE model for the detailed investigation of the behaviour of end plate connections. Their model accounted for different types of nonlinearities, such as elastoplasticity and contact. They made a thorough description of the contact regions of the joints with increasing loading. 1.1.2 Scope of the work, objectives and research approach The research work reported herein focuses on the characterization of the rota-tional behaviour of bolted beam-to-column joints with an extended end plate,


8

similar to that shown in Fig. 1.1. In this joint type, the main source of deform-ability is the tension zone that can be idealized by means of equivalent T-stubs, which correspond to two T-shaped elements connected through the flanges by means of one or more bolt rows. This idealization is also adopted in modern design codes, as the Eurocode 3. The models for the column and the end plate sides are different. The T-stub elements on the column flange side are gener-ally hot rolled profiles, whilst on the end plate side such elements comprise two welded plates, the end plate and the beam flange, and a further additional stiff-ener that corresponds to the beam web (Fig. 1.2c). The first model (HR-T-stub) has been extensively studied over the past years and was the aim of several re-search programmes that are reported in technical literature. The current ap-proach to account for the behaviour of T-stubs made up of welded plates (WP-T-stub) consists in a mere extrapolation of the existing rules for the other as-sembly type. This assumption can be erroneous and can lead to unsafe estima-tions of the characteristic properties. To deal with this problem, a research pro-ject was devised to increase the knowledge and understanding of end plate be-haviour and contribute towards the improvement of its design. Simultaneously, the issue of available ductility is also addressed in this work. The knowledge of the plastic rotation capacity of beams is particularly important in the case of full strength beam-to-column joints, because yielding occurs at the member ends. In the case of partial strength joints, there is sig-nificant yielding of the connection and the evaluation of its ductility becomes crucial. The ductility of a joint reflects the length of the yield plateau of the M-Φ response and is intrinsically linked to the rotational capacity of the joint. The research described in this dissertation is divided into experimental, numerical (FE modelling) and analytical works. Reliable test results are essen-tial and support the validity of analytical and numerical work. Numerical analyses are important as they provide a means of carrying out wide-ranging parametric studies to complement existing experimental results. Analytical work allows the development of relatively simple design models that can be used in practice.

The experimental programme was conducted at the Delft University of Technology and included the (monotonic) testing of thirty-two individual T-stub connections made up of welded plates and eight full-scale single sided beam-to-column joints. The primary intent of the first series of tests on isolated T-stubs was to provide insight into the actual behaviour of this type of connec-tion, failure modes and deformation capacity. The parameters affecting the de-formation response of bolted T-stubs were identified and their influence on the overall behaviour of the connection was qualitatively and quantitatively as-sessed. In addition, the role of the welding and the presence of transverse stiff-eners were tackled. For the follow up study on extended end plate connections, the main objective was the analysis of the ultimate behaviour of the assembly end plate in bending-bolts and eventually the proposal of sound design rules for this elemental part within the framework of the so-called component method [1.1,1.40].


9

The numerical part of the work included the assessment of the load-carrying behaviour of single T-stubs and the exploration of other model fea-tures, namely the prying effect and the variation of contact flange surfaces within the course of loading. In this context, two T-stub connections represen-tative of HR- and WP-T-stubs were modelled and calibrated against experi-mental results. Having validated a three-dimensional FE model for the individ-ual T-stubs, a parametric study was conducted in order to provide a better un-derstanding of the overall behaviour and to evaluate the influence of the main parameters on the connection deformability.

The analytical approach of the research involved: (i) The proposal of a simplified beam model for the characterization of the T-stub

response. With this simplification some information and features of the T-stub model may be lost. However, this methodology overcomes the complexity of the above approaches and is less time-consuming.

(ii) The assessment of the global M-Φ response of an end plate connection based on the component method. A software tool developed at the University of Coimbra [1.41] was used for this assessment. The outcomes were validated through comparison with experimental evidence.

1.1.3 Outline of the dissertation The dissertation is divided into three parts. Part 1 (Chapter 1) presents background material on extended end plate con-nections. References to previous research work on the characterization of the rotational behaviour of this joint type are made. Special emphasis is given to the component method for the evaluation of the M-Φ response. Part 2 contains five chapters and includes further developments on the T-stub model. Chapter 2 is a brief introduction. Chapter 3 describes the experi-mental programme on isolated T-stub connections made up of welded plates. Chapter 4 includes the numerical evaluation of the force-deformation (F-∆) re-sponse of T-stubs. A three-dimensional FE model is recommended for that purpose. In both chapters, detailed results are given for benchmark specimens and the approaches are validated. A parametric study is described in Chapter 5. It provides insight into the main behavioural features of T-stub connections and highlights the parameters that affect their deformability. A two-dimensional simplified model that provides analytical solutions for the F-∆ response is pro-posed in Chapter 6. Because ductility is such an important characteristic of connection performance, this chapter emphasises the evaluation of deformation capacity of isolated T-stubs. Part 3 contains Chapters 7 and 8. Chapter 7 is entirely dedicated to the ex-periments on extended end plates connections. All the test details are provided and the results are thoroughly analysed. Chapter 8 presents a ductility analysis where the experimental results for the overall end plate connection are con-


10

fronted with the component tests. For that purpose, a procedure based on the component methodology is recommended. Comparisons with other proposals from the literature are also drawn. Finally, conclusions and recommendations are summarized in Chapter 9. 1.2 DEFINITIONS Beam-to-column joints consist of a web panel and one or two connections (sin-gle- or double-sided joint configuration) – Fig. 1.3. The web panel zone in-cludes the column web and the flange(s) of the column for the height of the connected beam profile(s). The connection is the location where two members are interconnected and the means of interconnection, i.e. the set of physical components that mechanically fasten the connected elements. The behaviour of a steel beam-to-column joint is represented by a M-Φ curve, as already explained. The rotational deformation of a joint, Φ, results from the in-plane bending, M, and is the sum of the shear deformation of the column web panel zone, γ, and the connection deformation, φ. The deformation of the connection includes the deformation of the fastening elements (bolts, end plate, etc.) and the load-introduction deformation of the column web. It results in a relative rotation between the beam and column axes, θb and θc, which is equal to:

b cφ θ θ= − (1.1) according to Fig 1.4, and provides a flexural deformability curve M-φ. This de-formability is only due to the couple of forces Fb transferred by the flanges of the beam that are statically equivalent to the bending moment M acting on the beam. In this figure, z is the lever arm. The shear deformation of the column web panel is associated with the force Vwp acting in this panel and leads to a relative rotation γ between the beam and column axes. A shear deformability curve Vwp-γ may then be established. For a

Joint

Connection

Web panel

Fig. 1.3 Parts of a beam-to-column joint (single-sided configuration).


11

single-sided joint configuration (see Fig. 1.5), the shear action in the panel is related to the internal actions on the joint as follows:

( )1 212

β = + + = wp c c

M z MV V Vz M z

(1.2)

The transformation parameter β relates the web panel shear force, Vwp, with the internal actions. Conservative values for the transformation parameter β, ne-glecting the effect of the shear force in the column, are suggested in Eurocode 3: (i) β = 1, in the case of single-sided joints, (ii) β = 2, in the case of double-sided joints with equal but unbalanced end bending moments and (iii) β = 0, in the case of double-sided joints with balanced end bending moments. The global M-Φ response of the joint is obtained by summing the contribu-

θb

θc

zM

Fb

Fb

Fig. 1.4 Sources of connection deformability.

Vb

Nb Mb = M

Nc2 Vc2

Mc2

Nc1

Vc1 Mc1

Fig. 1.5 Internal forces acting on the joint (single-sided configuration).


12

tions of rotation of the connection (φ) and of the shear panel (γ), as illustrated in Fig. 1.6. The M-γ curve is obtained from the Vwp-γ by means of the transfor-mation parameter β.

M

φ

Mi M

γ

Mi

+ φi γi

M

Φ

Mi

= Φi (= φi + γi)

Fig. 1.6 Global moment-rotation response of a joint. 1.3 METHODS FOR MODELLING THE ROTATIONAL BEHAVIOUR OF BEAM-

TO-COLUMN JOINTS 1.3.1 Generality The characterization of the M-Φ curve can be ascertained by experimental test-ing or mathematical models based on the geometrical and mechanical proper-ties of the joint. Full-scale experimental tests are naturally the most reliable method of description of the rotational behaviour of structural joints. However, they are time consuming, expensive and cannot certainly be regarded as a de-sign tool. In addition, the data gathered from tests of prototype joints are few and generally limited to displacement and surface measurements, as strain measurements, for instance. Therefore the results cannot be extended to differ-ent joint configurations. Nonetheless, tests provide accurate information on the joint response that is used to validate mathematical models of prediction of the M-Φ curve. Mathematical models for representation of the curve include: (i) curve fitting to test results by regression analysis, (ii) simplified analytical models, (iii) mechanical models that take into account the various sources of joint deformability and (iv) numerical models. For a review of different meth-ods, the reader should refer to Nethercot and Zandonini [1.42]. Mechanical models are the most effective solution for an accurate descrip-tion of the complex nature of bolted joint behaviour. These models use a set of rigid and flexible elements to simulate the overall joint. The interplay between these elements results in different mechanical models, as explained below. 1.3.2 The component method Current design practice adopts the so-called component method for the predic-tion of the rotational behaviour of beam-to-column joints. For the purposes of


13

simplicity, any joint can be subdivided into three different zones: tension, compression and shear. Within each zone, several sources of deformability can be identified, which are simple elemental parts (or “components”) that contrib-ute to the overall response of the joint. From a theoretical point of view, this methodology can be applied to any joint configuration and loading conditions provided that the basic components are properly characterized. Essentially, the method comprises three basic steps: (i) identification of the active components for a given structural joint, (ii) characterization of the individual component F-∆ response and (iii) assembly of those elements into a mechanical model made up of extensional springs and rigid links. This spring assembly is treated as a structure, whose F-∆ behaviour is used to generate the M-Φ curve of the full joint. The method is illustrated in Fig. 1.7 for the particular case of a bolted ex-tended end plate connection (with two bolt rows in tension). For the computa-tion of the joint rotational stiffness, the active joint components for this con-figuration, according to Eurocode 3, are: column web in shear (cws), column web in compression (cwc), column web in tension (cwt), column flange in bending (cfb), end plate in bending (epb), and bolts in tension (bt). The welds connecting the end plate and the beam are not taken into account for computa-tion of the rotational stiffness, as well as components beam web and flange in compression (bfc) and beam web in tension (bwt). Each component is charac-terized by a nonlinear F-∆ response, which can be obtained by means of ex-perimental tests or analytical models. These individual components are assem-bled into a mechanical model in order to evaluate the M-Φ response of the whole joint. The Eurocode 3 spring model is represented in Fig. 1.7 [1.40]. Al-ternative spring and rigid link models are proposed in literature, as the “Inns-bruck model” proposed by Huber and Tschemmernegg [1.43]. Essentially, they share the same basic components but assume different component interplay.

M Φ

(cwc) (cws)

(cwt.2) (cfb.2) (epb.2) (bt.2)

(cwt.1) (cfb.1) (epb.1) (bt.1)

M

Fig. 1.7 Component method: active components and mechanical model adopted

by Eurocode 3 for characterization of the joint rotational stiffness.


14

1.4 CHARACTERIZATION OF BASIC COMPONENTS OF BOLTED JOINTS IN TERMS OF PLASTIC RESISTANCE AND INITIAL STIFFNESS

Within the framework of the component method, the basic joint components are modelled by means of nonlinear extensional springs (Fig. 1.8a; K: spring axial stiffness). This complex behaviour can be approximated with simple rela-tionships without significant loss of accuracy. The elastic-perfectly plastic re-sponse is one of the simplest possible idealizations. Following the Eurocode 3 approach for idealization of the flexural joint spring nonlinear behaviour, this response is characterized by a secant stiffness, ke/η, and a full plastic resis-tance, FRd (Fig. 1.8b). ke is the initial stiffness of the component and η is a stiffness modification coefficient. Eurocode 3 defines this coefficient for dif-ferent types of connections. For a single component, similar values can be adopted. The post-limit stiffness, kp-l is taken as zero, which means that strain hardening and geometric nonlinear effects are neglected. Regarding the com-ponent ductility, i.e. the extension of the plastic plateau, the code [1.1] presents some qualitative principles that are however insufficient. For instance, the component column web in shear has very high ductility and therefore the de-formation capacity is taken as infinite; on the other hand, the bolts in tension are brittle components with no plastic plateau. The following sections present the formulations adopted in Eurocode 3 for prediction of the plastic resistance and initial stiffness of the basic components of bolted joints. Particular attention is devoted to the T-stub model that is used to idealize the tension zone of this joint typology. Then, the remaining compo-nents are briefly analysed.

K

F

∆

(a) Extensional spring representing a generic component.

F

∆

Actual behaviour

Elastic-plastic approximationFRd

ke/η

(b) Actual behaviour and elastic-plastic response. Fig. 1.8 Modelling of a component subjected to compression.


15

1.4.1 T-stub model for characterization of the tension zone of bolted joints The equivalent T-stub corresponds to two T-shaped elements connected through the flanges by means of one or more bolt rows as depicted in Fig. 1.9. The main behavioural aspects of the T-stub as a standalone configuration have been widely investigated over the past thirty years, both experimentally and theoretically. As a result, the structural response of this kind of connection is thoroughly known in elastic and plastic ranges, and appropriate design rules for prediction of the elastic-plastic F-∆ curve have been assessed.

Lateral viewTransverse view

bf

Plan b e1

0.5p

ewe

Fig. 1.9 Equivalent T-stub (one bolt row). 1.4.1.1 Plastic resistance of single T-stub connections The evaluation of the plastic (design) resistance of bolted T-stub connections is based on the well-known yield line principle. The works of Zoetemeijer [1.2], Packer and Morris [1.3] and Mann and Morris [1.11] form the basis of the pro-cedure presented below. Zoetemeijer suggests that the determination of the plastic resistance of such a connection type is based on the plastic behaviour of


16

the flanges and the bolts, and assumes that the yielding is large enough to allow the adoption of the most favourable static equilibrium [1.2]. For the purposes of simplicity, consider a bolted T-stub with one bolt row only. This simple con-nection can fail according to three possible “plastic” collapse mechanisms, as illustrated in Fig. 1.10. Type-1 mechanism is characterized by the formation of four plastic hinges: two hinges are located at the bolt axes, due to the bending moment caused by the prying forces, Q and the other two hinges are located at the flange-to-web connection. The formation of two plastic hinges at the flange-to-web connection and the failure of the bolts typify type-2 mechanism. The third collapse mechanism involves bolt failure only. A fourth supplemen-tary mechanism corresponds to the metal shear tearing around the bolt head or washer but is not relevant in most cases. The resistance corresponding to each collapse mechanism is easily computed by establishing the equilibrium equa-tions in the plastic conditions (cf. Appendix A). The plastic (design) resistance of the T-stub, FRd.0, corresponds to the smallest value among the examined “plastic” modes, i.e. FRd.0

= min (F1.Rd.0, F2.Rd.0, F3.Rd.0), where: .

1. .0

4 f RdRd

MF

m= (1.3)

( )( )

. .2. .0

2 2 2 21

1f Rd Rd f Rd Rd

RdRd

M B n MF

m n mβ λ

β λ+ −

= = + + +

(1.4)

3. .0 2Rd RdF B= (1.5) The plastic flexural resistance of the T-flanges, Mf.Rd, is given by:

2

. .4f

f Rd y f eff

tM f b= (1.6)

where beff is the effective width tributary to one bolt row, tf is the flange thick-

Q Q B

(=F1.Rd.0/2+Q)

F1.Rd.0

n m n m

Mf.Rd Mf.Rd

b

B

F2.Rd.0

n m nm

Mf.Rd ξMf.Rd

BRd

Q Q

b

BRd

n m nm

ξMf.Rd

BRd

b

BRd

F3.Rd.0

(a) Type-1:

22 1Rd

λβλ

≤ + .

(b) Type-2 (ξ ≤ 1.0): 2 2

2 1 Rdλ β

λ < ≤ +

.

(c) Type-3 (ξ ≤ 1.0): ( )2Rdβ > .

Fig. 1.10 “Plastic” collapse mechanisms of bolted T-stubs.


17

ness and fy.f is the yield stress of the flanges. The length m represents the dis-tance between the bolt axis and the section corresponding to the plastic hinge at the flange-to-web connection. According to Eurocode 3, m d sζ= − , where d represents the length between the bolt axis and the face of the T-stub element web, ζ is a coefficient taken as 0.8 and s r= or 2 ws a= , for hot rolled pro-files or welded plates as T-stub, respectively; r is the fillet radius of the flange-to-web connection and aw is the throat thickness of the fillet weld. The geomet-rical parameter λ is defined as the ratio n/m, being n the effective edge dis-tance. In Eurocode 3, n is taken as the minimum value of e (distance between the bolt axis and the tip of the flanges) and 1.25m, i.e. n = min (e,1.25m). BRd is the “plastic” (design) resistance of a single bolt in tension. The β-ratio is the relation between the flexural resistance of the flanges and the axial resistance of the bolts and governs the occurrence of a given (“plas-tic”) collapse mode (Fig. 1.11). At plastic conditions, this parameter, βRd, is the ratio between the plastic resistances corresponding to type-1 mechanism and that corresponding to a type-3 mechanism:

.2 f RdRd

Rd

MB m

β = (1.7)

The basic formulations presented above do not cater for the influence of the moment-shear interaction on the resistance of bolted T-stubs that can lead to a decrease in the plastic resistance. Faella et al. [1.44] assume that such interac-tion can be taken into account under the Von Mises yield criterion. Analytical expressions for type-1 and type-2 mechanisms allowing for moment-shear in-teraction are derived in Appendix A and are reproduced below:

( )

2

.1. .0 2

8 31 13

f RdRd

f f

MmFt mm t

= + −

(1.8)

Type-2

Type-1

Type-3

βRd

1

1 2

Non-circular yield line patterns Circular yield line patterns Rd

F2B

22 1

λλ +

22 1

λλ +

Fig. 1.11 Influence of βRd on the “plastic” collapse mechanism of bolted T-stubs.


18

and:

( )( ) ( )

2

.2. .0 2 2

2 116 31 1 13 4 1

f RdRdRd

f f

MmFt mm t

λ βλ

λ

+ = + + − +

(1.9)

Naturally, the “plastic” resistance for mechanism type-3 is not affected by this interaction. Regarding mechanism type-1, a significant increase in resistance can be ex-pected due to the influence of the bolt action on a finite contact area. Jaspart [1.20] suggests an alternative formulation to cater for this effect (Appendix A):

( )( ) ( )

. .1. .0

32 2 32 28 8 1

w f Rd f RdwRd

w w

n d M Mm dF

mn d m n m d mλ

λ λ− −

= =− + − +

(1.10)

whereby dw is the bolt head, nut or washer diameter, as appropriate. By combining both effects for type-1 mechanism, the plastic resistance can be expressed as:

( )

2

.1. .0 22

16 31 13 4

f RdRd

f f

MmFt mm t

= Γ + − Γ

(1.11)

( )8 1

16 1

w

w

md

md

λ λ

λ

− +Γ =

− (1.12)

as derived in Appendix A. The effective width of the T-element flange, beff, that appears explicitly in the above formulae is a notional width and does not necessarily represent any physical length of the flange. beff represents the width of the flange plate that contributes to load transmission. Zoetmeijer has successfully introduced this concept in [1.2]. It accounts for all possible yield line mechanisms of the T-stub flange and cannot exceed the actual flange width. This effective length has to be defined by establishing the equivalence, in the plastic collapse condition, between the beam model and the actual plate behaviour where collapse occurs due to the development of a yield line mechanism [1.44]. In the case of a bolted T-stub with one bolt row, three possible yield line mechanisms are considered: (i) circular pattern (Fig. 1.12a): beff.1 = 2πm, (ii) non-circular pattern (Fig. 1.12b): beff.2 = 4m + 1.25e and (iii) beam pattern (Fig. 1.12c): beff.3 = b. Regarding the circular pattern, beff is determined from the equivalence be-tween the failure load that corresponds to the collapse mechanism of a simply supported plate (P = 2πmpl, mpl

= tf2fy/4) and that of the equivalent beam model.

By equating both relationships, the following expression is determined for beff.1: 2

.1 2

4 12 24f

eff yyf

t mb f mft

π π= = (1.13)


19

(a) Circular pattern. (b) Non-circular pattern. (c) Beam pattern. Fig. 1.12 Yield line mechanisms of bolted T-stubs with one bolt row.

(a) Circular pattern. (b) Non-circular pattern. (c) Beam pattern.

(d) “Circular” pattern. (e) “Non-circular” pattern. Fig. 1.13 Yield line mechanisms of bolted T-stubs with two bolt rows. Referring now to the non-circular pattern, Zoetemeijer provides a simplified expression for the evaluation of the effective width associated to this mecha-nism [1.2]. For the beam pattern, the computation of this length is quite straightforward. The effective width of the equivalent T-stub corresponds to the smallest value of the above, i.e., beff = min (beff.1, beff.2, beff.3). Now, consider the case of multiple bolt rows. Depending on the pitch of the bolts, p, they may behave as a single bolt row or as a bolt group. For the par-ticular case of two bolt rows illustrated in Fig. 1.13, the behaviour is such of a group in cases c, d and e and of an individual bolt in the remaining. The effec-tive width of each bolt row is taken as the minimum among the five cases: (i) individual bolt (Fig. 1.13a): beff.1 = 2πm, (ii) individual bolt (Fig. 1.13b): beff.2 = 4m + 1.25e, (iii) bolt group (Fig. 1.13c): beff.3 = b, (iv) bolt group, (Fig. 1.13d): beff.4 = πm + 0.5p and (v) bolt group (Fig. 1.13e): beff.5 = 2m + 0.625e + 0.5p. Again, beff = min (beff.1, beff.2, beff.3, beff.4, beff.5). 1.4.1.2 Initial stiffness of single T-stub connections The evaluation of the initial stiffness of a T-stub, ke.0, is based on the analysis


20

of the elastic response of the connection, which has been analysed for the first time by Aggerskov [1.7] and later by Holmes and Martin [1.45] to accommo-date the effect of the prying forces on the bolt behaviour. Yee and Melchers [1.5] adopted a similar procedure for the evaluation of the elastic deformation of this type of connection. The single T-stub element is modelled as a simply supported beam, the supports corresponding to the location of the prying forces. This system is loaded by a concentrated force applied at the mid-span section, equivalent to the force applied on the T-stub through the web and the bolt force acting at the bolt axes (Fig. 1.14). The analysis of the T-stub is car-ried out by taking the interaction of the two T-stub elements and the bolts into consideration as well as the compatibility requirements to cater for the bolt de-formation. Jaspart [1.20] applies the same approach with a slight modification concerning the position of the prying forces. This location depends on the rela-tive stiffness of the flange and the bolts, i.e. the flange cross-section dimen-sions and the bolt diameter, as well as the degree of bolt preloading. Yee and Melchers [1.5] assume that the prying forces are located at the edge of the flange (n = e ≤ 1.25m). Jaspart [1.20] uses the distribution proposed by Douty and McGuire [1.6] (n = 0.75e ≤ 0.75×1.25m = 0.9375m). The elastic deformation of the bolted T-stub is determined from the follow-ing expressions (subscripts u and l refer to the upper and lower T-stub element, respectively):

. / 3/.0. / . / . /

1 3 24 2 2

f u l u le u l f u l f u l

Z qF

Eα α ∆ = − −

(1.14)

E is the Young modulus of steel. The other parameters that appear explicitly in the above expression are defined as follows:

( )( )

3

2 3

2 1.5 2

2 6 82

α α

α α

−=

− +

f f f

bf f f

s

Zq

LZA

(1.15)

( ) 3

3

2f

eff f

m nZ

b t+ =

′ (1.16)

( )2fn

m nα =

+ (1.17)

effb′ is the effective width for stiffness calculations, computed per bolt row. As is the bolt tensile stress area and Lb is the conventional bolt length. Eurocode 3 defines this length as:

( ). . 2 0.5b f u f l wsh n hL t t t t t= + + + + (1.18) where th, twsh and tn represent the bolt head, washer and nut thickness, respec-tively (Fig. 1.15). Aggerskov [1.7] defines a different conventional bolt length and distinguishes between the cases of snug-tightened and preloaded bolts. Ac-cording to the author:


21

F B

n n m m

B

Fig. 1.14 Equivalent (half-) model for the flange flexural elastic behaviour.

Ls

Ltg

th

tf.l

tn

tf.u twsh

twsh

Fig. 1.15 Bolt geometrical properties (including washer).

( )

1 4

2 3

2 3

2 snug-tightened bolts

preloaded boltsb

k kL k k

k k

+ ⇐= ⇐ +

(1.19)

where (see Fig. 1.15): . .

1 3

2 4

1.43 0.715

1.43 0.91 0.8 0.1 0.4

f u f ls tg n

s tg n wsh n wsh

t tk L L t k

k L L t t k t t

+= + + =

= + + + = + (1.20)

The initial stiffness coefficients ke.0.u and ke.0.l, which include the bolt de-formation, are defined as the ratio between the applied force F and the corre-sponding deformation:

.0. /3.0. / /

. / . / . /1 3 24 2 2

e u le u l u l

f u l f u l f u l

F EkqZ α α

= =∆ − −

(1.21)

The initial stiffness of the bolted T-stub is then given by:

.0

.0. .0.

11 1e

e u e l

k

k k

=+

(1.22)

The expressions presented above are lengthy and therefore they are not suitable for practical design. Jaspart proposes a simplified approach for the prediction of the axial stiffness of bolted T-stubs in [1.46]. This approach relies


22

on two major assumptions: (i) the distance n is taken as 1.25m (see Fig. 1.16b) and (ii) the bolt deformability is dissociated from that of the T-stub (Fig. 1.16c). Under these assumptions, the initial stiffness coefficients of the single T-stub elements may be simplified to the following expressions (cf. Appendix A; subscripts f and b refer to the flange and the bolt, respectively):

3. /

. . / 3/

eff f u le T u l

u l

Eb tk

m′

= (1.23)

and the axial stiffness of a snug-tightened bolt row is equal to:

. 1.6 se bt

b

EAk

L= (1.24)

The stiffness coefficient ke.bt from Eq. (1.24) characterizes the deformation of a snug-tightened bolt row in tension and is determined assuming that the bolt force is increased from 0.5F to 0.63F due to the prying effect (cf. Appendix A). The initial stiffness of the overall connection is computed by means of the fol-

b

B

n n m mQQ

B

F

(a) Actual behaviour.

B = 0.63F

n m n m Q = 0.13FQ

B

F

n = 1.25m

(1)

BB = 0.63F

F

(b) T-stub element alone. (c) Bolts alone. Fig. 1.16 Elastic deformation of the T-stub.


23

lowing relationship:

.

. . . . .

11 1 1e o

e T u e T l e bt

k

k k k

=+ +

(1.25)

Referring to Eqs. (1.16) and (1.23), the effective length effb′ represents a new effective length for stiffness calculations, slightly different from the effec-tive width beff for resistance calculations defined above. This new length may be taken as (cf. Appendix A):

0.9eff effb b′ = (1.26) Faella et al. [1.44] also adopt a procedure that neglects the compatibility requirements between the axial deformation of the bolts and the deformation of the T-stub flanges and neglects the effect of prying action. The bolt deform-ability is again separated from that of the T-stub. They derive the initial stiff-ness of the single T-stub by means of a flexible beam model, i.e. the bolt re-straining action is modelled as simple supports at the bolt axis (Fig. 1.17a). In this case (I: moment of inertia of the beam section):

( )( )3 3

3

2 248flex

eff f

F m FmEI Eb t

δ = =′

(1.27)

δ(flex)

m m

B B

F

(a) Flexible beam approach.

F

δ(rig)

m m

B B

(b) Rigid beam approach. Fig. 1.17 Behavioural schemes of the equivalent T-stub modelled as a beam.


24

and so:

( )( )

3

3

0.5 eff fflex

flex

Eb tFKmδ

′= = (1.28)

If the bolt acts as a fixed edge (Fig. 1.17b), then the beam is fully restrained at the bolt line (rigid beam approach) and the displacement is evaluated as fol-lows:

( )

3 3

3

0.524rig

eff f

Fm FmEI Eb t

δ = =′

(1.29)

( )( )

3

3

2 eff frig

rig

Eb tFKmδ

′= = (1.30)

In reality, the restraining action of the bolts lies in between these two limit situations. In fact, the Eurocode 3 adopts an expression that yields results in be-tween these two boundaries (see Eq. (1.23)). By adopting a nomenclature simi-lar to the above, according to Faella et al. [1.44], the axial stiffness of a single T-element is determined from the following relationship:

3

. . / ( ) 30.5 eff fe T u l flex

Eb tk K

m′

= = (1.31)

For the bolt stiffness they propose the expression from Eurocode 3 – Eqs. (1.18) and (1.24). The effective width effb′ is now defined as follows:

2eff hb m d b′ = + ≤ (1.32) where dh is the bolt head diameter and b is the actual width of the T-stub. This new width is obtained by consideration of a 45º spreading of the bolt action starting from the bolt head edge (Fig. 1.18) [1.47]. The accuracy of such an as-sumption is confirmed by experimental evidence in [1.44].

dh45º

b’eff

m

Fig. 1.18 Effective width for stiffness calculations [1.47]. 1.4.2 Characterization of the several joint components The Eurocode 3 formulations for prediction of the full plastic resistance and initial stiffness for each component are summarized in Table 1.1. In this table, fy is the yield stress, fu is the ultimate stress, Avc is the shear area of the column


25

profile, dc is the clear depth of the column web, β is the transformation parame-ter defined in §1.2, γM are partial safety factors, Mb.Rd is the moment resistance of the beam cross-section and subscripts b, c, ep, f and w refer to the beam, the column, the end plate, the flange and the web, respectively. The partial safety factors for design purposes are taken as γM0

= 1.1 = γM1 and γM2 = 1.25, for the

resistance of cross-sections and bolts, respectively [1.1]. The geometric pa-rameters are defined in Table 1.2 and Fig. 1.19 for bolted joints. Regarding the evaluation of the plastic resistance of the components col-umn web in compression and column web in tension, the reduction factors for plate buckling and interaction with shear in the column web panel, ρ and ω, re-spectively, are defined below:

( ) 2

1.0 if 0.72

0.2 if 0.72

λρ

λ λ λ

≤= − >

p

p p p

with . .20.932 eff cwc c y wc

pwc

b d fEt

λ = (1.33)

( )( )( ) ( )

1 1

1 2 1

1 if 0 0.52 1 1 if 0.5 1

1 if 1 2

βω ω β ω β

ω β ω ω β

≤ ≤

= + − − < ≤ − − − < ≤

(1.34)

with:

1 2.

1

1 1.3 eff cwc wc

vc

b tA

ω =

+

and 2 2.

1

1 5.2 eff cwc wc

vc

b tA

ω =

+

(1.35)

aep.w rc

ec

mc 0.8rc

eep

0.8 2 aep.wmep

( )min , ,1.25c c ep cn e e m=

( )min , ,1.25ep c ep epn e e m=

.0.8 2X X X ep fm L e a= − −

bep

LX

eepeep w eX

p2-3

mX

mep

aep.w

aep.f p1-2

m2

(a) Column flange and end plate between beam flanges.

(b) End plate extension.

Fig. 1.19 Definition of the geometric parameters m and n for the column flange and end plate (particular case of a hot rolled column section).


26

κwc is a reduction factor to account for the effect of an axial force in the col-umn. Generally, this reduction factor is unitary [1.1]. As mentioned above, the components column flange in bending and end plate in bending are modelled by means of equivalent T-stubs, provided that the effective width is properly defined (Table 1.2 – e1 is the end distance from the centre of the bolt hole to the adjacent edge, α is a coefficient obtained from an abacus provided by Eurocode 3). The design resistance associated to each of the three possible failure modes is thus obtained from Eqs. (1.3-1.5) by intro-duction of the appropriate geometric and mechanic parameters and partial safety coefficients, γM0, γM1 or γM2. Table 1.1 Synthesis of the code formulations for evaluation of the properties

of basic bolted joint components.

Component Plastic resistance Initial stiffness

cws .

.0

0.9

3y wc vc

cws RdM

f AF

γ= .

0.38 vce cws

EAk

zβ=

cwc

. ..

0

wc eff cwc wc y wccwc Rd

M

b t fF

ωκγ

= , but:

. ..

1

wc eff cwc wc y wccwc Rd

M

b t fF

ωκ ργ

≤

..

0.7 eff cwc wce cwc

c

Eb tk

d=

cwt . ..

0

wc eff cwt wc y wccwt Rd

M

b t fF

ωκγ

= ..

0.7 eff cwt wce cwc

c

Eb tk

d=

cfb ( ). .1. .2. .3.min , ,cfb Rd cfb Rd cfb Rd cfb RdF F F F=

3.

. 3eff cfb fc

e cfbc

Eb tk

m′

=

and: . .0.9eff cfb eff cfbb b′ =

epb ( ). .1. .2. .3.min , ,cfb Rd epb Rd epb Rd epb RdF F F F=

3.

. 3eff epb ep

e epbep

Eb tk

m′

=

and: . .0.9eff epb eff epbb b′ =

bfc ..

b Rdbfc Rd

b fb

MF

h t=

− .e bfck = ∞

bwt . ..

0

eff bwt wb y wbbwt Rd

M

b t fF

γ= .e bwtk = ∞

bt ..

2

0.9 u b sbt Rd Rd

M

f AF B

γ= = .

1.6 se bt

b

Ak

L=


27

With respect to the effective widths of the two above components, the equivalence between the column flange in transverse bending and the T-stub model is quite straightforward. In fact, for an unstiffened column flange, the ef-fective width is obtained directly from Fig. 1.13 by changing the geometry ac-cordingly, except for the beam pattern that is unlikely to develop. In the case of a stiffened column flange or the end plate in bending, the groups of bolt rows at each side of a stiffener are treated as separate equivalent T-stubs. The exten- Table 1.2 Definition of geometric parameters that appear explicitly in the

above formulae.

Component Geometric parameters

cwc

( ). 2 2 5

hot-rolled profile column section

2 welded profile column section

eff cwc fb ep fc p

c

c

b t a t s s

rs

a

= + + + +

⇐= ⇐

sp: length obtained by dispersion at 45º through the end plate cwt . .eff cwt eff cfbb b=

. . ..

.

but type-1 mechanism

type-2 mechanismeff nc eff cfb eff cp

eff cfbeff nc

b b bb

b

≤ ⇐= ⇐, whereby

subscript cp refers to circular yield line patterns and nc to non-circular yield line patterns. Bolt row location Circular pattern Non-circ. pattern

Bolt row considered individually Inner bolt row

2 cmπ 4 1.25c cm e+

End bolt row ( )1min 2 , 2c cm m eπ π +

()1

min 4 1.25 ,

2 0.625c c

c c

m e

m e e

+

+ +

Bolt row adjacent to a stiffener

2 cmπ cmα

Bolt row as part of a group of bolt rows Inner bolt row

2 p p

End bolt row ( )1min , 2cm p e pπ + +

()

1min 0.5 ,

2 0.625 0.5c c

e p

m e p

+

+ +

cfb

Bolt row adjacent to a stiffener

cm pπ + ( )

0.52 0.625

c

c c

p mm e

α+ −

− +


28

Table 1.2 Definition of geometric parameters that appear explicitly in the above formulae (cont.).

Component Geometric parameters

. . ..

.

but type-1 mechanism

type-2 mechanismeff nc eff epb eff cp

eff epbeff nc

b b bb

b

≤ ⇐= ⇐

Bolt row location Circular pattern Non-circ. pattern

Bolt row considered individually Bolt row outside tension flange of beam

()

min 2 , ,

2X X

X ep

m m w

m e

π π

π

+

+

(

)

min 4 1.25 ,0.5 ,

2 0.625 ,

2 0.625 0.5

X X ep

X X ep

X c

m e b

m e e

m e w

+

+ +

+ +

1st row below tension flange of beam

2 epmπ epmα

Other in-ner bolt row

2 epmπ 4 1.25ep epm e+

Other end bolt row

2 epmπ 4 1.25ep epm e+

Bolt row as part of a group of bolt rows Bolt row outside tension flange of beam

1st row below tension flange of beam

epm pπ + ( )

0.5

2 0.625ep

ep ep

p m

m e

α+ −

− +

Other in-ner bolt row

2 p p

epb

Other end bolt row epm pπ + 2 0.625 0.5ep epm e p+ +

bwt . .eff bwt eff epbb b=


29

sion of an end plate and the portion between the beam flanges are also mod-elled as two separate equivalent T-stubs [1.1] and the resistance and plastic failure modes are determined separately. 1.5 CHARACTERIZATION OF THE POST-LIMIT BEHAVIOUR OF BASIC

COMPONENTS OF BOLTED JOINTS Design codes as the Eurocode 3 do not give an accurate description of the post-limit response of the individual joint components and their deformation capac-ity, in particular. Within the framework of the component method, the overall joint behaviour is determined by the behaviour of its elementary parts. As a consequence, the rotation capacity of a joint is bound by the deformation ca-pacity of the single components. In terms of characterization of the post-limit component behaviour with a bilinear approximation, two main properties have to be fully described: the post-limit stiffness, kp-l and deformation capacity, ∆u. Jaspart [1.20] and Jaspart and Maquoi [1.48] assume that this behaviour can be approximated by a linear relationship (Fig. 1.20) and propose a general, simple methodology for characterization of both properties for all components. kp-l is taken as the strain hardening stiffness since the effects of material strain hardening after yielding of the component are dominant. It is defined below:

hp l e

Ek k

E− = (1.36)

for components column web in compression, column web in tension, column flange in bending, end plate in bending and:

( )2 13

hp l e

Ek k

Eυ

−

+= (1.37)

for component column web in shear. Eh is the strain hardening modulus of the material and υ is the Poisson’s ratio. For the bolts in tension kp-l is taken as zero since this is a brittle component. Components beam web in tension and beam flange and web in compression are disregarded since they only provide a limi-tation to the joint flexural resistance [1.1]. They also suggest expressions for computation of the ultimate resistance, Fu, and, consequently, ∆u. Fu is readily determined by formally equivalent expressions to those listed in Table 1.1, by replacing fy with fu, the ultimate stress of the structural steel. The deformation capacity is determined from the intersection of the post-limit behaviour with Fu:

Rd u Rdu

e p l

F F Fk k −

−∆ = + (1.38)

From a qualitative point of view, the basic components can be grouped ac-cording to three ductility classes that reflect this post-limit behaviour [1.49]. The component ductility reflects the “length” of the post-limit response and can be quantified by means of an index ϕ i for each component i. The author


30

Actual behaviour

Elastic-plastic approximation (using the component initial stiffness)

Post-limit linear approximation

kp-l

F

∆

FRd

ke

Fu

∆u Fig. 1.20 Bilinear approximation of the component behaviour as proposed by

Jaspart [1.20] and Jaspart and Maquoi [1.48]. proposes the following expression for the definition of ϕ i:

Rd

ui

F

ϕ∆

=∆

(1.39)

whereby ∆u is the component deformation capacity and RdF Rd eF k∆ = is the

deformation value corresponding to the component plastic resistance, FRd. Kuhlmann et al. [1.49] propose three ductility classes: (i) components with high ductility (ϕi ≥ α) (e.g. cws, cfb, epb), (ii) components with limited ductil-ity (β ≤ ϕi < α) (e.g. cwc, cwt) and (iii) components with brittle failure (ϕi < β) (e.g. bt, welds). α and β represent ductility limits. Simões da Silva et al. [1.50] propose α = 20 and β = 3. The ductility behaviour of the several joint compo-nents is analysed in the following sub-sections according to alternative proce-dures from the literature. 1.5.1 Column web in shear (component with high ductility) For the web panel subjected to shear, literature proposes an alternative model, the Krawinkler et al. model that can be used to predict the contribution of this component to the overall joint response [1.44]. This model was developed based on experimental observations regarding the significant post-yield resis-tance of the panel zone. Fig. 1.21 illustrates this model in terms of a global Vwp-γ response. This curve is easily converted into a Vwp-∆wp response by means of the following simplified relationship:

( )cwswp

z zγ

∆∆= = (1.40)


31

Vwp

γ

Vwp.y

Vwp.p

4γy

Kcws.h

γy Fig. 1.21 Krawinkler et al. trilinear model. This relation was derived by idealizing the web panel as a short column stub of height z, subjected to a shear force Vwp [1.5,1.20]. There are two swivel points in the model corresponding to: (i) first yielding of the panel zone, (Vwp.y,γy) and (ii) first yielding of the column flanges, (Vwp.p,4γy). According to Krawinkler et al., the rotational behaviour of the panel after yielding can be attributed to the bending of the column flanges [1.44]. The co-ordinates at the swivel points of this curve are given by [1.44]:

. 3y wc c b

wp y

f t h hV

zβ= (1.41)

( )2 1 3y

y

fEγυ

=+

(1.42)

2

. . 1 3.12 c fcwp p wp y

c b wc

b tV V

h h t

= +

(1.43)

whereby hc and hb are the column and beam depth, respectively. The residual stiffness Kcws.h is given by [1.44]:

( ). 2 1h wc c b

cws hE t h h

Kzυ β

=+

(1.44)

This model imposes no limits on the deformation capacity of this compo-nent. Beg and co-workers [1.51-1.52] present some expressions to limit the ul-timate shear panel rotation, γu:

( )

128 0.38 if 0 0.10

[%]1 128 0.38 55 0.81 0.1 if 0.10

cN

wcu

c cN

wc wc

dn

t

d dn n

t t

κε

γ

κε ε

− ≤ ≤

= − − − − >

(1.45)


32

with:

pl

NnN

= (1.46)

235

yfε = (1.47)

The influence of the level of axial force in the column, N, can be assessed by means of the following parameter κN:

1 if 1N b ch hκ = ≥ and 2

1 if 10.4

cN b c

b

h n h hh

κ = − < (1.48)

1.5.2 Column flange in bending, end plate in bending and bolts in tension

(T-stub idealization) These three components can be idealized with the equivalent T-stub approach. The deformation capacity of a T-stub mainly depends on the plate/bolt resis-tance ratio. It has been well established that the best way to accomplish defor-mation capacity is a design in the type-1 situation [1.8], i.e. βRd < 2λ/(2λ+1). In type-1, the deformation can be regarded as “indefinitely large” because yield-ing occurs in the flange. The only limitations are the membrane stresses in the plate that develop with large deformations. In a type-3 mechanism (βRd > 2), the deformations are mainly determined by bolt tension elongation, which leads to a brittle failure. A thorough analysis of the post-limit behaviour of isolated T-stub connections is carried out later in the text (Chapter 6). Previous research work of several authors on this subject is also reviewed, namely the work of Jaspart [1.20], Faella et al. [1.44], Beg et al. [1.51-1.52] and Swanson [1.53]. 1.5.3 Column web in compression (component with limited ductility) Aribert and co-workers opened up the elastoplastic studies of web profiles sub-jected to local compression forces [1.54-1.56]. Their studies mainly focused on resistance evaluation rather than a full description of the overall deformation behaviour. The component column web in compression in particular was extensively studied by Kuhlmann and Kühnemund [1.57-1.58] and Kühnemund [1.59]. They performed numerous tests on this component and characterized its F-∆ behaviour in detail (model depicted in Fig. 1.22). The elastic-plastic response is easily determined from the Eurocode 3 proposals (see Tables 1.1-1.2). The post-limit behaviour is described by two distinct branches. The first branch is defined between the plastic resistance and the maximum resistance, Fcwc.u. The second (softening) branch follows on until fracture. In this phase they redefine


33

Fcwc

∆cwc

2/3Fcwc.Rd

Fcwc.Rd

4.5∆e.cwc

ke.cwc

∆e.cwc

Fcwc.u

∆u.cwc Fig. 1.22 Kuhlmann and Kühnemund model. the effective width beff.cwc. The procedure for assessment of the relevant ordi-nates of the curve in this post-limit regime can be found in [1.59]. Other authors also looked into the behaviour of this component. Huber and Tschemmernegg [1.60] suggested values for the deformation capacity for this component for different standard shapes of the column section (Table 1.3). Beg and co-workers [1.51-1.52] carried out a numerical analysis of the component and proposed expressions for evaluation of the deformation capacity that de-pend on the level of axial force in the column. These expressions are defined in a non-dimensional form below:

.

1 118.5 0.75 if 20

1 1[%] 5.7 0.11 if 20 33

12.07 if 33

c c

wc wc

c cu cwc

wc wc

c

wc

d dt t

d dt t

dt

ε ε

δε ε

ε

− <

= − ≤ <

≥

for n = 0 (1.49)

and:

( )

.

1 1 19.4 0.34 15 0.75 0.5 if 20

1 1[%] 4.8 0.11 if 20 33

11.17 if 33

c c c

wc wc wc

c cu cwc

wc wc

c

wc

d d dn

t t td dt t

dt

ε ε ε

δε ε

ε

− + − − <

= − ≤ <

≥

(1.50)


34

for n ≥ 0.1. n and ε are defined in Eqs. (1.46) and (1.47), respectively. The de-formation capacity of the column web in compression, ∆u.cwc, is computed from:

. .u cwc u cwc cdδ∆ = (1.51) Table 1.3 Deformation capacity of the column web in compression accord-

ing to Huber and Tschemmernegg [1.60].

Column profile IPE HEA HEB HEM ∆u.cwc (mm) 1.5 3.0 5.0 7.5

1.5.4 Column web in tension (component with limited ductility) Witteveen et al. [1.61] suggest a very simple expression for evaluation of the deformation capacity of the column web subjected to tension:

. 0.025u cwt ch∆ = (1.52) This limit value is also adopted in Eurocode 3. Beg et al. [1.51-1.52] also studied this component and derived an analytical expression for evaluation of the ultimate deformation:

. .u cwt u cwt cdδ∆ = (1.53) with:

22

. 0

4 32

x xu cwt

s sδ ε

− − =

(1.54)

sx is defined below (σx: transverse stress):

.

xx

y wc

sfσ

= (1.55)

and ε0 is the ultimate transverse strain in the case that the axial force in the col-umn is absent. They suggest that this value should be set as equal to 0.1. 1.6 EVALUATION OF THE MOMENT-ROTATION RESPONSE OF BOLTED

JOINTS BY MEANS OF COMPONENT MODELS Mechanical (component) models use a set of rigid and flexible parts (springs) to simulate the interaction between the various sources of joint deformation. The springs are combined in series or in parallel depending on the way they in-terplay with each other. Springs in series are subjected to the same force whilst parallel springs undergo the same deformation. The active components of a joint are grouped according to their type of loading (tension, compression or shear). They can also be distinguished be-


35

tween those linked to the web panel, the load-introduction into the column web panel and the connection. A sophisticated component interplay as assumed in the Innsbruck model [1.43,1.62] allows separate representation of the behav-iour of the web panel in shear, the load-introduction and the connection ele-ments (Fig. 1.23). Due to its complexity, this model is not easily implemented in a design code, as it requires successive iterations within the component as-sembly [1.62]. This is why a simplified component model, as depicted in Fig. 1.7, is desirable. Huber highlights the two main differences between the two component models [1.62]. In the Eurocode 3 model there is no separation between the panel and connecting zone, which may lead to a non-straight deformation of the column front in contradiction with experimental evidence. Also, the stiff separation bar between tension and compression components (see Fig. 1.7) prevents the interaction between these components within the web panel that exists in reality. However, this simplified model yields analytical solutions rather than iterative, making it a simpler tool for daily design practice. Aribert et al. propose an alternative component model that is yet restricted to the case of internal flush end plate joints under balanced loading, i.e. the web panel is not subjected to shear forces [1.63]. Basically they assume the same components but introduce an additional component at the level of the ten-sion beam flange. This new component corresponds to the part of the end plate located between the tension beam flange and the first bolt row and is subjected to longitudinal bending. Also, they assume a sophisticated component interplay since they do not separate the components under compression from the tensile zone as in Eurocode 3 (see Fig. 1.7). Finally, reference is made to the component model commonly used at the University of Coimbra (Fig. 1.24). This model assumes a sophisticated compo-nent interplay since it does not establish the equivalence of all tensile compo-nents into a single equivalent spring as in Eurocode 3. This equivalence is ex-plained below. For further reference, this model is designated by UC model. For illustration of the differences between the alternative spring models, consider the evaluation of the initial stiffness of a (single-sided) bolted ex-

(cws) (cwc) (bfc)

Φ

(cfb.2)

(epb.2)

(bwt.2)

(bt.2)

(cfb.1) (epb.1) (bt.1)

M

(cwt.1)

(cwt.2)

Fig. 1.23 Innsbruck spring model (single-sided steel joint configuration).


36

(cwc) (bfc) (cws)

Φ

(cwt.2)

(cfb.2)

(epb.2)

(bwt.2)

(bt.2)

(cwt.1) (cfb.1) (epb.1) (bt.1)

M

Fig. 1.24 UC spring model (single-sided steel joint configuration).

30 30 100

30

165

35

35

50

Fig. 1.25 Illustrative example: connection geometry. tended end plate connection. The column is made up of a HE240B profile and the beam profile is IPE240. Bolts M20 fasten the elements. The end plate di-mensions are 315×160 mm2 and 15 mm thickness. The continuous fillet welds between the beam and the end plate have a throat thickness aw = 8 mm. The ge-ometry of the connection is depicted in Fig. 1.25. For all components, E = 210 GPa. The characterization of the elastic stiffness of the single components is based on the Eurocode 3 proposals (Tables 1.1-1.2). The following results are obtained: (i) Eurocode 3 model: Sj.ini = 23.82 kNm/mrad, (ii) Innsbruck model: Sj.ini = 23.12 kNm/mrad (difference of -2.94% in comparison with the Eurocode 3 model) and (iii) UC model: Sj.ini = 24.25 kNm/mrad (difference of 1.81% in comparison with the Eurocode 3 model).


37

1.6.1 Eurocode 3 component model Fig. 1.26a depicts the Eurocode 3 mechanical model for the particular case of a bolted extended end plate connection, with two bolt rows in tension, which al-lows for characterization of the rotational behaviour of such a joint type. The model assumes that the compressive springs are located at the centre of com-pression, which corresponds to the centre of the lower beam flange, and the tensile springs are positioned at the corresponding bolt row level. The bolt row deformations are proportional to the distance to the centre of compression and the forces acting in each row depend on the component stiffness [1.40]. 1.6.1.1 Model for stiffness evaluation For evaluation of the initial rotational stiffness, Sj.ini, the model (Fig. 1.26a) is simplified by replacing each assembly of springs in series with an equivalent spring, which retains all the relevant characteristics (Fig. 1.26b). Weynand fur-ther simplifies this model by establishing the equivalence between the parallel spring assembly t.1 and t.2 and the spring t (Fig. 1.26c) [1.64]. By means of simple equilibrium considerations and compatibility requirements, the follow-

(cwc) (bfc) (cws)

Φ

(cwt.2)

(cfb.2)

(epb.2)

(bwt.2)

(bt.2)

(cwt.1) (cfb.1) (epb.1) (bt.1)

M

(a) Eurocode 3 spring model: active components for a bolted extended end plate connection with two bolt rows in tension (see Fig. 1.7).

(c)

(t.2)

(t.1)

Φ M

(c)

(t)

Φ M

(b) Eurocode 3 equivalent model. (c) Eurocode 3 simplified model. Fig. 1.26 Eurocode 3 spring model and simplifications.


38

ing expression for initial stiffness is derived in [1.64]: 2

.ec et

j iniec et

k kMS zk k

= =Φ +

(1.56)

The elastic stiffness of each equivalent spring t.i and c, corresponding to a spring assembly in series, are readily obtained as follows (Fig. 1.26a):

. . .

11 1 1ec

e cws e cwc e bfc

k

k k k

=+ +

(1.57)

and:

.

. . . . . . . . . .

11 1 1 1 1et i

e cwt i e cfb i e epb i e bwt i e bt i

k

k k k k k

=+ + + +

(1.58)

The lever arm z is given by [1.64]: 2

.1

.1

=

=

=∑

∑

n

et i ii

n

et i ii

k zz

k z (1.59)

whereby zi is the distance from bolt row i to the centre of compression. 1.6.1.2 Model for resistance evaluation For evaluation of the joint flexural resistance, Mj.Rd, simple equilibrium equa-tions yield:

. .1

n

j Rd ti Rd ii

M F z=

= ∑ (1.60)

in the absence of an axial force. Fti.Rd is the potential resistance of bolt row i in the tension zone and zi is the distance of the i-th bolt row from the centre of compression. Fti.Rd is taken as the least of the following values:

( ). . . . . . . . . . .min , , , ,ti Rd cwt i Rd cfb i Rd epb i Rd bwt i Rd bt i RdF F F F F F= (1.61) The values of Fti.Rd are calculated starting at the top row and working down. Bolt rows below the current row are ignored. Each bolt row is analysed first in isolation and then in combination with the successive rows above it. The pro-cedure can be summarized as follows [1.1]: (i) Compute the plastic resistance of bolt row 1 omitting the bolt rows below:

( )1. . . . .1. .1. .1. .1.min , , , , , ,t Rd cws Rd cwc Rd bfc Rd cwt Rd cfb Rd epb Rd bt RdF F F F F F F Fβ= (1.62) (ii) Compute the plastic resistance of bolt row 2 omitting the bolt rows below:

2. mint RdF = ( . 1. . 1. . 1. .2., , , ,cws Rd t Rd cwc Rd t Rd bfc Rd t Rd cwt RdF F F F F F Fβ − − −


39

( ).2. .2. .2. .2. 1.. 1 2 ., , , , ,cfb Rd epb Rd bwt Rd bt Rd t Rdcwt RdF F F F F F+ −

( ) ( ) )1. 1.. 1 2 . . 1 2 .,t Rd t Rdcfb Rd bt RdF F F F+ +− − (1.63)

(iii) Compute the plastic resistance of bolt row 2 omitting the bolt rows below: 3. mint RdF = ( . 1. 2. . 1. 2., ,cws Rd t Rd t Rd cwc Rd t Rd t RdF F F F F Fβ − − − −

. 1. 2. .3. .3. .3. .3. .3., , , , , ,bfc Rd t Rd t Rd cwt Rd cfb Rd epb Rd bwt Rd bt RdF F F F F F F F− − ( ) ( ) ( )2. 2. 2.. 2 3 . . 2 3 . . 2 3 ., , ,t Rd t Rd t Rdcwt Rd cfb Rd epb RdF F F F F F+ + +− − −

( ) ( ) )2. 1. 2. 1.. 1 2 3 . . 1 2 3 .,t Rd t Rd t Rd t Rdcwt Rd cfb RdF F F F F F+ + + +− − − − (1.64)

and so forth. 1.6.1.3 Idealization of the moment-rotation curve The conversion of the F-∆ curves of the individual active joint components into a global M-Φ curve is based on the spring model so that the compatibility and equilibrium requirements are met. Depending on the desired level of accuracy and available software, the rotational joint behaviour can be fully characterized (full nonlinear shape) or approximated by nonlinear or multilinear simplifica-tions. The characterization of the actual nonlinear M-Φ curve is not easily open to simple analytical formulations and therefore the simplified approximations are preferred for hand calculations. Recently, the author proposed an energy approach for evaluation of the multilinear M-Φ response from component models in closed-form solutions. It also allows the identification of the yielding sequence of the individual components and the corresponding levels of defor-mation [1.65-1.70]. The Eurocode 3 adopts two possible idealizations of the M-Φ curve, bilin-ear (elastic-plastic curve) and nonlinear, as depicted in Fig. 1.27. The stiffness modification factor, η, (Fig. 1.27a) depends on the joint type and configuration and is defined in the code [1.1]. For bolted end plate beam-to-column joints this factor is taken as 2. The stiffness ratio µ that is used to define the nonlinear part of the idealized M-Φ curve in Fig. 1.27b is defined as follows:

.. . .

.

1.5 2 for 3

j Edj Rd j Ed j Rd

j Rd

MM M M

Mµ

Ψ

= < ≤

(1.65)

and Ψ is a coefficient that depends on the type of connection. For bolted end plate connections, this coefficient is taken as 2.7. 1.6.2 Guidelines for evaluation of the ductility of bolted joints

The ductility of a joint can be defined as the amount of a plastic rotation that


40

0

30

60

90

120

150

180

0 15 30 45 60 75 90 105 120

Joint rotation (mrad)

Ben

ding

mom

ent (

kNm

) Actual response

Mj.Rd

Sj.ini/η

(a) Bilinear idealization of the moment-rotation curve.

0

30

60

90

120

150

180

0 15 30 45 60 75 90 105 120

Joint rotation (mrad)

Ben

ding

mom

ent (

kNm

)

2/3Mj.Rd

Actual response

Mj.Rd

Sj.ini

Sj.ini/µ

(b) Nonlinear idealization of the moment-rotation curve. Fig. 1.27 Eurocode 3 idealizations of the actual rotational response. can be sustained while maintaining a certain percentage of its ultimate resis-tance [1.53]. It reflects the length of the yield plateau of the M-Φ response. This property can be quantified by means of an index ϑj that relates the rotation capacity of the joint, ΦCd to the rotation value corresponding to the joint plastic resistance [1.46,1.50]. In this work, the following relationship is proposed:

Rd

Cdj

M

ϑΦ

=Φ

(1.66)

similarly to Eq. (1.39). This index allows a direct classification of a joint in terms of ductility, similarly to the basic joint components (§1.5).

RdMΦ is the “analytical” rotation value corresponding to Mj.Rd and is given by the ratio

. .j Rd j iniM S (Fig. 1.28). Fig. 1.28 presents other distinctive rotation values. ΦXd


41

M

Φ

Mj.Rd

ΦXd

Mmax

RdMΦ maxMΦ ΦCd

*CdΦ

Fig. 1.28 Definitions of joint rotation. is the rotation at which the moment first reaches Mj.Rd and *

CdΦ is the rotation at which the moment deteriorates back to Mj.Rd after reaching a moment above Mj.Rd through deformation beyond ΦXd. maxMΦ is the rotation at which the mo-ment resistance is maximum. Jaspart [1.46] classifies the joints in terms of available rotation capacity. He groups structural joints into three classes: (i) class 1 joints, which have a suffi-ciently good rotation capacity to allow a plastic frame analysis (high ductility), (ii) class 2 joints, with a limited rotation capacity (limited ductility) and (iii) class 3 joints, for which brittle failure or instability phenomena limits the rota-tion capacity. Literature reports several procedures for characterization of the available rotation capacity. Zoetemeijer [1.10] proposes some criteria and simple empiri-cal expressions for the estimation of a joint deformation capacity based on a se-ries of experiments. He concluded that considerable rotation capacity was ob-tained from the tension side of a joint if βRd < 2λ/(2λ+1), within the T-stub ide-alization of the region. This means that the tension zone fails according to a type-1 mechanism, with complete yielding of one of the plate components (column flange or end plate). If βRd > 2, then the joint behaves elastically up to failure of the bolts without deformation of the plate(s). In this case, the bolt elongation mainly supplies the joint deformation. To prevent this situation, Zoetemeijer suggests that the condition βRd < 1.75 should always be satisfied [1.10]. For the intermediate situations, i.e. 2λ/(2λ+1) < βRd ≤ 1.75, the joint ro-tational deformation remains limited since the bolt is also engaged in the col-lapse mode. In the latter situation, Zoetemeijer suggests an expression for evaluation of the rotation capacity, ΦCd [1.10]:


42

1

10.6 41.3

RdCd z

β−Φ = (1.67)

whereby z1 is the distance in [mm] between the first bolt row from the tension flange and the centre of compression. Later, Jaspart [1.46] extended the above criteria for inclusion in Eurocode 3. The code states that a bolted end plate joint may be assumed to have suffi-cient rotation capacity for plastic analysis, provided that both of the following conditions are satisfied: (i) the moment resistance of the joint is governed by the resistance of either the column flange in bending or the end plate in bend-ing and (ii) the thickness t of either the column flange or the end plate (not nec-essarily the same basic component as in (i)) satisfies:

.0.36φ≤ u b

y

ft

f (1.68)

where φ is the bolt diameter, fu.b is the tensile strength of the bolt and fy is the yield strength of the relevant basic component. This expression is derived in [1.46]. These guidelines are yet insufficient to ensure adequate ductility in par-tial strength joints. More recently, Adegoke and Kemp [1.23] performed an experimen-tal/analytical study on thin extended end plates and realized that most of the connection rotation in these cases came from the end plate deformation. From these observations, they proposed a simple expression for evaluation of the connection ultimate rotation:

2. .1.4 40f y ep f X y ep

Cdep yfn ep mrn

m f m m fEt h Et h

φ = + (1.69)

In this expression, the first part corresponds to the connection rotation when the yield lines in the extended and flush zones of the end plate (above and be-low the top tension beam flange, respectively) are fully developed and is based on the elastic flexibility of the stronger flush bolt lines [1.23]. mf represents the average distance from each bolt to the adjacent web and flange welds below the tension flange, i.e.:

2

2ep

f

m mm

+= (1.70)

(see Fig. 1.19). hyfn is the height of the flush bolt row above the neutral axis at yield and hmrn is the height of the resultant tension force above the neutral axis at maximum strain. They assumed that the rotation capacity was attained when fracture of the end plate occurred. This would happen when the maximum strain would be thirty times the yield strain. In the context of the component method, several researchers have devel-oped simplified approaches to quantify the overall rotation capacity. Since in many cases the most important sources of deformability in bolted joints can be idealized by means of the equivalent T-stub in tension, special attention has been devoted to the evaluation of the deformation capacity of this individual


43

component. Swanson [1.53] developed a methodology for characterization of the ductility of T-stub connections. Faella and co-workers [1.44,1.71-1.72] set up a procedure for computation of the deformation capacity of the isolated T-stub and the overall joint. Other components have also been studied within this framework. Kuhlmann and Kuhnemund [1.57] performed tests on the compo-nent column web under transverse compression and proposed design rules for this component from the point of view of resistance and deformation capacity. The researchers also conducted a series of full-scale tests that are reported in [1.58-1.59]. The study was restricted to joints under balanced loading. The dominant component of all tests was the column web in compression. They also developed a procedure based on the component method to determine the rotation capacity of the joint for those cases where the critical component was the column web under compression. Beg et al. [1.51] set up a methodology based on a simplified component model for characterization of the rotational response to include the evaluation of rotation capacity. They analysed different components, the column web, the bolts in tension, the column flange and the end plate in bending, and proposed simple expressions for evaluation of their deformation capacity based on numerical evidence, as already mentioned above (see §1.5). Yet, there was no calibration of this work. For this reason, this methodology is questionable and should be used with special care. They then established a simple mechanical model to mimic the joint rotational be-haviour (Fig. 1.29). This model is composed of bilinear springs that represent the response of all relevant components. The overall joint rotation results from the contribution of all components and can be readily determined as follows (Fig. 1.29):

0cwt cwc

zγ

∆ + ∆ + ∆Φ = + (1.71)

The rotation capacity mainly depends on the deformation capacity of the weak-est component, i.e. the component with lower resistance. In Fig. 1.29 the T-stub that represents the tension zone is the governing component. Conse-quently:

. 0. .cwt R u cwc RCd Rz

γ∆ + ∆ + ∆

Φ = + (1.72)

It is worth mentioning that this procedure is identical to the proposals of Faella and co-workers [1.44]. They also proposed a similar expression for evaluation of the rotation capacity, though they mainly focused on the study of the tension zone idealized as a T-stub. Finally, and in the framework of the component method, the author’s pro-posals for evaluation of the rotation capacity are also referred [1.65-1.70]. By means of an elastic analogy of the nonlinear behaviour, the author proposed an elastic equivalence for the spring models mentioned above (Figs. 1.7 and 1.24). The basic building block of an equivalent model corresponds to replacing each nonlinear spring with an equivalent elastic spring consisting of a set of linear elastic springs with specific properties. Such equivalent models provide closed-


44

form solutions to an otherwise numerical problem and allow for the identifica-tion of the yielding sequence of the components and, ultimately, the computa-tion of the joint rotation capacity.

cwt

F

∆

FR

∆cwt.R T-stub

F

∆ ∆0.u cwc

F

∆ ∆cwc.R

cws

F

γ γR

Φ(cwc)

M

Φ

(cwt) (T-stub) (cws)

M

cws

cwc

cwt T-stub idealization

z

Fig. 1.29 Computation of the joint rotation capacity according to Beg et al.

[1.51]. 1.7 REFERENCES [1.1] European Committee for Standardization (CEN). prEN 1993-1-8:2003,

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[1.40] Weynand K, Jaspart JP, Steenhuis M. The stiffness model of revised Annex J of Eurocode 3. In: Proceedings of the Third International Workshop on Connections in Steel Structures III, Behaviour, Strength and Design (Eds.: R. Bjorhovde, A. Colson and R. Zandonini), Trento, Italy; 441-452, 1995.

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[1.43] Huber G, Tschemmernegg F. Modelling of beam-to-column joints. Journal of Constructional Steel Research; 45:199-216, 1998.

[1.44] Faella C, Piluso V, Rizzano G. Structural semi-rigid connections – the-ory, design and software. CRC Press, USA, 2000.

[1.45] Holmes M, Martin LH. Analysis and design of structural connections: reinforced concrete and steel. Ellis Horwood Limited, Chichester, UK, 1983.

[1.46] Jaspart JP. Contributions to recent advances in the field of steel joints – column bases and further configurations for beam-to-column joints and beam splices. Aggregation thesis. University of Liège, Liège, Belgium, 1997.

[1.47] Ballio G, Mazzolani FM. Theory and design of steel structures. Chap-man and Hall, London, UK, 1983.

[1.48] Jaspart JP, Maquoi R. Prediction of the semi-rigid and partial strength properties of structural joints. In: Proceedings of the Annual Technical Meeting on Structural Stability Research, Lehigh, USA; 177-191, 1994.

[1.49] Kuhlmann U, Davison JB, Kattner M. Structural systems and rotation capacity. In: Proceedings of the International Conference on Control of the Semi-Rigid Behaviour of Civil Engineering Structural Connections (Ed.: R. Maquoi), Liège, Belgium; 167-176, 1998.

[1.50] Simões da Silva L, Santiago A, Vila Real P. Post-limit stiffness and ductility of end plate beam-to-column steel joints. Computers and Struc-tures; 80:515-531, 2002.

[1.51] Beg D, Zupančič E, Vayas I. On the rotation capacity of moment con-


48

nections. Journal of Constructional Steel Research; 60:601-620, 2004. [1.52] Zupančič E, Beg D, Vayas I. Deformation capacity of components of

moment resistant connections. European Convention for Constructional Steelwork – Technical Committee 10, Structural Connections (ECCS-TC10), Document ECCS-TWG 10.2-02-005, 2002.

[1.53] Swanson JA. Characterization of the strength, stiffness and ductility be-havior of T-stub connections. PhD dissertation, Georgia Institute of Technology, Atlanta, USA, 1999.

[1.54] Aribert JM, Lachal A. Étude élasto-plastique par analyse des contraintes de la compression locale sur l’âme d’un profilé. Construction Métal-lique; 4:51-66, 1977.

[1.55] Aribert JM, Lachal A, El Nawawy O. Modélisation élasto-plastique de la résistance d’un profilé en compression locale. Construction Métal-lique; 2:3-26, 1981.

[1.56] Aribert JM, Lachal A, Moheissen M. Interaction du voilement et de la résistance plastique de l’âme d’un profilé laminé soumis à une double compression locale (nuance dácier allant jusqu’à FeE460. Construction Métallique; 2:3-23, 1990.

[1.57] Kuhlmann U, Kühnemund F. Rotation capacity of steel joints: verifica-tion procedure and component tests. In: Proceedings of the NATO Ad-vanced Research Workshop: The paramount role of joints into the reli-able response of structures (Eds.: C.C. Baniotopoulos and F. Wald), Nato Science series, Kluwer Academic Publishers, Dordrecht, The Ne-therlands; 363-372, 2000.

[1.58] Kuhlmann U, Kühnemund F. Ductility of semi-rigid steel joints. In: Proceedings of the International Colloquium on Stability and Ductility of Steel Structures (SDSS 2002) (Ed.: M. Ivanyi), Budapest, Hungary; 363-370, 2002.

[1.59] Kühnemund F. On the verification of the rotation capacity of semi-rigid joints in steel structures. PhD Thesis (in German), University of Stutt-gart, Stuttgart, Germany, 2003.

[1.60] Huber G, Tschemmernegg F. Component characteristics, Chapter 4 in Composite steel-concrete joints in braced frames for buildings (Ed.: D. Anderson), COST C1, Brussels, Luxembourg; 4.1-4.49, 1996.

[1.61] Witteveen J, Stark JWB, Bijlaard FSK, Zoetemeijer P. Welded and bolted beam-to-column connections. Journal of the Structural Division ASCE; 108(ST2):433-455, 1982.

[1.62] Huber G. Nicht-lineare berechnungen von verbundquerschnitten und biegeweichen knoten. PhD Thesis (in English), University of Innsbruck, Innsbruck, Austria, 1999.

[1.63] Aribert JM, Lachal A, Dinga ON. Modélisation du comportement d’assemblages métalliques semi-rigides de type pouter-pouteau boulon-nés par platine d’extremité. Construction Métallique; 1:25-46, 1999.

[1.64] Weynand K. Sicherheits-und Wirtsschaftlichkeitsuntersuchungen zur anwendung nachgiebiger anschlüsse im stahlbau. PhD thesis (in Ger-


49

man). University of Aachen, Aachen, Germany, 1996. [1.65] Girão Coelho AM. Equivalent elastic models for the analysis of steel

joints. MSc thesis (in Portuguese). University of Coimbra, Coimbra, Portugal, 1999.

[1.66] Simões da Silva LAP, Girão Coelho AM, Neto EL. Equivalent post-buckling models for the flexural behaviour of steel connections. Com-puters and Structures; 77:615-624, 2000.

[1.67] Simões da Silva LAP, Girão Coelho AM. A ductility model for steel connections. Journal of Constructional Steel Research; 57:45-70, 2001.

[1.68] Simões da Silva LAP, Girão Coelho AM, Simões RAD. Analytical Evaluation of the moment-rotation response of beam-to-column com-posite joints under static loading. Steel and Composite Structures; 1(2):245-268, 2001.

[1.69] Simões da Silva LAP, Girão Coelho AM. Mode interaction in non-linear models for steel and steel-concrete composite structural connec-tions. In: Proceedings of the Third International Conference on Coupled Instabilities in Metal Structures (CIMS’2000) (Eds.: D. Camotim, D. Dubina and J. Rondal), Lisbon, Portugal; 605-614, 2000.

[1.70] Simões da Silva L, Calado L, Simões R, Girão Coelho A. Evaluation of ductility in steel and composite beam-to-column joints: analytical evaluation. In: Proceedings of the Fourth International Workshop on Connections in Steel Structures IV: Steel Connections in the New Mil-lennium (Ed.: R. Leon), Roanoke, USA; 2000 (available on CD).

[1.71] Piluso V, Faella C, Rizzano G. Ultimate behavior of bolted T-stubs. I: Theoretical model. Journal of Structural Engineering ASCE; 127(6):686-693, 2001.

[1.72] Piluso V, Faella C, Rizzano G. Ultimate behavior of bolted T-stubs. II: Model validation. Journal of Structural Engineering ASCE; 127(6):694-704, 2001.


50

APPENDIX A: DESIGN PROVISIONS FOR CHARACTERIZATION OF RESIS-TANCE AND STIFFNESS OF T-STUBS

A.1 Basic formulations for prediction of plastic resistance of bolted T-

stubs The equilibrium conditions of the mechanisms illustrated in Fig. 1.10 provide the equations for the evaluation of the corresponding “plastic” resistance, FRd.0. A.1.1 Type-1 mechanism Regarding type-1 mechanism, the three equilibrium equations yield the follow-ing relationships:

1. .002Rd

vF

F Q B= ⇔ = −∑ (A.1)

( )(1). .f Rd f RdM M Bm Q n m M= ⇔ − + =∑ (A.2)

and: (2)

. .f Rd f RdM M Qn M= ⇔ =∑ (A.3) Section (1) corresponds to the critical section at the flange-to-web connection and section (2) is the section at the bolt axis. By equating Eqs. (A.2-A.3), the bolt force is computed as:

.2

f Rdn mB Mmn+

= (A.4)

Eqs. (A.1), (A.3) and (A.4) provide: . .

1. .0

42 f Rd f Rd

Rd

M MF B

n m

= − =

(A.5)

A.1.2 Type-2 mechanism For the second mechanism, the equilibrium equations are written as follows:

2. .002Rd

vF

F Q B= ⇔ = −∑ (A.6)

and: ( )(1)

. .f Rd f RdM M Bm Q n m M= ⇔ − + =∑ (A.7) This mechanism involves bolt fracture. Therefore, the bolt force at “plastic” conditions is equal to

RdB B= (A.8)


51

The plastic resistance F2.Rd.0 is then calculated by means of the following rela-tionship:

.2. .0 .

2 22 22 f Rd RdRd Rd f Rd

M nBmF B Mm n m n m n

+ = − + = + + + (A.9)

By inserting the parameter n mλ = in Eq. (A.9), F2.Rd.0 is re-written as fol-lows:

( )( )

. .2. .0

212 2 21

1 1f Rd f Rd RdRd

RdRd

M MF

m m

λβ λβ

λ β λ

+ −

= = + + +

(A.10)

A.1.3 Type-3 mechanism Type-3 mechanism is characterized by bolt fracture only. The force equilib-rium equation yields:

3. .00 2v Rd RdF F B= ⇔ =∑ (A.11) A.1.4 Supplementary mechanism Supplementary plastic mechanisms corresponding to the metal shear tearing around the bolt head or the washer should also be taken into account, though they are not relevant in most cases. The yielding condition in this case provides the following relationship:

*.2Rd w y f fF d tπ τ= (A.12)

whereby τy.f is the yield shear stress of the flanges. A.2 Influence of the moment-shear interaction on resistance formula-

tions The moment-shear interaction can be approximately assessed by assuming that the external fibres take the bending moment stresses and the internal ones the shear stresses, as illustrated in Fig. A.1 (see reference [1.44]). The reduced plastic flexural resistance of the flanges is given by:

( ) .f f eff y fS

M dS x t x b fσ= = −∫ (A.13)

and the reduced plastic shear resistance is defined as: ( ) .2f f eff y f

S

V dS t x bτ τ= = −∫ (A.14)

Under of the Von Mises criterion, the yield shear stress is computed as:


52

tf

x

xMf

fy

Vf

τy

S

Fig. A.1 Distribution of internal stresses in the plastic condition under com-

bined bending moment and shear force.

3y

y

fτ = (A.15)

From Eqs. (A.13-A.15), the distance x is derived:

.

32 2f f

eff y f

t Vx

b f= − (A.16)

Substitution of x into Eq. (A.13) yields: 2

..

34

ff f Rd

eff y f

VM M

b f= − (A.17)

where Mf.Rd is given by Eq. (1.6). The pure plastic shear resistance, Vf.Rd, is ex-pressed by the following relationship:

. .3f

f Rd eff y f

tV b f= (A.18)

Therefore, Mf.Rd and Vf.Rd are related by means of: 2

. . .3

4 4f

f Rd eff y f f f Rd

tM b f t V= = (A.19)

Eqs. (A.17-A.19) provide the following yielding condition: 2

. .

1f f

f Rd f Rd

M VM V

+ =

(A.20)

The shear force in the T-stub flange is given by: .0

2Rd

fF

V B Q= − = (A.21)

A.2.1 Type-1 mechanism Regarding type-1 mechanism, the equilibrium condition provides (§A.1.1):

1. .0

4 fRd

MF

m= (A.22)


53

Mf and Vf are related by means of the following relationship that derives from Eqs. (A.21-A.22):

1. .0

2 2Rd

f f fF mV M V= ⇔ = (A.23)

The yielding condition brings: 2 2

. . . .

21 1 03

f f f f

f Rd f Rd f f Rd f Rd

M V V VmM V t V V

+ = ⇔ − − =

(A.24)

that has the positive solution:

( )2.

1 31 13

f

f Rd f f

V mV t m t

= + −

(A.25)

By equating Eqs. (1.6), (A.18) and (A.23-A.25), the following relationship is obtained for the plastic resistance associated with type-1 mechanism:

( ) ( )

2

.1. .0 .2 2

2 3 8 31 1 1 13 3

f RdRd eff y f

ff f

MmF m b ft mm t m t

= + − = + −

(A.26)

A.2.2 Type-2 mechanism With reference to type-2 mechanism (§A.1.2), the plastic resistance F2.Rd.0 is given by Eqs. (A.9-A.10). From Eq. (A.21), Mf and Vf are correlated by means of the following relationship:

( )2. .0

2Rd

f f f RdF

V M m n V nB= ⇔ = + − (A.27)

The yielding condition provides: 2 2

. . . . .

4 41 1 03 3

f f f fRd

f Rd f Rd f Rd f f Rd f Rd

M V V VB m nM V V t V V

++ = ⇔ + − − =

(A.28)

The ratio .Rd f RdB V can be written:

.

3 12

fRd

f Rd Rd

tBV m β

= (A.29)

by taking Eqs. (1.6-1.7) and (A.18) into account. Thus, from Eq. (A.28), the following condition is obtained:

( )2

. .

12 41 03

f f

Rd f f Rd f Rd

V Vmt V V

λλβ

++ − − =

(A.30)

The positive solution for Eq. (A.30) is:


54

( )( )

2. 2

2 12 31 1 1

431

f Rd

f Rd f

f

V mV t m

t

λβ

λ

λ

+

= + + −

+

(A.31)

and, therefore, by means of Eqs. (A.7), (A.18) and (A.21), F2.Rd.0 is given by:

( )( )

2

.2. .0 2

2

2 116 31 1 13 4

1

f RdRdRd

f

f

MmFt mm

t

λβ

λ

λ

+

= + + − +

(A.32)

A.3 Influence of the bolt dimensions on resistance formulations To cater for the influence of the bolt finite size on the plastic resistance for type-1 mechanism, Jaspart (see reference [1.20]) provides an alternative formu-lation that assumes that the bolt action is uniformly distributed under the washer, the bolt head or the nut, as appropriate. Consider the half T-stub repre-sented in Fig. A.2, whereby qb is the uniformly distributed bolt action, which is statically equivalent to B, and dw is the diameter of the washer, the bolt head or nut, as suitable. Equilibrium conditions provide the following relationships:

1. .002Rd

v wF

F Q qd= ⇔ = −∑ (A.33)

( )(1). .f Rd b w f RdM M q d m Q n m M= ⇔ − + =∑ (A.34)

and: 2

(2). .8

wf Rd b f Rd

dM M Qn q M= ⇔ − =∑ (A.35)

By solving this system of equations, the prying force, the bolt force and the plastic resistance are obtained:

( )( )

.88

w f Rd

w

m d MQ

mn m n d+

=− +

(A.36)

( )( )

.8 28

f Rdw

w

m n MB qd

mn m n d+

= =− +

(A.37)

and:

( ) ( )( )

.1. .0

32 22

8w f Rd

Rdw

n d MF B Q

mn m n d−

= − =− +

(A.38)

The previous relationships do not allow for moment-shear interaction. By


55

tf

e m

Q

n qb

F1.Rd.0

B

Q

B

(2)

(1) Mf.Rd

Mf.Rd

Fig. A.2 Influence of the bolt-finite size on the T-stub resistance. applying a similar procedure to §A.2.1, the following relationships are derived for the plastic resistance. The plastic T-stub resistance is now given by:

( )( )1. .0

32 28

w fRd

w

n d MF

mn m n d−

=− +

(A.39)

From Eq. (A.21), ( )1. .0 8

2 16wRd

f f fw

mn m n dFV M V

n d− +

= ⇔ =−

(A.40)

By re-arranging the equations, the positive solution for the yielding condi-tion is written as follows:

( )22.

2 31 13 4

f

f Rd f f

V mV t m t

= Γ + − Γ

(A.41)

with:


56

( )8 116

w

w

m dm d

λ λλ− +

Γ =−

(A.42)

Thus:

( )

2

.1. .0 22

16 31 13 4

f RdRd

f f

MmFt mm t

= Γ + − Γ

(A.43)

A.4 Formulations for prediction of elastic stiffness of bolted T-stubs A.4.1 Elastic theory for evaluation of the elastic stiffness of a bolted T-stub The elastic stiffness of a bolted T-stub can be computed by means of the theo-retical model shown in Fig. 1.14. From simple elastic bending theory, applying the double integration method, the deflection of the T-stub web, ∆e.0.u/l is com-puted as:

.0. /e u l∆ ( )2y x m= = ( )2

212 3f

n nm n BEI

= − + − −

( )2 3

22

2 3 3 2n n m Fm m n

− − + − −

(A.44)

If represents the flange inertia and is defined as follows: 3

12eff f

f

b tI

′= (A.45)

where effb′ is the effective width for stiffness calculations, computed per bolt row. Eq. (A.44) can be re-written in a simpler form by bringing in the parame-ters Zf and αf defined in Eqs. (1.16-1.17):

3.0. /

324 4

fe u l f f

Z F BE

α α ∆ = − − (A.46)

The bolt-elastic deformation, ∆e.bt, is given by:

.b

e bts

BLEA

∆ = (A.47)

The compatibility requirement between the bolt and the flange deformation at the bolt centreline yields:

( ) ( )3 2 3.1

3 2 6 82 2 2 2

fe b bf f f f

s

Z BLFy x n BE EA

α α α α∆ = = = − − − =

(A.48)

Thus:


57

( )

3

2 3

3 22

22 6 82

f f f

bf f

s

Z FqB F

LA

α α

α α

− = =

− + (A.49)

where q is defined by Eq. (1.15). From Eqs. (A.46-A.49) the deformation of the upper (u) or the lower (l) T-stub element is given by:

3.0. /

1 3 24 2 2

fe u l f f

Z q FE

α α ∆ = − − (A.50)

and the total deformation: 3

.0 .0. .0.1 3 22 2

fe e u e l f f

Zq F

Eα α ∆ = ∆ + ∆ = − −

(A.51)

The elastic axial stiffness of the bolted T-stub is then defined as:

.03.0 1 3 2

2 2

ee

f f f

F EkZ q α α

= =∆ − −

(A.52)

A.4.2 Simplification of the stiffness coefficients for inclusion in design codes As already mentioned above, Jaspart (see reference [1.46]) simplifies the com-plex above formulae for inclusion in Eurocode 3 – cf. §1.4.1.2. By supposing that the analysis of the T-elements and the bolts is carried out separately, ke.T is derived by means of Eq. (A.52) adopting ≈ ∞sA in Eq. (1.16). Further, if n is taken as equal to 1.25m, as explained in the text, then:

( ) ( )1.25 0.278

2 2 1.25fn m

m n m mα = = =

+ + (A.53)

( ) ( )3 33

3 3 3

2 2 1.2591.125f

eff f eff f eff f

m n m m mZb t b t b t

+ + = = =′ ′ ′

(A.54)

and:

( ) ( )

3 3

2 3 2 3

3 32 2 2 0.278 2 0.2782 2

2 6 8 2 6 0.278 8 0.2782

f f f f

b bf f f f

s

Z Zq

L LZ ZA

α α

α α

− × − × = = =

− + × − × +∞

1.28(A.55)

By replacing the above results in Eq. (A.52): 3 3

.0.5

1.936e Teff f eff f

E m E mkb t b t

= ≈ ′ ′

(A.56)


58

The effective width for stiffness calculations, effb′ , is related to the effective width for strength calculations, beff as explained below. With reference to Fig. 1.16, the elastic bending moment at the T-stub flange (section (1)) is evaluated as follows:

( )1 (0.63 0.13 2.25) 0.3375M Fm Fm= − × = (A.57) If the maximum elastic load corresponds to the formation of a plastic hinge at section (1), then from internal equilibrium conditions and Eq. (A.57) the fol-lowing relationship is obtained:

2(1)max .

1 10.3375 0.3375 4

eff fel y f

b tF M f

m m′

= = (A.58)

The maximum elastic load, Fel, corresponds to 2/3 of the plastic resistance, FRd, as in Eurocode 3 (see reference [1.1]), being FRd given by Eq. (A.5). As the T-stub flange is fixed at the bolt centreline, the only possible collapse mode of the T-stub is that of the complete yielding of the flange (type-1 mechanism). Then, the maximum elastic moment is given by:

2

1. .0 .2 23 3

eff fel Rd y f

b tF F f

m= = (A.59)

by taking Eqs. (A.5) and (1.6) into consideration. By equating Eqs. (A.57-A.58), effb′ is computed as follows:

2 2

. .1 2 2 0.3375 4 0.9

0.3375 4 3 3′

′= ⇔ = × × =eff f eff fy f y f eff eff eff

b t b tf f b b b

m m (A.60)

The bolt elastic deformation can be computed by means of Eq. (A.47) and assuming that the prying effect increases the bolt force from 0.5F to 0.63F. The bolt elastic stiffness is expressed as the ratio between the tensile force F and ∆e.bt:

..

1.60.63

se bt

be bt b

s

EAF FkFL L

EA

= = ≈∆

(A.61)

59

PART II: FURTHER DEVELOPMENTS ON THE T-STUB MODEL

61

2 IMPROVEMENTS ON THE T-STUB MODEL: INTRODUCTION 2.1 INTRODUCTION The T-stub model is widely accepted as a simplified model for the characteri-zation of the behaviour of the tension zone of a bolted joint, which is often the most important source of deformability of the whole joint. Within the frame-work of the component method, this connection behaviour is modelled by means of a F-∆ response that is intrinsically nonlinear, due to mechanical and geometrical nonlinearities and contact phenomena. Current design specifica-tions based on the T-stub model rely on pure plastic yield line mechanisms and do not allow for a complete characterization of the deformation capacity at ul-timate conditions. Modern design codes, as the Eurocode 3 [2.1], approximate the nonlinear component behaviour by means of a linearized response, characterized by a full “plastic” resistance, FRd.0 and initial stiffness, ke.0. The design rules for the pre-diction of both parameters are given in §1.4.1. Fig. 2.1 illustrates the bilinear approximation of the actual behaviour of an isolated T-stub connection tested by Bursi and Jaspart [2.2]. In this particular case, the plastic mechanism of the connection is of type-1, which corresponds to double curvature of the flange, owing to the formation of plastic hinges at the bolt axes and at the flange-to-web connection. Therefore, the connection has considerable deformation ca-pacity [2.3]. No quantitative guidance is given in the code to evaluate this property though, and therefore no limits are imposed to the extension of the plastic plateau. This part of the research work is devoted to the characterization of the full nonlinear (monotonic) behaviour of isolated T-stub connections, in order to provide insight into the actual component behaviour, failure modes and defor-mation capacity. Tests, both experimental and numerical, were hence carried out at the Delft University of Technology and at the University of Coimbra to fulfil those objectives. Additionally, this test programme clarified some aspects related to the differences between the assembly types. The T-stub assemblage may comprise hot rolled profiles or welded plates as T-stub elements, denomi-nated HR-T-stubs and WP-T-stubs, respectively. The current approach to ac-count for the behaviour of T-stubs made up of welded plates consists in a mere extrapolation of the existing rules for the other assembly type. This assumption can be erroneous and can lead to unsafe estimations of the characteristic prop-erties, as reported earlier by the author in technical literature [2.4-2.5]. The following sections present and discuss the results of thirty-two experi-mental tests and three numerical tests on WP-T-stubs, and twenty-six numeri-cal tests on HR-T-stubs. The experimental programme is described in Chapter

Further developments on the T-stub model

62

0

30

60

90

120

150

180

210

0 1 2 3 4 5 6 7 8 9 10

Total deformation (mm)

Tot

al a

pplie

d lo

ad (k

N)

Actual response

Eurocode 3 bilinear approximation

Fig. 2.1 Simplified approximations of the response of a bolted T-stub con-

nection. 3. Detailed results for the benchmark specimen are also provided. The numeri-cal model is fully described in Chapter 4 where the calibration procedure is also explained. Chapter 5 is completely devoted to the parametric study that highlights the main parameters affecting the deformation capacity of bolted T-stubs and assesses, both qualitatively and quantitatively, their influence on the overall behaviour of the connection. Moreover, this study adds further exam-ples to a database for future validation of a simplified analytical (beam) model that is addressed in Chapter 6. This model attempts at filling in some code gaps on the characterization of the component behaviour, namely post-limit stiff-ness, kp-l.0 and deformation capacity, ∆u.0. 2.2 FAILURE MODES In this work, two categories of failure modes are considered: plastic mecha-nisms, that rely on pure “plastic” conditions and ultimate conditions, which correspond to cracking of the material. “Plastic” failure mechanisms indicate the strength of connections for design purposes whereas ultimate conditions in-dicate failure of the connection after certain deformation. The three possible “plastic” failure mechanisms have been briefly described in §1.4.1 and corre-spond to: (i) type-1: complete yielding of the flange, with the development of four plastic hinges (double curvature bending), (ii) type-2: partial yielding of the flange with bolt “plastic” failure, with the development of two plastic hinges at the flange-to-web connection (single curvature bending) and (iii) type-3: bolt “plastic” failure without yielding of the flanges (the flange remains virtually undeformed). With respect to ultimate conditions, four different ty-pologies for the failure modes of a bolted T-stub connection are defined: (i) type-11, characterized by a plastic type-1 mode and cracking of the flange ma-

Improvements on the T-stub model: introduction

63

terial at ultimate conditions, (ii) type-13, also a type-1 plastic mechanism but with fracture of the bolt at limit conditions, (iii) type-23, where the plastic mode involves both flange and bolt and the deformation capacity is governed by the bolt itself and (iv) type-33, a type-3 “plastic” mode and deformation ca-pacity determined by bolt fracture.

The failure mechanism typology, in both cases, is governed by the β-ratio, which is a resistance-based parameter that expresses the ratio between the flex-ural resistance of the flanges and the axial strength of the bolts. It depends ex-clusively on geometric and mechanic characteristics of the connection. In plas-tic conditions, this parameter is defined by Eq. (1.7). With reference to ultimate conditions, the β-ratio, βu, is given by:

.2 f uu

u

MB m

β = (2.1)

whereby Mf.u is the ultimate flexural resistance of the T-stub flanges and Bu is the tensile strength of the bolts. According to Piluso et al. [2.6], the limit value for this parameter to have a collapse failure mode governed by cracking of the flange material is:

( ).lim2 1 1

2 1 8w

udn

λβ λλ

= − + + (2.2)

Otherwise (βu > βu.lim), bolt fracture is likely to determine the ultimate condi-

tions. Table 2.1 summarizes the various failure mechanism types. The ultimate tensile strength of a bolt subjected to an axial loading is evalu-ated by assuming that tension fracture of the bolt occurs before stripping of the threads. Therefore, the axial ultimate strength is calculated as the ultimate strength of the bolt material multiplied by the effective tensile area:

.u u b sB f A= (2.3) For computation of the ultimate flexural resistance of the flange, two alterna-tive expressions are suggested. Gioncu et al. [2.7] propose the following rela-tionship:

..

f Rdf u

y

MM

ρ= (2.4)

Table 2.1 Failure mechanisms typologies.

“Plastic” conditions Ultimate conditions Typology βRd Typology βu

Type-11 .limuβ≤ Type-1 2

2 1λ

λ≤

+

Type-13

Type-2 2

2 1λ

λ>

+ but 2≤ Type-23

Type-3 2> Type-33

.limuβ>


64

being y y uf fρ = the yield ratio and Mf.Rd the full plastic flexural resistance of the T-stub flanges, defined in Eq. (1.6). Faella and co-authors [2.6,2.8] present an alternative expression that derives from simple equilibrium equations be-tween internal stresses and the external moment. They assumed that the flange behaves as a rectangular compact section of width beff and thickness tf. The flange constitutive law is approximated by means of a quadrilinear model (Fig. 2.2). The following relationship is therefore obtained:

( )

( )

.. 2

13 1 22

1 2

f y h h hf u u h

u uu

h u m mu m

u u

M EM

E

E EE

µ µµ µ

µ µµ

µ µµ µ

µ µ

= − + − − + −

−

− − − +

(2.5)

whereby Mf.y is the bending moment corresponding to first yielding of the flange:

2

. . .2

6 3f

f y y f eff f Rd

tM f b M= = (2.6)

and: h m u

h m uy y y

ε ε εµ µ µ

ε ε ε= = = (2.7)

The ratio . .f u f yM M is a parameter that also depends exclusively on the me-chanical properties of the flange material and can be written in a formally equivalent expression to Eq. (2.4):

.. *

f Rdf u

y

MM

ρ= (2.8)

σ

ε

fy

εy εh εm εu

E

Eh

Eu

Fig. 2.2 Flange piecewise material constitutive law (qaudrilinear approxima-

tion proposed by Faella and co-authors [2.8]).

Improvements on the T-stub model: introduction

65

with (cf. Eqs. (2.5-2.6)):

( )

( )

*2

1

4 13 1 23

1 2

h h hy u h

u uu

h u m mu m

u u

EE

E EE

µ µρ µ µ

µ µµ

µ µµ µ

µ µ

−

= − + − − + −

−− − − +

(2.9)

2.3 REFERENCES [2.1] European Committee for Standardization (CEN). PrEN 1993-1-8:2003,

Eurocode 3: Design of steel structures, Part 1.8: Design of joints, Stage 49 draft, May 2003, Brussels, 2003.

[2.2] Bursi OS, Jaspart JP. Benchmarks for finite element modelling of bolted steel connections. Journal of Constructional Steel Research; 43(1):17-42, 1997.

[2.3] Zoetemeijer P. Summary of the research on bolted beam-to-column connections. Report 25-6-90-2. Faculty of Civil Engineering, Stevin Laboratory – Steel Structures, Delft University of Technology. 1990.

[2.4] Girão Coelho AM, Bijlaard F, Simões da Silva L. On the deformation capacity of beam-to-column bolted connections. Document ECCS-TWG 10.2-02-003, European Convention for Constructional Steelwork – Technical Committee 10, Structural connections (ECCS-TC10), 2002.

[2.5] Girão Coelho AM, Bijlaard F, Simões da Silva L. On the behaviour of bolted end plate connections modelled by welded T-stubs. In: Proceed-ings of the Third European Conference on Steel Structures (Eurosteel) (Eds.: A. Lamas and L. Simões da Silva), Coimbra, Portugal, 907-918, 2002.

[2.6] Piluso V, Faella C, Rizzano G. Ultimate behavior of bolted T-stubs – I: Theoretical model. Journal of Structural Engineering ASCE; 127(6):686-693, 2001.

[2.7] Gioncu V, Mateescu G, Petcu D, Anastasiadis A. Prediction of available ductility by means of local plastic mechanism method: DUCTROT computer program, Chapter 2.1 in Moment resistant connections of steel frames in seismic areas (Ed.: F. Mazzolani). E&FN Spon, London, UK; 95-146, 2000.


67

3 EXPERIMENTAL ASSESSMENT OF THE BEHAVIOUR OF T-

STUB CONNECTIONS 3.1 INTRODUCTION A series of thirty-two tests on bolted T-stub connections made up of welded plates is presented in this chapter. Although T-stubs have been used for many years to model the tension zone of bolted joints, the research was mainly con-centrated on rolled profiles as T-stub elements. To extend this model to the case of welded plates as T-stub elements, a test programme was undertaken at the Delft University of Technology. It provides insight into the behaviour of this different type of assembly, in terms of resistance, stiffness, deformation capacity and failure modes, in particular. The key variables tested include the weld throat thickness, the size of the T-stub, the type and diameter of the bolts, the steel grade, the presence of transverse stiffeners and the T-stub orientation. The results show that the welding procedure is particularly important to en-sure a ductile behaviour of the connection. Most of the T-stubs failed by ten-sion fracture of the bolts after significant yielding of the flanges. However, some of the specimens have shown early damage of the plate material near the weld toe due to the effect of the welding consumable that induced premature cracking and reduced the overall deformation capacity. A solution to this prob-lem was given by setting requirements to the weld metal to be used. This chapter describes the main collapse modes observed and gives detailed information on the benchmark specimen WT1 (eight tests). The remaining re-sults are discussed in Chapter 5, as part of the parametric study presented. 3.2 DESCRIPTION OF THE EXPERIMENTAL PROGRAMME 3.2.1 Geometrical properties of the specimens The basic configuration of the test specimens comprised two plates of 10 mm thickness. The plates were welded together by means of a continuous 45º-fillet weld with similar plate characteristics. Snug-tightened high-strength bolts fas-tened the T-stub elements. The unstiffened specimens were designed to fail ac-cording to a plastic collapse mode 1 that ensures a good ductility of the connec-tion [3.1]. The general characteristics of the specimens are given in Table 3.1. For no-tation the reader should refer to Fig. 3.1. Both nominal and actual properties are reported. The actual geometry was measured before testing the specimens and is listed in Table 3.1 as an average value of the several T-elements from


68

each series. In series WT7 and WT57, the T-stub elements were fastened by means of one bolt row only due to equipment limitation. The two T-stubs for most series were symmetrical. For series WT64A and WT64B, the T-stub ele-ments included a stiffener only on one side of the connection. Table 3.1 Tests description [dimensions (nominal and averaged actual values

– in bold) in mm; //: T-stub elements parallel, ⊥: T-stub elements orientated at right angles].

Geometry Test ID # tf tw b p e1 w e aw 10 10 45 50 20 90 30 WT1 8 10.32 10.93 45.05 49.8 20.1 89.7 30.0 5 10 10 45 50 20 90 30 WT2A 2 10.27 10.54 45.0 49.9 20.0 89.9 29.9 3 10 10 45 50 20 90 30 WT2B 2 10.29 10.69 44.95 49.9 20.0 89.9 29.9 7 10 10 75 90 30 90 30 WT4A 2 10.39 11.00 74.85 89.6 30.1 89.7 30.0 5 10 10 75 90 30 90 30 WT4B 1 10.37 10.92 74.8 89.8 29.9 89.8 29.9 5 10 10 45 50 20 90 30 WT51 2 9.98 10.01 45.0 50.7 19.6 90.1 30.2 5 10 10 45 50 20 90 30 WT53C 1 10.09 10.10 45.05 50.0 20.0 90.1 30.0 5 10 10 45 50 20 90 30 WT53D 1 10.14 10.22 45.0 49.9 20.0 90.0 30.0 5 10 10 45 50 20 90 30 WT53E 1 10.09 10.17 44.65 49.2 20.0 90.0 30.1 5 10 10 45 50 20 90 30 WT61 2 10.31 10.93 45.1 49.9 20.1 89.8 29.4 5 10 10 75 90 30 90 30 WT64A 1 10.28 10.94 74.95 90.0 29.9 89.7 29.8 5 10 10 75 90 30 90 30 WT64B 2 10.42 10.82 74.9 89.9 29.9 89.7 30.0 5 10 10 75 90 30 90 30 WT64C 1 10.30 10.84 75.1 89.7 30.2 89.8 29.9 5 10 10 75 30 90 30 WT7_M12 1 10.33 10.84 75.6 30.0 89.9 29.9 5 10 10 75 30 90 30 WT7_M16 1 10.33 10.81 74.9 30.0 89.9 29.8 5 10 10 75 30 90 30 WT7_M20 1 10.33 10.87 75.2 29.9 89.8 29.7 5 10 10 75 30 90 30 WT57_M12 1 10.09 10.18 75.0 30.0 89.7 30.2 5 10 10 75 30 90 30 WT57_M16 1 10.16 10.18 75.3 30.0 90.0 30.1 5 10 10 75 30 90 30 WT57_M20 1 10.15 10.15 75.1 30.0 90.0 30.2 5

Experimental assessment of the behaviour of T-stub connections

69

3.2.2 Mechanical properties of the specimens 3.2.2.1 Tension tests on the bolts In order to characterize the mechanical properties of the M12, grade 8.8 and 10.9 bolts, two series of experiments were performed. In the first series, the ac- Table 3.1 Tests description (cont.).

Bolt Materials Test ID # φ # Type Plate Bolt

Stiff. Orient.

12 WT1 8 11.8 4 ST S355 8.8 No // 12 WT2A 2 11.8 4 ST S355 8.8 No // 12 WT2B 2 11.8 4 ST S355 8.8 No // 12 WT4A 2 11.8 4 ST S355 8.8 No // 12 WT4B 1 11.8 4 ST S355 8.8 No ⊥ 12 WT51 2 11.8 4 ST S690 8.8 No // 12 WT53C 1 4 FT S690 8.8 No // 12 WT53D 1 11.9 4 ST S690 10.9 No // 12 WT53E 1 4 FT S690 10.9 No // 12 WT61 2 11.9 4 ST S355 8.8 Yes // 12 WT64A 1 11.8 4 ST S355 8.8 Yes // 12 WT64B 2 11.8 4 ST S355 8.8 Yes ⊥ 12 WT64C 1 11.8 4 ST S355 8.8 Yes // 12 WT7_M12 1 11.9 2 ST S355 8.8 No // 16 WT7_M16 1 2 FT S355 8.8 No // 20 WT7_M20 1 2 FT S355 8.8 No // 12 WT57_M12 1 2 FT S690 8.8 No // 16 WT57_M16 1 2 FT S690 8.8 No // 20 WT57_M20 1 2 FT S690 8.8 No //


70

Section xx

Plan

b e1

p

x x

ee wbf

aw

b e1

Fig. 3.1 T-stub geometry: notation. tual bolt (short-threaded, ST or full-threaded, FT) was tested under tension (Fig. 3.2a). Failure always occurred in the threaded region. This type of test did not provide enough data to determine the Young modulus and the proof strength of the bolt. Then, in a second test series, the bolts were machined so that the threads within the bolt grip were removed and a constant diameter was obtained (Fig. 3.2b). This procedure was not expected to introduce major influ-ences on the bolt behaviour since the removal of the material was limited to the threads, even though the bolt mechanical properties were not uniform. Both specimen types were tested in tension under displacement control in a special test rig as shown in Fig. 3.3. The elongation behaviour of the bolt was measured by means of a measuring bracket (or horseshoe device, also illus-trated in Fig. 3.3) in the first series of tests and by means of internal strain gauges in the second. The strain gauges (TML-BTM-6C) could measure strains up to 6000 µm/m. The graphs from Fig. 3.4a plot the bolt elongation curve for


71

one of the short-threaded M12 grade 8.8 (group 2) tested bolts. The graph in-cludes the load cell displacement results and the measuring bracket data up to its removal in the elastic range. Clearly, the results obtained from the measur-ing bracket are stiffer since the displacement of the actuator also includes the slippery of the clamps. Fig. 3.4b traces the force-strain results obtained for an identical bolt type (now chosen from the second test series). Naturally, the

Group 1 (8.8) Group 3 (10.9) Group 2 (8.8) Group 4 (10.9) (i) Full-threaded specimens. (ii) Short-threaded specimens. (a) First series: “actual” bolts.

Group 1 (8.8) Group 3 (10.9) Group 2 (8.8) Group 4 (10.9) (i) Full-threaded specimens. (ii) Short-threaded specimens. (b) Second series: machined bolts. Fig. 3.2 Bolt specimens: two series of tests.

Fig. 3.3 Test rig for the bolt tensile testing and horseshoe device.


72

0

10

20

30

40

50

60

70

80

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0

Deformation (mm)

App

lied

load

(kN

)

Results from the load cell

Results from the measuring bracket

(a) Bolt elongation behaviour.

0

10

20

30

40

50

60

70

80

0 600 1200 1800 2400 3000 3600 4200 4800 5400 6000 6600

Strain (µm/m)

App

lied

load

(kN

) Maximum load

(b) Bolt strain behaviour.

0

10

20

30

40

50

60

70

80

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

Bolt elongation (mm)

App

lied

load

(kN

)

Results from the measuring bracket

Results from the strain gauge

(c) Comparison of the bolt experimental (elastic) results for both test series. Fig. 3.4 Bolt tensile response (e.g. bolt from group 2).


73

maximum load level in the latter case decreases, as the bolt tensile area is smaller. To verify the accuracy of the special bolt-measuring bracket, Fig. 3.4c compares the elastic deformation of two bolts from the same group 2, repre-senting each bolt series. The bolt elongation is given by ∆b = εbLg, in the case of the strain measurements, being εb the bolt strain and Lg the grip length. The results are identical for both series, which means that the measuring bracket that is much simpler to attach can be used to assess the bolt elongation behav-iour in future tests. Bolts M16 and M20 used in series WT7 and WT57 have not been tested. Table 3.2 summarizes the average relevant characteristics for the tested bolts. Usually, for the bolts, the following parameters are measured: Young modulus, E, ultimate or tensile stress, fu and ultimate strain, εu. Table 3.2 Average characteristic values for the bolts.

Bolt grade Type Group E (MPa) fu (MPa) εu FT 1 216942 968.36 0.20 8.8 ST 2 221886 919.91 0.13 FT 3 217060 1196.37 0.14 10.9 ST 4 217824 1165.97 0.11

3.2.2.2 Tension tests on the structural steel The test programme included two different steel grades: S355 and S690. Ac-cording to the European Standards EN 10025 [3.2] and EN 10204 [3.3], the steel qualities were S355J0 (ordinary steel) and N-A-XTRA M70 (high-strength steel for plates), respectively. Table 3.3 summarizes the chemical composition for the two steel grades. The coupon tension testing of the structural steel material was performed according to the RILEM procedures [3.4]. The plate coupon specimens were of a standard type for flat materials and were of full thickness of the product [3.4]. Fig. 3.5 depicts the test arrangements for the standard tensile test. The experi-ments were driven under displacement control. The engineering stress-strain relation for the web and flange strips is represented in Fig. 3.6, for one of the tested strip-coupons. The four typical regions of the stress-strain curve of a low carbon structural steel are very clear: linear elastic region, yield plateau, strain hardening region and strain softening or necking portion, after reaching the maximum load. The average characteristics are set out in Table 3.4. In this table the values for the Young modulus, E, the strain hardening modulus, Eh, the static yield and tensile stresses, fy and fu, the yield ratio, ρy the strain at the strain hardening point, εh, the uniform strain, εuni, and the ultimate strain, εu, are given. The stress values indicated in the table correspond to the static stresses, which are the stress values obtained at zero strain rate, i.e. during a hold on of the defor-


74

(a) Test set up. (b) Detail of the

extensometer. (c) Coupon necking.

(d) S355 coupons af-ter failure.

Fig. 3.5 Tensile coupon tests.

0

150

300

450

600

750

900

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Strain (m/m)

Stre

ss (M

Pa)

Web strip coupon

Flange strip coupon

(a) Steel grade S355.

0

150

300

450

600

750

900

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Strain (m/m)

Stre

ss (M

Pa)

Web strip coupon

Flange strip coupon

(b) Steel grade S690. Fig. 3.6 Engineering stress-strain relation.


75

mation driven experiment. It has been observed that the static stresses were reached after a hold on of circa one minute. The total hold on lasted for three minutes. The yield ratio gives an idea on the material ductility. Gioncu and Mazzolani suggest that a good ductility is ensured if 0.5 ≤ ρy ≤ 0.7 [3.5]. High strength steel grades with ρy > 0.9 show a rather poor structural ductility [3.5]. That is the case of the steel grade S690 (Table 3.4). In the author’s opinion, these values are rather conservative. Eurocode 3 indicates that a good material ductility is guaranteed if ρy ≤ 0.83 (recommended value for steel grades up to S460). The assurance of a good material ductility does not necessarily imply that the whole structure is ductile. The structural ductility depends on the yield ratio but especially on the structural discontinuities. Table 3.3 Chemical composition of the structural steels according to the

European standards.

%C max.

%Mn max.

%Si max.

%P max.

%S max.

%N max.

%CEV max.

S355J0 0.20 1.60 0.55 0.040 0.045 0.009 0.40 N-A-XTRA M70

0.20 1.60 0.80 0.020 0.010 0.48

Table 3.4 Average characteristic values for the structural steels.

Steel grade

Strip # E (MPa)

Eh (MPa)

fy (MPa)

fu (MPa)

ρy

Web 2 209211 2145 391.54 493.80 0.793 S355 Flange 2 209856 2264 340.12 480.49 0.708 Web 2 208895 2201 706.31 742.96 0.950 S690 Flange 2 204462 2495 698.55 741.28 0.940

Steel grade

Strip # εh εuni εu

Web 2 0.019 0.163 0.300 S355 Flange 2 0.015 0.224 0.361 Web 2 0.018 0.082 0.160 S690 Flange 2 0.014 0.075 0.174

3.2.3 Testing procedure The specimens were subjected to monotonic tensile force, which was applied to the webs that were clamped to the testing machine (Schenck, maximum test load 600 kN, maximum piston stroke ±125 mm) as shown in Fig. 3.7. The tests were carried out under displacement control with a speed of 0.01 mm/s up to collapse of the specimens. Two different ultimate failure modes were observed,


76

as explained below in the text: (i) fracture of the bolts and (ii) cracking of the flange near the weld toe. The gap of the flanges was measured at opposite sides of the specimen, in the centreline of the webs by means of Linear Variable Displacement Trans-ducers (LVDTs). The bolt elongation was measured with a measuring bracket that was removed prior to collapse, as before, so that it was not damaged. In some of the specimens, internal strain gauges similar to those used in the ten-sion tests were attached to the bolts. Strain gauges TML (maximum strain 30000 µm/m) were used to monitor strains in the flange. Due to cost restric-tions not all specimens have been instrumented with strain gauges. For illustra-tion, Fig. 3.8 shows the instrumentation of some of the specimens. Before installation of the specimens into the testing machine, the dimen-sions of the plates were recorded and the bolts were hand-tightened and meas-ured. The specimen was next placed into the machine and aligned, so that the clamping devices were centred with respect to the webs. The bolts were subse-quently fastened by using an ordinary spanner (45º turn) and measured. After that, the measurement devices and strain gauges, if any, were connected. The test itself then started with loading of the specimen up to 2/3FRd.0, which corre-sponded to the theoretical elastic limit. FRd.0 was determined according to Eurocode 3. Complete unloading followed on and the specimen was then re-loaded up to collapse. In this third phase the test was interrupted at the load

(a) Unstiffened specimen. (b) Stif. spec. (T-stubs orientated at 90º).

(c) Detail of the measuring devices. Fig. 3.7 Test set up for testing WP-T-stubs.


77

20.0 mm

50.0 mm

20.0 mm

30.0 mm 90.0 mm 30.0 mm

Left side Right side5.0 mm SG5

SG4

SG3

SG2

SG1

SG6

SG7

20.0 mm

50.0 mm

20.0 mm

30.0 mm 90.0 mm30.0 mm

Left side Right side

SG4

SG3

SG2

SG6xSG6z

SG7xSG7z

(a) Specimens WT1b/c/d/h, WT51a. (b) Specimen WT51b.


LB RB

LF RF

SG3

SG2

SG1

90.0 mm

30.0 mm

30.0 mm

30.0 mm90.0 mm 30.0 mm


LB RB

LF RF

SG5

SG4

SG6

HP1

HP4

HP2

HP3

30.0 mm 90.0 mm30.0 mm

90.0 mm

30.0 mm

30.0 mm

(i) Upper profile. (ii) Lower profile.

(c) Specimen WT64Bb. Left side Right side

30.0 mm90.0 mm30.0 mm

90.0 mm

30.0 mm

30.0 mm

16.0

SG7

SG1SG2

SG6

SG3

48.0

40.0

14.0

SG5SG4

HP3

HP2

HP1

(d) Specimen WT64C (upper profile; lower profile not instrumented). Fig. 3.8 Instrumentation of some of the tested specimens (SG: strain gauge;

L: left; R: right; B: back; F: front; HP: LVDT).


78

levels of 2/3FRd.0, FRd.0, at the knee-range and after this level each six minutes, equivalent to an actuator displacement of 3.6 mm. The knee-range of the F-∆ curve (K-R) corresponds to the transition from the stiff to the soft part. The hold on of the test lasted for three minutes and intended to record the quasi-static forces. Regarding the stiffened specimens, the former load levels were taken as equal to the parent unstiffened cases. For the rotated configurations, the lower hydraulic actuator was rotated 90º so that the T-stub element was orientated at a right angle to the upper element (Fig. 3.7b). 3.2.4 Aspects related to the welding procedure In this type of T-stub assembly, two plates, web and flange, are welded to-gether by means of a continuous 45º-fillet weld. The fillet welds were done in the shop in a down-hand position. The procedure involved manual metal arc welding in which a consumable electrode was used. Three main zones could be identified after the welding process [3.6]: the weld metal (WM), the heat af-fected zone (HAZ) and the base metal (BM), which is the part of the parent plate that is not influenced by the heat input. The HAZ is the portion of the plate on either side of the weld affected by the heat in which metal suffers thermal disturbances and therefore structural modifications that may include re-crystallization, refining and grain growth [3.7]. The hot WM causes the plate to bend up due to shrinkage during cooling down and so considerable force is exerted to do this [3.7]. Residual stresses can then be expected in the HAZ. Obviously, this will influence the overall behaviour of the connection. The composition of the WM deposited with the electrode compared to that of the BM is of great importance, since this will naturally alter the properties of the steel at and near the weld toe [3.7]. For each steel quality there are often a large number of electrode types to choose from. In this test programme two dif-ferent types of carbon steel covered electrodes were used: rutile and basic (Ta-ble 3.5). The distinction between them lies in the type of covering that result in different performances. Rutile electrodes have high titanium oxide content and produce easy striking with a stable arc and low spatter. They are commonly known as general-purpose electrodes. The mechanical strength is generally classed as moderate. This type of consumable normally has high hydrogen con-tent (higher than 10 ml/100 g all-WM). Basic electrodes offer improved me-chanical properties and superior weld penetration. The mechanical strength is generally classed as good to high and the resistance to cracking is enhanced. They have a high proportion of calcium carbonate and calcium fluoride in the coating, which makes it more fluid than rutile coatings and also fast freezing. The hydrogen content is generally lower, which reduces the cracking problem. Table 3.5 summarizes the main characteristics of the various electrodes. The classification indicated in the table complies with the European standard EN 499 [3.8]. Regarding this classification standard, the first two digits desig-


79

nate the minimum yield strength of the deposited WM and also refer to the limit boundaries of the tensile strength and the minimum elongation of the WM. For instance, the Kardo electrode (E35) has a minimum yield stress of 350 MPa (measured value: 396 MPa), the tensile strength varies between 440 and 570 MPa (measured value: 453 MPa) and a minimum elongation of 22%. The latter value decreases as the strength of the WM increases [3.8], thus re-ducing the deformation capacity of the weld. Table 3.6 lists the electrodes used in the welding of each specimen. Clearly, the Kardo and the Conarc 70G were the most utilized electrode types. These are soft, low hydrogen electrodes. The experiences on the consumable per-formance were carried out in test series WT1. Fig. 3.9 shows the influence of the deposited WM on the global behaviour of the eight specimens from series WT1. Essentially, such behaviour mainly depends on the mismatch in me-chanical properties between the three different zones and the hydrogen content [3.6-3.7]. In the elastic range, the deformation behaviour is not too much de-pendent on the WM properties. However, when the connection is plastically deformed, the choice of the electrode type becomes crucial. The graphs show that the deformation capacity of the joint was greatly influenced by the depos- Table 3.5 Characteristics of the electrodes and mechanical properties of the

deposited weld metal.

Brand Type Classif. Actual mech. prop. name (EN 499) fy (MPa) fu (MPa) Cumulo Rutile E38 O R12 Not provided. Conarc 51 Basic E42 4 B12 H5 Not provided. Kardo Basic E35 4 B32 H5 396 453 Conarc 70G Basic E55 4 B32 H5 600 655 Brand Chemical composition name %C %Mn %Si %P %S %Ni Cumulo 0.06 0.50 0.30 Conarc 51 Not provided. Kardo 0.016 0.30 0.21 0.010 0.008 0.03 Conarc 70G 0.06 1.2 0.4 0.014 0.009 1.0 Table 3.6 Types of electrode used in the tests.

Test ID Electrode Test ID Electrode WT1a/b/c Cumulo Series WT51 Conarc 70G WT1d Conarc 51 Series WT53 Conarc 70G WT1e/f Cumulo Series WT61 Kardo WTg/h Kardo Series WT64 Kardo Series WT2 Kardo Series WT7 Kardo Series WT4 Kardo Series WT57 Conarc 70G


80

0

30

60

90

120

150

180

210

0 2 4 6 8 10 12 14 16 18 20 22

Deformation (mm)

Tota

l app

lied

load

(kN

)WT1b

WT1c

WT1e WT1a

WT1f

(a) Cumulo electrode (rutile).

0

30

60

90

120

150

180

210

0 2 4 6 8 10 12 14 16 18 20 22

Deformation (mm)

Tota

l app

lied

load

(kN

)

WT1c

WT1d

(b) Conarc 51 and Cumulo electrodes (basic and rutile, respectively).

0

30

60

90

120

150

180

210

0 2 4 6 8 10 12 14 16 18 20 22

Deformation (mm)

Tota

l app

lied

load

(kN

)

WT1h

WT1d

WT1c

WT1g

(c) Kardo electrode (basic). Fig. 3.9 Performance of the different electrode types for steel grade S355.


81

(i) Deformation at failure (WT1e). (ii) Detail: typical crack (WT1a). (a) Cumulo electrode (rutile).

(i) Deformation at failure. (ii) Detail: crack.

(iii) Detail: bolts (no bending deforma-tions).

(b) Cumulo electrode (rutile) and aw = 8.0 mm (WT1f).

(c) Conarc 51 electrode (basic) (WT1d). (d) Kardo electrode (basic) (WT1h). Fig. 3.10 Illustration: specimens (series WT1) after failure for comparison of the

effect of the deposited WM with different electrode-types.


82

ited WM mechanical properties. Both Cumulo and Conarc 51 electrodes in-duced an early cracking of the plates at the HAZ, limiting the deformation ca-pacity of the T-stub and did not allow for the effective use of the bolts (Figs. 3.9 and 3.10a-c). Also, the scatter of the responses was not acceptable (Fig. 3.9a). Note that the load-carrying behaviour of WT1f deviates even more from the remaining tests because, by mistake, the actual weld throat thickness was 8.0 mm instead of the specified value of 5.0 mm. The electrode that provided the best ductility to the overall connection (steel grade S355) is the Kardo (Figs. 3.9-3.10). Therefore, it was the most suitable consumable and it was used in the rest of the specimens to weld the plates. This electrode is classified as an evenmatch electrode as the nominal properties of the WM and the BM are identical. Finally, regarding the welding of the plates made up of S690, the electrode Conarc 70G, specified by the distributor as the proper electrode type for that steel quality, guaranteed a performance identical to the Kardo for S355. 3.3 EXPERIMENTAL RESULTS 3.3.1 Reference test series WT1 Test series WT1 includes eight specimens that differ in the electrode type used in the welding procedure, as explained above. It has been shown previously that the Kardo electrode seemed to be the most suitable in terms of overall connection performance (Fig. 3.9). For further analysis consider specimens WT1g/h whose collapse was determined by bolt fracture with some damage of the plate in the HAZ in the first case, as well (Figs. 3.10d and 3.11). The load-carrying behaviour of the above specimens is compared with the Eurocode 3 [3.1] predictions for elastic stiffness and plastic resistance in Fig. 3.12 (results in Tables 3.7-3.8). Notice that the experimental F-∆ curves de-picted throughout the text correspond to the third part of the test – reloading up to collapse (cf. §3.2.3). Eurocode 3 often underestimates both properties – see also Table 3.8. The experimental global elastic stiffness is computed by means of a regression analysis of the unloading portion of the F-∆ curve (which is not traced in the graphs). By comparing the results, there is a ratio between the ex-periments and the code predictions of 1.58 and 1.48 for WT1g and WT1h, re-spectively. In addition, if the lower bound of the knee-range of the F-∆ curve is compared with FRd.0 predicted by Eurocode 3, deviations of 0.84 (WT1g) and 0.81 (WT1h) are observed. The remaining characteristics of the F-∆ response (post-limit stiffness and deformation capacity) cannot be compared with any code provisions since it does not cover the post-limit behaviour. Table 3.8 sets out the values of maxi-mum load, Fmax, post-limit stiffness (also determined by means of a regression analysis of the post-limit response) and deformation capacity, taken as the de-


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formation level corresponding to Fmax. Table 3.8 summarizes the results for the eight tests, corroborating the above-mentioned scatter of results between the six initial specimens. Figs. 3.13-3.14 show other results also obtained in this test series. The results of the

(a) Detail of WT1h.

(b) WT1g: front view. (c) WT1g: top view. Fig. 3.11 Specimens WT1g/h after failure. Table 3.7 Eurocode 3 predictions of (global) initial stiffness and plastic resis-

tance (evaluated using the average real dimensions of the speci-mens).

Test ID ke.0 (kN/mm)

FRd.0 (kN)

Test ID ke.0 (kN/mm)

FRd.0 (kN)

WT1 217.28 96.66 WT53D 194.16 190.66 WT2A 175.78 88.88 WT53E 189.84 187.23 WT2B 254.24 102.18 WT57_M12 151.69 107.49 WT4A 343.86 163.47 WT57_M16 163.02 159.18 WT7_M12 168.64 81.00 WT57_M20 166.89 158.41 WT7_M16 179.58 80.22 WT61 380.92 153.19 WT7_M20 186.44 80.73 WT64A 388.02 172.85 WT51 184.16 178.90 WT64C 425.46 182.45 WT53C 190.10 187.35


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bolt elongation behaviour for specimen WT1h are given in Fig. 3.13. The graph does not apply up to collapse since the bolt deformation was measured by means of the horseshoe device that was removed before the maximum load was reached. The graph shows that the results are alike for the four bolts and, consequently, the four curves are nearly indistinguishable. Specimen WT1h was also instrumented with strain gauges (see Fig. 3.8a). These were attached close to the fillet weld, near the theoretical location of the expected yield line. Table 3.8 Main characteristics of the force-deformation curves for the un-

stiffened specimens [The elastic stiffness is quantified by using the average deformation values. The deformation capacity here is taken as the average deformation, from the two opposite LVDTs, corresponding to the maximum load. Italic values for def. capacity refer to the readings of HP1].

Resistance (kN) Stiffness (kN/mm) Test ID K-R Fmax ke.0 kp-l.0 ke.0/ kp-l.0

∆u.0 (mm)

WT1a 125-140 157.65 96.28 5.97 16.13 6.24 WT1b 140-155 182.08 109.88 4.63 23.73 10.37 WT1c 135-145 166.37 128.63 4.89 26.30 8.12 WT1d 137-145 150.08 120.42 7.91 15.22 4.46 WT1e 140-150 168.12 134.25 6.80 19.74 4.97 WT1f 168-180 184.99 118.46 2.37 50.00 4.90 WT1g 115-135 182.66 137.16 4.22 32.50 14.10 WT1h 119-139 184.99 147.17 4.14 35.55 14.55 WT2Aa 103-124 162.01 128.63 6.47 19.88 WT2Ab 106-130 173.64 123.65 3.80 32.54 17.98 WT2Ba 118-156 191.97 127.15 6.47 19.65 10.09 WT2Bb 123-160 195.75 159.49 4.43 36.00 13.09 WT4Aa 118-209 216.40 150.15 5.50 27.30 5.35 WT4Ab 140-196 206.51 173.91 8.74 19.90 4.33 WT7_M12 60-96 100.64 91.18 3.78 24.12 4.60 WT7_M16 80-104 132.34 116.09 5.08 22.85 11.47 WT7_M20 88-118 145.72 137.70 5.61 24.55 9.12 WT51a 155-188 193.71 119.24 3.47 34.36 4.10 WT51b 158-189 194.59 123.67 3.98 31.07 3.82 WT53C 166-192 197.79 128.46 4.75 27.04 4.24 WT53D 185-218 234.72 105.79 9.52 11.11 5.54 WT53E 178-215 230.07 129.63 8.25 15.71 5.26 WT57_M12 75-119 121.87 85.78 1.14 75.25 4.33 WT57_M16 104-165 173.64 110.43 6.99 15.80 5.88 WT57_M20 126-204 241.71 150.96 6.32 23.89 15.98 WT61a 128-180 203.89 164.65 9.75 16.89 6.18 WT61b 119-177 213.20 152.05 11.09 13.71 7.96 WT64A 121-200 220.47 164.04 9.39 17.47 4.60 WT64C 118-214 236.47 172.45 8.84 19.51 4.59


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comparison with Eurocode 3 (EC3) predictions.

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Fig. 3.13 Experimental results for the bolt elongation behaviour (WT1h).

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(a) Strain gauges SG1, SG3 and SG5. Fig. 3.14 Experimental results for the flange strain behaviour (WT1h).


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The force-strain results are shown in Fig. 3.14. In particular, Fig.3.14a com-pares the results for the edge strain gauges (SG1, SG5) and the one attached at mid length of the specimen (SG3). The results are very similar. Figs. 3.14b-c depict symmetry. The results are analogous but not exactly the same for sym-metric strain gauges, since they may not be placed exactly on the same spot. Finally, Fig. 3.14d compares the strain results for the remaining strain gauges that are also alike. 3.3.2 Failure modes and general characteristics of the overall behaviour

of the test specimens The deformation capacity of a bolted T-stub connection made up of welded plates primarily depends on the plate/bolt strength ratio and the weld resistance that is associated to the consumable type and properties. Collapse is eventually governed by brittle fracture of the bolts or the welds, or cracking of the plate material near the weld toe. Most of the tested specimens failed by tension rup-ture of the bolts after bending deformation of the flange. The degree of plastic deformation of the flange depends first and foremost on the geometric charac-teristics of the connection and the mechanical properties of the elements. How-ever, the collapse of some specimens was due to cracking of the plate material in the HAZ. In this T-stub assembly type, the collapse mode involving rupture of the plate was also affected by residual stresses and modified microstructure in the HAZ. This could lead to a reduction of the ultimate material strain with respect to the unaffected material and thus to an earlier failure of the whole connection. It was also observed that the extent of the properties variations in the HAZ, which were inherent to the welding procedure, was highly dependent on the electrode type and the hydrogen content, in particular. The observed failure modes involved combined bending and tension bolt fracture (type-13 or -23) in nineteen specimens, stripping of the nut threads bolt fracture (type-23B) in one specimen (WT57_M16), cracking of the plate material in the HAZ (type-11) in ten specimens and combined collapse modes 11 and 13 (type-1(1+3)) in the remaining cases. Notice that the stiffened speci-mens failed in a combined bending and tension bolt fracture mode. Table 3.9 summarizes the collapse modes of the several tests. Depending on the failure mode and naturally on the connection configura-tion, a similar behaviour was observed between related specimens. The most significant characteristic describing the overall behaviour of the connection is the F-∆ response. Fig. 3.15 plots the load-carrying behaviour of six selected examples that illustrate the five above-mentioned collapse modes. For the par-allel T-stub elements specimens, the deformation corresponds to the average value measured by the two opposite LVDTs at each specimen. For specimen WT64B that includes a stiffener and where the two T-stubs are orientated at right angles, the results for LVDTs HP1 and HP2 (see Fig. 3.8c) are shown.


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Table 3.9 Observed failure modes.

Test ID Failure mode



Type # Type # Type # 13 1 WT4A 13 2 WT64B “23” 2 11 6 WT4B “13” 1 WT64C 23 1 WT1 1(1+3) 1 WT51 23 2 WT7_M12 13 1 13 1 WT53C 23 1 WT7_M16 11 1 WT2A 1(1+3) 1 WT53D 13 1 WT7_M20 11 1 13 1 WT53E 13 1 WT57_M12 23 1 WT2B 1(1+3) 1 WT61 23 2 WT57_M16 23B 1

WT64A 23 1 WT57_M20 1(1+3) 1

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Fig. 3.15 Force-deformation characteristics of some of the tested specimens. Figs. 3.11b-c and 3.16 depict the six above specimens at collapse conditions [WT1g: collapse type-1(1+3)]. First, consider specimens WT1g, WT7_M16, WT2Ba and WT61b, which exhibit failure modes type-1(1+3), type-11, type-13 and type-23, respectively (Figs. 3.11b-c and 3.16a-c). An elastic branch, with slope ke.0, that develops un-til yielding of the flange begins, characterizes the F-∆ curves. A loss of stiff-ness then follows on and at a certain load level a quasi-linear branch with slope kp-l.0 arises. This post-limit region is longer for specimens WT1g and WT7_M16 that develop large bending deformations of the flange, when com-pared to tests WT2Ba and WT61b. For these latter specimens, fracture of the bolts determined collapse. In the specific case of WT61b, which was stiffened on one side, the bolts at the stiffened side fractured. Therefore, at failure, there was a sudden drop of load with constant deformation, which characterizes a brittle failure type. Regarding specimen WT7_M16, the failure mechanism was


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(a) Specimen WT7_M16 (collapse type-11).

(b) Specimen WT2Ba (collapse type-13).

(c) Specimen WT61b (collapse type-23). (d) Specimen WT64Bb (collapse

type-“23”).

(e) Specimen WT57_M16 (collapse type-23B). Fig. 3.16 Specimens at failure. very ductile and after the maximum load was reached, at a deformation of about 12 mm, the drop of load was very smooth and proceeded with increasing deformation between the flanges. This test was stopped at 16∆ ≈ mm because the webs started to bend and twist excessively and that would damage the equipment. If the test had continued, the behavioural tendency would have been the same. Finally, with respect to specimen WT1g that exhibits a com-bined failure mechanism, the maximum load was reached for a deformation of 14 mm, after which it started decreasing. This decrease was smooth and corre-sponded to the beginning of cracking of the flange plate close to the weld toe.


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Eventually, at 20∆ ≈ mm, there was a sudden drop of load that coincided with the bolt fracture. Notice that bolt rupture took place at opposite side of plate cracking. Apparently, a larger deformation capacity would be expected for specimen WT7_M16, when compared to WT1g, because the bolt did not gov-ern the collapse. However, since the T-stub width tributary to a bolt row was higher in specimen WT7_M16, smaller deformation capacity was expected [3.9]. Specimen WT64Bb tried to mimic the actual configuration of the tension side of a bolted connection: elements orientated at right angles, one T-element stiffened and the other unstiffened. When this assembly was subjected to a ten-sile force, the plates became in contact except at the stiffener-web contact, as clearly shown in Fig. 3.16d. Therefore, the F-∆ response depicted in Fig. 3.15 shows that the two flanges are opening at the stiffener side (HP2) and closing at the opposite side (HP1). The characteristics of the curve for LVDT HP2 are very similar to those described for WT61b, where bolt fracture at the stiffener side also governs the ultimate condition. Finally, type-23B failure that occurs in specimen WT57_M16 (and is not common) is a brittle rupture mode. The specimen at collapse is illustrated in Fig. 3.16e and the corresponding load-carrying behaviour is shown in the graph from Fig. 3.15. 3.4 CONCLUDING REMARKS The experiments presented above can be regarded as a reliable database for the characterization of the behaviour of the T-stub assembly made up of welded plates. The test procedure and the instrumentation set up adopted for the pro-gramme were satisfactory, as evidenced by the identical results obtained from the various sets of tests from one series (Figs. 3.17-3.18, for illustration). De-tailed information on the experimental results is given in Table 3.8, which set out the main characteristics of the load-carrying behaviour of the various speci-mens, and later in Chapter 5. Reference [3.10] also provides a thorough de-scription of this experimental programme. The programme provides insight into the assessment of failure modes and available deformation capacity of bolted T-stub connections. The major contri-butions of the overall T-stub deformation are the flange deformation and the tension bolt elongation. Usually, a higher deformation capacity of the T-stub is expected if the flange cracking governs the collapse instead of bolt fracture. However, in this type of assembly this statement is not so straightforward. The cracking associated to the flange mechanism, in this case, depends on structural constraint conditions and modifications in the mechanical properties in the HAZ, particularly those linked to the presence of residual stresses. Therefore, the selection of the electrodes and welding procedures is of the utmost impor-tance in this connection type to ensure a ductile behaviour. It has been found out that soft, low hydrogen, mismatch (or evenmatch) electrodes prevent an


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Fig. 3.17 (Experimental) load-carrying behaviour for test series WT2.

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Removal of measuringbrackets

Fig. 3.18 Experimental results for the bolt elongation behaviour (specimen

WT4Ab). early cracking of the flange thus enhancing the overall deformation capacity. Regarding the definition of “deformation capacity”, some clarification seems appropriate: “Which criterion should be considered to define the defor-mation capacity?”. This question has been addressed previously by the author [3.11] since the designation adopted so far (deformation capacity taken as the deformation level at maximum load) seems very conservative (e.g.: WT1g, WT7_M16, among others). In many examples, there is a long plateau in the F-∆ response after the maximum load level is reached that cannot be disregarded. Then, some guidelines on this specific issue are desirable.


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3.5 REFERENCES [3.1] European Committee for Standardization (CEN). prEN 1993-1-8:2003,


[3.2] European Committee for Standardization (CEN). prEN 10025:2000E: Hot rolled products of structural steels, September 2000, Brussels, 2000.

[3.3] European Committee for Standardization (CEN). EN 10204:1995E: Me-tallic products, October 1995, Brussels, 1995.

[3.4] RILEM draft recommendation. Tension testing of metallic structural materials for determining stress-strain relations under monotonic and uniaxial tensile loading. Materials and Structures; 23:35-46, 1990.

[3.5] Gioncu V, Mazzolani FM. Ductility of seismic resistant steel structures. Spon Press, London, UK, 2002.

[3.6] Rodrigues DM, Menezes LF, Loureiro A, Fernandes JV. Numerical study of the plastic behaviour in tension of welds in high strength steels. International Journal of Plasticity; 20:1-18, 2004.

[3.7] Davies AC. The science and practice of welding – welding science and technology – Vol. I. Cambridge University Press, Cambridge, UK, 1992.

[3.8] European Committee for Standardization (CEN). EN 499:1994E: Weld-ing consumables – Covered electrodes for manual metal arc welding of non alloy and fine grain steels - Classification, December 1994, Brus-sels, 1994.

[3.9] Girão Coelho AM, Simões da Silva L. Numerical evaluation of the duc-tility of a bolted T-stub connection. In: Proceedings of the third interna-tional conference on advances in steel structures (ICASS’02) (Eds.: S.L. Chan, F.G. Teng and K.F.Chung), Hong Kong, China, 277-284, 2002.

[3.10] Girão Coelho AM, Bijlaard F, Gresnigt N, Simões da Silva L. Experi-mental assessment of the behaviour of bolted T-stub connections made up of welded plates. Journal of Constructional Steel Research; 60:269-311, 2004.

[3.11] Girão Coelho AM, Bijlaard F, Simões da Silva L. Experimental re-search work on T-stub connections made up of welded plates. Docu-ment ECCS-TWG 10.2-217, European Convention for Constructional Steelwork – Technical Committee 10, Structural Connections (ECCS-TC10), 2002.

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4 NUMERICAL ASSESSMENT OF THE BEHAVIOUR OF T-STUB

CONNECTIONS 4.1 INTRODUCTION The behaviour of a bolted T-stub connection can be predicted with numerical simulations. The numerical modelling of this type of problem is complex since it requires an adequate representation of the connection geometry, the materials constitutive laws, boundary and load conditions [4.1]. Today, the FE method is widely accepted as the most expedite technique for obtaining numerical solu-tions for structural mechanical problems [4.2]. The basic steps of this method are [4.3]: (i) the continuum is divided into non-overlapping discrete elements, over which the main variables are interpolated, (ii) these elements are intercon-nected at a number of points along their periphery (the nodal points), (iii) the solution strategy can be obtained using implicit or explicit solvers and (iv) sub-sidiary quantities, such as stresses and strains, are evaluated for each element. Regarding the solution technique, the implicit method is based on static equi-librium and is characterized by the assembly of a global stiffness matrix, fol-lowed by simultaneous solution of the set of linear equations [4.4]. The result-ing system of equations is solved for the nodal variables and so the nodal dis-placements are computed directly, i.e. implicitly. The explicit method is based on dynamic equilibrium. The FE model allows complex geometry to be modelled fairly accurately. Material and geometrical nonlinearities are also adequately simulated, as well as the boundary and load conditions. In terms of geometry modelling, the nu-merical model must reproduce the global behaviour of the connection. Such behaviour is three-dimensional in nature. The choice of elements must then be made among three-dimensional elements: solid or shell elements. Several at-tempts of a two-dimensional approach were made in the past but proved to be unsatisfactory. Shell elements behave in a three-dimensional fashion and are able to reproduce the collapse mechanisms but are not suitable for element in-terfacing, in particular for bolt/plate contact simulation. For that purpose, solid elements are accurate and therefore this type of elements was used in the nu-meric simulations. Regarding the material properties, for steel components the modelling of elastoplasticity is fundamental. In elasticity type problems, no permanent deformations occur. The plastic behaviour is characterized by a time-independent irreversible straining that can only be sustained once a cer-tain stress level has been reached [4.5]. The elastoplastic material response is taken into account through dissociation of the elastic and plastic deformations (ε e and ε p, respectively). The total strain ε is thus defined as the sum


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e pε ε ε= + . In general, plasticity is modelled with strain hardening, i.e. once the yield stress is reached, the stress continues to increase with strain but with a reduced modulus of elasticity. The plasticity formulations are based on three fundamental concepts [4.6]: (i) a yield condition to specify the onset of plastic deformation, (ii) a flow rule to define the plastic straining and (iii) a hardening rule to define the evolution of the yield surface with plastic straining. For steel components the yield condition is usually defined under the Von Mises yield criterion. The flow rule defines the direction of the plastic straining. In most cases, the direction of the plastic strain vector is orthogonal to the yield surface (associated flow). With respect to the element-interfacing phenomenon, in FE analysis the element penetration in contact zones is avoided by adding special interface or contact elements. Generally, it is not possible to define a priori the zones that come into contact because of the different load stages and corresponding de-formations. This means that contact may not be attained for the same element under different loading conditions. As a result, the simulation of contact behav-iour between the connection components is rather complex. Contact phenome-non is intrinsically nonlinear: the contact zones are very stiff (compression) whilst non-contacting zones are very flexible (tension). The interfacing forces that are developed when two parts come into contact transmit the applied forces. These contact forces are normal to the interface direction and the fric-tional forces are developed along the tangential direction of the interface. The distribution of the interface stresses and the contact conditions (sticking or slid-ing) are also unknown. Most FE packages offer some facilities for dealing with the unilateral contact problem with friction. The modelling of a bolted T-stub connection is therefore highly nonlinear, involving complex phenomena such as material plasticity, second-order effects and unilateral contact boundary conditions. In the following sections, the pro-cedures for the implementation of a FE model using the commercial FE pack-age LUSAS [4.7-4.8] for the analysis of this type of problem are described. This numerical model is validated through comparison with experimental evi-dence. 4.2 PREVIOUS RESEARCH The FE modelling of an individual T-stub connection has been performed by a number of authors from different research centres. In the framework of the Numerical Simulation Working Group of the European Research Project COST C1 “Civil Engineering Structural Connections”, this task was proposed as a benchmark for FE modelling of bolted steel connections. Jaspart provided the necessary experimental data for those simulations (T-stub T1) [4.9]. Bursi [4.10] and Bursi and Jaspart [4.11-4.12] developed and calibrated a three-dimensional nonlinear model to mimic the experimental load-carrying response

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of the given example (T1). Later, they extended the method to another T-stub configuration to investigate another connection representative of a different collapse mode [4.11-4.12]. The models are proposed as benchmarks in the validation process of FE software packages. The simulations of the individual connections were performed by means of solids and contact elements. The F-∆ response as well as the bolt behaviour (elongation, preloading effects) and pry-ing effects were addressed. The proposed model was satisfactory, in general. Gomes et al. [4.13] implemented a three-dimensional model as well for the simulation of the test T1 but used shell elements instead of solids. Their model allowed for the assessment of second order effects and nonlinear material be-haviour with strain hardening. The agreement between results was rather poor. Mistakidis et al. proposed a two-dimensional FE model capable of describing plasticity, large displacements and unilateral contact effects [4.14-4.15]. Al-though the model encompasses all the essential characteristics and dominant plastification mechanisms, the numerical results are much stiffer than the ac-tual response. In general, the FE results do not compare well to the experi-ments. Zajdel also carried out a three-dimensional FE analysis of the bench-mark problem and proposed a reliable model that accounted for most of the T-stub features [4.16]. Wanzek and Gebbeken [4.17] validated a three-dimensional numerical model against experimental results performed in Munich [4.18]. They used other experimental results (e.g. strain results, bolts measurements) for calibra-tion of the model. The agreement between responses was very good. More recently, Swanson [4.19] and Swanson et al. [4.20] performed tests on individual T-stubs and proposed a robust FE model to supplement their re-search. This sophisticated model provided insight into the characteristics of the T-stub behaviour and stress distributions (namely, contact stresses). The results of this robust model were used to validate a simpler two-dimensional model. The main criticism to their approach lies in the input of the material properties. They used nominal properties instead of actual properties. This procedure is questionable. Naturally, this validation process was only applicable within the range analysis, which was limited to a single example. The authors explored many features of the T-stub model, as the bolt response and the prying effect. They discussed the conclusions drawn from the FE analyses but they did not broaden the scope of their analysis to conclude about the mechanisms and pa-rameters that influence (and how) the T-stub behaviour. The main concern of all above models was the accomplishment of a reliable FE model that was calibrated against experiments to obtain the F-∆ response. Furthermore, only the case of HR-T-stubs was addressed. These models af-forded some basis for the implementation of the FE models described below. Several model features have already been highlighted by these authors. How-ever, some aspects still have to be looked into. Additionally, this research also proposes a FE model for WP-T-stubs that necessarily includes specific aspects, namely the influence of the welding of the plates.


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4.3 DESCRIPTION OF THE MODEL The T-stub connection was generated with three-dimensional elements, solid and joint elements. In particular, the solid elements were hexahedral bricks and were used to model the continuum. The joint elements were employed in the simulation of element contact. In the FE library of the commercial package LUSAS [4.21] there are three types of hexahedral solid elements: the eight-node brick (HX8(M)), the six-teen-node brick (HX16) and the twenty-node brick (HX20). These elements belong to a family of serendipity isoparametric elements, i.e. they have no in-side nodes and the geometry and displacement interpolation are carried out by means of the same shape functions. The elements have three degrees-of-freedom per node (u, v and w) and are numerically integrated. The complete formulation of this element type is detailed in the literature [4.2-4.3]. The choice of one or another type of brick depends on their application. In linear elastic problems, the higher order elements (sixteen and twenty nodes) are more accurate than the eight-node brick. For nonlinear problems, involving plasticity and contact phenomenon, in particular, the eight-node element, which has no mid-side nodes, leads to improved numerical solutions since they allow for a better representation of the discontinuities at element edges and of the strain field. The FE code LUSAS implements two eight-node bricks, HX8 e HX8M, as already pointed out. The element HX8M exhibits improved accu-racy in coarse meshes when compared with the parent element HX8, particu-larly in bending dominated problems [4.22]. In addition, the element does not suffer from shear locking in the nearly incompressible limit. The element for-mulation is based on the works of Simo and Rifai [4.23]. It includes an as-sumed “enhanced” strain field related to the internal degrees-of-freedom that are eliminated at the element level before assembly of the structure stiffness matrix. Thereby, the eight-node brick with enhanced strains, HX8M, and full integration (2×2×2 Gauss points) was chosen for the numerical analysis. The kinematic description of solid elements in nonlinear geometrical analy-sis is based on three different formulations: (i) the total Lagrangian formula-tion, that accounts for large displacements and small strains; in this formulation all variables are referred to the undeformed configuration, (ii) the updated La-grangian formulation that accounts for large displacements and moderately large strains; all variables are referred to the last converged solution configura-tion and (iii) the Eulerian formulation catering for large displacements and large strains; in each iteration at the same load increment, the deformed con-figuration is updated and it corresponds to the reference configuration for the subsequent iteration. In the total Lagrangian formulation stresses and strains are output in terms of the “second Piola-Kirchoff stresses” and “Green-Lagrange strains”, with reference to the undeformed configuration [4.22]. The stresses and strains output for the updated Lagrangian and Eulerian formula-tions are the “Cauchy (or true) stresses” and the “natural (or logarithmic) strains” [4.22]. For elements with no rotational degrees-of-freedom of the


97

nodes, in which the internal displacement field is defined in terms of nodal displacements only, the three types of formulations yield identical results for arbitrarily large displacements, provided that the strains are small. In the FE code LUSAS, the Lagrangian approach is preferred in structural problems [4.22]. Consequently, the updated Lagrangian nonlinear formulation was em-ployed. For the material nonlinearity, an elastoplastic constitutive law based on the Von Mises yield criterion was adopted. A plastic potential defined the flow rule [4.5]. The constitutive model was integrated by means of the explicit for-ward Euler algorithm [4.5]. For this algorithm the hardening data and direction of the plastic flow are evaluated at the point at which the elastic stress incre-ment crosses the yield surface. In order for an element formulation to be applicable to a specific response prediction, both kinematic and constitutive descriptions must be appropriate [4.2]. In a materially nonlinear only analysis, the configuration and volume of the body under consideration are constant. In this type of analysis, both dis-placements and rotations are assumed infinitesimally small. Then, the engi-neering stress-strain constitutive law describes the material behaviour in a proper way. In a large displacement and strain elastoplastic analysis, the con-figuration and volume of the body do not remain constant. The Lagrangian formulation includes the kinematic nonlinear effects due to large displacements and strains, but whether the large strain behaviour is modelled accurately de-pends on the constitutive relations specified. This requires the use of a true stress-logarithmic strain measure (σn – εn) for the definition of the uniaxial ma-terial response, instead of the classic engineering constitutive law (σ – ε). These quantities are defined with respect to the current length and cross-sectional area of the coupon and are related to the engineering values by means of the follow-ing relationships:

( )1nσ σ ε= + and ( )ln 1nε ε= + (4.1) Node-to-node nonlinear contact friction elements simulated the interface boundary conditions. The contact between two bodies was modelled with a joint mesh interface, which used a “master” and “slave” connection to tie the two surfaces together at their nodes. The sliding and sticking conditions are reproduced with the classic isotropic Coulomb friction law. The selected ele-ment from LUSAS FE package for the contact analysis was the three-dimensional joint element JNT4 [4.21] that connected two adjacent nodes by means of springs with adequate properties. This element is compatible with the brick HX8M, comprising three nodal degrees-of-freedom (u, v and w). The element has four nodes: two active nodes, a third and fourth auxiliary nodes for definition of the local xy-plane. The two active nodes are connected with ex-tensional springs in the three local directions x, y and z. The friction model was able to represent frictional and gap connections between adjacent nodes whereby on the closure of a specified initial gap, fric-tional forces were allowed to develop. In the proposed numerical model, this


98

initial gap was set as equal to zero. In order to model the nonlinear relation between stresses F and relative displacements δ, the linear stiffness moduli have to be specified. The local element stiffness matrix is formulated directly from user input stiffness coefficients and is then transformed to the global Car-tesian system. The normal stiffness modulus, k1, should be set as equal to infin-ity, i.e. its value should be the biggest possible. However, a stiffness value too large could induce poor conditioning of the stiffness matrix. The optimum value was found when the change in the results for an additional increase in the stiffness value was negligible or when the penetration between the bodies in contact reached a certain limit [4.11]. Concerning the tangential stiffness moduli, k2 and k3, their value must be non-zero, otherwise the bodies in contact would have an unrealistic infinite movement in these directions at the com-mencement of loading. The location and magnitude of the contact forces can be ascertained by the joint elements arrangement, since a zero force means separa-tion of the flanges whilst a compressive force implies contact between the plates at that location. The joint element possesses no geometrically nonlinear terms in its formu-lation. However, it may be used in geometrically nonlinear analysis but it re-mains geometrically linear. To determine the structural response of the nonlinear problem an implicit solution strategy was used, which is suitable for problems involving smooth nonlinear analyses. A load stepping routine was hence used. There was no re-striction on the magnitude of the load step as the procedure was uncondition-ally stable. The increment size followed from accuracy and convergence crite-ria. Within each increment, the equilibrium equations were solved by means of the Newton-Raphson iterations, which is stable and converges quadratically. In the Newton-Raphson method, for each load step, the residuals are eliminated by an iterative scheme. In each iteration, the load level remains constant and the structure is analysed with a redefined tangent stiffness matrix. The accuracy of the solution is measured by means of appropriate convergence criteria. Their selection is of the utmost importance: too tight convergence criteria may lead to an unnecessary number of iterations and a consequent waste of computer resources, whilst a loose tolerance may result in incorrect solutions. Generally speaking, in nonlinear geometrical analysis relatively tight tolerances are re-quired, while in nonlinear material problems slack tolerances are admitted, since high local residuals are not easy to eliminate. The FE code LUSAS dis-poses of six different convergence criteria [4.22]: (i) Euclidean residual norm, γψ, defined by the norm of the residuals, ψ , as a percentage of the norm of the

external forces, R : 22

100Rψγ ψ= × , (ii) Euclidean displacement norm,

γd, defined by the norm of the iterative displacements, aδ , as a percentage of the total displacements, a :

2 2100d a aγ δ= × , (iii) Euclidean iterative dis-

placement norm, γdt, defined by the norm of the iterative displacements, aδ , as


99

a percentage of the total displacements a∆ for a certain increment: dtγ =

2 2100a aδ= ∆ × , (iv) work norm, γw, corresponding to the work done by

the residuals forces on the current iteration as the percentage of the work done

by the external forces on iteration zero, ( )( ) ( ) (0) (0)T Ti iw a R aγ ψ δ δ = ×

100× , (v) root mean square of residuals and (vi) maximum absolute residual. Based on nonlinear numerical analysis from literature [4.24-4.26], it was concluded that establishing displacement-based convergence criteria was enough. Nevertheless, Crisfield [4.25] suggests that any displacement con-straint must be coupled with a force limitation. The following convergence criteria were hence used. LUSAS [4.22] suggests the following values as limit tolerances:

(i) Euclidean displacement norm

−−−

0.11.0:reasonable001.01.0:tight0.10.5:slack

(ii) Euclidean incremental norm

−−−

0.11.0:reasonable001.01.0:tight0.10.5:slack

(iii) Work norm

−

−

−

−

−−

6

96

10001.0:reasonable

1010:tight

001.01.0:slack

For predominantly materially nonlinear problems, where high local residuals have to be tolerated, slack convergence criteria are usually more effective [4.22]. As a consequence, the following slack tolerance values were used: γd

=

3.0, γdt = 3.0 and γw= 0.05.

With respect to the incremental method, a load curve was defined. Loads were applied to the specimen in a displacement-control fashion that enforced a better conditioning of the tangent stiffness matrix when compared to the classi-cal load-control procedure. 4.4 CALIBRATION OF THE FINITE ELEMENT MODEL The FE model for both T-stub assembly-types was identical. The only differ-ence lied in the representation of the flange-to-web connection. For the HR-T-stub, flange and web were connected by means of a fillet radius, r, that ensured the continuity between both plates. In the case of WP-T-stubs, a continuous 45º-fillet weld (throat thickness aw) linked the flange and the web, though the two plates were not necessarily in contact.


100

The calibration of the FE model for the HR-T-sub was based on the ex-perimental test programme carried out by Bursi and Jaspart [4.11-4.12]. The specimen T1, which was obtained from an IPE300 beam profile, with snug-tightened bolts was selected for the following study. Regarding the WP-T-stub, the approach was validated with experimental evidence from the series of tests WT1 reported in Chapter 3. 4.4.1 Geometry The general geometrical characteristics of the specimens are specified in Table 4.1 for the two specimens reported herein. By adopting the adequate boundary conditions only one eighth of the T-stub was modelled, owing to symmetry considerations (Fig. 4.1). The xy and yz planes are geometrical planes of sym-metry. Although the xz plane does not meet such criterion, since the bolt elon-gation behaviour is not symmetrical along the y direction, some authors pro-pose numerical models that account for a symmetric behaviour of an “equiva-lent bolt” complying with the requirements for symmetry in the xz plane [4.11,4.17]. If the “equivalent bolt” is defined in such a way that its geometri-cal stiffness is identical to that of the actual bolt, i.e. the elongation of the “equivalent bolt” represents half of the elongation behaviour of the actual bolt, only one eight of the T-stub has to be considered. This approach can be very useful in terms of FE analysis, since the number of elements is significantly reduced. In this case, the xz symmetry plane between the two flanges was mod-elled by contact elements on a rigid foundation (Fig. 4.1). The interface boundaries between flanges and washer or bolt head and between web and flange plates in the case of WP-T-stubs were also modelled by means of con-tact elements. In order to reduce the number of contact planes the bolt head or nut and the washer, if any, were assumed fully connected. This simplification led to slightly stiffer deformation behaviour, but the overall response was not greatly influenced, as already shown in the literature [4.12,4.16-4.17]. The bolt modelling in this type of connection is very important since the overall response of the T-stub is greatly influenced by the bolt behaviour. The bolt is composed of head, nut and shank (threaded and non-threaded part). Each of these components constitutes a source of flexibility that must be taken into account when modelling the bolt. Bursi and Jaspart [4.12-4.16] defined the above-mentioned “equivalent bolt” by means of the Aggerskov model [4.27] Table 4.1 Nominal geometrical properties of the various specimens (dimen-

sions in [mm]; ST: short-threaded, FT: full-threaded).

T-elements geometry Bolt characteristics Test ID tf tw w e p/2 e1 r/aw φ Washer Type #

T1 10.7 7.1 90 30 20 20 15 12 Yes ST 4 WT1 10.0 10.0 90 30 25 20 5 12 No ST 4


101

z

x

External load

External load

y

‘Equivalent’ bolt M12, short-threaded

Section xx

150.0 mm

3.55

mm

15.0

mm

d0 = 14.0 mm

20.0 mm

20.0 mm

30.0 mm 26.45 mm

Plan x x

y

x

10.7 mm

Rigid foundation Contact plane

¼ external load

Boundary geometric conditions

Fig. 4.1 Finite element geometry model assuming symmetry in the xy, xz and

yz planes: particular specimen T1. and reproduced the bolt shank with a cylinder of cross-sectional area As (tensile stress area of the bolt). In the proposed numerical model, a different approach was implemented. The “equivalent bolt” had half of the conventional bolt length, as defined in Eurocode 3 [4.28] and the “equivalent shank” has a threaded part (cross-sectional area As) and a non-threaded portion (actual bolt diameter). The length of these parts was proportional to that of the real bolt. 4.4.2 Boundary and load conditions The nodes in the symmetry planes xy and yz were fixed with symmetric geo-


102

metrical boundary conditions (Fig. 4.1): in plane xy, the nodes were fixed in the z direction on one side and in plane yz in the x direction, along the back of the half of the web. The nodes in plane xz between the two flanges and between the washers and the flanges were restrained with contact elements. The boundary conditions for the second model were identical except for the symmetry plane xz between the two flanges. This plane was modelled by contact elements on a rigid foundation (Fig. 4.1). The nodes on this rigid base were fully restrained. Complying with geometrical symmetry, the bottom bolt nodes were also fixed in the y direction. No friction was assumed between the flanges interface because of the T-elements symmetric behaviour. For the flange-washer and flange-web (in the case of welded profiles) interfaces a non-zero friction coefficient, µ, was as-sumed. A value of 0.25 for this type of contact surface was suggested by Va-sarhelyi and Chiang [4.29], who carried out an experimental study for supply-ing reliable values for this parameter. This value was adopted in the model. A uniform total prescribed displacement of 0.1 mm was applied at the top of the upper T-element in positive y direction (Fig. 4.1). In the nonlinear analy-sis, the total load factor was increased from 1.0 to the collapse, as explained below. A final remark concerning the nodal restraints must be made: in LU-SAS FE package, when applying total prescribed displacements in a certain direction, the corresponding nodes must be fixed in the same direction. 4.4.3 Mechanical properties of steel components For a good correlation with experimental results, the full actual stress-strain relationship of the materials must be adopted in the numerical simulation. For both models a rate and temperature independent plasticity law with hardening was used for the T-stub profile and the high strength bolt. The constitutive laws were reproduced with a piecewise linear model [4.11-4.12,4.30]. As already pointed out, to perform realistic simulations, the conventional constitutive law had to be converted into a constitutive true law (Fig. 4.2). The material proper-ties for the rigid foundation were also defined. Since it is a rigid element, a linear elastic material was assumed, with E = 1015 MPa and υ = 0.45. 4.4.4 Specimen discretization A FE mesh must be sufficiently refined to produce accurate results but the number of elements and nodes should be kept as small as possible in order to limit the processing time needed for the analysis. The behaviour of a bolted T-stub connection is dominated by the flexural deformation of the flange. Particular attention must then be devoted to the dis-cretization of this part. Based on the study performed by Wanzek and Geb-beken [4.17], the flange discretization with HX8M elements was analysed with


103

0

200

400

600

800

1000

1200

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Logarithmic strain

True

str

ess

(MPa

)

Bolt (fy=893MPa)

T-flange (fy=431MPa)

T-web (fy=496MPa)

(a) HR-T-stub specimen T1 [4.11].

0

200

400

600

800

1000

1200

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Logarithmic strain

True

str

ess

(MPa

)

Bolt (fy=803MPa)

T-flange (fy=340MPa)

T-web (fy=391MPa)

(b) WP-T-stub specimen WT1. Fig. 4.2 Stress-strain true laws for specimens T1 and WT1. respect to two parameters: (i) degree of discretization in order to represent the bending dominated problem and (ii) number of elements through thickness to check the capability of representing the yielding lines. The FE mesh depicted in Fig. 4.3a complies with the requirements for a reliable simulation and satisfies the mesh convergence study that was per-formed within the framework of this research work (cf. Appendix B). For the bolt discretization, in order to simulate the complex state of stress in the bolt, a reasonably refined mesh was essential. In a T-stub connection the bolt works in tension and bending due to the deformation of the T-stub flanges. The overall response of the T-stub is greatly influenced by the bolt behaviour. As a result, the bolt modelling is crucial. The bolt is composed of head, nut and shank (threaded and non-threaded part). Each of these components constitutes a source of flexibility that has to be taken into account when modelling the bolt. The number of elements was determined decisively by the discretization


104

(a) Flange discretization. (b) Bolt discretization. (c) Global mesh. Fig. 4.3 Flange discretization adopted for further analysis (e.g. T1). of the circumference of the bolt. Technical literature suggests a minimum of 12 to 16 nodes around the circular hole [4.1]. The bolt mesh represented in Fig. 4.3b also complied with the requests for an accurate modelling. In the proposed numerical model, a different approach was implemented. As already explained, the “equivalent bolt” has half of the conventional bolt length, Lb, defined in Eq. (1.18) and the “equivalent shank” a threaded part (cross-sectional As) and a non-threaded part (actual bolt diameter), whose lengths were proportional to those of the real bolt. Fig. 4.3c shows specimen T1 global mesh that comprises 3588 elements and 5680 nodes. For the welded specimen, similar discretization was adopted. The global mesh in this case included 4164 elements and 6618 nodes. 4.4.5 Contact analysis Appendix B describes the models used for the calibration of the joint elements stiffness coefficients ki in the interface behaviour. 4.5 FAILURE CRITERIA The deformation capacity of a T-stub is related to the plate/bolt resistance ratio and is eventually determined by bolt fracture or cracking of the plate material, as already mentioned. In both situations, the modelling of the failure condition


105

can be ascertained by assuming that cracking occurs when the ultimate strain εu is attained, either at the bolt or at the T-stub critical sections [4.31-4.32]. Due to the nature of the materials, the bolt deformation supply is substantially less than the plate. Whilst for high strength bolts the ultimate strain is circa 5%-6%, for constructional steels, ultimate strains of 25%-30%, at least, can be expected [4.33]. As a result, bolt fracture is likely to govern most ultimate conditions and its assessment is of primary importance. The potential failure mechanisms of a bolt under axial loading are: (i) ten-sion failure, (ii) stripping of the bolt threads and (iii) stripping of the nut threads. Swanson [4.19] points out that high-strength fasteners are designed so that tension failure of the bolt occurs before stripping of the threads. The strip-ping phenomena should not be expected in most cases. Additionally, such a failure type is not easily opened to a numerical or analytical implementation. Therefore, the ultimate deformation of the bolt is frequently governed by ten-sion failure. A comprehensive numerical study on the behaviour of a single bolt in tension was hence carried out to evaluate its maximum deformation capacity and has been recently reported by the author [4.34]. Based on the study of a single bolt in tension, a failure criterion for the as-sessment of the T-stub collapse is now proposed. As a component of the T-stub connection, the bolt is subjected to combined tension and bending. In this case, the strain distribution at the bolt critical section changes from the symmetric case depicted in Fig. 4.4a to the case illustrated in Fig. 4.4b. The bolt axis di-rection is no longer a principal direction. However, if a similar failure criterion

ε11.av = εy.av

Bolt cross-section

εmin

εmax

Bolt cross-section

εmin

εmax

(a) Bolt under pure axial tension. (b) Bolt under combined tension and

bending. Fig. 4.4 Sketch of the strain distribution within a bolt cross-section.


106

is adopted to the single bolt in tension respecting to the maximum average principal strain, i.e. 11. .av u bε ε= , the deformation capacity of the bolt in com-bined tension and bending can be determined. It has been concluded that since the bolt, as a T-stub element, is subjected to combined tension and bending deformations, failure should be assessed by comparison of the maximum aver-age principal strain, ε11.av.b with εu.b. Should the flange section be critical, a similar criterion based on the maxi-mum principal strain, i.e. ε11.av.f = εu.f, seems appropriate. 4.6 NUMERICAL RESULTS FOR HR T-STUB T1 The most significant characteristic describing the overall behaviour of the mo-del is the F-∆ curve. The implementation of the above FE model yields the results shown in Fig. 4.5. The numerical results are compliant with the experi-mental response [4.12] showing that the proposed model is rather accurate. The end of the numerical curve, i.e. the deformation capacity of the connection is established by application of the above failure criteria. For the T-stub specimen T1, experimental observations indicated that the collapse is due to inelastic phenomena in the bolts and significant flange yield-ing [4.11]. Under the above failure criterion, the ultimate conditions are gov-erned by bolt fracture. The maximum average bolt strain ε11.av.b equals εu.b for a global deformation of 9.20 mm. This value is very close to the experiments (9.49 mm; ratio = 0.97). Fig. 4.5 compares the actual T-stub behaviour with the FE model. The curves in this case include the web deformation. However, the real gap be-tween the two flanges does not account for the web deformation. This response is depicted in Fig. 4.6a. The “real” flange deformation, ∆ is smaller than the total deformation due to the contribution of the web. Therefore, the “real” F-∆

0

30

60

90

120

150

180

210

0 1 2 3 4 5 6 7 8 9 10


Tota

l app

lied

load

(kN

)

Experimental results

Numerical results LUSAS

Fig. 4.5 Global response of specimen T1: numerical and experimental re-

sults.


107

0

30

60

90

120

150

180

210

0 1 2 3 4 5 6 7 8 9 10

Deformation (mm)

Tota

l app

lied

load

(kN

)

Num. - Total def.

Num. - Real def.

Def. capacity (bolt)Bolt elongation

(a) Load-deformation behaviour.

0.0

0.2

0.4

0.6

0.8

1.0

0 15 30 45 60 75 90 105

Applied load F, per bolt row (kN)

Rat

io

Prying force, Q Bolt force, B

(b) Bolt and prying force. Fig. 4.6 Numerical results for specimen T1. curve is stiffer, showing a deviation of nearly 25% in the elastic regime and 8% in the post-limit range. Fig. 4.6a also plots the bolt elongation behaviour against the applied load until collapse. The evolution of the ratios of prying and bolt forces with the applied load per bolt row, F, Q/F and B/F, respectively, is illustrated in Fig. 4.6b showing an increase of such ratios with plastic straining in the flange. The yielding of the flange starts at a load level of 96.29 kN (Fig. 4.7a). The ratio Q/F for this load level is 0.22; at collapse (2F = 207.98 kN) it increases to 0.34, which means an enlargement of 1.5 times. This information on the contact pressures provided by the FE model is very useful and cannot be obtained from experiments. Furthermore, the model gives detailed results for the bolt behaviour, particularly in terms of bolt elongation behaviour (curve B-δb) – Fig. 4.8. The evolution of the flange yielding is represented in Fig. 4.7. The beam pattern governs the kinematic mechanism: two yield lines develop in the flange, one near the bolt hole and another close to the flange-to-web connec-tions (see also Appendix C). This means that the flange is in double curvature,


108

(a) 2F=96.29kN; ∆=0.61mm.

(b) 2F=117.44kN; ∆=0.76mm.

(c) 2F=134.20kN; ∆=0.93 mm.

(d) 2F=146.45kN; ∆=1.10mm.

(e) 2F=159.55 kN; ∆=1.43mm.

(f) 2F=166.25kN; ∆=1.69mm.

(g) 2F=179.08kN; ∆=3.04mm.

(h) 2F=190.85kN; ∆=4.82mm.

(i) 2F=207.98kN; ∆=8.70mm.

Fig. 4.7 Flange yielding evolution with the applied load. as Fig. 4.7c clearly shows. The location of the prying forces changes during the course of loading. Fig. 4.9 shows the evolution of the contact area with the applied load. Clearly, as the load increases, the contact area spreads to the bolt axis. Let n be the dis-tance between the prying forces and the bolt axis. The ratio n/e is plotted against the external load in Fig. 4.10. Two cases are taken into account: (i) the overall contact area and (ii) the flange cross-section at the horizontal bolt axis x. In both cases, n is computed as follows:

influence.

influence.

number of active joints

number of active joints

Qi i ii

Qi ii

F L xn e

F L

× ×= −

×

∑∑

(4.2)


109

0

15

30

45

60

75

90

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bolt deformation (mm)

Bol

t int

erna

l for

ce (k

N)

Fig. 4.8 Bolt elongation behaviour.

(a) 2F = 96.29kN; ∆ = 0.61mm.

(b) 2F = 159.55kN; ∆ = 1.43mm.

(c) 2F = 171.00kN; ∆ = 2.06mm.

(d) 2F = 174.46kN; ∆ = 2.45mm.

(e) 2F = 179.08kN; ∆ = 3.04mm.

(f) 2F ≥ 183.40kN; ∆ ≥ 3.63mm.

Fig. 4.9 Evolution of the contact area with the applied load.

0.72

0.76

0.80

0.84

0.88

0.92

0.96

1.00

0 15 30 45 60 75 90 105


Rat

io n

/e

Whole contact area

Joint elements at the bolt x axis

Fig. 4.10 Evolution of the ratio n/e with the applied load per bolt row.


110

where FQi is the force associated to a joint row (in the y direction), Linfluence.i is the influence length of each of those joint rows and xi is the distance of the joint row to the tip of the flanges. Clearly, as the load increases, Q is shifted inside, from the tip of the flanges. Such situation is even more evident in the second case. 4.7 NUMERICAL RESULTS FOR WP T-STUB WT1 The first series of tests WT1 included eight different specimens for analysis of the adequate electrode for the welding procedure (cf. §3.3.1). The criterion for the choice of one or another electrode type was based on the ductility provided to the connection. It was seen experimentally that basic electrodes with low hydrogen content ensured enhanced deformation capacity of the T-stub con-nection. Two tests (WT1g/h) from this series were performed with this elec-trode type and are used for further comparisons. Fig. 4.11a compares the load-carrying behaviour from the numerical model with the experiments. In the FE modelling, the average real dimensions are used. Exception is made for the fillet weld throat thickness, as it was not measured. Therefore, the nominal value (aw = 5 mm) was used in the model. This can lead to some differences since the F-∆ response is sensitive to the value of m. The measurement of the gap between the two flanges in the test was per-formed by means of two LVDTs at opposite sides of the web. The numerical results that appear in the graph correspond to the location of those LVDTs. Fig. 4.11b shows the bolt elongation response for specimen WT1h, for the broken bolts (LB: left back and LF: left front) – see also Figs. 3.10d and 3.11. The graph of Fig. 4.11b does not display the experimental results of the bolt elonga-tion behaviour up to collapse since the measuring device was removed before the collapse. The FE model yields stiffer results than the experiments, though the agree-ment is good. The differences may derive from the insufficient geometrical and mechanical characterization of the fillet weld and also because of the model-ling of the HAZ, near the weld toe. In fact, some authors [4.35] have already highlighted the fact that due to the welding process, the connection behaviour and the cracking of material, in particular, are influenced by the presence of residual stresses and modified microstructures in the HAZ. It is very difficult to quantify these effects and therefore they were not included in the simulations. However, it should be borne in mind that if cracking of material governs the collapse model, a reduction of the ultimate strain with respect to the unaffected material is advised. For both specimens WT1g/h, bolt fracture determines the failure mode. Yet, for specimen WT1g there was a combined failure type in-volving cracking of the flange in the HAZ and bolt fracture. Figs. 3.11b-c illus-trate the specimen at failure. The graph from Fig. 4.11a also shows this type of fracture: at a deformation level of circa 14 mm there is a smooth drop of load that follows on until fracture of the bolt at 20.5 mm. Numerically and under the


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sults. above proposed failure criterion, it was established that bolt determines col-lapse. This was in line with experimental observations and the numerical pre-diction (13.98 mm) matches the experimental results for WT1h (15.11 mm at maximum load). The average maximum principal strain level in the HAZ is 6.8% with a local maximum of 14% (FE results). For the flange plate, the maximum (natural) strain measured in standard material tensile testing was 30.8%. Finally, Fig. 4.12 plots the strains in the flange, close to the fillet weld. Specimen WT1h was instrumented with five strain gauges on one side of the connection near the weld toe (Figs. 3.8a and 3.10d): (i) SG1 and SG5 are lo-cated near the flange edge, (ii) SG2 and SG4 are placed at the bolt x axis cross-section and (iii) SG3 is attached at the T-stub half width. The good correspon-dence between results is a valid statement of the reliability of the procedure. Regarding the ratios Q/F and n/e for the specimen WT1, Fig. 4.13 shows


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(b) Evolution of the ratio n/e with the applied load per bolt row. Fig. 4.13 Numerical results for the prying forces (specimen WT1). their evolution with the external load. In the elastic regime Q/F = 0.26 and n/e = 0.73 at the bolt horizontal axis; at failure, Q/F = 0.37 and n/e = 0.58 at the same section. There is an amplification in Q/F of 1.42 and the prying force shifts to the bolt vertical axis as the load increases. 4.8 CONSIDERATIONS ON THE NUMERICAL MODELLING OF THE HEAT

AFFECTED ZONE IN WP T-STUBS The numerical results presented above for WP-T-stub specimen WT1 do not account for the specific behaviour of the HAZ. The characterization of the me-chanical properties of this zone is very complex and uncertain due to its het-erogeneity and small size. Nevertheless, most of the high strength steels that


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have a high carbon content and various alloying elements as chromium, cop-per, nickel, etc. present a marked loss of hardness and strength in the HAZ that may affect the performance of the T-stub connection [4.36]. Moreover, local-ized heating from the welding process and subsequent rapid cooling induce a local triaxial residual tensile stress field in the HAZ and the constraint condi-tions affect the failure ductility of the metal in the zone. These effects are not easily modelled. However, there is evidence that when soft electrodes are used (cf. §3.2.4), the strength of the weld is slightly affected and the installed residual stress field is not significant [4.36]. As the fine microstructure is lost during the weld ther-mal cycle, the HAZ strength and toughness are expected to decrease below those of the BM [4.37], i.e. the HAZ softens. This softening effect, which de-pends on the heat input, can be so severe that fracture can occur in the HAZ instead of the BM, as seen in the previous chapter 3. Bang and Kim [4.37] es-timate the degree of HAZ softening in 20% at 6 kJ/mm. This means that the strength properties in this zone should be reduced to a maximum of 80% in relation to those of the BM. Another aspect that must be considered in the modelling of the HAZ is the width itself, lHAZ. Rodrigues et al. show that the ratio lHAZ/tf is an important parameter in the characterization of the change of strength in the zone [4.38]. They studied the influence of the HAZ size in the geometrical constraint effect and consequent influence behaviour of the joint. The study covered the range of lHAZ/tf 1/6-1 and it demonstrated that this influence is negligible if the WM tensile strength evenmatches the BM. This is the case of the tested specimens. Taking these considerations into account, a FE model was implemented for specimen WT1 in order to analyse the influence of the HAZ properties on the overall behaviour. The width of the zone was taken as 5 mm, which corre-sponds to lHAZ/tf

= 0.5. The model assumed a degree of softening of 15%, slightly below the maximum, as there was no information on the heat input during the welding process. The results are illustrated in Fig. 4.14. Compari-

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Fig. 4.14 Numerical results for specimen WT1 accounting for a reduction of

15% in the strength properties of the HAZ.


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sons with the original numerical model and experimental results are also set. The correspondence between the FE results and the experiments improves when compared to the model described earlier. There is a slight drop in the load-carrying behaviour in the post-limit domain (circa 7% in the load and 10% in the deformation capacity). These differences, however, can be considered insignificant. Therefore, for future analyses, the influence of the softening of the HAZ is disregarded. 4.9 CONCLUDING REMARKS The three-dimensional FE model presented above provides accurate deforma-tion predictions (up to fracture) of the T-stub response. It allows for a complete characterization of the load-carrying behaviour of both types of T-stub assem-blies. Table 4.2 compares the main characteristics of the F-∆ curve as ascer-tained numerically and experimentally. The results are very close, which means that the FE model is valid and reliable. The characterization of the T-stub collapse failure modes and correspond-ing ductility levels can be performed by means of this numerical procedure in order to clarify some code deficiencies. Additionally, the numerical model al-lows the evaluation of the prying forces, thus opening the way to more reliable design rules. Further, the parameters affecting the deformation capacity of bolted T-stubs can be highlighted and their influence on the overall behaviour of the connection can be assessed both qualitatively and quantitatively. It is easy to recognize that the deformation capacity of isolated bolted T-stub con-nections mainly depends on the mechanical properties of the materials and on some geometrical parameters. The next logical step forward is the implementa-tion of a parametric study based on the above procedures, in order to get in-sight on this particular aspect. The following chapter is devoted to such a study, presenting an experimental/numerical investigation that allows for a complete understanding of the main influences on the T-stub ultimate behav-iour. Table 4.2 Results for the two specimens [values in bold correspond to aver-

aged experimental results; underlined values include the web de-formation; K-R refers to the knee-range of the curve]. Stiffness (kN/mm) Strength (kN) ∆u Q/F Spec.

ke.0 kpl.0 ke.0/kpl.0 K-R Fu (mm) K-R Ult. 83.54 2.68 29.60 65-85 103.99 8.70 0.24 0.34 T1 49.00 1.73 28.32 58-87 102.81 9.49 69.29 1.57 44.24 55-76 94.98 14.20 0.27 0.37 WT1 71.09 2.09 34.01 58-69 183.83 14.33


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4.10 REFERENCES [4.1] Virdi KS. Guidance on good practice in simulation of semi-rigid con-

nections by the finite element method. In: Numerical simulation of semi-rigid connections by the finite element method (Ed.: K.S. Virdi). COST C1, Report of working group 6 – Numerical simulation, Brussels; 1-12, 1999.

[4.2] Bathe KJ. Finite element procedures in engineering analysis. Prentice-Hall, Englewood Cliffs, New Jersey, USA, 1982.

[4.3] Hinton E, Owen DR. An introduction to finite element computations. Pineridge Press Limited, Swansea, UK, 1979.

[4.4] van der Vegte GJ, Makino Y, Sakimoto T. Numerical research on sin-gle-bolted connections using implicit and explicit solution techniques. Memoirs of the Faculty of Engineering Kumamoto University; XXXXVII(1):19-44, 2002.

[4.5] Owen DRJ, Hinton E. Finite elements in plasticity, theory and practice. Pineridge Press Limited, Swansea, UK, 1980.

[4.6] Bathe KJ, Wilson EL. Numerical methods in finite element analysis. Prentice-Hall, Englewood Cliffs, New Jersey, USA, 1976.

[4.7] Lusas 13. Modeller reference manual. Finite element analysis Ltd, Ver-sion 13.2. Surrey, UK, 2001.

[4.8] Lusas 13. Solver reference manual. Finite element analysis Ltd, Version 13.2. Surrey, UK, 2001.

[4.9] Jaspart JP. Numerical simulation of a T-stub – experimental data. Cost C1, Numerical simulation group, Doc. C1WD6/94-09, 1994.

[4.10] Bursi OS. A refined finite element model for T-stub steel connections. Cost C1, Numerical simulation group, Doc. C1WD6/95-07, 1995.

[4.11] Bursi OS, Jaspart JP. Benchmarks for finite element modelling of bolted steel connections. Journal of Constructional Steel Research; 43(1):17-42, 1997.

[4.12] Bursi OS, Jaspart JP. Basic issues in the finite element simulation of extended end-plate connections. Computers and Structures; 69:361-382, 1998.

[4.13] Gomes FCT, Neves LFC, Silva LAPS, Simões RAD. Numerical simula-tion of a T-stub. Cost C1, Numerical simulation group, Doc. C1WG6/95-, 1995.

[4.14] Mistakidis ES, Baniotopoulos CC, Bisbos CD, Panagiotopoulos PD. A 2-D numerical model for the analysis of steel T-stub connections. Cost C1, Numerical simulation group, Doc. C1WD6/96-09, 1996.

[4.15] Mistakidis ES, Baniotopoulos CC, Bisbos CD, Panagiotopoulos PD. Steel T-stub connections under static loading: an effective 2-D numeri-cal model. Journal of Constructional Steel Research; 44(1-2):51-67, 1997.

[4.16] Zajdel M. Numerical analysis of bolted tee-stub connections. TNO-Report 97-CON-R-1123, 1997.


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[4.17] Wanzek T, Gebbeken N. Numerical aspects for the simulation of end plate connections. In: Numerical simulation of semi-rigid connections by the finite element method (Ed.: K.S. Virdi). COST C1, Report of working group 6 – Numerical simulation, Brussels; 13-31, 1999.

[4.18] Gebbeken N, Wanzek T, Petersen, C. Semi-rigid connections, T-stub model – Report on experimental investigations. Report 97/2. Institut für Mechanik und Static, Universität des Bundeswehr München, Munich, Germany, 1997.

[4.19] Swanson JA. Characterization of the strength, stiffness and ductility behavior of T-stub connections. PhD dissertation, Georgia Institute of Technology, Atlanta, USA, 1999.

[4.20] Swanson JA, Kokan DS, Leon RT. Advanced finite element modelling of bolted T-stub connection components. Journal of Constructional Steel Research; 58:1015-1031, 2002.

[4.21] Lusas 13. Element reference manual. Finite element analysis Ltd, Ver-sion 13.2. Surrey, UK, 2001.

[4.22] Lusas 13. Theory manual. Finite element analysis Ltd, Version 13.2. Surrey, UK, 2001.

[4.23] Simo JC, Rifai MS. A class of mixed assumed strain methods and the method of incompatible modes. International Journal for Numerical Methods in Engineering; 29:1595-1638, 1990.

[4.24] Crisfield M. Large deflection elasto-plastic buckling analysis of plates using finite elements. TRRL Report LR 593, Transport and Road Re-search Laboratory, Department of the Environment, Crowthorne, UK, 1973.

[4.25] Crisfield M. Non-linear finite element analysis of solids and structures, Volume 1 – Essentials. John Wiley & Sons Ltd., Chichester, UK, 1997.

[4.26] Crisfield M. Non-linear finite element analysis of solids and structures, Volume 2 – Advanced topics. John Wiley & Sons Ltd., Chichester, UK, 1997.

[4.27] Aggerskov H. High-strength bolted connections subjected to prying. Journal of Structural Division ASCE; 102(ST1):161-175, 1976.

[4.28] European Committee for Standardization (CEN). prEN 1993-1-8:2003, Part 1.8: Design of joints, Eurocode 3: Design of steel structures. Stage 49 draft, May 2003, Brussels, 2003.

[4.29] Vasarhelyi DD, Chiang KC. Coefficient of friction in joints of various steel. Journal of Structural Division ASCE; 93(ST4):227-243, 1967.

[4.30] Girão Coelho AM. Material data of the plate sections of the welded T-stub specimens. Internal report, Steel and Timber Section, Faculty of Civil Engineering, Delft University of Technology, 2002.

[4.31] Faella C, Piluso V, Rizzano G. Structural semi-rigid connections – the-ory, design and software, CRC Press, USA, 2000.

[4.32] Gioncu V, Mateescu G, Petcu D, Anastasiadis A. Prediction of available ductility by means of local plastic mechanism method: DUCTROT computer program, Chapter 2.1 in Moment resistant connections of steel


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frames in seismic areas (Ed.: F. Mazzolani). E&FN Spon, London, UK; 95-146, 2000.

[4.33] Hirt MA, Bez R. Construction métallique – Notions fondamentales et methods de dimensionnement. Traité de Génie Civil de l’École polytechnique fédérale de Lausanne, Volume 10. Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, 1994.

[4.34] Girão Coelho AM, Bijlaard F, Simões da Silva L. On the deformation capacity of beam-to-column bolted connections. Document ECCS-TWG 10.2-02-003, European Convention for Constructional Steelwork – Technical Committee 10, Structural connections (ECCS-TC10), 2002.

[4.35] Piluso V, Faella C, Rizzano G. Ultimate behavior of bolted T-stubs. II: model validation. Journal of Structural Engineering ASCE; 127(6):694-704, 2001.

[4.36] Loureiro AJR. Effect of heat input on plastic deformation of under-mathed welds. Journal of Materials Processing Technology; 128:240-249, 2002.

[4.37] Bang KS, Kim WY. Estimation and prediction of the HAZ softening in thermomechanically controlled-rolled and accelerated-cooled steel. Welding Journal; 81(8):174S-179S, 2002.

[4.38] Rodrigues DM, Menezes LF, Loureiro A, Fernandes JV. Numerical study of the plastic behaviour in tension of welds in high strength steels. International Journal of Plasticity; 20:1-18, 2004.


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APPENDIX B: PRELIMINARY STUDY FOR CALIBRATION OF THE FINITE ELE-MENT MODEL (E.G. HR-T-STUB T1)

B.1 Mesh convergence study As mentioned in §4.4.4, the flange mesh discretization with HX8M elements is analysed with respect to two parameters: (i) x: degree of discretization in order to represent the bending dominated problem and (ii) y: number of elements through thickness to check the capability of representing the yielding lines. The various discretizations are labelled ‘FxTy’, concerning the two above parame-ters, respectively (Fig. B1). The material properties adopted in these simula-tions are those from Fig. 4.2a. Fig. B2 depicts the F-∆ response of discretization F0Ty, with y = 1, 2, 3, 4 and 5. The deformation behaviour of F0T1 is stiffer than the other models be-cause shear locking occurs. The remaining models yield identical solutions in the elastic domain but slightly different solutions in the plastic domain. Model F0T2 is more flexible than F0T3, F0T4 and F0T5, which show very small de-viations. For future analysis, the model with three layers of HX8M is adopted.

(a) F0T1. (b) F0T4. (c) F1T2.

(d) F1T3. (e) F2T3. Fig. B1 Flange discretization: analysed models FxTy.


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To assess the influence of the degree of the flange discretization, the F-∆ be-haviour of models F0T3, F1T3 and F2T3 is compared (Fig. B3). In the elastic domain, the three models yield identical stiffness. The model F1T3 is stiffer than model F2T3 in the plastic regime, but the post-limit stiffness is identical for both models. For model F0T3 the slope of the F-∆ curve is smaller than the corresponding value for the finer meshes. Fig. B4 compares the curves for models F1Ty, y = 2, 3 and 4 and F2T3. Again, the model with two layers of elements shows a weaker response than the remainders. This situation, again, is due to the shear locking effect, which is compensated in this particular case by a weaker plastic response. Comparison of models F1T3 and F1T4 shows that the lesser the number of elements across flange thickness, the stiffer the re-sponse. F1T4 and F2T3, however, yield similar results. Model F2T3 (11910 nodes and 7722 elements) satisfies convergence re-quirements but demands greater computation effort. Model F1T3 (5680 nodes and 3588 elements) shows small deviations from F2T3 and is not as time-consuming. Therefore, it will be used extensively in future comparisons.

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F1T4 and F2T3. B.2 Influence of the definition of the constitutive law and element for-

mulation on the overall behaviour Fig. B5a compares the F-∆ curve for the different material stress-strain rela-tionships. As expected, this relation does not influence the elastic behaviour, but the post-limit behaviour is rather stiffer in the case of true stress-logarithmic strain relation and closer to the experimental behaviour. Fig. B5b depicts the F-∆ characteristics for two different element formulations: total Lagrangian and updated Lagrangian formulations. Again, the elastic part of the curve is not affected by the different element HX8M formulation. However, in the plastic range, the updated Lagrangian formulation yields closer results to the experimental curve. Therefore, to perform realistic simulations, a true stress-logarithmic strain relation must describe the constitutive material laws and the updated Lagrangian element formulation must be used. B.3 Calibration of the joint element stiffness Regarding the joint element stiffness, three cases are analysed in Fig. B6. The stiffer the elements, the stiffer the global T-stub response. From the stiffness values, it can be concluded that the results for the stiffness coefficient k1 = 8000 N/mm/mm2 are more realistic than the higher values. In terms of elastic stiffness and ultimate resistance, the three curves fit each other. However, in the knee-range of the global response, this model is more compliant and accu-rate than the remaining. The joint element stiffness k1 is hence taken as equal to 8000 N/mm/mm2 and the tangential stiffness coefficients are taken as k2 = k3 = 1000 N/mm/mm2.


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APPENDIX C: STRESS AND STRAIN NUMERICAL RESULTS FOR HR-T-STUB T1 C.1 Load steps for stress and strain contours To illustrate the stress and strain contour results, four load steps are chosen (Fig. C1) as follows: (i) 2F = 96.29 kN (load case 4) for the elastic regime, (ii) 2F = 159.55 kN (load case 9) for the knee-range, (iii) 2F = 190.85 kN (load case 25) for the subsequent linear part (in the post-limit regime) and (iv) 2F = 207.98 kN (load case 46) for the collapse (maximum deformation). C.2 Von Mises equivalent stresses, σeq The Von Mises equivalent stress, σeq, combines the individual component stresses at a node according to the classical Von Mises failure criterion. The stress distribution within the T-stub flange is well reproduced with the general-ized stress σeq. Fig. C2 illustrates the σeq contours in the three-dimensional view, for the four load levels. The bending of the flange is well reproduced. The peak equivalent stress values are located at the bolt axis and at the flange-to-web connection, where the yield lines develop. Fig. C3 depicts the equiva-lent stresses in xy cross-section corresponding to the bolt axis. Regarding the bolt behaviour, Fig. C4 shows the equivalent stresses for the chosen load levels. The bending of the bolt is clearly present from the com-mencement of loading. For the first load stage, no yielding occurs. As the load increases, the bolt stresses and strains magnify and so do the yielded portions. The compression and tension zones of the bolt are also noticeable: the bolt area near the web is subjected to tension whilst the zone near the tips of the flange is in compression.

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(a) Elastic regime. (b) Knee-range.

(c) Post-limit regime. (d) Collapse. Fig. C2 Von Mises equivalent stresses in the T-stub flange.

(a) Elastic regime. (b) Knee-range. Fig. C3 Von Mises equivalent stresses in xy cross-section. C.3 Stresses σxx and strains εxx The stress component in the x direction, σxx, represents the bending of the T- element flanges along this axis. The development of σxx is illustrated in Fig.


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C5. The double curvature bending of the T-stub flange is clear. The high values of tension (positive values) occur in the upper part of the flange-to-web con-nection and on the lower part of the flange near the bolt axis. Conversely, the high values of compression are located on the opposite parts. Fig. C6 presents the strain results for the same load levels.

(c) Post-limit regime. (d) Collapse. Fig. C3 Von Mises equivalent stresses in xy cross-section (cont.).


(c) Post-limit regime. (d) Collapse. Fig. C4 Von Mises equivalent stresses in the bolt.


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C.4 Stresses σyy The stress component along the y direction, σyy, is depicted in Figs. C.7-C.8 for the T-stub flange, three-dimensional and bottom xz plane views, respectively.


(c) Post-limit regime. (d) Collapse. Fig. C5 Stresses σxx in the T-stub flange.

(a) Elastic regime. (b) Knee-range. Fig. C6 Strains εxx in the T-stub flange.


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(c) Post-limit regime. (d) Collapse. Fig. C6 Strains εxx in the T-stub flange (cont.).


(c) Post-limit regime. (d) Collapse. Fig. C7 Stresses σyy in the T-stub flange. The positive stress σyy is quite uniform in the flange in the elastic areas. The stress uniform transfer between the flange and the web is also clear in Fig. C7 (red contour). The concentration of negative stress σyy occurs at the contact


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areas: the washer/flange and the flange/rigid foundation contact planes (Figs. C7-C8, respectively). In the latter, the stress concentration is quite distinct in the middle of the flange-to-web connection, due to the bending of the flanges and at the tips of the flanges, where the prying effect takes place. C.5 Stresses σzz The stress component on the z-axis, σzz, represents the deformation behaviour along the T-stub width. The distribution of the stress is not uniform and the peak values occur at the washer/flange contact plane, due to sliding (Fig. C9). C.6 Principal stresses and strains, σ11 and ε11 The principal stress σ11 and the principal strain ε11 represent the maximum stress and strain values, respectively. The maximum values of stress in the T-stub flange (Fig. C10) occur at the bolt axis and at the flange-to-web connec-tion. Fig. C11 shows the corresponding strain contours. Figs. C12-C13 illustrate the principal stresses and strains in the bolt, respec-tively. The maximum strain ε11 in the bolt corresponds to the maximum al-lowed strain and therefore once it is attained, collapse occurs. The distribution


(c) Post-limit regime. (d) Collapse. Fig. C8 Stresses σyy in the T-stub flange/rigid foundation contact plane.


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(c) Post-limit regime. (d) Collapse. Fig. C9 Stresses σzz in the T-stub flange.

(a) Elastic regime. (b) Knee-range. Fig. C10 Principal stresses σ11 in the T-stub flange. of principal stresses in the bolt shank is not uniform as the applied load in-creases. The maximum contour area enlarges with the increasing of load. The principal strain contours in the xy cross-section are illustrated in Fig. C14.


130

(c) Post-limit regime. (d) Collapse. Fig. C10 Principal stresses σ11 in the T-stub flange (cont.).


(c) Post-limit regime. (d) Collapse. Fig. C11 Principal strains ε11 in the T-stub flange.


131


(c) Post-limit regime. (d) Collapse. Fig. C12 Principal stresses σ11 in the bolt.

(a) Elastic regime. (b) Knee-range. Fig. C13 Principal strains ε11 in the bolt.


132

(c) Post-limit regime. (d) Collapse. Fig. C13 Principal strains ε11 in the bolt (cont.).


(c) Post-limit regime. (d) Collapse. Fig. C14 Principal strains ε11 in xy cross-section. C.7 Displacement results in xy cross-section Finally, the displacement contour in the x and y directions are illustrated in Figs. C15-C16 for the middle xy cross-section. The results are presented in the deformed configuration (magnification factor = 1.0). Fig. C15 shows that no penetration occurs between the bolt and the flange at collapse conditions.


133


(c) Post-limit regime. (d) Collapse. Fig. C15 Horizontal displacement contours in xy cross-section.

(a) Elastic regime. (b) Knee-range. Fig. C16 Vertical displacement contours in xy cross-section.


134

(c) Post-limit regime. (d) Collapse. Fig. C16 Vertical displacement contours in xy cross-section (cont.).

135

5 PARAMETRIC STUDY 5.1 DESCRIPTION OF THE SPECIMENS A parametric analysis was undertaken in order to identify the dependence of the T-stub behaviour on the main geometrical and mechanical variables. The basic HR specimen is T1 from the previous section and the study on this as-sembly type was performed numerically. For the WP specimens, an experimen-tal programme was devised along with a FE analysis. Three supplementary WP-T-stubs derived from HR-T-stub T1 were also considered to compare the behaviour of the two assembly types. The main geometric parameters that were varied in the study are: (i) weld throat thickness, aw, for WP-T-stubs (ii) gauge of the bolts, w, (iii) pitch of the bolts, p, (iv) end distance, e1, (v) edge distance, e, and (vi) flange thickness, tf. The influence of the bolt is analysed by varying: (i) the diameter, φ, (ii) the thread length (spec. P24: FT bolt) and (iii) the pre-load, S0 (spec. P25). The steel constitutive law is the mechanical variable in the study. Additionally, the question of the number of bolt rows is also tackled. Tables 3.1 and 5.1 sum up the main characteristics of the several specimens. In the following sections the load-carrying behaviour of the several speci-mens is compared to assess the influence of the main parameters. For some specimens other results are also included to get insight into other behaviour pa-rameters. 5.2 INFLUENCE OF THE ASSEMBLY TYPE AND THE WELD THROAT THICK-

NESS Having validated the numerical procedure for both T-stub assemblies, in this section the influence of welding and of the size of the fillet weld on the overall behaviour are analysed. For that purpose, HR-T-stub specimen T1 is selected. The equivalent WP-T-stub (generally labelled Weld_T1 hereafter) is identical to T1 in terms of geometrical and mechanical properties. The flange-to-web connection radius is thus replaced with a continuous 45º-fillet weld of throat thickness (i) 0.5 3.55w wa t= = mm [Weld_T1(i)], (ii) 7.1w wa t= = mm [Weld_T1(ii)] and (iii) 10wa = mm [Weld_T1(iii)]. The values for aw are cho-sen to meet the Eurocode 3 requirements [5.1]. The first value, 0.5w wa t= , complies with the minimum dimension prescribed in the code (3.0 mm) and is commonly used in practice. The value w wa t= can be regarded, in design prac-tice, as an upper value for the size of the fillet weld. Finally, the latter value


136

yields a distance m similar to the one in the HR specimen T1. For T1, m = 41.45 0.8 15 29.45− × = mm and for Weld_T1(iii), 41.45 0.8 2 10m = − × = 30.14 mm. The numerical results for both specimens T1 and Weld_T1 are compared in Figs. 5.1-5.3. Concerning the overall behaviour, the connections clearly yield different responses. Bolt fracture determines collapse of all T-stubs. Compari-son of the F-∆ responses of the welded specimens shows that as the weld throat thickness increases, the stiffness and resistance improve but the deformation capacity significantly decreases (Fig. 5.1). It should be noted that the increase Table 5.1 Details of the parametric numerical study [dimensions in mm;

yield stress in MPa] (see Figs. 1.9 and 3.1 for notation).

T-stub elements geometry Type Test ID

Profile b p e1 w e r/aw tf tw

P1 IPE300 40 40 20 100 25 15 10.7 7.1 P2 IPE300 40 40 20 80 35 15 10.7 7.1 P3 IPE300 35 30 20 90 30 15 10.7 7.1 P4 IPE300 52.5 65 20 90 30 15 10.7 7.1 P5 IPE300 60 80 20 90 30 15 10.7 7.1 P6 IPE300 35 40 15 90 30 15 10.7 7.1 P7 IPE300 45 40 25 90 30 15 10.7 7.1 P8 HEA220 40 40 20 90 65 18 11.0 7.0 P9 HEB180 40 40 20 90 45 15 14.0 8.5 P10 HEAA160+ 40 40 20 90 30 15 7.0 4.5 P11 HEB180 40 40 20 90 30 15 14.0 8.5 P12 IPE300 40 40 20 90 30 15 10.7 7.1 P13 IPE300 40 40 20 90 30 15 10.7 7.1 P14 IPE300 40 40 20 90 30 15 10.7 7.1 P15 IPE300 40 40 20 80 35 15 10.7 7.1 P16 IPE300 70 70 35 90 30 15 10.7 7.1 P17 IPE300 70 70 35 90 30 15 10.7 7.1 P18 IPE300 70 70 35 90 30 15 10.7 7.1 P19 IPE300 70 90 25 90 30 15 10.7 7.1 P20 HEB180 70 70 35 90 30 15 14.0 8.5 P21 IPE300 92.5 115 35 90 30 15 10.7 7.1 P22 UB457×

152×67 70 70 35 90 30 10.2 15.0 9.0

P23 UB457× 152×82

70 70 35 90 30 10.2 18.9 10.5

P24 IPE300 40 40 20 90 30 15 10.7 7.1

HR

spec

imen

s

P25 IPE300 40 40 20 90 30 15 10.7 7.1 Weld_T1(i) 40 40 20 90 30 3.55 10.7 7.1 Weld_T1(ii) 40 40 20 90 30 7.1 10.7 7.1 W

P sp

ec.

Weld_T1(iii) 40 40 20 90 30 10 10.7 7.1

Parametric study

137

Table 5.1 Details of the parametric numerical study (cont.). Bolt Material (fy) Type Test

ID Profile

φ # Type Flange Bolt P1 IPE300 12 4 ST 431 893 P2 IPE300 12 4 ST 431 893 P3 IPE300 12 4 ST 431 893 P4 IPE300 12 4 ST 431 893 P5 IPE300 12 4 ST 431 893 P6 IPE300 12 4 ST 431 893 P7 IPE300 12 4 ST 431 893 P8 HEA220 12 4 ST 431 893 P9 HEB180 12 4 ST 431 893 P10 HEAA160+ 12 4 ST 431 893 P11 HEB180 12 4 ST 431 893 P12 IPE300 16 4 ST 431 893 P13 IPE300 12 4 ST 355 893 P14 IPE300 12 4 ST 275 893 P15 IPE300 16 4 ST 431 893 P16 IPE300 12 4 ST 431 893 P17 IPE300 16 4 ST 431 893 P18 IPE300 20 4 ST 431 893 P19 IPE300 16 4 ST 431 893 P20 HEB180 16 4 ST 431 893 P21 IPE300 20 4 ST 431 893 P22 UB457×

152×67 20 4 ST 431 893

P23 UB457× 152×82

20 4 ST 431 893

P24 IPE300 12 4 FT 431 893

HR

spec

imen

s

P25 IPE300 12 4 ST 431 893 Weld_T1(i) 12 4 ST 431 893 Weld_T1(ii) 12 4 ST 431 893 W

P sp

ec.

Weld_T1(iii) 12 4 ST 431 893 Table 5.2 Numerical results (per bolt row) for T1 and weld-equiv. Weld_T1.

Stiffness (kN/mm) Resistance (kN) ∆u.0 Q/F Test ID ke.0 kp-l.0 ke.0/kp-l.0 K-R Fu.0 (mm) K-R Ult.

T1 83.54 2.68 29.60 65-85 103.99 8.70 0.24 0.34 Weld _T1(i) 73.50 1.70 43.12 50-78 92.02 10.85 0.34 0.45 Weld _T1(ii) 88.04 2.51 35.07 60-87 102.75 8.01 0.27 0.36 Weld _T1(iii) 107.29 3.31 32.41 75-97 113.10 6.22 0.22 0.28


138

0

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(kN

)

T1 Weld_T1(i)

Weld_T1(ii) Weld_T1(iii)

Fig 5.1 Overall response of specimens T1 and Weld_T1.

0.0

0.2

0.4

0.6

0.8

1.0

0 15 30 45 60 75 90 105 120


Rat

io B

/F

T1 Weld_T1(i)Weld_T1(ii) Weld_T1(iii)

(a) Ratio B/F.

0.0

0.2

0.4

0.6

0.8

1.0

0 15 30 45 60 75 90 105 120


Rat

io Q

/F


b) Ratio Q/F. Fig. 5.2 Bolt and prying force ratios for specimens T1 and Weld_T1.

Parametric study

139

0.72

0.76

0.80

0.84

0.88

0.92

0.96

1.00

0 15 30 45 60 75 90 105 120


Rat

io n

/e


(a) Whole contact area.

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 15 30 45 60 75 90 105 120


Rat

io n

/e


(b) Joint elements at the bolt x axis. Fig. 5.3 Evolution of the ratios n/e for specimens T1 and Weld_T1.

0

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240

0 1 2 3 4 5 6 7 8 9 10 11

Deformation (mm)

Tota

l app

lied

load

(kN

)

Weld_T1(i) Weld_T1(ii)

157.95 kN(Ld cs 6)

205.50 kN(Ld cs 29)

140.11 kN(Ld cs 9)

184.05 kN(Ld cs 57)

Fig. 5.4 Selected load levels for stress and strain analyses of specimens

Weld_T1.


140

(i) Knee-range. (ii) Collapse. (a) Specimen T1.

(i) Knee-range. (ii) Collapse. (b) Specimen Weld_T1(i).

(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.5 Von Mises equivalent stresses in the T-stub flange.

Parametric study

141



(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.6 Von Mises equivalent stresses in xy cross-section. in aw leads to a decrease in m. To conclude about the influence of welding it-self, the comparisons have to be made between specimen T1 and Weld_T1(iii) that have similar values of the distance m. Clearly, if the flange-to-web connec-tion radius is replaced with a fillet weld, the stiffness and the resistance of the connection improve, but the deformation capacity is greatly reduced: it drops


142



(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.7 Stresses σxx in the T-stub flange.

Parametric study

143

(i) Knee-range. (ii) Collapse. (a) Specimen T1. Fig. 5.8 Strains εxx in the T-stub flange. from a gap between flanges of 8.70 mm to 6.22 mm. For specimens T1 and Weld_T1(ii), the F-∆ curves are surprisingly coincident. However, in the welded case, the ductility is smaller. Table 5.2 sets out the main characteristics of the four F-∆ curves. Regarding the bolt force and the prying forces, their magnitude in relation to the applied load is higher for smaller weld throat thickness, in the case of the welded specimen for all course of loading (Fig. 5.2). The location of the con-tact forces also changes with the increasing of loading (Fig. 5.3). For the WP specimen Weld_T1(i), with smaller aw, Q is closer to the bolt axis than in the remaining specimens. Concerning the influence of the assembly type, Fig. 5.2 shows that in the elastic regime the ratios B/F and Q/F for specimens T1 and Weld_T1(iii) are coincident but as the load increases the same ratios decrease in the welded case. Worth mentioning is the fact that even if T1 and Weld_T1(ii) yield identical F-∆ behaviour, the evolution of B/F and Q/F with the course of loading is different (Fig. 5.2). With respect to the location of the prying forces, Fig. 5.3 shows that for the welded specimen Weld_T1(iii) there is a almost constant relationship of n/e from the commencement of loading to failure, with a slight increase near collapse. For specimen T1, the variation of


144

(i) Knee-range. (ii) Collapse. (b) Specimen Weld_T1(i). Fig. 5.8 Strains εxx in the T-stub flange (cont.). Q with the applied load is more evident and Q is shifted to the bolt axis near collapse failure. The magnitude of the difference in performance of the HP-T-stub T1 and the welded equivalents Weld_T1 is rather surprising. In terms of the overall de-formation behaviour, the differences can arise due to the redefinition of the length m that slightly increases in the welded case. This accounts for the de-crease in stiffness and resistance. Regarding the deformation capacity, as it will also be shown in the following section, the increase in the same distance m im-proves the ultimate deformation of the connection, ∆u. With respect to the pry-ing effect, the disparity of results was not expected. To compare the stress and strain contour results for the above specimens, two load steps are chosen (Fig. 5.4), corresponding to the knee-range of the curves and collapse (maximum deformation). For specimen T1 the reader should refer to Fig. C1 from Appendix C for indication of the analogous levels. As the contour results for the welded specimens are identical, only the results for specimens Weld_T1(i-ii) are shown. For the two selected load steps, Figs. 5.5-5.6 show the Von Mises equivalent stress contours in the T-stub flange and in xy cross-section at the bolt axis. The figures show that the higher stress val-ues in the flange concentrate at the bolt axis and near the flange-to-web con-

Parametric study

145

(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.8 Strains εxx in the T-stub flange (cont.). nection. In particular, in the case of WP specimens, such concentration takes place at the “potential” HAZ rather than at the weld toe. The stress distribution in the bolt is identical in all three cases. Figs. 5.7-5.10 illustrate the stress and strain contours in the x direction. They clearly show the double curvature of the flange in the three cases and confirm the previous conclusions related to the location of yield lines and po-tential fracture lines. Finally, Figs. 5.11-5.14 display identical results with re-spect to the principal direction 1. The experimental programme also included the analysis of the influence of the fillet weld throat thickness, aw on the overall behaviour (series WT2 – cf. Table 3.1). Bolt fracture is still the determinant factor of collapse, though some damage in the HAZ has been observed in specimens WT2Aa and WT2Ba. In these specimens the weld quality was inferior to the expected and this may ex-plain such plate damage. Fig. 5.15a shows that if aw decreases, the resistance slightly decreases, whilst the deformation capacity improves, with little varia-tion of stiffness. On the other hand, if aw increases, the deformation capacity is reduced, resistance increases and there is still small change in the slope of the two characteristic branches of the F-∆ curve (Fig. 5.15b; see also Table 3.8). The main (experimental) characteristics of the tests also confirm the above


146



(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.9 Stresses σxx in xy cross-section. statements related to the influence of the fillet weld throat thickness. In series WT2Aa there was a malfunctioning of the LVDTs and there is only a record of the deformation behaviour until 4.5∆ ≈ mm.

Parametric study

147



(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.10 Strains εxx in xy cross-section. 5.3 INFLUENCE OF GEOMETRIC PARAMETERS The main geometric connection parameters that were varied in this parametric study are indicated in §5.1.


148

(i) Knee-range. (ii) Collapse (a) Specimen T1.


(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.11 Principal stresses σ11 in the T-stub flange.

Parametric study

149

(i) Knee-range. (ii) Collapse. (a) Specimen T1. Fig. 5.12 Principal strains ε11 in the T-stub flange. 5.3.1 Gauge of the bolts To assess the influence of the variation of the distance w, two HR specimens, P1 and P2, were obtained from T1 by shifting the bolt axis centreline (Table 5.1). Naturally, this will also alter the distance m between yield lines. Note that in all three cases the determinant plastic failure mechanism was of type-1. Yet, the collapse condition of the several specimens was determined by bolt fracture (black circles). Fig. 5.16 shows that if the gauge of the bolts increases, consequently in-creasing the distance m between plastic hinges, the connection strength and stiffness decrease but the deformation capacity improves. These results can be found in Table 5.3. 5.3.2 Pitch of the bolts and end distance The enlargement of the pitch of the bolts and/or the end distance implies larger T-stub widths and therefore higher stiffness and resistance values but reduced deformation capacity (Figs. 5.17-5.18). As the T-stub width increases, the ef-


150

(i) Knee-range. (ii) Collapse. (b) Specimen Weld_T1(i). Fig. 5.12 Principal strains ε11 in the T-stub flange (cont.). fective width also increases, since the beam pattern governs the plastic mecha-nisms. Therefore, the flexural resistance of the flanges is enhanced and so βRd assumes larger values. For a certain ratio λ, then the starting governing plastic mechanism type-1 changes into type-2 and eventually into type-3, as beff grows. This transition of plastic modes is associated with the increase in resistance and initial stiffness and the reduction of deformation capacity. For all the analysed specimens, bolt determined collapse and the plastic mechanism was of type-1 (flange yielding). For specimen WT4A, however, βRd was very close to the boundary limit of type-2. This is rather evident in Fig. 5.19a that shows WT4Ab at collapse conditions. Apparently, the flange is in single bending cur-vature. Table 5.3 sets out the main characteristics of the above F-∆ responses and confirms the above statements concerning the major influences of the T-stub width on the overall behaviour of T-stubs. For better understanding, part of Ta-ble 3.8 for the welded specimens is included here. The results in Table 5.3 are presented for a bolt row. This means that the previous experimental results for stiffness and resistance are divided by 2. With respect to the experiments, Fig. 5.18c compares the results for the two tests in this series with WT1h. The connection ductility clearly decreases.

Parametric study

151

(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.12 Principal strains ε11 in the T-stub flange (cont.). When comparing specimens WT1g/h and WT4Aa/b, the reduction of ∆u.0 is, on average, 66%. Bolt determined collapse in all cases. In particular, for WT4Aa, only bolt RB did not fail and for specimen WT4Ab, for which there is a record of the bolt elongation behaviour up to collapse (Fig. 5.20), the bolts on the left hand side were broken (Fig. 5.19 – the specimen is rotated in this figure). In fact, the graph from Fig. 5.20 shows that the bolts on the right hand side yielded smaller deformation than the others. Fig. 5.19b shows that the bolts were highly deformed at collapse. In this figure, an unbroken bolt is shown and the combined bending and tension deformations are very clear. It should be stressed that the bolt measurement up to collapse has only been carried out in this specific specimen as an experiment. Unfortunately, it was observed that the measuring brackets were damaged in the end and therefore they had to be replaced prior to collapse. 5.3.3 Edge distance and flange thickness The variation of the edge distance e is analysed in Fig. 5.21 that depicts the F-∆ behaviour of specimens T1, P8, P9 and P11. For these specimens not only


152



(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.13 Principal stresses σ11 in xy cross-section. the edge distance was varied, but also the flange thickness and the distance m were slightly different, as the beam profiles changed (P8: tf = 11.0 mm; P9-11: tf = 14.0 mm). To conclude about the single effect of e, only specimens P9 and P11 can be compared. Clearly, both stiffness and resistance are identical and the deformation capacity does not vary significantly either. If e is bigger, ∆u.0 is

Parametric study

153



(i) Knee-range. (ii) Collapse. (c) Specimen Weld_T1(ii). Fig. 5.14 Principal strains ε11 in xy cross-section. somewhat improved. Fig. 5.22 depicts the influence of the flange thickness on the overall behaviour (the distance m also varies as the profile changes). From the graphs it can be concluded that as the flange thickness decreases and all the


154

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0 2 4 6 8 10 12 14 16 18 20 22

Deformation (mm)

Tota

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(kN

)

WT1h WT2Aa WT2Ab

(a) Series WT2A: smaller weld throat thickness.

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Tota

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(kN

)

WT1h WT2Ba WT2Bb

(b) Series WT2B: smaller weld throat thickness. Fig. 5.15 Experimental load-carrying behaviour of specimen series WT2 and

comparison with WT1h.

0

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80

120

160

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240

280

0 1 2 3 4 5 6 7 8 9 10 11Deformation (mm)

Tota

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load

(kN

)

T1 P1 P2

Fig. 5.16 Influence of the gauge of the bolts on the overall behaviour.

Parametric study

155

0

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0 1 2 3 4 5 6 7 8 9 10 11

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(kN

)

T1 P3 P4 P5

(a) Pitch of the bolts, p.

0

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0 1 2 3 4 5 6 7 8 9 10 11

Deformation (mm)

Tota

l app

lied

load

(kN

)

T1 P6 P7

(b) End distance, e1. Fig. 5.17 Influence of the T-stub width on the overall behaviour: single effects

of the pitch and end distance.

0

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280

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Tota

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lied

load

(kN

)

T1 P16

(a) Numerical results for HR specimens. Fig. 5.18 Influence of the T-stub width on the overall behaviour: combined ef-

fects of the pitch and end distance.


156

0

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0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0


Tota

l app

lied

load

(kN

)

Experimental results: WT4Aa

Experimental results: WT4Ab


(b) Experimental and numerical load-carrying behaviour of WP specimen WT4A.

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Deformation (mm)

Tota

l app

lied

load

(kN

)

WT1h WT4Aa WT4Ab

(c) Comparison of the responses of the original specimen WT1 and WT4A: ex-perimental assessment.

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0 2 4 6 8 10 12 14 16

Deformation (mm)

Tota

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lied

load

(kN

)

WT1 WT4A

(d) Comparison of the responses of the original specimen WT1 and WT4A: numerical assessment. Fig. 5.18 Influence of the T-stub width on the overall behaviour: combined ef-

fects of the pitch and end distance (cont.).

Parametric study

157

(a) Deformation at failure. (b) Detail of an unbroken bolt after failure of the connection. Fig. 5.19 Specimen WT4Ab at collapse conditions.

0

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

Bolt deformation (mm)

Tota

l app

lied

load

(kN

)

Experimental results: bolt LF

Experimental results: bolt LB


(a) Comparison of the numerical results with experimental evidence.

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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8


Tota

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lied

load

(kN

)

Bolt RB Bolt LB

Bolt LF Bolt RF

(b) Experimental results. Fig. 5.20 Bolt elongation behaviour for specimen WT4Ab.


158

Table 5.3 Synthesis of the characteristic results (per bolt row) of the curves comparing the effect of the geometric parameters on the overall behaviour [underlined values correspond to experimental results]. Stiffness (kN/mm) Resistance (kN) ∆u.0 Q/F Test

ID ke.0 kp-l.0 ke.0/kp-l.0 K-R Fu.0 (mm) K-R Ult. T1 83.54 2.68 29.60 65-85 103.99 8.70 0.24 0.34 P1 63.27 2.01 31.48 60-73 91.76 10.77 0.33 0.44 P2 117.06 3.49 33.54 70-100 116.72 6.18 0.18 0.25 P3 72.62 2.30 31.62 60-75 95.41 10.17 0.26 0.40 P4 97.86 4.24 23.08 70-103 115.97 4.68 0.20 0.26 P5 101.23 6.00 16.88 90-120 130.20 3.63 0.18 0.18 P6 76.56 2.36 32.44 55-75 95.53 10.06 0.26 0.42 P7 91.45 2.97 30.79 70-93 111.34 7.56 0.22 0.29 P8 95.12 3.06 31.04 75-93 112.71 8.08 0.19 0.28 P9 128.47 6.19 20.76 85-120 131.43 3.31 0.13 0.16 P10 43.88 0.90 48.63 40-47 76.79 32.75 0.56 0.77 P11 122.80 6.82 18.01 90-110 121.15 2.94 0.20 0.21 P16 111.23 8.44 13.18 85-112 125.69 3.06 0.19 0.20 WT1 69.29 1.57 44.24 55-76 94.98 14.20 0.27 0.37 WT1g 68.58 2.11 32.50 58-68 91.33 14.10 WT1h 73.58 2.07 35.55 60-70 92.50 14.55 WT4A 85.95 3.84 22.38 67-99 107.95 5.18 0.22 0.27 WT4Aa 75.08 2.75 27.30 59-105 108.20 5.35 WT4Ab 86.96 4.37 19.90 70-98 103.26 4.33 other parameters remain the same, the component ductility improves consid-erably, whilst stiffness and resistance decrease. In this case, the flange-bolt stiffness decreases and, consequently, the degree of plastic deformation in the flange increases. 5.4 INFLUENCE OF THE BOLT AND FLANGE STEEL GRADE Fig. 5.23 illustrates the influence of the bolt diameter on the overall behaviour of HR-T-stubs. Essentially, if the bolt diameter is bigger, the initial stiffness, the strength and the ductility improve greatly but the post-limit stiffness de-creases. For a given geometry, the bolt ceases to be the determining factor of collapse. The bolt-threaded length has an effect on the overall response if the bolt governs the specimen collapse. In that case, if the threaded portion of the bolt is longer, the deformation capacity of the whole connection increases. The remaining properties do not change much (Fig. 5.24). The effect of a bolt pre-loading is the enhancement of the initial stiffness (Fig. 5.25). The quantifica-tion of the observed behaviour variations is summarized in Table 5.4.

Parametric study

159

0

40

80

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T1 P8 P9 P11

Fig. 5.21 Influence of the edge distance on the overall behaviour.

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T1 P9 P10 P11

Fig. 5.22 Influence of the flange thickness on the overall behaviour. Fig. 5.26 compares the behaviour of the specimens with larger bolts (M16 and M20), confirming the previous considerations on the influence of the bolt diameter. This conclusion is also supported by experimental evidence (Fig. 5.27). Surprisingly, if the deformation capacity of the connection is evaluated at the maximum load level, specimen WT7_M16 yields a higher value when compared to WT7_M20 (Fig. 5.27). However, the ductile branch after collapse starts is longer in the latter case. These conclusions can also be taken from Ta-ble 5.4 that repeats part of the information contained in Table 3.8 for this test series. For illustration, Fig. 5.28 shows the specimens from the experiments at failure conditions. For specimen WT7_M20 whose failure mode is of type-11 (cracking of the material at the HAZ – Fig. 5.28c), a numerical model was also implemented.


160

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T1 P12

(a) Geometry from T1.

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P2 P12 P15

(b) Geometry from P2.

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P16 P17 P18

(c) Geometry from P16. Fig. 5.23 Influence of the bolt diameter on the overall behaviour: numerical

results.

Parametric study

161

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T1 P24

Fig. 5.24 Influence of the bolt threaded length on the overall behaviour.

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T1 P25

Fig. 5.25 Influence of a bolt preloading on the overall behaviour. Again, this model does not cater for the specific behaviour of the HAZ, as al-ready mentioned in §4.8. The differences between the two F-∆ responses shown in Fig. 5.29 derive from this simplification. In particular, the failure ductility of the metal in this HAZ is clearly reduced. Experimentally, the de-formation of the T-stub flange at maximum load is 9.12 mm, whilst numeri-cally a total deformation of 25.37 mm is reached. However, since the softening branch is sufficiently large, this numerical value is comparable to the maxi-mum deformation of 18.70 mm that was reached in the experiments. At this displacement level, the test was stopped to prevent damage of the equipment. The numerical deformation capacity was established by setting the maximum average principal strain, at the critical zone, ε11.av.f, as equal to the ultimate strain of the flange material, εu.f (cf. §4.5). In this specific case, since the criti-cal section is located at the HAZ, the deformation of the flanges was also


162

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P17 P19 P20

(a) Specimens with bolt M16.

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P18 P21 P22 P23

(b) Specimens with bolt M20. Fig. 5.26 Influence of some geometric variations for bolts M16 and M20 and

the corresponding geometries P17 and P18 (HR-T-stub series).

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WT7_M20 WT7_M16 WT7_M20

Fig. 5.27 Influence of the bolt diameter on the overall behaviour: experimental

results (series WT7).

Parametric study

163

(a) Spec. WT7_M12. (b) Spec. WT7_M16. (c) Spec. WT7_M20. Fig. 5.28 Deformation of specimens WT7 at failure. Table 5.4 Synthesis of the characteristic results (per bolt row) of the curves

comparing the effect of the bolt on the overall behaviour [under-lined values correspond to experimental results].

Stiffness (kN/mm) Strength (kN) ∆u Q/F Test ID ke.0 kpl.0 ke.0/kpl.0 K-R Fu (mm) K-R Ult.

T1 83.54 2.68 29.60 65-85 103.99 8.70 0.24 0.34 P12 102.05 2.19 46.68 80-103 154.06 24.22 0.29 0.57 P24 80.46 1.89 42.51 65-87 108.14 13.80 0.24 0.31 P25 127.66 2.59 49.29 65-85 104.26 8.72 0.32 0.34 P2 117.06 3.49 33.54 70-100 116.72 6.18 0.18 0.25 P12 102.05 2.19 46.68 80-103 154.06 24.22 0.29 0.57 P15 128.41 2.71 47.47 80-120 171.08 18.02 0.21 0.48 P16 111.23 8.44 13.18 85-112 125.69 3.06 0.19 0.20 P17 138.31 3.86 35.81 115-165 192.01 9.29 0.22 0.36 P18 171.57 2.56 66.96 150-200 266.57 26.07 0.27 0.44 P19 127.27 3.93 32.36 115-160 186.52 9.29 0.22 0.38 P20 181.68 9.73 18.67 160-200 225.94 4.06 0.18 0.23 P21 168.61 3.22 52.38 155-230 281.33 17.67 0.23 0.42 P22 250.69 7.08 32.07 190-270 305.21 6.40 0.21 0.34 P23 322.09 9.61 33.53 275-305 346.01 5.22 0.20 0.24 WT4Aa 75.08 2.75 27.30 59-105 108.20 5.35 WT4Ab 86.96 4.37 19.90 70-98 103.26 4.33 WT7_M12 91.18 3.78 24.12 60-96 100.64 3.86 WT7_M16 116.09 5.08 17.54 80-104 132.34 5.88 WT7_M20 142.80 2.86 49.93 90-131 177.53 25.37 0.30 0.45 WT7_M20 137.70 5.61 16.33 88-118 145.72 15.98 evaluated for other flange strain levels, as indicated in Fig. 5.29 (e.g.: for an ε11.av.f = 0.15, corresponding to 0.5εu.f, ∆ = 12.21 mm). Another comparison that can be performed with the experimental test series


164

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Numerical resultsExperimental resultsE11.p = 0.15 (def. = 12.21 mm)E11.p = 0.20 (def. = 16.60 mm)E11.p = 0.25 (def. = 20.98 mm)

Fig. 5.29 Global response of specimen WT7_M20: numerical and experimen-

tal results.

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lied

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t row

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WT4Aa WT4Ab WT7_M12

Fig. 5.30 Experimental load-carrying behaviour of specimen WT7_M12 and

comparison with series WT4Aa (per bolt row): assessment of the in-fluence of number of bolt rows for identical geometries.

WT7 (specimen WT7_M12, more specifically) and series WT4A relates to the influence of the number of bolts fastening the T-stub elements. Fig. 5.30 com-pares the F-∆ response, per bolt row, for the three specimens, and shows a good agreement. This means that the symmetric behaviour is valid. This graph also shows that for specimen WT7_M12 at a load level of 58 kN some slippage occurred, resulting in a sharp decrease of stiffness in the response. Identical situation is observed in WT4Aa. Regarding the effect of the flange steel grade, Fig. 5.31 shows that the ini-tial stiffness is not affected by the steel properties (as long as the Young

Parametric study

165

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T1 P13 P14

(a) Numerical results.

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WT1h WT51a WT51b

(b) Experimental results. Fig. 5.31 Influence of the flange steel grade on the overall behaviour. modulus is constant) but as the yield stress of the flange, fy.f increases the resis-tance and the post-limit stiffness also increase and the deformation capacity decreases. Table 5.5 confirms these conclusions. The FE models of P13 and P14 were obtained from the original specimen T1 by reducing the stress values of the flange mechanical properties and main-taining the strain ordinates. Both new specimens exhibit a type-1 plastic failure mechanism. The flexural resistance of the flanges increases with the flange yield stress and so βRd becomes greater. This explains the improvement in the resistance properties despite a reduction in the deformation capacity. In the above case, specimen P14, whose flange is steel grade S275, is typified by a failure type-11, i.e. cracking of the flange material is the determining factor of collapse. For the other two specimens, bolt failure governs the ultimate conditions. Test series WT51 comprises the testing of two specimens geometrically


166

Table 5.5 Synthesis of the characteristic numerical results (per bolt row) of the curves comparing the effect of the flange steel grade.

Stiffness (kN/mm) Strength (kN) ∆u Q/F Test ID ke.0 kpl.0 ke.0/kpl.0 K-R Fu (mm) K-R Ult.

T1 83.54 2.68 29.60 65-85 103.99 8.70 0.24 0.34 P13 81.31 2.19 37.17 55-72 93.71 11.38 0.24 0.42 P14 81.97 1.07 76.69 45-65 86.57 24.15 0.24 0.53 identical to the original test series WT1 and whose T-stub elements are made up of high-strength steel S690. According to Eurocode 3, these specimens ex-hibit a type-2 plastic mechanism. Bolt governs collapse in the three cases and the deformation capacity is far reduced in series WT51 because the bolts are engaged in collapse of the specimen at an earlier stage. The knee-range of the F-∆ response of specimens WT51 develops for higher loads in comparison to WT1h. The slope of the post-limit part of these curves is lower than in the original case. Here, the single curvature of the flange is evident (Fig. 5.32) and the deformation of the flanges is far less than in series WT1 (see Fig. 3.11, for instance). This is also clear in Fig. 5.33a where the strains for WT1h and WT51b are compared for SG3, on the same location in both specimens (Figs. 3.8a-b). Fig. 5.33b plots the force-strain results for the two T-rosettes attached to specimen WT51b. It shows that the flange strain level there at collapse is rather low. Symmetry of results is also rather obvious. Now consider test series WT53 to assess the influence of the bolt type on the overall response. Naturally, since the actual bolt properties also vary (Table 3.2), the global results will include not only the effect of the bolt type (short- or full-threaded) but also their mechanical properties. Fig. 5.34 depicts the F-∆ re-sponse of identical T-stub elements connected by means of the four different M12 bolt types tested (cf. §3.2.2.1). The graphs show that if higher strength bolts are used (WT53D/E), since bolt determines failure in the four cases, the maximum load reached is also higher (see Table 3.8). For the four specimens compared in this figure, the initial stiffness is identical because the Young modulus, which is one of the main parameters used in the computation of ke.0, is identical for the four bolt types (Table 3.2). If bolt governs the failure mode of the T-stub, the overall deformation ca-pacity mainly depends on the maximum elongation of the bolt, or, in other words, on the ultimate strain values. Table 3.2 shows that full-threaded bolts exhibit higher values of εu (though for bolt grade 10.9 that difference is smaller) and higher bolt grades exhibit smaller deformations, i.e. the failure type is more brittle. When taking into account the T-stubs WT51b and WT53C/D/E, the above considerations are still valid. Specimen WT53C is more ductile than the remaining since the fasteners are full-threaded M12 grade 8.8, even though the deformation level at Fmax is lower (Table 3.8). For this specimen, the plateau that follows Fmax is far longer than in the other cases

Parametric study

167

(a) Deformation at failure (WT51b). (b) Detail of a broken bolt (WT51a). Fig. 5.32 Deformation of specimens WT51 at failure.

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WT1h WT51b

(a) Comparison of the results for SG3 in specimens WT1h and WT51b.

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SG6x SG6zSG7x SG7z

(b) Results for the rosettes. Fig. 5.33 Experimental results for the flange strain behaviour (specimen

WT51b).


168

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WT51b WT53C

WT53D WT53E

Fig. 5.34 Experimental load-carrying behaviour of specimen series WT53 and

comparison with WT51b.

Fig. 5.35 Comparison of the deformation of specimens WT51, WT53C,

WT53D and WT53E (from left to right) at failure. (Fig. 5.34). Surprisingly, the deformation capacity for both tests WT53D/E that use bolt grade 10.9 is identical. These conclusions are also indicated in Table 3.8. Fig. 5.35 illustrates the four specimens after failure. Having analysed the influence of the steel grade on the overall T-stub be-haviour (mainly: increase of strength and decrease of ductility for higher strength steel grades), the response obtained for series WT57 and WT7 can be compared. In series WT57, when using bolts M12 and M16, the plastic resis-tance of the specimens, as determined according to Eurocode 3 (Table 3.7) cor-responds to that of a plastic mechanism type-2 whilst for M20 it corresponds to a type-1 plastic mechanism. This is evident in the graphs from Fig. 5.36 where the responses of the three specimens are shown. For comparison, WT7_M20 is also included. It is worth mentioning that the bolt is also engaged in collapse in the case of the high-strength steel (WT57_M20) since the specimen fails in a combined failure mode (type-13), whilst in WT7_M20 collapse is governed by plate cracking near the weld toe only. Basically, the conclusions drawn above are supported with this series of experiments (summary in Table 3.8).

Parametric study

169

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WT7_M20 WT57_M12WT57_M16 WT57_M20


comparison with WT7_M20. 5.5 EXPERIMENTAL RESULTS FOR THE STIFFENED TEST SPECIMENS AND

THE ROTATED CONFIGURATIONS The experimental programme included the test of some transversely stiffened specimens and T-stub connections with the elements orientated at right angles, in order to simulate the actual behaviour in tension of the components model-ling the end plate side. The results obtained for those cases are discussed in this section. For complete description of the specimens and the characteristic re-sults of the load-carrying behaviour, the reader should refer to Chapter 3. 5.5.1 Influence of a transverse stiffener If a transverse stiffener is added to a T-stub connection, stiffness and resistance properties improve and deformation capacity decreases. To support this state-ment, first consider series WT61 that is obtained from the original WT1 by in-cluding a transverse stiffener in order to simulate the T-stub model for the end plate side (Fig. 1.8c). The load-carrying behaviour of the two specimens in-cluded in this series is compared with specimen WT1h and the code predictions [5.1] in Fig. 5.37 and Tables 3.7-3.8. The collapse of the specimens is deter-mined by bolt fracture at the stiffener side (Fig. 3.16c) – labelled “left side”. Fig. 5.38 plots the bolt elongation behaviour against the overall deformation for specimens WT61a and WT1h. Whilst for WT61a the record of the bolt elongation was carried out nearly until collapse, for WT1h, the measuring brackets were removed at an earlier stage. Therefore, the loss of stiffness in this response, which is evident for WT61a, is not plotted in the graph. This loss of stiffness does not occur for the unbroken bolt RB (unstiffened side), of which the response is very close to the bolts from WT1h.


170

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WT1h WT61a WT61b

EC3: Plastic resistance

EC3: Init. stiffness


comparison with WT1h and Eurocode 3 predictions.

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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2


Def

orm

atio

n (m

m)

Bolt RB (WT61a) Bolt LF (WT61a)Bolt RB (WT1h) Bolt LF (WT1h)

WT1h: max. deformation

WT61a: max. deformation

Fig. 5.38 Comparison of the overall deformation-bolt elongation response for

bolts LF and RB in specimens WT1h and WT61a. Now consider the stiffened specimen WT64C that derives from series WT4A by inclusion of the stiffeners. The above conclusions are not so obvious in this case. Fig. 5.39 and Table 3.8 show that both the initial and the post-limit stiffness values are identical for the two series. Nevertheless, resistance is still higher in the stiffened case. With respect to the ductility properties, if the abso-lute maximum deformation is taken into account, then WT64C shows im-proved ductility. If the deformation capacity is assumed as the level corre-sponding to Fmax instead, the same conclusion applies. For specimen WT64C some strain results are given in Fig. 5.40 (see Fig. 3.8d for an indication of the strain gauges nomenclature) and they prove that the flange is not engaged in collapse, as the strain level is low at failure conditions. Fig. 5.41 compares the strains at equivalent strain gauges in WT4A and WT64C. At the bolt axis, the

Parametric study

171

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WT4Ab WT64C

Fig. 5.39 Experimental load-carrying behaviour of specimen series WT64C

and comparison with WT4Ab.

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SG1 SG6 SG7

Lim

it of

the

stra

in g

auge

s

(a) Strain gauges SG1, SG6 and SG7.

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SG2 SG3SG4 SG5

Lim

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auge

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(b) Strain gauges SG2, SG3, SG4 and SG5. Fig. 5.40 Experimental results for flange behaviour (specimen WT64C).


172

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SG1 SG2

(c) Strain gauges SG1 and SG2. Fig. 5.40 Experimental results for flange behaviour (specimen WT64C) (cont.).

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Strain (µm/m)

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orm

atio

n (m

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SG2 (WT64C)SG4 (WT64C)SG6z (WT4Aa)

(a) Strain gauges SG2 and SG2 from WT64C and SG6 from WT4Aa.

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SG1 (WT64C)SG7 (WT64C)SG2 (WT4Ab)

(b) Strain gauges SG1 and SG7 from WT64C and SG2 from WT4Aa. Fig. 5.41 Experimental results for the flange behaviour: comparison between

specimens WT64C and WT4Ab.

Parametric study

173

strain level is higher in WT64C (Fig. 5.41a), whilst at the weld toe the strains are higher in WT4Aa (Fig. 5.41b). Finally, series WT64A is identical to WT64C but only one of the T-stub elements is stiffened. The results for both specimens are analogous (Fig. 5.42, Table 3.8). The deformation behaviour is illustrated at two different load stages in Fig. 5.43. The main effect of the transverse stiffness is in fact the increase of stiffness and resistance and decrease of ductility of the connection. The two stiffened specimen series also indicate that a trilinear curve best fits the experiments rather than a bilinear approximation as suggested for the other cases. A final remark concerns the evaluation of ke.0 and FRd for these specimens, according to Eurocode 3. A simplification has been introduced: both properties are evaluated for full stiffened and unstiffened specimens and then the average value is taken (Table 3.7).

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WT64A WT64C

(a) Average gap (LVDTs HP1 and HP2).

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-1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0

Deformation measured by HP3 (mm)

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WT64A WT64C

(b) LVDT HP3. Fig. 5.42 Experimental load-carrying behaviour of specimen series WT64A

and comparison with WT64C.


174

(i) WT64A (F = 214 kN; ∆ = 4.46 mm).

(ii) WT64C (F = 231 kN; ∆ = 4.49 mm).

(a) After removal of the measuring brackets.

(i) WT64A. (ii) WT64C. (b) After failure. Fig. 5.43 Deformation of the specimens WT64A and WT64C at two different

load stages. 5.5.2 Influence of the T-stub orientation To assess the influence of the T-stub orientation, consider series WT4B and WT64B. They are identical to specimens WT4A and WT64A, respectively, by rotating 90º one of the T-stubs. In these tests the flanges are bent in two direc-tions with double curvature in the plan (x and z directions) – Figs. 3.16d and 5.44. Contrary to the previous tests, here there is no gap between the flanges, except at the stiffener side in WT64B (Fig. 3.16d). For WT4B, both flanges are bent as a whole until the bolt starts deforming excessively. At this time, at the bolt centrelines, the flanges start opening and the maximum deformation at the web is nearly equal to the bolt deformation capacity. Fig. 5.45 illustrates the F-∆ response. It shows results for HP1, HP2 and HP3. HP1 (at the back, from the eye position) shows that the two plates on this side are compressed and their displacement is negative. HP2 and HP3, located at the front and left sides, re-spectively, show that the plates are compressed until a certain load level is reached, but then they start “opening” and there is an inversion of the deforma-

Parametric study

175

tion. That inversion starts at a lower load level for HP3 and becomes positive closer to the maximum load. For comparison, Fig. 5.45 also plots the deforma-tion of WT4B against the average gap of specimen WT4Ab. Clearly, no re-semblance between results is observed. The maximum load for specimen WT4B (223.67 kN) is close to the maximum load of WT4Aa, but a bit higher though. For specimen WT64B similar conclusions are drawn (Figs. 5.46a-b) except at the stiffener side where the two flange plates “open” from the commence-ment of loading. The results for LVDTs HP2 and HP3 for both specimens WT4B and WT64B are compared in Fig. 5.46c. They are identical apart from the influence of the stiffener. The F-∆ response, as given by HP2, for WT64Bb and WT64A is compared in Fig. 5.46d. A similar behaviour is observed. It should be noted though that some perturbations might have occurred in the measurement by means of the LVDTs in these series since the devices are not so easily attached here.

Fig. 5.44 Deformation of specimen WT4B at failure (two different views).

5.6 SUMMARY OF THE PARAMETRIC STUDY AND CONCLUDING REMARKS The experimental/numerical investigation presented in this chapter provides accurate deformation predictions (up to failure) of the T-stub response. Particu-lar emphasis on the identification of the main parameters affecting the defor-mation capacity of bolted T-stubs has been given. Their influence on the over-all behaviour of the connection has been assessed both qualitatively and quanti-tatively. The main conclusions drawn from this study are listed below and sum-marized in Table 5.6: 1. The enlargement of the weld throat thickness improves stiffness and resis-tance but decreases the deformation capacity; 2. The effect of the width of the T-stub is identical to the above; 3. The increase of the distance m leads to lower stiffness and resistance values and improves the deformation capacity; 4. Long-threaded bolts increase the overall deformation capacity of a T-stub


176

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-2.1 -1.4 -0.7 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6

Deformation (mm)

Tot

al a

pplie

d lo

ad (k

N)

WT4B (HP1)WT4B (HP2)WT4B (HP3)WTAb (av.def.)

Fig. 5.45 Experimental load-carrying behaviour of specimen series WT4B and

comparison with WT4A.

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-2.1 -1.4 -0.7 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6

Deformation (mm)

Tot

al a

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d lo

ad (k

N)

WT64Ba (HP1)WT64Ba (HP2)WT64Bb (HP1)WT64Bb (HP2)

(a) Results measured by LVDTs HP1 and HP2 for the two tests WT64B.

0

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-2.1 -1.4 -0.7 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6

Deformation (mm)

Tot

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d lo

ad (k

N)

HP1 HP2

HP3 HP4

(b) Results measured by the four LVDTs for test WT64Bb. Fig. 5.46 Experimental load-carrying behaviour of specimen series WT64B

and comparison with other test series.

Parametric study

177

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240

-2.1 -1.4 -0.7 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6

Deformation (mm)

Tot

al a

pplie

d lo

ad (k

N)

WT64Bb (HP2)WT64Bb (HP3)WT4B (HP2)WT4B (HP3)

(c) Comparison of the results from WT64Bb with WT4B (measured with LVDT HP2 and HP3).

0

30

60

90

120

150

180

210

240

0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0

Deformation (mm)

Tot

al a

pplie

d lo

ad (k

N)

WT64Bb (HP2)

WT64A (HP2)

(d) Comparison of the results from WT64Bb with WT64A (measured with LVDT HP2). Fig. 5.46 Experimental load-carrying behaviour of specimen series WT64B

and comparison with other test series (cont.). connection when compared to short-threaded equivalent bolts, if collapse is governed by bolt fracture; 5. Higher bolt diameters increase the strength of the bolt and therefore enhance the three characteristic properties of the load-carrying behaviour of the connec-tion: resistance, stiffness and ductility; 6. Identical T-stubs yield higher resistance and lower deformation capacity for higher steel grades. Regarding the influence of the stiffener, its main effect is the decrease of the deformation capacity (note that, for the stiffened specimens, a trilinear ap-proximation for simplified calculations best fits the experimental results rather


178

than the classical bilinear approximation. Moreover, for stiffened T-stubs, the influence of the elements orientation is not relevant at the stiffener side; in the case of unstiffened specimens it has been shown that the two plates become in contact when the connection is subjected to tension. Table 5.6 Summary of the main conclusions drawn from the parametric

study [Notation: x↑ ⇒ y↑ means that if x increases then y also in-creases; similarly, x↑ ⇒ y↓ means that if x increases then y de-creases].

Strength Stiffness Ductility FRd Fu ke.0 kpl.0 ∆u Assembly type WP RdF⇒ ↑ WP uF⇒ ↑

.0WP ek⇒ ↑ .0WP plk⇒ ↑ WP u⇒ ∆ ↓ Throat thickness (WP-T-stubs only) aw w Rda F↑⇒ ↑

w ua F↑⇒ ↑ .0w ea k↑⇒ ↑ .0w pla k↑⇒ ↑ w ua ↑⇒ ∆ ↓

Connection geometry w Rdw F↑⇒ ↓

uw F↑⇒ ↓ .0ew k↑⇒ ↓ .0plw k↑⇒ ↓ uw ↑⇒ ∆ ↑

p Rdp F↑⇒ ↑ up F↑⇒ ↑

.0ep k↑⇒ ↑ .0plp k↑⇒ ↑ up ↑⇒ ∆ ↓

e1 1 Rde F↑⇒ ↑ 1 ue F↑⇒ ↑ 1 .0ee k↑⇒ ↑ 1 .0ple k↑⇒ ↑

1 ue ↑⇒ ∆ ↓ tf f Rdt F↑⇒ ↑

f ut F↑⇒ ↑ .0f et k↑⇒ ↑

.0f plt k↑⇒ ↑ f ut ↑⇒ ∆ ↓

Bolt characteristics φ RdFφ ↑⇒ ↑ uFφ ↑⇒ ↑

.0ekφ ↑⇒ ↑ .0plkφ ↑⇒ ↓ uφ ↑⇒ ∆ ↑

Ltg No influence t uL ↑⇒ ∆ ↑ S0 No influence 0 .0eS k↑⇒ ↑ No influence Plate material .y f Rdf F↑⇒ ↑

.y f uf F↑⇒ ↑ No influ-ence

. .0y f plf k↑⇒ ↑ .y f uf ↑⇒ ∆ ↓



179

6 SIMPLIFIED METHODOLOGIES FOR ASSESSMENT OF THE

BEHAVIOUR OF SINGLE T-STUB CONNECTIONS 6.1 INTRODUCTION Previous chapters deal with the characterization of the overall behaviour of single T-stub connections by means of experimental tests or numerical three-dimensional models. Both approaches provide a complete definition of the F-∆ response up to collapse of the connection. From a practical point of view, nei-ther of the above methods seems appropriate. Therefore, a simple methodology for prediction of the connection response up to collapse is desired. As already pointed out, the collapse is governed by fracture of the bolts and/or cracking of the T-stub material. Because of the emphasis placed on connection ductility, the methodology must be able to predict the response of the T-stub well into its plastic and strain hardening range with a reasonable degree of accuracy. This chapter presents simplified methods for determining the monotonic de-formation response of T-stubs. First, existing models from the literature are discussed. Next, a two-dimensional beam model is proposed and calibrated against test results from the database compiled previously. Some recommenda-tions for modifications are also given. Finally, conclusions are drawn. 6.2 PREVIOUS RESEARCH The analytical prediction of the overall response of bolted T-stub connections is very complex. The behaviour of this type of connections is intrinsically three-dimensional and involves both geometrical and material nonlinearities. It includes the bending deformations of the flange and the combined axial and bending deformations of the bolts. Several theoretical approaches for the characterization of the behaviour of T-stubs have already been proposed in the literature. Essentially, they use the same basic prying mechanism, which is also the model implemented in Euro-code 3 [6.1] (Fig. 6.1). The model is two-dimensional, i.e. the three-dimensional effects are not accounted for. The system is statically indetermi-nate to the first degree. It is loaded by applying a vertical force F/2 to the sup-port (1), which corresponds to the critical section at the flange-to-web connec-tion. Only one quarter-model is taken into account due to symmetry considera-tions. The contact points at the tips of the flange are modelled with a pinned support and reproduce the effect of the prying forces. The T-stub flange be-haves as a rectangular cross-section of width beff and depth tf. Such width, beff, represents the flange plate width tributary to a bolt row that contributes to load transmission. This width varies with increasing loading but cannot exceed the ac-


180

(2)

(1)B Q

F2 2

∆

2F

2F

∆

Fig. 6.1 Typical T-stub prying model. tual flange width, b. At pure plastic conditions and for evaluation of the plastic (design) resistance, it accounts for all possible yield line mechanisms of the T-stub flange. Despite these major simplifications, the nonlinear analysis of this prying model is still very complex and requires an incremental procedure. Therefore, it is not intended for hand computation unless some simplifications that reduce the model complexity to a reasonable level are assumed. In this section three alternative simplified models developed by Jaspart [6.2], Faella and co-workers [6.3-6.5] and Swanson [6.6] are briefly addressed. These models yield a piecewise F-∆ relationship for characterization of the connection response. In addition, the proposals of Beg et al. [6.7] for assess-ment of the deformation capacity are also reviewed. 6.2.1 Jaspart proposal (1991) Jaspart approximates the nonlinear T-stub behaviour to a bilinear response [6.2]. The characteristics of this bilinear behaviour are summarized as follows: (i) The initial elastic region has a slope ke.0 that is evaluated by application of

Eqs. (1.21-1.22) (later, Jaspart simplified this expression for inclusion in Eurocode 3 in [6.8] – cf. Eqs. (1.23-1.25), in Chapter 1).

Simplified methodologies: assessment of the behaviour of T-stub connections

181

(ii) The swivel point in the bilinear relationship represents the full development of the yield lines and the correponding force is FRd.0. This “plastic” resistance is is determined from the above prying model as explained in Chapter 1 – Eqs. (1.3-1.5,1.10).

(iii) In the plastic region, above the swivel point, the effects of material strain hardening are dominant. The slope of this second linear region is given by:

.0 .0h

p l eE

k kE− = (6.1)

whereby Eh is the strain hardening modulus of the flange material. (iv) The point of maximum force, Fu.0, is determined by formally equivalent

expressions to FRd.0, by replacing the plastic conditions (index Rd) with ultimate conditions (index u). This means that these expressions are based on the same geometric characteristics but the plastic moment of the flange, Mf.Rd is replaced with:

2. .0.25f u f u f effM t f b= (6.2)

which is an identical expression to Eq. (1.6) and BRd is replaced with: .u u b sB f A= (6.3)

The following expressions are then obtained for Fu.0: ( ).0 1. .0 2. .0 3. .0min ; ;u u u uF F F F= (6.4)

and:

( )

( )( ) ( )

.1. .0

.1. .0

4 basic formulation

32 2formulation accounting for the bolt

8

=

−=

− +

f uu

w f uu

w

MF

mn d M

Fmn d m n

(6.5)

( )( )

. .2. .0

2 2 2 21

1f u u f u u

uu

M B n MF

m n mβ λ

β λ+ −

= = + + +

(6.6)

3. .0 .2 2u u u b sF B f A= = (6.7) The deformation capacity is readily determined by intersection of the plastic region, with slope kp-l.0, with the maximum resistance, Fu.0, i.e.:

.0 .0 .0.0

.0 .0−

−∆ = +Rd u Rd

ue p l

F F Fk k

(6.8)

This methodology can be easily extended to a nonlinear idealization of the F-∆ response, similar to that proposed in Eurocode 3 for the joint overall M-Φ response (cf. §1.6.1.3), provided that the transition portion of the two straight curves is well established. 6.2.2 Faella and co-workers model (2000) Faella and co-workers [6.3-6.5] developed a procedure based on the resem-


182

blance of the distribution of internal forces at plastic and ultimate conditions (Figs. 1.10 and 6.2). They assumed the following simplifications [6.4]: (i) geometrical nonlinearities are neglected, (ii) compatibility between bolt and flange deformation is not considered, (iii) the shear interaction is disregarded, (iv) prying forces are located at the tip of the flanges, (v) bending of the bolts is neglected and (vi) cracking of the material is modelled by assuming the crack-ing condition as the occurrence of the ultimate strain in the extreme fibres of the T-stub flanges. The plastic deformation of the flange is computed from the corresponding moment-curvature (M-χ) diagram. This is obtained from simple internal equilibrium conditions of the section and by assuming that the material constitutive law can be approximated by a quadrilinear relationship (see Fig. 2.2). This stress-strain relationship is defined in natural coordinates. The basic formulations for computation of plastic deformations are derived from the inte-gration of the M-χ diagram over a certain length, the cantilever length, L, that remains unchanged during the loading process and equal to that occurring at ul-timate conditions [6.3]. This simple model yields a multilinear F-∆ curve for the behaviour of the T-stub. The characteristic coordinates of this curve are determined according to the potential failure mode (Fig. 6.2). In particular, for the evaluation of the characteristic force coordinates, they use the same expression as the Eurocode 3 for plastic conditions (cf. Chapter 1 and [6.3-6.5]).

Q Q B B

(=F1.u/2+Q)

F1.u.0

n m n m

Mf.u Mf.u

b

F2.u.0

n m nm

Mf.u ξMf.u

Bu

Q Q

b

Bu

n m nm

ξMf.u

Bu

b

Bu

F3.u.0

(a) Flange fracture mechanism: ( ).limu uβ β≤ .

(b) Combined bolt/flange mechanism: ( ).lim 2u uβ β< ≤ .

(c) Bolt fracture mecha-nism: ( )2uβ > .

Fig. 6.2 Collapse mechanism typologies of a single T-stub prying at ultimate conditions according to Faella et al. [6.3].

6.2.3 Swanson model (1999) Swanson developed a prying model that uses the geometrical properties de-fined in Fig. 6.3a, which is consistent with the strength model proposed by Ku-


183

lak et al. [6.6]. The author uses the following dimensions: 0.5 0.5b d r φ′ = − − (6.9)

( )min 1.25 ; 0.5a b e φ′ ′= + (6.10) For comparison, Fig. 6.3b shows the dimensions used in Eurocode 3:

0.8m d r= − (6.11) ( )min 1.25 ;n m e= (6.12)

The model includes: (i) nonlinear material properties, (ii) a variable bolt stiffness that captures the changing behaviour of the bolts as a function of the loads they are subjected to, (iii) partially plastic hinges in the flange and (iv) second order membrane behaviour of thin flanges [6.6]. The bolt behaviour is incorporated by means of an extensional spring lo-cated at the inside edge of the bolt shank. This spring is characterized by a piecewise linear force-deformation, B-δb, response. Swanson [6.6] proposes an analytical model for the characterization of the bolt deformation behaviour similar to that depicted in Fig. 6.4. The multilinear curve refers to hand-tightened bolts and its characteristic coordinates are set out in Table 6.1. The bolt elastic stiffness, Kb is evaluated as follows [6.6]:

bb

s b tg s

EK

L A L A=

+ (6.13)

whereby Ls and Ltg are the shank and threaded lengths of the bolt included in

B-δb

b’ a’

2F

0.5r

0.5φ

B-δb

m n

2F

0.8r

(a) Dimensions used by Swanson. (b) Dimensions used in Eurocode 3. Fig. 6.3 T-stub prying model proposed by Swanson [6.6].


184

δb.1 δb.2 δb.fract δb

B

Elastic, Kb

Bolt fracture

Yielding, 0.1Kb

Plastic,0.02Kb

0.85Bu 0.90Bu

Fig. 6.4 Bolt force-deformation model according to Swanson. the grip length, respectively, Ab is the nominal area of the bolt shank, Ab = πφ2/4 and φ is the bolt nominal diameter. Based on mechanistic considerations, the deformation capacity of the single bolt in tension, δb.fract, is easily assessed as follows [6.6]:

. .0.90 2u s

b fract u b tgb b th

B LL

A E nδ ε

= + +

(6.14)

being nth the number of threads per unit length of the bolt. These predictions are based on the assumption that the bolt shank remains elastic with inelastic deforma-tion concentrated in the threads that are included in the grip length [6.6]. It is also recognized that a portion of the bolt inside the nut will deform inelastically. As a result, two of the threads within the nut are included in the predictions. The flange mechanistic model assumes an elastic-yielding-plastic constitu-tive relationship for the steel. It also accommodates the shear deformations as well as the membrane effect, which can be particularly relevant for flexible flanges. Plastic hinges will develop at the flange-to-web connection and at the bolt axis and their length is taken as equal to the flange thickness. Strain hard-ening is assumed to start immediately following the formation of a plastic hinge and was modelled with rotational springs [6.6]. The partially plastic states were incorporated in the model in a simplified way, as reported in [6.6]. Swanson derived the stiffness coefficients and corresponding prying gradi-ents, qij,k = ∂Q/∂∆, by using the direct stiffness method. Both parameters are used in an incremental solution technique. First, the initial stiffness and the ini-tial prying gradient, qee, are determined:

( )3.0

12 3 be

ee

EI EI Kk

γγ

+= (6.15)

( )29 2b bee

ee

EI K a b EIq

β

γ

′ ′ −= (6.16)


185

Table 6.1 Characteristic coordinates of the bolt deformation response.

Bolt force, B Bolt stiffness, Kb Bolt elongation, δb

0 0.85 uB B≤ < bK .10.85 u

bb

BK

δ =

0.85 0.90u uB B B≤ < 0.10 bK .2 .10.050.10

ub b

b

BK

δ δ= +

0.90 u fractB B B≤ ≤ 0.02 bK . .0.90 u s

b fract u b tgb b

B LL

A Eδ ε= +

whereby:

3

12eff fb t

I = (6.17)

is the inertia of the flange cross-section and the remaining coefficients are de-fined below:

1 212ee bEI Kγ γ γ= + (6.18)

( )3 2 2 31 3 3a ba a b a b bγ β β′ ′ ′ ′ ′ ′= + + + (6.19)

3 3 2 4 22 4 3a b ba b a bγ β β β′ ′ ′ ′= + (6.20)

3 23 3a ba a bγ β β′ ′ ′= + (6.21)

2 2

12 121 1a beff f eff f

EI EIGb t a Gb t b

β β= + = +′ ′ ′ ′

(6.22)

The coefficients βa and βb account for shear deformations. Next, several checks are made to determine which limit is reached first (bolt force or flange internal stresses limits). Incremental deformations are then calculated for each of the potential limits with the smallest value governing. The F-∆ curve can yield up to nine linear branches, with different stiffness, be-fore failure. Swanson states that the strength and the deformation capacity of the flange are not always predicted accurately because of sensitivities of the model to strain hardening parameters and bolt ductility [6.6]. It should be stressed that this model is not intended for hand calculations and it will not be used for further comparisons. 6.2.4 Beg and co-workers proposals for evaluation of the deformation ca-

pacity (2002) Beg et al. developed a set of simple analytical expressions for evaluation of the deformation capacity of single T-stub connections [6.7]. They also assumed two alternative cracking conditions: (i) attainment of the ultimate strain at the


186

outer fibre of the flange section and (ii) fracture of the bolt. The maximum strain allowed at the flange section is 0.20 and the fracture of the bolt is as-sessed as follows:

*. .2u b u b bLδ ε= (6.23)

whereby εu.b is taken as 0.10 for full-threaded bolts and 0.02-0.05 for small-threaded bolts, and *

bL is the clamping length of bolts, i.e. thickness of clamped plates including thickness of washers [6.7]. Factor 2 results from symmetry. For each potential plastic failure mode (see Fig. 1.10) the authors propose the following relationships (δu is the deformation capacity of a half-T-stub): (i) Mode 1: .00.4 2 0.8u u um mδ δ= ⇒ ∆ = = (6.24) (ii) Mode 2:

..0 .1 2 1

2u b

u u u u bm mk kn n

δδ δ δ = + ⇒ ∆ = = +

(6.25)

whereby k is an empirical factor varying from 3.0 to 4.0 [6.7]. (iii) Mode 3:

*..0 .2

2u b

u u u b bLδ

δ ε= ⇒ ∆ = (6.26)

These expressions account for the dependence of the deformation capacity of a T-stub on the fracture elongation of the bolts, on the ultimate strain of the steel and on the geometrical parameters m and n. However, the dependence on the flange thickness is neglected. In Chapter 5 it has been shown evidenced the strong dependence of the deformation behaviour on this geometrical parameter. For identical geometry connections that fail according to a plastic mechanism of type-1, the T-stub with thicker flanges exhibits a lower deformation capacity than the thinner flange. Eq. (6.24) does not account for this effect. 6.2.5 Examples To illustrate the alternative procedures, the T-stub (unstiffened) specimens re-ferred in Chapters 3, 4 and 5 are used. These specimens constitute a database for exemplification of the procedures presented in this section. Later, the same specimens will be used for validation of an alternative model for characteriza-tion of the behaviour of isolated T-stub connections. 6.2.5.1 Evaluation of initial stiffness Chapter 1 already presented some procedures for evaluation of the initial stiff-ness of single T-stub connections, namely, the Yee and Melchers standard pro-posals [6.9-6.10] and the subsequent modifications suggested by Jaspart in [6.2]. The Eurocode 3 simple expressions were also derived. These are the


187

same expressions adopted by Jaspart in his simple methodology [6.2] (§6.2.1). Faella et al. presented an alternative formulation for the definition of ke.0 in [6.3] (cf. Chapter 1). Table 6.2 compares the initial stiffness predictions (per bolt row) of some T-stub specimens by application of two of the above procedures: Faella et al. formulation and Eurocode 3. Identical tables for the other methodologies are shown in Appendix D. The specimens were grouped according to the assembly type (hot rolled profiles or welded plates). The results do not differ substan-tially. The ratio value in the tables is given by: Ratio = Predicted value Actual value (6.27) As a general conclusion, it can be stated that the procedures proposed by Faella et al. provide the best prediction for evaluation of ke.0 (third column in Table 6.2). Eurocode 3 overestimates ke.0 (fifth column Table 6.2). Regarding the remaining methods, the following conclusions are also drawn (Appendix D): (i) the location of the prying forces for application of the Yee and Melchers procedures does not introduce major differences within the limits analysed and (ii) the Swanson model is more accurate in the elastic domain if the geometri-cal dimensions from Eurocode 3 are used. In both cases, however, the average error is systematically over 100%. 6.2.5.2 Evaluation of plastic resistance The alternative methodologies analysed in this work use the same approach as the Eurocode 3 for evaluation of the “plastic” resistance, FRd.0 (see Chapter 1). Table 6.3 summarizes the predictions for FRd.0 using the expressions from Eurocode 3. The potential plastic mode is also indicated. For those specimens failing according to a plastic collapse mode 1, both results from application of the basic formulation and the formulation accounting for the bolt action are given. In some cases, if the latter formulation is taken into account, the collapse type-2 may become critical (specimens P4, WT4, for instance). In those cases, the values for type-2 are shown in bold/italic. These predictions are compared with the knee-range of the actual F-∆ (nu-merical or experimental) response since the definition of FRd.0 in this case is not straightforward. The predictions of FRd.0 are within these limits, which means that the Eurocode proposals are accurate. 6.2.5.3 Piecewise multilinear approximation of the overall response and evalua-

tion of the deformation capacity and ultimate resistance The global F-∆ response of a T-stub is characterized in this section by using the bilinear approximation suggested by Jaspart and the multilinear model pro-posed by Faella and co-workers. The same examples from above are used for


188

Table 6.2 Prediction of axial stiffness by application of the Faella, Piluso and Rizzano procedures and the Eurocode 3.

Faella et al. predic-tions

Eurocode 3 predic-tions

Test ID Num./Exp. stiffness

ke.0 Ratio ke.0 Ratio T1 83.54 87.77 1.05 144.36 1.73 P1 63.27 57.38 0.91 97.27 1.54 P2 117.06 140.95 1.20 220.29 1.88 P3 72.62 77.95 1.07 129.45 1.78 P4 97.86 111.10 1.14 178.63 1.83 P5 101.23 124.32 1.23 197.37 1.95 P9 128.47 173.54 1.35 255.91 1.99 P10 43.88 23.85 0.54 42.08 0.96 P12 102.05 92.09 0.90 156.40 1.53 P14 81.97 87.77 1.07 144.36 1.76 P15 128.41 152.41 1.19 249.63 1.94 P16 111.23 141.11 1.27 220.51 1.98 P18 171.57 158.73 0.93 266.81 1.56 P20 181.68 300.54 1.65 441.03 2.43 P23 322.09 487.19 1.51 675.18 2.10

Average 1.13 1.80 Coefficient of variation

0.24

0.18

Weld_T1(i) 73.50 45.50 0.62 78.07 1.06 Weld_T1(ii) 88.04 62.40 0.71 105.25 1.20 Weld_T1(iii) 107.29 82.56 0.77 136.49 1.27 WT1g 68.58 63.38 0.92 108.64 1.58 WT1h 73.58 63.38 0.86 108.64 1.48 WT2Aa 64.32 50.79 0.79 87.89 1.37 WT2Ab 61.75 50.79 0.82 87.89 1.42 WT2Ba 63.58 74.80 1.18 127.12 2.00 WT2Bb 79.75 74.80 0.94 127.12 1.59 WT4Aa 75.08 103.38 1.38 171.93 2.29 WT4Ab 86.96 103.38 1.19 171.93 1.98 WT7_M12 91.18 101.21 1.11 168.64 1.85 WT7_M16 116.09 104.59 0.90 179.58 1.55 WT7_M20 137.70 107.38 0.78 186.44 1.35 WT51a 59.62 53.27 0.89 92.08 1.54 WT51b 61.84 53.27 0.86 92.08 1.49 WT53C 64.23 55.13 0.86 95.05 1.48 WT53D 52.90 56.36 1.07 97.08 1.84 WT53E 64.82 55.05 0.85 94.92 1.46 WT57_M12 42.89 90.34 2.11 151.69 3.54 WT57_M16 55.22 94.71 1.72 163.02 2.95 WT57_M20 75.48 95.72 1.27 166.89 2.21


0.34 0.33


189

Table 6.3 Prediction of the plastic resistance Eurocode 3 (per bolt row).

Eurocode 3 predictions FRd.0 (kN)

Test ID Num./Exp. knee-range

Potential plas-tic mode Basic for-

mulation Formul. account-ing for the bolt

T1 65 - 85 1 67.02 79.78 P1 60 - 73 1 57.29 67.92 P2 70 - 100 1 80.73 98.53 P3 60 - 75 1 58.64 69.80 P4 70 - 103 1 or 2 87.97 96.37 P5 90 - 120 2 99.48 P9 85 - 120 2 108.23 P10 40 - 47 1 27.47 32.52 P12 80 - 103 1 67.02 84.04 P14 45 - 65 1 42.76 50.90 P15 80 - 120 1 80.73 104.67 P16 85 - 112 2 103.63 P18 150 - 200 1 117.29 157.16 P20 160 - 200 2 190.88 P23 275 - 305 2 296.71 Weld_T1(i) 50 - 78 1 57.23 61.10 Weld_T1(ii) 60 - 87 1 59.07 69.26 Weld_T1(iii) 75 - 97 1 65.50 77.73 WT1g 58 - 68 1 48.33 55.77 WT1h 60 - 70 1 48.33 44.44 WT2Aa 52 - 62 1 44.44 50.94 WT2Ab 53 - 65 1 44.44 50.94 WT2Ba 59 - 78 1 51.09 59.35 WT2Bb 62 - 80 1 51.09 59.35 WT4Aa 89 - 105 1 or 2 81.62 87.34 WT4Ab 70 - 98 1 or 2 81.62 87.34 WT7_M12 60 - 96 1 or 2 81.00 86.96 WT7_M16 80 - 104 1 80.22 96.14 WT7_M20 88 - 118 1 80.73 100.79 WT51a 78 - 94 2 89.48 WT51b 79 - 95 2 89.48 WT53C 79 - 94 1 or 2 93.19 93.38 WT53D 83 - 96 1 or 2 94.31 107.76 WT53E 93 - 109 1 92.59 106.67 WT57_M12 75 - 119 2 110.48 WT57_M16 104 - 165 1 or 2 158.52 163.78 WT57_M20 126 - 204 1 157.72 196.15


190

comparison with the simple methodology of Jaspart. For application of the Faella et al. procedures only six examples are considered. As already stated above, the Swanson proposals are not illustrated herein. a) Methodology recommended by Jaspart Fig. 6.5 illustrates the bilinear approximation of the F-∆ response of some se-lected specimens, as proposed by Jaspart. The graphs compare the actual re-sponse with four alternative approaches of the methodology, regarding the me-chanical properties of the T-stub flange, the resistance formulation and a com-bination of both. The two alternative resistance formulations are the basic for-mulations (BF) and the formulation accounting for the bolt action (FBA) when applicable. The complete characterization of the actual material properties of the various specimens from the database was given in Chapters 3, 4 and 5. The

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22 24

Deformation, ∆ (mm)

Loa

d, F

(kN

)

Actual responseBilinear response (actual Eh and BF)Bilinear response (actual Eh and FBA)Bilinear response (nominal Eh and BF)Bilinear response (nominal Eh and FBA)

(a) HR-T-stub T1 (fy.f = 430 MPa).

0

10

20

30

40

50

60

70

80

90

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Loa

d, F

(kN

)


(b) HR-T-stub P14 (fy.f = 275 MPa). Fig. 6.5 Illustration of the methodology proposed by Jaspart.


191

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)


(c) WP-T-stub WT1 (fy.f = 340 MPa).

0

15

30

45

60

75

90

105

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual responseBilinear response (actual Eh and FBA)

(d) WP-T-stub WT51 (fy.f = 698 MPa). Fig. 6.5 Illustration of the methodology proposed by Jaspart (cont.). actual strain hardening modulus, Eh, for these specimens however is always lower than the nominal properties [6.3,6.11]. For steel grade S355, Eh = E/48.2 and for S275, Eh = E/42.8. No quantitative guidance is given in neither refer-ences for steel grade S690. Hence, both actual and nominal values for Eh are taken into account for those specimens where steel grade S355 and S275 was employed (S275 was used in specimen P14). For further details on this methodology, the reader should refer to Appendix D. Generally speaking, the bilinear approximation proposed by this author re-produces well the actual behaviour for those specimens made up of S690, with an overestimation of the deformation capacity. For the remaining cases, the predictions are fine provided that the nominal value of the strain hardening modulus is used. If the actual value of Eh is used instead, then the predictions are not so good.


192

b) Methodology recommended by Faella, Piluso and Rizzano Fig. 6.6 shows the overall F-∆ response for some T-stub specimens that were chosen to illustrate the different failure modes. The graphs trace the response for actual and nominal flange mechanical properties and include the compati-bility of the deformations of the flange and the bolt (Appendix D). In general, this model does not provide an accurate modelling of the deformation behav-iour. c) Methodology recommended by Beg, Zupančič and Vayas The procedures for a direct computation of the deformation capacity from Beg et al. [6.7] are illustrated in Appendix D for the various specimens. The predic-tions are not satisfactory though, particularly for those specimens that fail ac-

0

15

30

45

60

75

90

105

120

135

0 1 2 3 4 5 6 7 8 9 10 11


Loa

d, F

(kN

)

Actual response

Quadrilinear approximation (BF: type-2governs failure)Quadrilinear approximation(FBA: (BF: type-1governs failure)

(a) HR-specimen T1 (actual material properties).

0

30

60

90

120

150

180

210

240

270

0 3 6 9 12 15 18 21 24 27 30


Loa

d, F

(kN

)

Actual responseQuadrilinear approximation (actual mat. prop.)Quadrilinear approximation (nominal mat. prop.)

(b) HR-specimen P18. Fig. 6.6 Illustration of the methodology proposed by Faella, Piluso and Riz-

zano.


193

0

15

30

45

60

75

90

105

120

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual responseQuadrilinear approximation (actual mat. prop.)Quadrilinear approximation (nominal mat. prop.)

(c) WP-T-stub specimen WT4A.

0

15

30

45

60

75

90

105

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5


Loa

d, F

(kN

)

Actual response

Quadrilinear approximation (actual mat. prop.)

(d) WP-T-stub specimen WT51. Fig. 6.6 Illustration of the methodology proposed by Faella, Piluso and Riz-

zano (cont.). cording to a plastic mode 1. The average ratios of the actual numerical or ex-perimental predictions are 2.10 for HR-T-stubs and 3.55 for WP-T-stubs, with coefficients of variation of 0.50 and 0.48, respectively. Again, it is noted that this methodology only gives an estimation of the T-stub deformation capacity, rather than a description of the full nonlinear behaviour. 6.2.5.4 Summary This section described and illustrated several methodologies for the assessment of the F-∆ response of T-stubs (or some of its characteristics). Table 6.4 com-pares the different methods from a statistical point of view, in terms of average ratios of the sample of examples and coefficient of variation. The examples are divided according to the assembly type (HR-T-stub or WP-T-stub). The pa-


194

Table 6.4 Summary of the different proposals from a statistical point of view (average ratios and coefficients of variation, the latter in italic) for evaluation of the force-deformation characteristics.

Stiffness Ultimate re-sistance

Deformation capacity

Methodology T-stub assembly

ke.0 Fu.0 ∆u.0 HR 2.10 (0.50) Beg et al. WP 3.55 (0.48) HR 1.80 (0.18) Eurocode 3 WP 1.75 (0.33) HR 1.13 (0.24) 0.80 (0.46) 0.73 (0.54) Faella et al. WP 1.03 (0.34) 0.95 (0.23) 1.16 (0.36) HR 2.30 (0.21) 0.96 (0.11) 0.91 (0.35) Jaspart WP 2.30 (0.31) 0.93 (0.08) 1.13 (0.30) HR 2.35 (0.26) Swanson WP 3.08 (0.37) HR 2.32 (0.18) Yee and

Melchers WP 2.23 (0.32) rameters chosen for comparison are the initial stiffness, the ultimate resistance and the deformation capacity. The best approach for characterization of the initial stiffness is that pro-posed by Faella et al. [6.3], though the coefficient of variation of the sample is slightly higher than for the Eurocode 3. Regarding the evaluation of the defor-mation capacity, Jaspart [6.2] gives accurate predictions. The scatter of results for the deformation capacity is rather high when compared to the other proper-ties as shown by the coefficient of variation. The results shown for the method-ology proposed by Faella and co-authors, in terms of ultimate resistance and deformation capacity, are merely illustrative since the sample is not big enough for a statistical analysis of this type. 6.3 PROPOSAL AND VALIDATION OF A BEAM MODEL FOR CHARACTERI-

ZATION OF THE FORCE-DEFORMATION RESPONSE OF T-STUBS 6.3.1 Description of the model The models described above afford some basis for the development of an ana-lytical method for the evaluation of the deformation capacity and the load-carrying capacity of single T-stub connections. The mechanical model is simi-lar to that depicted in Fig. 6.3b. A model that uses geometrical and mechanical


195

properties consistent with the Eurocode 3 prying model is desired so as to make implementation easier. The prying model is analysed up to collapse. From static equilibrium (Fig. 6.7):

2FB Q= + (6.28)

( )1

2FM Qn m= − (6.29)

( )2M Qn= (6.30) being (1) the section at the flange-to-web connection and (2) the flange section at the bolt line. Normally, M1

≥ M2. However, Swanson points out that in some cases this inequality may not be observed due to the effect of the removal of flange material at the bolt line when the holes are drilled for the bolts [6.6]. The deflected shape and the moment diagram of one side of the flange in pure elastic conditions and after separation at the bolt axis are as shown in Fig. 6.8a [6.12]. If the bolt is strong enough, a stage of loading will be reached when the plastic moment of the flange, Mf.p, is attained at the flange-to-web connection (Fig. 6.8b, [6.12]). Any additional load will cause further flange deflection that results in strain hardening of the flange and an increase in the internal moment at that section. The zone of full plastification spreads into the flange. With con-tinued loading, a similar condition may develop at the bolt axis. When the most

M2 = Qn

M1 = Qn - 0.5Fm

m n

Q

(1)

(2)

F2

B

(1*)

0.2r or 0.2 w2a

Fig. 6.7 Internal forces.


196

highly strained flange fibres are strained to the breaking point (εu) and fracture, the ultimate resisting moment of the flange, Mf.u, is also reached and the ulti-mate conditions (subscript u) are attained (Fig. 6.8c, [6.12]). In this figure, Bu is not necessarily the tensile strength of the bolt. For common steels, Mf.u is significantly higher than Mf.p [6.3,6.11,6.13]. If simple plastic theory is applied, the limit resistance and the deformation would be determined by Mf.p. Therefore, consideration of strain hardening is crucial to carry out an ultimate analysis of the system up to a fracture condition.

0.5F

Q

B

M(1)

M(2) = Qn m n

M(1) = Qn-0.5Fm

0.5F

Q

B

Mf.p

M(2) = Qn m n

M(1) = Mf.p Mf.y

Qu

Bu

Mf.u

M(1) = Mf.u

M(2) = Qun

m n

Mf.y Mf.p

Mf.y Mf.p

0.5Fu

(a) Pure elastic condi-tions.

(b) Full plastification of section (1).

(c) Fracture of section (1).

Fig. 6.8 Effect of material strain hardening. 6.3.1.1 Fracture conditions The two possible ultimate fracture conditions are: (i) fracture of the bolt and (ii) cracking of material of the flange near the web as already explained in Chapter 2. In the context of a two-dimensional model, where the flange is modelled as a rectangular cross-section, this latter condition may be too severe. Critical sec-tion (1) is defined at a distance m from the bolt axis, where the flange thickness is higher than tf owing to the fillet weld or radius that provide some extra mate-rial thickness. Therefore, the imposition of cracking of the material should also be checked at the end of this fillet, section (1*) – Fig. 6.7, i.e. at a distance m* =

d – r or m* = d – 2 aw for HR- and WP-T-stubs, respectively. 6.3.1.2 Bolt deformation behaviour The bolt elongation response is based on the Swanson’s proposals. Its influence


197

on the overall response is accounted for by means of an extensional spring with similar characteristics to the Swanson bolt model, as just explained (Fig. 6.4 and Table 6.1). For the computation of the bolt deformation at fracture, how-ever, the parameter nth that appears in Eq. (6.14) is disregarded. For design calculations, the nominal material characteristics of the bolts have to be defined. The parameters given in Table 6.5 are suggested by Hirt and Bez [6.14] for high-strength bolts. Table 6.5 Minimum mechanical properties of high-strength bolts.

Bolt grade fy fu εu (MPa) (MPa)

8.8 640 800 0.12 10.9 900 1000 0.09

6.3.1.3 Flange constitutive law The flange material constitutive law is modelled by means of a piecewise linear curve, that accounts for the strain hardening effects. This law is a true stress-logarithmic strain relationship, i.e. it is defined in natural coordinates in order to capture the actual material behaviour. Faella et al., in fact, adopted the same approach since the prediction of the plastic deformation capacity of compact sections is more accurate if natural stress-strain coordinates are used [6.3]. The above model is not suitable for a hand calculation. Instead, a numerical FE method is used to determine the structural response. Consequently, the piecewise constitutive law may contain numerous branches. It should be stressed that many FE codes do not allow for an elastoplastic analysis with strain hardening for beam elements. The FE code LUSAS [6.15] implements a beam element that belongs to the Kirchoff beams group (with quadrilateral cross-section) [6.15]. Shear deformations are excluded in this element formula-tion. It has a quadrilateral cross-section. From a design point of view, the constitutive law should be of a standard type though. The stress-strain curve can be idealized by means of a multilinear model with a straight line for hardening range, as suggested in [6.13] (Fig. 6.9). The maximum stress is reached for a strain value:

u yhs h

h

f fE

ε ε−

= + (6.31)

With reference to Fig. 6.9, Gioncu and Mazzolani give no guidance on the characterization of the softening branch of this curve [6.13]. For current struc-tural steel grades, the characteristic coordinates are set out in Table 6.6, for plate thickness smaller than 40 mm [6.13]. These coordinates are transformed into a true stress-logarithmic strain curve by means of Eq. (4.1), which is re-produced below:


198

( )1nσ σ ε= + and ( )ln 1nε ε= + (6.32) The stress-strain characteristics in natural coordinates for the three above steel grades are depicted in Fig. 6.10. The fifth linear part of the curve in natural co-ordinates has a slope of Eu = fu, according to Faella et al. [6.3].

σ

ε

fy

εy εh εhs εu

E

Eh

εuni

fu

Fig. 6.9 Idealization of the stress-strain diagram with a multi-linear model.

0

100

200

300

400

500

600

700

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24

Logarithmic strain

Tru

e st

ress

(MPa

)

S355S275

S235

Fig. 6.10 True stress-logarithmic strain characteristics for steel grades S235,

S275 and S355. Table 6.6 Characteristics of the stress-strain curve (stress values in [MPa]).

Steel grade

fy fu εy εh εhs εuni εu Eh Eu

S235 235 360 0.001 0.014 0.037 0.140 0.250 5500 360 S275 275 430 0.001 0.015 0.047 0.120 0.220 4800 430 S355 355 510 0.002 0.017 0.053 0.110 0.200 4250 510


199

6.3.2 Analysis of the model in the elastic range First, the proposed model is analysed and validated in the elastic domain by us-ing the specimens from the author’s database. The initial stiffness of a T-stub connection is evaluated and compared with the actual predictions, correspond-ing to experimental or (three-dimensional) numerical values. In order to assess the importance of the shear deformability on the flange rectangular cross-section, two beam elements are tested: thin beam and thick beam. The thick beam element includes the shear deformations in its formula-tion but does not allow for a material nonlinear analysis with strain hardening. The results are compared in Table 6.7 for fifteen selected examples that repre-sent the three different failure types. Clearly, the thick beam model provides a better agreement with the actual results (second column of Table 6.7), which indicates that the shear deformation in the elastic domain may be significant. From the analysis of the average ratio to the actual values, the model with a thin beam element was re-analysed with a reduced Young modulus for the flange material. This reduction was taken as half of the actual E since the ear-lier results were nearly twice as much as the actual. This factor of reduction may be slightly increased in order to best fit the actual results (unitary average ratio). However, the calibration of this factor should be based on a larger sam- Table 6.7 Influence of shear deformations on the initial stiffness of some of

the tested T-stubs (stiffness values in [kN/mm]).

(Thin) Beam model predictions Thick beam model Actual Young

modulus Reduced Young modulus (0.5E)

Test ID

Num. res.

ke.0 Ratio ke.0 Ratio ke.0 Ratio T1 83.54 149.35 1.79 175.06 2.10 99.68 1.19 P1 63.27 108.03 1.71 124.27 1.96 69.86 1.10 P2 117.06 215.72 1.84 259.43 2.22 153.84 1.31 P3 72.62 133.92 1.84 157.91 2.17 88.80 1.22 P4 97.86 184.95 1.89 214.04 2.19 125.31 1.28 P5 101.23 204.49 2.02 235.17 2.32 139.71 1.38 P9 128.47 262.74 2.05 292.60 2.28 185.87 1.45 P10 43.88 49.35 1.12 54.57 1.24 28.14 0.64 P12 102.05 162.51 1.59 194.79 1.91 106.02 1.04 P14 81.97 149.35 1.82 175.06 2.14 99.68 1.22 P15 128.41 240.45 1.87 298.29 2.32 167.26 1.30 P16 111.23 228.71 2.06 261.17 2.35 157.91 1.42 P18 171.57 279.74 1.63 333.71 1.95 183.33 1.07 P20 181.68 422.05 2.32 509.64 2.81 328.23 1.81 P23 322.09 605.32 1.88 775.43 2.41 522.49 1.62

Average 1.83 2.16 1.27 Coeff. variation

0.15

0.16

0.21


200

ple of examples, which also includes other connections beyond those from the author. The results for this reduced modulus of elasticity are set out in Table 6.7 as well. As expected, the initial stiffness values decrease. These values will improve further if an additional correction of the Young modulus for shear is introduced. It is desirable to obtain this correction by means of a simple for-mula rather than an empirical correction. In Appendix A of Chapter 1, the shear interaction was already taken into consideration for resistance purposes. In mode-1 plastic failure types the ratio between the design resistance of mechanism type-1 accounting for shear and that corresponding to the basic formulation is given by (cf. Appendix A):

( )( )

2

.2 2

1. .0 2.

42 31 13 2 31 1

4 3

f Rd

f f

Rdf Rd f f

Mmt mm t mF

M t m tm

+ −

= = + −

(6.33)

The above relationship depends exclusively on the ratio m/tf. Fig. 6.11 plots that relationship and shows that 1. .0lim 1

fRdm t

F→∞

= . The value of 1. .0RdF is signifi-

cant for ( )1. .02.5 0.9f Rdm t F≤ < . If the analysis of the T-stub elements and the bolts is carried out separately, the T-flange is fixed at the bolt centreline. Therefore, the only possible collapse mode is of type-1 and so the above rela-tionship applies. The following expression is then proposed for determining the reduced modulus to employ in the beam model:

( ) ( )

2 2

2 2

2 3 30.5 1 1 1 13 3red

f ff f

m E mE Et tm t m t

= × + − = + −

(6.34)

This reduction does not have much influence at ultimate conditions. In fact, the effect of shear on the moment resistance of the flange is apparently beneficial [6.12].

0.00.10.20.30.40.50.60.70.80.91.0

0 1 2 3 4 5 6 7 8 9 10 11 12

m/tf

1.Rd.0F

Fig. 6.11 Interaction 1. .0RdF vs. fm t .


201

Table 6.8 shows the results for initial stiffness obtained through application of the above expression, as well as the reduced Young modulus. In this table both the ratio to the actual stiffness values (eighth column – Act.) and to the beam model with the simple reduction (ninth column – 0.5E) are calculated. The difference between the two approaches is 8% on average. The results ob-tained with Ered as in Eq. (6.34) show a good agreement with the actual predic-tions (error of 17%, on average). This value of Ered is then used onwards. It is important to stress that this value has little influence at ultimate conditions, as already explained. The beam model referred hereafter is hence the model that employs the thin beam model and Ered for the flange material. Table 6.9 evaluates the initial stiffness for other T-stubs from the database. The results are in line with the previous predictions. Additionally and in the elastic behaviour domain, a set of sixteen T-stub connections tested by Faella et al. [6.3] are considered. Unfortunately, these specimens cannot be used for further comparisons due to lack of data. The geometrical and mechanical char-acteristics of the latter specimens are set out in Table 6.10. The initial stiffness predictions by application of this model are summarized in Table 6.11 along with the reductions to be applied. In this table these results are also compared with the experiments and the beam model with the simple reduction, as before. Table 6.8 Values of the reduced Young modulus accounting for shear.

E Ered Beam model predictions Reduced Young modulus

( 1. .00.5red RdE EF= ) ke.0 Ratio

Test ID (MPa)

m/tf 1. .0RdF 1. .00.5 RdF (MPa)

kN/mm Act. 0.5E T1 2.75 0.92 0.46 95416 92.48 1.11 0.93 P1 3.22 0.94 0.47 97472 65.96 1.04 0.94 P2 2.29 0.89 0.44 92315 139.46 1.19 0.91 P3 2.75 0.92 0.46 95416 82.29 1.13 0.93 P4 2.75 0.92 0.46 95416 116.60 1.19 0.93 P5 2.75 0.92 0.46 95416 130.20 1.29 0.93 P9 2.05 0.87 0.43 90180 167.39 1.30 0.90 P10 4.39 0.96 0.48 100318 27.16 0.62 0.96 P12 2.75 0.92 0.46 95416 97.94 0.96 0.92 P14 2.75 0.92 0.46 95416 92.48 1.13 0.93 P15 2.29 0.89 0.44 92315 154.94 1.21 0.93 P16 2.75 0.92 0.46 95416 147.44 1.33 0.93 P18 2.75 0.92 0.46 95416 169.58 0.99 0.93 P20 2.05 0.87 0.43 90180 296.64 1.63 0.90 P23

2081

53

1.67 0.82 0.41 85306 461.22 1.43 0.88 Average 1.17 0.92

Coeff. variation 0.20 0.02


202

Table 6.9 Validation of the approach with further examples from the data-base: comparison of initial stiffness predictions (Young modulus in [MPa] stiffness values in [kN/mm]).

Beam model predictions 0.5E 1. .00.5red RdE EF=

Ratio

Test ID

E m/tf 1. .0

2RdF

Actual

ke.0 ke.0 Ra-

tio ke.0

Act. 0.5E HR-T-stubs

P6 2.75 0.46 76.68 88.80 1.16 82.29 1.07 0.93 P7 2.75 0.46 91.72 110.19 1.20 102.35 1.12 0.93 P8 2.46 0.45 95.19 127.28 1.34 116.44 1.22 0.91 P11 2.05 0.43 122.93 188.83 1.54 170.56 1.39 0.90 P13 2.75 0.46 83.46 99.68 1.19 92.48 1.11 0.93 P17 2.75 0.46 138.60 172.19 1.24 159.89 1.15 0.93 P19 2.75 0.46 130.47 172.19 1.32 159.89 1.23 0.93 P21 2.75 0.46 174.25 228.11 1.31 211.79 1.22 0.93 P22

2081

53

2.16 0.44 253.74 328.89 1.30 296.85 1.17 0.90 Average 1.29 1.19 0.92

Coeff. variation

0.09

0.08 0.01 WP-T-stubs

Weld_ T1(i) 3.50 0.47 73.77 55.10 0.75 52.36 0.71 0.95

Weld_ T1(ii) 3.12 0.47 89.12 73.19 0.82 68.82 0.77 0.94

Weld_ T1(iii)

2081

53

2.82 0.46 107.29 94.26 0.88 87.68 0.82 0.93

WT1 3.27 0.47 71.08 75.96 1.07 71.61 1.01 0.94 WT2A 3.53 0.47 61.83 62.09 1.00 58.97 0.95 0.95 WT2B 3.08 0.47 79.75 88.19 1.11 82.62 1.04 0.94 WT4A 20

9856

3.24 0.47 86.96 122.66 1.41 115.86 1.33 0.94 WT51 3.45 0.47 60.73 64.11 1.06 60.73 1.00 0.95 WT53C 3.40 0.47 64.23 65.60 1.02 62.12 0.97 0.95 WT53D 3.38 0.47 52.90 67.77 1.28 64.08 1.21 0.95 WT53E 20

4462

3.40 0.47 64.82 65.44 1.01 61.94 0.96 0.95 WT7_ M12 3.28 0.47 91.18 120.34 1.32 113.79 1.25 0.95

WT7_ M16 3.28 0.47 116.09 124.29 1.07 117.27 1.01 0.94

WT7_ M20

2098

56

3.27 0.47 137.70 128.08 0.93 120.60 0.88 0.94

WT57_ M12 3.38 0.47 85.78 105.35 1.23 99.95 1.17 0.95

WT57_ M16 3.37 0.47 110.43 113.07 1.02 106.94 0.97 0.95

WT57_ M20

2044

62

3.38 0.47 150.96 115.51 0.77 109.09 0.72 0.94

Average 1.04 0.99 0.94 Coeff. variation

0.18

0.18 0.01


203

Table 6.10 Geometric characteristics of the specimens tested by Faella et al.

Test ID Geometric characteristics (dimensions in [mm]) Kb b beff tf m n λ φ (kN/mm) TS1 189.0 90.70 11.40 28.35 35.44 1.25 20 1.79×106 TS2 189.0 116.03 11.00 41.01 49.85 1.22 20 1.84×106 TS3 189.0 64.05 9.10 30.03 34.10 1.14 20 2.13×106 TS4 189.0 96.33 9.35 31.16 33.10 1.06 20 2.09×106 TS5 189.0 98.13 13.75 32.06 32.21 1.00 20 1.54×106 TS6 189.0 95.50 13.45 30.75 32.98 1.07 20 1.57×106 TS7 188.0 80.28 14.85 29.99 37.48 1.25 12 5.10×105 TS8 189.0 77.12 14.90 28.41 35.51 1.25 12 5.09×105 TS9 189.0 78.65 16.50 29.18 36.47 1.25 12 4.66×105 TS10 189.0 78.15 15.50 28.93 36.16 1.25 12 4.92×105 TS11 189.5 101.55 11.05 40.63 50.25 1.24 12 6.53×105 TS12 189.5 76.50 10.70 28.10 35.13 1.25 12 6.71×105 TS13 189.0 79.45 12.65 29.58 36.97 1.25 12 5.84×105 TS14 189.0 80.65 12.70 30.18 37.72 1.25 12 5.82×105 TS15 189.0 84.20 13.80 31.95 39.94 1.25 12 5.43×105 TS16 189.5 82.65 13.45 31.18 38.97 1.25 12 5.55×105 Table 6.11 Validation of the approach with further examples tested by Faella,

Piluso and Rizzano [6.3]: comparison of initial stiffness predic-tions (Young modulus in [MPa]; stiffness values in [kN/mm]).

Beam model predictions 0.5E

1. .00.5red RdE EF=

Ratio

Test ID

E m/tf 1. .00.5 RdF Exp. ke.0

ke.0 Ra-tio

ke.0 Act. 0.5E

TS1 2.49 0.45 167 290.59 1.74 265.65 1.59 0.91 TS2 3.73 0.48 112 122.13 1.09 116.51 1.04 0.95 TS3 3.30 0.47 99 145.17 1.47 136.88 1.38 0.94 TS4 3.33 0.47 103 146.62 1.42 138.43 1.34 0.94 TS5 2.33 0.45 229 364.25 1.59 347.20 1.52 0.95 TS6 2.29 0.44 214 372.57 1.74 338.93 1.58 0.91 TS7 2.02 0.43 237 295.17 1.25 270.92 1.14 0.92 TS8 1.91 0.43 213 317.39 1.49 289.89 1.36 0.91 TS9 1.77 0.42 214 346.13 1.62 315.21 1.47 0.91 TS10 1.87 0.42 266 325.77 1.22 297.45 1.12 0.91 TS11 3.68 0.47 82 101.57 1.24 97.15 1.18 0.96 TS12 2.63 0.45 168 184.66 1.10 171.63 1.02 0.93 TS13 2.34 0.45 163 234.90 1.44 217.04 1.33 0.92 TS14 2.38 0.45 156 229.40 1.47 212.29 1.36 0.93 TS15 2.32 0.44 192 242.82 1.26 224.84 1.17 0.93 TS16

2100

00

2.32 0.44 179 241.14 1.35 223.10 1.25 0.93 Average 1.41 1.30 0.93

Coeff. variation

0.14 0.14 0.02


204

Regarding the latter specimens, the Young modulus of the flange cross-section was taken as equal to 210 GPa, since this particular parameter was not defined by the authors. And since the characteristics of the bolts were not pro-vided as well, the following assumptions were made to define the bolt elastic elongation behaviour: (i) Eb = 210 GPa, (ii) bolts are full-threaded, (iii) two washers per bolt with twsh = 2.95 mm for M20 bolts and twsh = 2.50 mm for M12 bolts, (iv) th = 13.10 mm for M20 bolts and th = 8.90 mm for M12 bolts and (v) tn = 16.00 mm for M20 bolts and tn = 11.90 mm for M12 bolts. Finally, a remark concerning the two values of b and beff that appear in Ta-ble 6.10 is required. The beam model employs beff for the definition of the cross-section width, which is kept constant throughout the load history. As mentioned earlier, this width accounts for all possible yield line mechanisms of the T-stub flange and cannot exceed the actual flange width, b. For the exam-ples from the database, beff = b; however, for the set of examples tested by Faella and co-authors, b > beff and so the smallest value governs. Physically, it is quite clear that the section width that contributes to load transmission expands with the loading provided that the actual flange width is not exceeded. The assumption of a constant flange width of beff seems appro-priate until strain hardening begins (the reduction at elastic conditions can be done indirectly with the reduction of the Young modulus, for instance), but it may be too conservative at ultimate conditions. 6.3.3 Analysis of the model in the elastoplastic range Having carried out the full nonlinear analysis of the beam model, numerous re-sults can be extracted, namely: (i) load-carrying behaviour, (ii) evolution of the prying forces, (iii) flange moment diagram, (iv) flange plastic strain diagram, etc. Thorough results for specimens T1 and WT1 are given in Appendix D to illustrate the capacities of the model. In this section, the F-∆ response of the specimens from the database is characterized up to collapse in the framework of the proposed methodology. This simplified load-carrying behaviour is com-pared with the actual response and the bilinear approximation proposed by Jas-part. Firstly, the collapse modes are defined. Table 6.12 sets out the actual de-termining fracture element: bolt or T-stub flange. The prediction of the failure modes as described in Chapter 2, which is based on a force criterion, is also given. These predictions are generally in line with the observed failure modes, except for specimens P13, WT53D and WT53E. Such situation may derive from the fact that the fracture criterion for the numerical three-dimensional model and for the beam model is based on a deformation condition. This may introduce some differences. This table also indicates the fracture element that is determinant in the two-dimensional model. Here, the differences are more fre-quent (underlined specimens). The critical section (1*) that is referred to in the table is the critical section right at the end of the fillet weld or radius. After


205

Table 6.12 Prediction of the failure modes.

Predicted poten-tial failure mode.

Test ID Actual deter-mining fracture

element Mf.u Eq. (2.4)

Mf.u Eq. (2.5)

Determining fracture element in the beam

model

T1 Bolt 13 13 Bolt P1 Bolt 13 13 Bolt P2 Bolt 13 13 Bolt P3 Bolt 13 13 Flange, at (1*) P4 Bolt 13 13 Bolt P5 Bolt 23 23 Bolt P6 Bolt 13 13 Bolt P7 Bolt 13 13 Bolt P8 Bolt 13 13 Bolt P9 Bolt 23 23 Bolt P10 Flange 11 11 Flange, at (1*) P11 Bolt 23 23 Bolt P12 Flange 11 11 Flange, at (1*) P13 Bolt 11 11 Flange, at (1*) P14 Flange 11 11 Flange, at (1*) P15 Flange 11 11 Flange, at (1*) P16 Bolt 23 23 Bolt P17 Bolt 13 13 Bolt and flange at

(1*) (simultaneously) P18 Flange 11 11 Flange, at (1*) P19 Bolt 13 13 Bolt and flange at

(1*) (simultaneously) P20 Bolt 23 23 Bolt P21 Bolt 13 13 Flange at the bolt axis P22 Bolt 13 13 Bolt P23 Bolt 23 23 Bolt Weld_T1(i) Bolt 11 11 Flange, at (1*) Weld_T1(ii) Bolt 13 13 Flange, at (1*) Weld_T1(iii) Bolt 13 13 Bolt WT1 Bolt and flange 11 13 Flange, at (1*) WT2A Bolt and flange 11 13 Flange, at (1*) WT2B Bolt and flange 11 13 Flange, at (1*) WT4A Bolt 13 13 Bolt WT51 Bolt 23 23 Flange, at (1*) WT53C Bolt 13 13 Flange, at (1*) WT53D Bolt 11 13 Flange, at (1*) WT53E Bolt 11 13 Flange, at (1*) WT7_M12 Bolt 13 13 Bolt WT7_M16 Flange 11 13 Flange, at (1*) WT7_M20 Flange 11 11 Flange, at (1*) WT57_M12 Bolt 23 23 Flange, at (1*) WT57_M16 Bolt (stripping) 13 13 Flange, at (1*) WT57_M20 Bolt and flange 11 11 Flange, at (1*)


206

application of the model and within the process of calibration of the model, it was observed that the imposition of cracking of the material at section (1), at the flange-to-web connection, as an ultimate condition was a too severe condi-tion indeed (§6.3.1.1). Therefore, collapse occurs when either or both of the following conditions are verified: (i) fracture of the bolt and (ii) cracking of material of the flange at the end of the fillet weld or radius, section (1*). Table 6.13 summarizes the predictions of deformation capacity and ulti-mate resistance of the proposed model. The predictions are compared with the actual results. The ultimate resistance is well estimated and the scatter of re-sults is lower since the variation is also low (coefficient of variation of 0.13 for HR-T-stubs and 0.15 for WP-T-stubs). Regarding the deformation capacity, the differences are more relevant. However, this disparity has to be accepted within reasonable limits due to all the simplifications inherent to this two-dimensional approach. In this table the specimens are separated according to their assembly type. Table 6.14 presents identical results but with specimens grouped according to the potential failure type (Mf.u from Eq. (2.4)). The speci-mens whose actual failure type was not well predicted were excluded from this table. If now the specimens are analysed in this context, the following conclu-sions may be drawn: (i) for specimens that fail according to type-23, both pre-dictions of deformation and resistance at ultimate conditions are good, with av-erage errors smaller than 10%, (ii) identical conclusions are valid for speci-mens of type-11 failure, regarding the evaluation of deformation capacity, (iii) for these latter specimens the ultimate resistance prediction is conservative, (iv) for those specimens failing according to a type-13 mode, the predictions for re-sistance are good and (v) the evaluation of deformation capacity for these specimens is rather weak. It should be noted that the deformation capacity, ∆u.0, that appears in the above tables corresponds to the deformation of the T-stub when the maximum load is reached. This definition may be quite conservative when the experimental results are taken for comparison since the softening branches sometimes can be quite extended (specimen WT7_M20, for exam-ple). Figs. 6.12-6.14 illustrate the load-carrying behaviour for some specimens that represent the various collapse modes. The curves are compared with the actual response and the bilinear approximation of Jaspart. This bilinear ap-proximation was defined using the formulation accounting for the bolt action for specimens of failure type-11 and the actual material strain hardening modulus. It becomes clear from these curves that the beam model provides a better agreement with the real F-∆ response and in general these two curves fit well. Finally, Fig. 6.15 compares the responses for those specimens whose failure mode was not well predicted by the beam model. Nonetheless, the agreement is surprisingly good. Fig. 6.15e traces the F-∆ behaviour of specimen WT57_M12. For this specimen the flange plates are fastened by means of two full-threaded bolts. At col-lapse, the bolt model estimates a fracture elongation of 4 mm – Eq. (6.14). Since


207

Table 6.13 Prediction of deformation capacity and ultimate resistance.

Actual results Beam model predictions Fmax ∆u.0 Fu.0 Ratio ∆u.0 Ratio

Test ID Potential failure type (kN) (mm) (kN) (mm)

T1 13 103.99 8.70 114.45 1.10 16.76 1.93 P1 13 91.76 10.77 103.25 1.13 25.34 2.35 P2 13 116.72 6.18 124.34 1.07 8.75 1.42 P3 13 95.41 10.17 111.96 1.17 24.17 2.38 P4 13 115.97 4.68 120.25 1.04 7.23 1.55 P5 23 130.20 3.63 123.76 0.95 4.63 1.28 P6 13 95.53 10.06 111.96 1.17 24.17 2.40 P7 13 111.34 7.56 116.43 1.05 11.66 1.54 P8 13 112.71 8.08 124.24 1.10 12.20 1.51 P9 23 131.43 3.31 137.48 1.05 3.67 1.11 P10 11 76.79 32.75 50.25 0.65 32.40 0.99 P11 23 121.15 2.94 134.34 1.11 3.47 1.18 P12 11 154.06 24.22 122.43 0.79 19.67 0.81 P13 11 93.71 11.38 97.44 1.04 18.57 1.63 P14 11 86.57 24.15 79.54 0.92 20.49 0.85 P15 11 171.08 18.02 153.17 0.90 16.48 0.91 P16 23 125.69 3.06 127.98 1.02 3.00 0.98 P17 13 192.01 9.29 212.18 1.11 20.67 2.22 P18 11 266.57 26.07 215.75 0.81 20.18 0.77 P19 13 186.52 9.29 212.18 1.14 20.67 2.23 P20 23 225.94 4.06 246.14 1.09 4.03 0.99 P21 13 281.33 17.67 285.70 1.02 20.61 1.17 P22 13 305.21 6.40 345.44 1.13 13.75 2.15 P23 23 346.01 5.22 408.62 1.18 5.24 1.00

Average 1.03 1.47 Coefficient of variation 0.13 0.38

Weld_T1(i) 11 92.02 10.85 84.80 0.92 18.27 1.68 Weld_T1(ii) 13 102.75 8.01 101.03 0.98 20.00 2.50 Weld_T1(iii) 13 113.10 6.22 114.21 1.01 18.82 3.02 WT1 11 91.91 14.32 85.60 0.93 18.51 1.29 WT2A 11 86.82 17.98 75.16 0.87 16.51 0.92 WT2B 11 97.88 13.09 92.51 0.95 19.18 1.47 WT4A 13 103.26 4.33 124.40 1.20 10.77 2.49 WT51 23 97.08 3.96 112.86 1.16 9.43 2.38 WT53C 13 98.90 4.24 114.45 1.16 8.50 2.00 WT53D 11 117.36 5.54 117.02 1.00 8.23 1.49 WT53E 11 115.04 5.26 116.04 1.01 9.03 1.72 WT7_M12 13 100.34 4.60 123.83 1.23 11.00 2.39 WT7_M16 11 132.34 11.47 140.79 1.06 17.78 1.55 WT7_M20 11 145.72 9.12 141.17 0.97 17.87 1.96 WT57_M12 23 121.87 4.33 174.51 1.43 8.07 1.86 WT57_M16 13 173.64 5.88 196.62 1.13 9.13 1.55 WT57_M20 11 241.71 15.98 196.77 0.81 8.36 0.52



208

Table 6.14 Prediction of deformation capacity and ultimate resistance: speci-mens organized by failure type group.



T1 13 103.99 8.70 114.45 1.10 16.76 1.93 P1 13 91.76 10.77 103.25 1.13 25.34 2.35 P2 13 116.72 6.18 124.34 1.07 8.75 1.42 P4 13 115.97 4.68 120.25 1.04 7.23 1.55 P6 13 95.53 10.06 111.96 1.17 24.17 2.40 P7 13 111.34 7.56 116.43 1.05 11.66 1.54 P8 13 112.71 8.08 124.24 1.10 12.20 1.51 P17 13 192.01 9.29 212.18 1.11 20.67 2.22 P19 13 186.52 9.29 212.18 1.14 20.67 2.23 P22 13 305.21 6.40 345.44 1.13 13.75 2.15 Weld_T1(iii) 13 113.10 6.22 114.21 1.01 18.82 3.02 WT4A 13 103.26 4.33 124.40 1.20 10.77 2.49 WT7_M12 13 100.34 4.60 123.83 1.23 11.00 2.39


P10 11 76.79 32.75 50.25 0.65 32.40 0.99 P12 11 154.06 24.22 122.43 0.79 19.67 0.81 P14 11 86.57 24.15 79.54 0.92 20.49 0.85 P15 11 171.08 18.02 153.17 0.90 16.48 0.91 P18 11 266.57 26.07 215.75 0.81 20.18 0.77 WT7_M16 11 132.34 11.47 140.79 1.06 17.78 1.55 WT7_M20 11 145.72 9.12 141.17 0.97 17.87 1.96 WT57_M20 11 241.71 15.98 196.77 0.81 8.36 0.52


P5 23 130.20 3.63 123.76 0.95 4.63 1.28 P9 23 131.43 3.31 137.48 1.05 3.67 1.11 P11 23 121.15 2.94 134.34 1.11 3.47 1.18 P16 23 125.69 3.06 127.98 1.02 3.00 0.98 P20 23 225.94 4.06 246.14 1.09 4.03 0.99 P23 23 346.01 5.22 408.62 1.18 5.24 1.00


bolt governs fracture of this specimen, the post-limit behaviour proceeds until this deformation of 4 mm is attained, leading to an overall deformation of 8.1 mm and ultimate resistance of 174.5 kN, corresponding to 1.43 times the maximum resis-


209

0

15

30

45

60

75

90

105

120

0 3 6 9 12 15 18 21 24 27 30 33


Loa

d, F

(kN

)

Actual response

Simplified response (Beam model)

Bilinear approximation (Jaspart)

(a) Specimen P1.

0

15

30

45

6075

90

105

120

135

0 1 2 3 4 5 6 7 8 9 10 11 12 13


Loa

d, F

(kN

)

Actual response



(b) Specimen P2.

0

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45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Loa

d, F

(kN

)

Actual response



(c) Specimen Weld_T1(iii). Fig. 6.12 Specimens that fail according to type-13: comparison of the actual

response with the beam model predictions.


210

0

10

20

30

40

50

60

70

80

0 3 6 9 12 15 18 21 24 27 30 33 36


Loa

d, F

(kN

)

Actual response


Bilinear approximation

(a) Specimen P10.

0

40

80

120

160

200

240

280

0 3 6 9 12 15 18 21 24 27 30


Loa

d, F

(kN

)

Actual response



(b) Specimen P18.

0

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60

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100

120

140

160

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual response



(c) Specimen WT7_M20. Fig. 6.13 Specimens that fail according to type-11: comparison of the actual



211

0

20

40

60

80

100

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140

0 1 2 3 4 5 6 7 8 9 10 11 12


Loa

d, F

(kN

)

Actual response



(a) Specimen P5.

0

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40

60

80

100

120

140

0 1 2 3 4 5 6 7 8 9 10 11


Loa

d, F

(kN

)

Actual response



(b) Specimen P16.

0

50

100

150

200250

300

350

400

450

0 1 2 3 4 5 6 7 8 9 10 11


Loa

d, F

(kN

)

Actual response



(c) Specimen P23. Fig. 6.14 Specimens that fail according to type-23: comparison of the actual



212

0102030405060708090

100

0 3 6 9 12 15 18 21 24 27 30 33 36 39


Loa

d, F

(kN

)

Actual response



(a) Specimen Weld_T1(i).

0

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30

45

60

75

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105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



(b) Specimen WT2B.

0

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60

75

90

105

120

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response



(c) Specimen WT51. Fig. 6.15 WP-T-stub specimens whose observed failure types are not coinci-

dent with the predictions.


213

0

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0 1 2 3 4 5 6 7 8 9 10


Loa

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(kN

)

Actual response



(d) Specimen WT53D.

0

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80100

120

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0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response



(e) Specimen WT57_M12.

0

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210

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response



(f) Specimen WT57_M16. Fig. 6.15 WP-T-stub specimens whose observed failure types are not coinci-

dent with the predictions (cont.).


214

tance from the tests. With reference to Eq. (6.14) derived for δb.fract, it was devel-oped for short-threaded bolts [6.6]. If a full-threaded bolt is considered instead, this expression seems to overestimate the bolt fracture deformation. Conse-quently, some guidelines concerning this matter are required. In terms of design calculations, the characterization of the behaviour of the above specimens should be based in nominal properties, as already mentioned. The results obtained by this procedure are fully described and analysed in Ap-pendix D. 6.3.4 Sophistication of the proposed method: modelling of the bolt action

as a distributed load Jaspart has shown that a significant increase in the resistance of single T-stubs that fail according to a plastic mechanism type-1 can be expected due to the in-fluence of the bolt action on a finite contact area [6.2]. This effect is taken into account in the beam model in this section. The bolt is then modelled as a spring assembly in parallel, as shown in Fig. 6.16. The length of this assembly is the bolt diameter. The behaviour of this spring assembly is the same as the original single spring, i.e. the spring stiffness and force values are divided by the num-ber of springs in the assembly. Eleven examples were chosen to illustrate this modification, five examples from type-11 group and three examples from each of the other two groups. The analysis of the first group is quite straightforward as the bolt is not engaged in the failure mode. For those specimens whose fracture is governed by the bolt itself, the fracture condition has to be redefined. Below, this condition is im-posed at two different sections and the results are then compared: (i) section (2) from above, at mid- bolt section and (ii) section (2*), located at ¼ of the inside bolt edge, from section (2). For those specimens that fail according to a type-23 mechanism, when the bolt fractures, (1)

.p p uε ε< , specimens belonging to type-

13 failure mode may exhibit (1).p p uε ε> when the bolt fracture but

(1*) (2).,ε ε ε<p p p u .

Table 6.15 summarizes the predicted resistance and deformation values at collapse when this modification is introduced. The examples are grouped ac-cording to the failure mode. The underlined connections, again, refer to those cases where the predicted failure mode does not match the observed type. For evaluation of ratios to actual values these examples are not taken into consid-eration. The fracture condition here was identical to the above. The application of this modified model provides a significant enhancement of results in terms of resistance, particularly for the evaluation of Fu.0. So are the pre-dictions of deformation capacity for specimens from type-11 group. Specimens that fail according to a type-23 mechanism show worse predictions of deforma-tion capacity. For specimens belonging to type-13 fracture mode, these predict-


215

B-δb

m n

F2

0.8r

φ

Fig. 6.16 Beam model accounting for the bolt action on a finite area. ions improve but are still distant from the actual deformation values. Further comparisons are carried out in Appendix D. It is worth mentioning that this sophistication enforced the “correct” frac-ture element to be critical in specimens P3 and Weld_T1(ii), for instance. Nev-ertheless, the beam model is not yet able to simulate the fracture of the bolt si-multaneously to cracking of the flange material in some WP-T-stubs. 6.3.5 Influence of the distance m for the WP-T-stubs Distance m is one of the geometrical parameters that most influences the de-formation behaviour of T-stub connections. For HR-T-stubs, this distance is well established and there is experimental and numerical evidence for its defi-nition. Common procedures for WP-T-stubs consisted in extrapolating the de-sign rules defined for HR-T-stubs. Parameter m defines the location of the po-tential yield line at the flange-to-web connection with respect to the bolt line. In previous chapters, it has been shown that this procedure may not be accurate enough. In fact, there is evidence that in some cases the yield line at the flange-to-web connection develops in the flange at the end of the fillet root, i.e. for a distance m defined as follows:

2 wm d a= − (6.35)


216

Table 6.15 Prediction of deformation capacity and ultimate resistance by ap-plying the sophisticated beam model accounting for bolt action: specimens organized by failure type group.

Actual results Sophisticated beam model pre-dictions

Fmax ∆ u.0 Fu.0 Ratio ∆ u.0 Ratio

Test ID Potential failure type

(kN) (mm) (kN) (mm) T1 13 103.99 8.70 113.87 1.09 12.74 1.46 P1 13 91.76 10.77 103.27 1.13 19.88 1.85 P3 13 95.41 10.17 108.18 1.13 16.06 1.58 Weld_T1(ii) 13 102.75 8.01 104.18 1.01 17.61 2.20 Weld_T1(iii) 13 113.10 6.22 113.33 1.00 14.16 2.27 WT57_M16 13 173.64 5.88 212.19 1.22 8.35 1.42


P10 11 76.79 32.75 59.17 0.77 27.79 0.85 P12 11 154.06 24.22 149.02 0.97 19.22 0.79 P14 11 86.57 24.15 88.79 1.03 19.87 0.82 P15 11 171.08 18.02 187.94 1.10 16.16 0.90 P18 11 266.57 26.07 280.34 1.05 18.69 0.72 Weld_T1(i) 11 92.02 10.85 88.57 0.96 17.17 1.58 WT1 11 91.91 14.32 90.37 0.98 15.67 1.09 WT2A 11 86.82 17.98 81.50 0.94 16.11 0.90 WT2B 11 97.88 13.09 102.67 1.05 19.83 1.10 WT7_M16 11 132.34 11.47 158.47 1.20 16.24 1.42 WT7_M20 11 145.72 9.12 177.93 1.22 14.71 1.61 WT57_M20 11 241.71 15.98 233.41 0.97 7.56 0.47


P5 23 130.20 3.63 121.71 0.93 3.37 0.93 P20 23 225.94 4.06 247.76 1.10 3.51 0.87 P23 23 346.01 5.22 399.89 1.16 4.00 0.77 WT51 23 97.08 3.96 119.08 1.23 9.00 2.27


The influence of this distance is further detailed in Appendix D. Generally speaking, if m from Eq. (6.35) is adopted, there is an increase on resistance and stiffness and decrease on ductility. 6.4 CONCLUDING REMARKS The proposed beam model yields an accurate prediction of the F-∆ response of bolted T-stub connections, despite the simplifications inherent to a two-


217

dimensional modelling of the behaviour. These reduced the model complexity to a more reasonable level, when compared to the three-dimensional FE model-ling. However, to obtain the F-∆ curve, a numerical incremental procedure is still required and, consequently, the model is not suitable for hand calculations. The dominant effects in both approaches are the strain hardening of the flange and the bolt elongation behaviour, as confirmed by experimental evidence. An-other important simplification of the beam model corresponds to the beam sec-tion width, which is kept constant with the course of loading. As the load in-creases, the flange width tributary to load transmission also increases. The analysis of this variation was not carried out and the implementation of such phenomenon is not straightforward neither. Nevertheless, for those specimens failing according to a type-11 mode, this issue can be particularly relevant. The applicability of the model was well demonstrated within the range of examples shown above. The behaviour predicted by this model is rather good in terms of resistance. With respect to ductility, it reflects an overestimation of test results that is within an acceptable error. Table 6.16 summarizes the statis-tical parameters (average and coefficient of variation) corresponding to the sample of connections that were analysed above. Two approaches in terms of material properties are taken into account: actual properties (Table 6.13) and nominal properties (Table D.17 in Appendix D). If the results are analysed in terms of failure types (cf. Table 6.14), the pre-dictions for resistance are accurate for those specimens whose fracture is deter-mined by the bolt. For those specimens failing according to a type-11, the results seem rather conservative. Concerning the predictions for deformation capacity, these are quite good for failure modes type-11 and -23, even though the scatter of results for specimens of failure type-11 is high (coefficient of variation of 0.45). For the remaining case (type-23 failure), there is an overestimation of results. The modification for inclusion of the bolt action provides an enhancement of results but introduces an additional complexity. From a design point of view, the methodology should be further simplified so that it can be used in an expedite way, as Jaspart’s simple proposal. This can be achieved by modelling plasticity phenom-ena in the flange by means of rotational springs at the critical sections that capture the overall behaviour. Table 6.16 Summary of the proposed beam model from a statistical point of

view (average ratios and coefficients of variation, the latter in italic) for evaluation of the force-deformation characteristics.

Ultimate resis-tance

Deformation capac-ity

T-stub assembly

Fu.0 ∆u.0 HR 1.03 (0.13) 1.47 (0.38) WP Actual material properties 1.05 (0.15) 1.81 (0.34) HR 0.98 (0.17) 1.73 (0.38) WP

Nominal material proper-ties 0.94 (0.23) 1.18 (0.53)


218





[6.4] Piluso V, Faella C, Rizzano G. Ultimate behavior of bolted T-stubs. I: theoretical model. Journal of Structural Engineering ASCE; 127(6):686-693, 2001.

[6.5] Piluso V, Faella C, Rizzano G. Ultimate behavior of bolted T-stubs. II: model validation. Journal of Structural Engineering ASCE; 127(6):694-704, 2001.

[6.6] Swanson JA. Characterization of the strength, stiffness and ductility be-havior of T-stub connections. PhD dissertation, Georgia Institute of Technology, Atlanta, USA, 1999.

[6.7] Beg D, Zupančič E, Vayas I. On the rotation capacity of moment con-nections. Journal of Constructional Steel Research; 60:601-620, 2004.

[6.8] Jaspart JP. Contributions to recent advances in the field of steel joints – col-umn bases and further configurations for beam-to-column joints and beam splices. Aggregation thesis. University of Liège, Liège, Belgium, 1997.

[6.9] Yee YL, Melchers RE. Moment-rotation curves for bolted connections. Journal of Structural Engineering ASCE; 112(3):615-635, 1986.

[6.10] Maquoi R, Jaspart JP. Moment-rotation curves for bolted connections: Discussion of the paper by Yee YL and Melchers RE. Journal of Struc-tural Engineering ASCE; 113(10):2324-2329, 1986.

[6.11] Gioncu V, Mateescu G, Petcu D, Anastasiadis A. Prediction of available ductility by means of local plastic mechanism method: DUCTROT computer program, Chapter 2.1 in Moment resistant connections of steel frames in seismic areas (Ed.: F. Mazzolani). E&FN Spon, London, UK; 95-146, 2000.

[6.12] McGuire W. Steel structures. Prentice-Hall International series in Theo-retical and Applied Mechanics (Eds.: NM Newmark and WJ Hall). Englewood Cliffs, N.J., USA, 1968.

[6.13] Gioncu V, Mazzolani FM. Ductility of seismic resistant steel structures. Spon Press, London, 2002.

[6.14] Hirt MA, Bez R. Construction métallique – Notions fondamentales et methods de dimensionnement. Traité de Génie Civil de l’École polytechnique fédérale de Lausanne, Volume 10. Presses Polytechniques et Universitaires Romandes, Lausanne, Switzerland, 1994.

[6.15] Lusas 13. Theory manual. Finite element analysis Ltd, Version 13.5. Surrey, UK, 2003.

219

APPENDIX D: DETAILED RESULTS OBTAINED FROM APPLICATION OF THE SIMPLIFIED METHODS FOR ASSESSMENT OF THE FORCE-DEFORMATION RESPONSE OF SINGLE T-STUB CONNECTIONS

D.1 Geometrical and mechanical characteristics of the specimens This appendix gives detailed results that were obtained from application of the simplified methods for assessment of the F-∆ response of single T-stub con-nections. For illustration of the various methodologies presented in Chapter 6, the specimens from the database compiled in Chapters 3-5 are employed. The relevant geometrical and mechanical characteristics of those specimens are summarized in Tables D.1 and D.2, respectively. D.2 Previous research: exemplification D.2.1 Evaluation of initial stiffness Table D.3 sets out the predictions of initial stiffness of the above specimens by application of the procedures proposed by Yee and Melchers [6.9] and subse-quently modified by Jaspart [6.2]. The results show that the two approaches (which differ essentially in the location of the prying forces) yield identical results, with consistent errors of 130%, which seem too high. The table also shows that the scatter of results is higher for the WP-T-stubs. This result how-ever may not be significant because in these cases most of the specimens were tested experimentally. Thus, the determination of the experimental initial stiff-ness is not as straightforward as for the specimens tested numerically. It should be noted that the values of ke.0 contained in this table were computed using the bolt conventional length as defined by Aggerskov – Eq (1.19). The results after application of the Swanson’s procedures are summarized in Table D.4. Again, the errors are excessive. Errors well above 100% are not acceptable. This table also shows that in spite of the deviations from the actual results, the Eurocode 3 prying model yields better results than the Kulak model, also adopted by Swanson. Both models are illustrated in Fig. 6.3. The two models differ in the geometry of the beam model. D.2.2 Piecewise multilinear approximation of the overall response and evalua-

tion of the deformation capacity and ultimate resistance a) Methodology recommended by Jaspart Table D.5 indicates the predictions of the different observed failure modes. The predicted potential failure type is defined according to the expressions pre-


220

Table D.1 Geometrical characteristics of the specimens. Geometric characteristics (dimensions in [mm]) Test ID Potential

failure type

beff tf m m/tf n λ φ

T1 13 40.0 10.7 29.45 2.75 30.00 1.02 12 P1 13 40.0 10.7 34.45 3.22 25.00 0.73 12 P2 13 40.0 10.7 24.45 2.29 30.56 1.25 12 P3 13 35.0 10.7 29.45 2.75 30.00 1.02 12 P4 13 52.5 10.7 29.45 2.75 30.00 1.02 12 P5 23 60.0 10.7 29.45 2.75 30.00 1.02 12 P6 13 35.0 10.7 29.45 2.75 30.00 1.02 12 P7 13 45.0 10.7 29.45 2.75 30.00 1.02 12 P8 13 40.0 11.0 27.10 2.46 33.88 1.25 12 P9 23 40.0 14.0 28.75 2.05 35.94 1.25 12 P10 11 40.0 7.0 30.75 4.39 30.00 0.98 12 P11 23 40.0 14.0 28.75 2.05 30.00 1.04 12 P12 11 40.0 10.7 29.45 2.75 30.00 1.02 16 P13 11 40.0 10.7 29.45 2.75 30.00 1.02 12 P14 11 40.0 10.7 29.45 2.75 30.00 1.02 12 P15 11 40.0 10.7 24.45 2.29 30.56 1.25 16 P16 23 70.0 10.7 29.45 2.75 30.00 1.02 12 P17 13 70.0 10.7 29.45 2.75 30.00 1.02 16 P18 11 70.0 10.7 29.45 2.75 30.00 1.02 20 P19 13 70.0 10.7 29.45 2.75 30.00 1.02 16 P20 23 70.0 14.0 28.75 2.05 30.00 1.04 16 P21 13 92.5 10.7 29.45 2.75 30.00 1.02 20 P22 13 70.0 15.0 32.34 2.16 30.00 0.93 20 P23 23 70.0 18.9 31.59 1.67 30.00 0.95 20 Weld_T1(i) 11 40.0 10.7 37.43 3.50 30.00 0.80 12 Weld_T1(ii) 13 40.0 10.7 33.42 3.12 30.00 0.90 12 Weld_T1(iii) 13 40.0 10.7 30.14 2.82 30.00 1.00 12 WT1 11 45.1 10.3 33.73 3.27 30.00 0.89 12 WT2A 11 45.0 10.3 36.29 3.53 29.90 0.82 12 WT2B 11 45.0 10.3 31.69 3.08 29.90 0.94 12 WT4A 13 74.9 10.4 33.69 3.24 30.00 0.89 12 WT51 23 45.0 10.0 34.39 3.45 30.20 0.88 12 WT53C 13 45.1 10.1 34.34 3.40 30.00 0.87 12 WT53D 11 45.0 10.1 34.24 3.38 30.00 0.88 12 WT53E 11 44.7 10.1 34.26 3.40 30.10 0.88 12 WT7_M12 13 75.6 10.3 33.87 3.28 29.90 0.88 12 WT7_M16 11 74.9 10.3 33.89 3.28 29.80 0.88 16 WT7_M20 11 75.2 10.3 33.81 3.27 29.70 0.88 20 WT57_M12 23 75.0 10.1 34.11 3.38 30.20 0.89 12 WT57_M16 13 75.3 10.2 34.26 3.37 30.10 0.88 16 WT57_M20 11 75.1 10.2 34.27 3.38 30.20 0.88 20


221

Table D.2 Mechanical characteristics of the specimens. Flange Bolt

fy.f εp.u.f fu.b δu.b Kb Test ID Potential

failure type (MPa) (MPa) (mm) (N/mm)

T1 13 431.0 0.284 974.0 0.97 6.92×105

P1 13 431.0 0.284 974.0 0.97 6.92×105

P2 13 431.0 0.284 974.0 0.97 6.92×105

P3 13 431.0 0.284 974.0 0.97 6.92×105

P4 13 431.0 0.284 974.0 0.97 6.92×105

P5 23 431.0 0.284 974.0 0.97 6.92×105

P6 13 431.0 0.284 974.0 0.97 6.92×105 P7 13 431.0 0.284 974.0 0.97 6.92×105 P8 13 431.0 0.284 974.0 0.98 6.78×105 P9 23 431.0 0.284 974.0 1.36 5.45×105

P10 11 431.0 0.284 974.0 0.75 9.37×105

P11 23 431.0 0.284 974.0 1.20 5.57×105 P12 11 431.0 0.284 974.0 1.01 1.19×106

P13 11 355.0 0.284 974.0 0.97 6.92×105 P14 11 275.0 0.284 974.0 0.97 6.92×105

P15 11 431.0 0.284 974.0 1.01 1.19×106

P16 23 431.0 0.284 974.0 0.97 6.52×105

P17 13 431.0 0.284 974.0 1.09 1.11×106 P18 11 431.0 0.284 974.0 1.09 1.73×106

P19 13 431.0 0.284 974.0 1.09 1.11×106 P20 23 431.0 0.284 974.0 1.37 9.54×105

P21 13 431.0 0.284 974.0 1.37 1.49×106 P22 13 431.0 0.284 974.0 1.52 1.40×106 P23 23 431.0 0.284 974.0 1.98 1.14×106

Weld_T1(i) 11 431.0 0.284 974.0 0.97 6.92×105 Weld_T1(ii) 13 431.0 0.284 974.0 0.97 6.92×105 Weld_T1(iii) 13 431.0 0.284 974.0 0.97 6.92×105 WT1 11 340.1 0.361 919.9 1.14 1.10×106 WT2A 11 340.1 0.361 919.9 1.14 1.10×106 WT2B 11 340.1 0.361 919.9 1.14 1.10×106 WT4A 13 340.1 0.361 919.9 1.14 1.10×106 WT51 23 698.6 0.174 919.9 1.14 1.10×106 WT53C 13 698.6 0.174 968.4 4.00 9.14×105 WT53D 11 698.6 0.174 1166.0 0.98 1.08×106 WT53E 11 698.6 0.174 1196.4 2.80 9.15×105 WT7_M12 13 340.1 0.361 919.9 1.14 1.10×106 WT7_M16 11 340.1 0.361 919.9 2.60 1.65×106 WT7_M20 11 340.1 0.361 919.9 2.60 2.57×106 WT57_M12 23 698.6 0.174 919.9 4.00 9.14×105 WT57_M16 13 698.6 0.174 919.9 2.60 1.65×106 WT57_M20 11 698.6 0.174 919.9 2.60 2.57×106


222

Table D.3 Prediction of axial stiffness by application of the standard Yee and Melchers procedures and the modified proposal of Jaspart.

Yee and Melchers standard procedure

Modified Yee and Melchers procedures




0.18

0.21



0.32

0.31


223

Table D.4 Prediction of axial stiffness by application of the Swanson proce-dures.

Prediction with b’ from Eq. (6.9)

Prediction with m from Eq. (6.11)




0.26 0.23



0.37

0.31


224

Table D.5 Prediction of failure modes.

Critical resis-tance formula

Critical resis-tance formula

Test ID

Pot. failure type Plst. Ultm.

Test ID Pot. failure type Plst. Ultm.

T1 13 1 1 or 2 P21 13 1 1 P1 13 1 1 or 2 P22 13 1 or 2 1 or 2 P2 13 1 1 P23 23 2 2 P3 13 1 1 or 2 Weld_T1(i) 11 1 1 P4 13 1 or 2 2 Weld_T1(ii) 13 1 1 P5 23 2 2 Weld_T1(iii) 13 1 1 or 2 P6 13 1 1 WT1 11 1 1 P7 13 1 1 or 2 WT2A 11 1 1 P8 13 1 1 or 2 WT2B 11 1 1 P9 23 2 2 WT4A 13 1 or 2 2 P10 11 1 1 WT7_M12 13 1 or 2 2 P11 23 2 2 WT7_M16 11 1 1 P12 11 1 1 WT7_M20 11 1 1 P13 11 1 1 WT51 23 2 1 or 2 P14 11 1 1 WT53C 13 1 or 2 P15 11 1 1 WT53D 11 1 or 2 1 P16 23 2 2 WT53E 11 1 1 P17 13 1 1 or 2 WT57_M12 23 2 2 P18 11 1 1 WT57_M16 13 1 or 2 1 or 2 P19 13 1 1 or 2 WT57_M20 11 1 1 P20 23 2 2 sented in Chapter 2. For computation of Mf.u, Eq. (2.4) recommended by Gioncu et al. is employed [6.11]. This table also indicates the critical resistance formula according to the Jaspart methodology (cf. 6.2.1). For application of the procedures, four different cases are considered regarding the resistance formu-lation (BF or FBA) and the mechanical properties of the T-stub material. The complete characterization of the actual material properties of the various specimens from the database was given in Chapters 3, 4 and 5. The actual strain hardening modulus, Eh, for these specimens however is always lower than the nominal properties [6.3,6.13]. For steel grade S355, Eh = E/48.2 and for S275, Eh = E/42.8. No quantitative guidance is given in any of the refer-ences for steel grade S690. Hence, both actual and nominal values for Eh are taken into account for those specimens where steel grade S355 and S275 was employed (S275 was used in specimen P14). This variation combined with the two alternative resistance formulations yields four different approaches that are summarized in Tables D.6-D.9. The specimens are divided according to the assembly type (HR-T-stubs and WP-T- stubs).


225

Table D.6 Prediction of ultimate resistance and deformation capacity by us-ing the actual strain hardening modulus of the flange material and the basic formulation for computation of resistance (HR-T-stubs).

Numerical results Jaspart methodology (Actual Eh; BF) Fmax ∆u.0 Fu.0 ∆u.0

Test ID

(kN) (mm) (kN) Ratio

(mm) Ratio

T1 103.99 8.70 92.53 0.89 21.54 2.47 P1 91.76 10.77 79.10 0.86 27.33 2.54 P2 116.72 6.18 111.45 0.95 17.00 2.75 P3 95.41 10.17 80.96 0.85 21.02 2.07 P4 115.97 4.68 112.95 0.97 17.17 3.67 P5 130.20 3.63 117.24 0.90 11.24 3.10 P6 95.53 10.06 80.96 0.85 21.02 2.09 P7 111.34 7.56 104.09 0.93 22.06 2.92 P8 112.71 8.08 106.27 0.94 19.18 2.37 P9 131.43 3.31 127.29 0.97 9.31 2.82 P10 76.79 32.75 37.93 0.49 30.29 0.93 P11 121.15 2.94 123.56 1.02 9.42 3.20 P12 154.06 24.22 92.53 0.60 19.88 0.82 P13 93.71 11.38 79.31 0.85 20.30 1.78 P14 86.57 24.15 66.87 0.77 20.21 0.84 P15 171.08 18.02 111.45 0.65 15.00 0.83 P16 125.69 3.06 122.97 0.98 10.93 3.57 P17 192.01 9.29 161.92 0.84 21.77 2.34 P18 266.57 26.07 161.92 0.61 20.39 0.78 P19 186.52 9.29 161.92 0.87 21.77 2.34 P20 225.94 4.06 225.65 1.00 9.84 2.42 P21 281.33 17.67 213.96 0.76 21.37 1.21 P22 305.21 6.40 298.54 0.98 20.76 3.25 P23 346.01 5.22 353.25 1.02 10.43 2.00


0.17

0.41

For identical assembly types, the tables show that the model yields identical results, in terms of average ratios. The ultimate resistance predictions are more accurate if the formulation accounting for the bolt finite size is employed. Note that for those specimens whose collapse mode is determined from Eq. (6.6) this formulation does not apply. With respect to the evaluation of deformation capacity, the best predictions are obtained through application of the basic formulation and assuming the nominal strain hardening modulus (Table D.8). The models where the actual strain hardening modulus was used provided an overestimation of this prop-erty. On the other hand, if the nominal strain hardening modulus is considered the approximation improves significantly.


226

Table D.6 Prediction of ultimate resistance and deformation capacity (WP-T-stubs) (cont.).

Num. or Exp. re-sults

Jaspart methodology (Actual Eh; BF)

Fmax ∆u.0 Fu.0 ∆u.0

Test ID


(mm) Ratio

Weld_T1(i) 92.02 10.85 72.79 0.79 31.33 2.89 Weld_T1(ii) 102.75 8.01 81.54 0.79 26.03 3.25 Weld_T1(iii) 113.10 6.22 81.54 0.72 22.26 3.58 WT1(h) 91.91 14.32 68.28 0.74 18.41 1.29 WT2A(b) 86.82 17.98 62.78 0.72 20.92 1.16 WT2B(b) 97.88 13.09 72.17 0.74 16.63 1.27 WT4A(b) 103.26 4.33 103.55 1.00 12.95 2.99 WT7_M12 100.34 4.60 103.11 1.03 13.30 2.89 WT7_M16 132.34 11.47 113.32 0.86 18.48 1.61 WT7_M20 145.72 9.12 114.05 0.78 17.92 1.96 WT51(b) 97.08 3.96 96.51 0.99 7.87 1.99 WT53C 98.90 4.24 98.89 1.00 6.40 1.51 WT53D 117.36 5.54 100.08 0.85 6.34 1.14 WT53E 115.04 5.26 98.25 0.85 6.36 1.21 WT57_M12 121.87 4.33 120.69 0.99 6.80 1.57 WT57_M16 173.64 5.88 168.21 0.97 6.34 1.08 WT57_M20 241.71 15.98 167.36 0.69 6.17 0.39


0.14

0.49

b) Methodology recommended by Faella, Piluso and Rizzano To illustrate the methodology proposed by Faella et al. [6.3-6.5], six examples were selected: T1, P16, P18, WT4A, WT7_M20 and WT51. These examples typify the three different failure modes (Type-11, -13 and -23) as well as the two assembly types (HR- and WP-T-stubs). In the framework of this method-ology, however, the potential collapse mode is defined differently. As already mentioned in §6.2.2, Faella et al. assume the occurrence of three alternative collapse modes, termed: (i) type-1 if cracking of the flange material occurs at the two critical sections, at the flange-to-web connection, (1) and at the bolt line, (ii) type-2 if cracking of the flange material occurs at critical section (1) and, simultaneously, bolt fracture also takes place and (iii) type-3 if a bolt frac-ture mechanism arises. The boundaries for the occurrence of a given mecha-nism are indicated in Fig. 6.2. The coefficient βu, which is defined in Eq. (2.1), with Mf.u given by Eq. (2.5), is compared with βu.lim. Faella et al. suggest two alternative expressions for βu.lim:


227

Table D.7 Prediction of ultimate resistance and deformation capacity by us-ing the actual strain hardening modulus of the flange material and the formulation accounting for the bolt for computation of resis-tance (HR-T-stubs).

Numerical results Jaspart methodology (Actual Eh; FBA) Fmax ∆u.0 Fu.0 ∆u.0

Test ID


(mm) Ratio



0.11

0.34

.lim2

2 1λβ

λ=

+u (D.1)

in [6.3] and later [6.4]:

( ).lim2 1 1

2 1 8w

udn

λβ λλ

= − + + (D.2)

which is the same as in Eq. (2.2). Table D.10 sets out the predictions of the failure modes by using the two above expressions. For further comparisons, reference is made to Eq. (D.2) – last column in Table D.10. Specimen T1 is the only case where a change in the collapse mode is observed. For compari-


228

Table D.7 Prediction of ultimate resistance and deformation capacity (WP-T-stubs) (cont.).

Num. or Exp. re-sults

Jaspart methodology (Actual Eh; FBA)

Fmax ∆u.0 Fu.0 ∆u.0

Test ID


(mm) Ratio

Weld_T1(i) 92.02 10.85 84.35 0.92 36.31 3.35 Weld_T1(ii) 102.75 8.01 95.61 0.93 30.52 3.81 Weld_T1(iii) 113.10 6.22 104.58 0.92 24.03 3.86 WT1(h) 91.91 14.32 78.79 0.86 21.24 1.48 WT2A(b) 86.82 17.98 71.97 0.83 23.98 1.33 WT2B(b) 97.88 13.09 83.84 0.86 19.32 1.48 WT4A(b) 103.26 4.33 103.55 1.00 9.73 2.25 WT7_M12 100.34 4.60 103.11 1.03 9.88 2.15 WT7_M16 132.34 11.47 135.81 1.03 22.15 1.93 WT7_M20 145.72 9.12 142.39 0.98 22.37 2.45 WT51(b) 97.08 3.96 98.21 1.01 9.54 2.41 WT53C 98.90 4.24 102.51 1.04 9.66 2.28 WT53D 117.36 5.54 115.33 0.98 8.15 1.47 WT53E 115.04 5.26 113.19 0.98 7.33 1.39 WT57_M12 121.87 4.33 120.69 0.99 6.80 1.57 WT57_M16 173.64 5.88 179.87 1.04 9.92 1.69 WT57_M20 241.71 15.98 208.15 0.86 7.67 0.48


0.07

0.44

son, the table also includes the critical modes for the same examples according to the author (second column) and Jaspart (third and fourth columns). The application of the method proposed by these authors requires, in the first place, the approximation of the steel flange constitutive law by a quad-rilinear relationship. The actual material properties for the example specimens were defined in terms of a piecewise σ-ε law earlier in Chapters 3 and 4. Fig. D.1 shows those laws in terms of natural coordinates and the corresponding quadrilinear approximations. Next, the F-∆ response may be fully characterized. In terms of ultimate conditions, the predictions of resistance and deformation capacity are summa-rized and compared with the actual results for the various specimens in Table D.11. These results are obtained by application of the basic formulation for evaluation of the resistance of specimens failing according to a type-1 mecha-nism. For those specimens whose failure mode is of type-2, the bolt is also subjected to plasticity. In this table, the bolt plastic deformations, δb.p.u, are evaluated. It should be noted that, so far, the compatibility requirements be-tween flange and bolt deformations have been disregarded. For specimen T1,


229

Table D.8 Prediction of ultimate resistance and deformation capacity by us-ing the nominal strain hardening modulus of the flange material and the basic formulation for computation of resistance.

Numerical results Jaspart methodology (Nominal Eh; BF) Fmax ∆u.0 Fu.0 ∆u.0

Test ID


(mm) Ratio



Weld_T1(i) 92.02 10.85 72.79 0.79 13.06 1.20 Weld_T1(ii) 102.75 8.01 81.54 0.79 10.85 1.35 Weld_T1(iii) 113.10 6.22 90.42 0.80 9.28 1.49 WT1(h) 91.91 14.32 68.28 0.74 9.29 0.65 WT2A(b) 86.82 17.98 62.78 0.72 10.56 0.59 WT2B(b) 97.88 13.09 72.17 0.74 8.40 0.64 WT4A(b) 103.26 4.33 103.55 1.00 6.62 1.53 WT7_M12 100.34 4.60 103.11 1.03 6.80 1.48 WT7_M16 132.34 11.47 113.32 0.86 9.33 0.81 WT7_M20 145.72 9.12 114.05 0.78 9.05 0.99


0.13

0.36


230

Table D.9 Prediction of ultimate resistance and deformation capacity by us-ing the nominal strain hardening modulus of the flange material and the formulation accounting for the bolt for computation of re-sistance.

Numerical results Jaspart methodology (Nominal Eh; FBA) Fmax ∆u.0 Fu.0 ∆u.0

Test ID


(mm) Ratio


Average 0.96 0.91 Coefficient of variation 0.11

0.35

Weld_T1(i) 92.02 10.85 84.35 0.92 15.14 1.39 Weld_T1(ii) 102.75 8.01 95.61 0.93 12.73 1.59 Weld_T1(iii) 113.10 6.22 104.58 0.92 10.05 1.61 WT1(h) 91.91 14.32 78.79 0.86 10.73 0.75 WT2A(b) 86.82 17.98 71.97 0.83 12.11 0.67 WT2B(b) 97.88 13.09 83.84 0.86 9.75 0.75 WT4A(b) 103.26 4.33 103.55 1.00 5.05 1.17 WT7_M12 100.34 4.60 103.11 1.03 5.13 1.12 WT7_M16 132.34 11.47 135.81 1.03 11.18 0.98 WT7_M20 145.72 9.12 142.39 0.98 11.29 1.24


0.08

0.30


231

0

150

300

450

600

750

900

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Strain

Stre

ss (M

Pa)

Actual piecewise true law

Quadrilinear true law

(a) Flange steel grade S355 (fy.f = 430 MPa) for specimens T1, P16 and P18.

0

150

300

450

600

750

900

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Strain

Stre

ss (M

Pa)



(b) Flange steel grade S355 (fy.f = 340 MPa) for specimens WT4A and WT7_M20.

0

150

300

450

600

750

900

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Strain

Stre

ss (M

Pa)



(c) Flange steel grade S690 (fy.f = 698 MPa) for specimen WT51. Fig. D.1 Quadrilinear approximation of the actual piecewise flange material

law for application of the method recommended by Faella, Piluso and Rizzano.


232

failing according to a type-2 mechanism, the application of the formulae pro-vided by Faella et al. yields a negative bolt plastic deformation, which has no physical meaning. If now the formulation accounting for the bolt action for type-1 is consid-ered, the ultimate resistance prediction improves (specimens P18 and WT7_M20) as well as the deformation capacity – Table D.12. In this table, the results for specimen T1 are computed by assuming that mode 1 governs col-lapse, as ascertained by Eq. (D.1). The results are not so different from those in Table D.11. With respect to the remaining specimens illustrating type-2 col-lapse mode, in Table D.12 the compatibility requirements between bolt and flange deformations are accounted for. The actual bolt deformation at fracture is known. If this value is imposed, a new value for the overall T-stub deforma-tion can be calculated by means of linear interpolation. By doing so, the predic-tions improve. So far, the actual material properties for the flange have been employed. As before, the nominal properties for the strain hardening range are also taken into consideration (Table D.13). For this analysis, specimen WT51 that uses S690 is Table D.10 Prediction of failure modes according to Faella and co-authors.

Crit. resistance formula according to Jaspart

Pot. failure type ac-cording to Faella et al.

Test ID Potential failure type

Plastic Ultimate Eq. (D.1) Eq. (D.2) T1 13 1 1 or 2 2 1 P16 23 2 2 2 2 P18 11 1 1 1 1 WT4A 13 1 or 2 2 2 2 WT7_M20 11 1 1 1 1 WT51 23 2 1 or 2 2 2 Table D.11 Prediction of ultimate resistance and deformation capacity by us-

ing the basic formulation for computation of resistance and ne-glecting the compatibility requirements between flange and bolt deformations.

Num. or Exp. results Faella et al. methodology Fmax ∆u Fu.0 ∆u.0 δb.p.u

Test ID


(mm) Ratio

(mm) T1 103.99 8.70 109.17 1.05 9.64 1.11 -12.79 P16 125.69 3.06 128.89 1.03 26.18 8.56 12.82 P18 266.57 26.07 106.19 0.40 9.86 0.38 0.00 WT4A 103.26 4.33 108.07 1.05 25.32 5.85 11.14 WT7_M20 145.72 9.12 130.96 0.90 12.15 1.33 0.00 WT51 97.08 3.96 100.90 1.04 6.46 1.63 1.27


233

Table D.12 Prediction of ultimate resistance and deformation capacity by us-ing the formulation accounting for bolt finite size for computation of resistance of specimens failing according to mode 1 and cater-ing for compatibility requirements between flange and bolt defor-mations (specimens from type-2 collapse mode).

Num. or Exp. results Faella et al. methodology Fmax ∆u.0 δb.p.fract Fu.0 ∆u.0

Test ID

(kN) (mm) (mm) (kN) Ratio

(mm) Ratio

T1 103.99 8.70 0.87 126.39 1.22 10.19 1.17 P16 125.69 3.06 0.87 83.24 0.66 1.96 0.64 P18 266.57 26.07 142.28 0.53 10.24 0.39 WT4A 103.26 4.33 1.08 71.89 0.70 2.95 0.68 WT7_M20 145.72 9.12 163.51 1.12 12.45 1.37 WT51 97.08 3.96 1.08 98.99 1.02 5.71 1.44 Table D.13 Actual properties of the flange steel grade and nominal properties

according to Faella and co-authors [S355(1) is the steel grade from specimens T1, P16 and P18; S355(2) is the steel grade from specimens WT4A and WT7_M20].

fy E Eh Eu Steel (MPa) (GPa) (GPa) (GPa)

εy εh εm εu

S355(1) 430 208 1.74 0.67 0.0021 0.0200 0.1545 0.2872 S355(2) 340 210 2.15 0.48 0.0016 0.0150 0.2240 0.3610 S355 355 210 4.36 0.51 0.0017 0.0166 0.0521 0.8100 Table D.14 Prediction of ultimate resistance and deformation capacity with

nominal steel properties by using the formulation accounting for bolt finite size for computation of resistance of specimens failing according to mode 1.

Fmax ∆u.0 δb.p.u Test ID Potential failure type (kN) (mm) (mm)

T1 2 112.08 28.23 -10.60 P16 2 64.83 1.70 0.87 P18 1 158.05 27.38 WT4A 2 60.60 2.53 1.08 WT7_M20 1 224.79 37.21 neglected. The results for this new approach are given in Table D.14 (bolt and flange compatibility is also accounted for). Again, for specimen T1 the results are poor. For the remaining cases, the results do not improve considerably.


234

c) Methodology recommended by Beg, Zupančič and Vayas for evaluation of the deformation capacity Beg and co-authors proposed simple formulae for the assessment of the defor-mation capacity of single T-stubs in [6.7]. For the examples under analysis, Table D.15 sets out the predictions of their proposals. For the specimens failing according to a type-2 plastic mechanism, a value of k = 3.5 is assumed (see Eq. (6.25)). In general, the predictions are not satisfactory. For those specimens failing according to a type-2 plastic mode, the predictions improve, but for the remaining cases the deviations are not acceptable. Table D.15 Prediction of deformation capacity by means of Beg et al. method-

ology. Num. or exp. re-

sults Beg et al. methodology

∆u.0 ∆u.0

Test ID

(mm) Potential plastic

failure mode (mm) Ratio

T1 8.70 1 23.56 2.71 P1 10.77 1 27.56 2.56 P2 6.18 1 19.56 3.17 P3 10.17 1 23.56 2.32 P4 4.68 1 23.56 5.03 P5 3.63 2 4.30 1.18 P6 10.06 1 23.56 2.34 P7 7.56 1 23.56 3.12 P8 8.08 1 21.68 2.68 P9 3.31 2 5.16 1.56 P10 32.75 1 24.60 0.75 P11 2.94 2 5.24 1.78 P12 24.22 1 23.56 0.97 P13 11.38 1 23.56 2.07 P14 24.15 1 23.56 0.98 P15 18.02 1 19.56 1.09 P16 3.06 2 4.32 1.41 P17 9.29 1 23.56 2.54 P18 26.07 1 23.56 0.90 P19 9.29 1 23.56 2.54 P20 4.06 2 5.95 1.46 P21 17.67 1 23.56 1.33 P22 6.40 1 25.87 4.04 P23 5.22 2 9.27 1.78

Average 2.10 Coefficient of variation

0.50


235

Table D.15 Prediction of deformation capacity by means of Beg et al. method-ology (cont.).

Num. or exp. re-sults

Beg et al. methodology

∆u.0 ∆u.0

Test ID

(mm) Potential plastic

failure mode (mm) Ratio

Weld_T1(i) 10.85 1 29.95 2.76 Weld_T1(ii) 8.01 1 26.73 3.34 Weld_T1(iii) 6.22 1 24.11 3.87 WT1 14.32 1 26.98 1.88 WT2A 17.98 1 29.03 1.61 WT2B 13.09 1 25.35 1.94 WT4A 4.33 1 26.96 6.23 WT51 3.96 2 5.67 1.43 WT53C 4.24 1 27.47 6.48 WT53D 5.54 1 27.39 4.94 WT53E 5.26 1 27.41 5.21 WT7_M12 4.60 1 27.10 5.89 WT7_M16 11.47 1 27.11 2.36 WT7_M20 9.12 1 27.05 2.97 WT57_M12 4.33 2 12.88 2.97 WT57_M16 5.88 1 27.40 4.66 WT57_M20 15.98 1 27.41 1.72

Average 3.55 Coefficient of variation

0.48

D.3 Application of the proposed model: results for HR-T-stub T1 Specimen T1 was selected for illustration of the results obtained when the pro-posed model was applied. The geometrical and mechanical characteristics for this connection are indicated in Tables D.1-D.2. First, the overall F-∆ response is shown in Fig. D.2. The actual behaviour, obtained from the three-dimensional numerical model, is plotted against the simplified response from the proposed beam model. Both curves fit well though the simplified curve yields larger ductility than the real behaviour. This graph also traces the bilinear approximation of Jaspart. For this approximation, the FBA was employed for resistance computation (this specimen fails accord-ing to a failure type-13). The actual value for the strain hardening modulus was used. This response deviates significantly from the real behaviour. The predic-tions from Faella and co-workers are also included (results from Table D.11). Fig. D.3 shows the bolt response. The trilinear model fits the numerical results well. The prying force is plotted against the total flange deformation in Fig. D.4 for both approaches. The two models yield different responses. The


236

beam model gives lower results for the prying force. The ratios B/F and Q/F are shown in Fig. D.5. Figs. D.6-D.9 trace the beam diagrams of bending mo-ment, flange deformation, flange rotation and plastic strain, respectively. Four load levels were chosen: (i) F = 36.0 kN, corresponding to yielding of the flange at the flange-to-web connection; (ii) F = 60.0 kN, corresponding to first yielding at the bolt axis (the section at the flange-to-web connection is not en-gaged in the strain hardening domain yet); (iii) F = 88.9 kN, corresponding to first yielding of the bolt and (iv) F = 114.5 kN, corresponding to fracture of the bolt. Fig. D.6 shows that, at ultimate conditions, the bending moments acting at sections (1) and (2) are similar. Again, the plastic deformation of the flange is restricted to an area close to the critical sections, as shown in Figs. D.8-D.9. Fig. D.10 traces the variation of the applied load with the parameter L/m.

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual responseSimplified response (Beam model)Bilinear approximation (Jaspart)Quadrilinear approximation (Faella and co-authors)

Fig. D.2 Specimen T1: force-deformation behaviour as ascertained by the

different approaches.

010

2030

4050

6070

8090

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bolt elongation, δ b (mm)

Bol

t for

ce, B

(kN

)

"Actual" response (3-dim. FE model)


Fig. D.3 Specimen T1: bolt elongation behaviour.


237

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18


Pryi

ng fo

rce,

Q (k

N)

"Actual" response (3-dim. FE model)


Fig. D.4 Specimen T1: prying force behaviour.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 15 30 45 60 75 90 105 120

Load, F (kN)

Rat

io B

/F, Q

/F

B/F (3-dim. FE model) Q/F (3-dim. FE model)B/F (Beam model) Q/F (Beam model)

Fig. D.5 Specimen T1: ratio B/F and Q/F.

-1.0E+06-8.0E+05-6.0E+05-4.0E+05-2.0E+050.0E+002.0E+054.0E+056.0E+058.0E+051.0E+06

0 5 10 15 20 25 30 35 40 45 50 55 60

Beam length (mm)

Mz (

Nm

m)

F = 36.0 kN F = 60.0 kNF = 88.9 kN F = 114.5 kN

Mp

Mp

Fig. D.6 Specimen T1: flange moment diagram.


238

01

23

45

67

89

0 5 10 15 20 25 30 35 40 45 50 55 60

Beam length (mm)

∆/2

(mm

)

F = 36.0 kN F = 60.0 kNF = 88.9 kN F = 114.5 kN

Fig. D.7 Specimen T1: flange (half-) deformation diagram.

-0.33-0.30-0.27-0.24-0.21-0.18-0.15-0.12-0.09-0.06-0.030.000.03

0 5 10 15 20 25 30 35 40 45 50 55 60

Beam length (mm)

θz (r

ad)

F = 36.0 kNF = 60.0 kNF = 88.9 kNF = 114.5 kN

Fig. D.8 Specimen T1: flange rotation diagram.

-0.5-0.4-0.3-0.2-0.10.00.10.20.30.40.5

0 5 10 15 20 25 30 35 40 45 50 55 60

Beam length (mm)

Plas

tic st

rain

F = 36.0 kNF = 60.0 kNF = 88.9 kNF = 114.5 kN

εp.u

εp.u

Fig. D.9 Specimen T1: flange plastic strain diagram.


239

0

15

30

45

60

75

90

105

120

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

L/m

Loa

d, F

(kN

)

Fig. D.10 Specimen T1: length of the equivalent cantilever. D.4 Application of the proposed model: results for WP-T-stub WT1 Specimen WT1 was also selected for further detail of the results extracted from the proposed model. The geometrical and mechanical characteristics for this connection are indicated in Tables D.1-D.2, as well. Identical results to the above are shown in this section. Fig. D.11 plots the alternative predictions for the deformation behaviour and compares those with the actual test (experimental) results. The agreement between the Faella et al. quadrilinear approximation and the real response is very good. As for the beam model and the simple approximation of Jaspart, the results are good but clearly underestimate the resistance predictions. The ductility is also overestimated. Figs. D.12-D.14 show the bolt response and the prying behaviour as ascer-tained by the beam model. No comparisons are established with the test results

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual response (WT1h)Simplified response (Beam model)Bilinear approximation (Jaspart)Quadrilinear approximation (Faella and co-authors)

Fig. D.11 Specimen WT1: force-deformation behaviour as ascertained by the

different approaches.


240

010

2030

4050

6070

8090

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Bolt elongation, δ b (mm)

Bol

t for

ce, B

(kN

)

Fig. D.12 Specimen WT1: bolt elongation behaviour.

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16 18 20


Pryi

ng fo

rce,

Q (k

N)

Fig. D.13 Specimen WT1: prying force behaviour.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 10 20 30 40 50 60 70 80 90 100

Load, F (kN)

Rat

io B

/F, Q

/F

B/F (Beam model) Q/F (Beam model)

Fig. D.14 Specimen WT1: ratio B/F and Q/F.


241

since these values were not determined experimentally. Fig. D.15-D.18 give results for the flange bending moment, flange defor-mation, flange rotation and plastic strain at four different load levels: (i) F = 26.0 kN, corresponding to yielding of the flange at the flange-to-web connec-tion, (ii) F = 42.0 kN, corresponding to first yielding at the bolt axis, (iii) F = 78.0 kN, corresponding to first yielding of the bolt and cracking of the flange material at section (1) and (iv) F = 85.6 kN, corresponding to cracking of the flange material at section (1*). Finally, Fig. D.19 shows the evolution of the non-dimensional parameter L/m with increasing loading. D.5 Prediction of the nonlinear response of the above connections using

the nominal stress-strain characteristics If the nominal mechanical properties of the bolt and flange plates are input, the

-8.0E+05

-6.0E+05

-4.0E+05

-2.0E+05

0.0E+00

2.0E+05

4.0E+05

6.0E+05

8.0E+05

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Beam length (mm)

Mz (

Nm

m)

F = 26.0 kN F = 42.0 kNF = 78.0 kN F = 85.6 kN

Mp

Mp

Fig. D.15 Specimen WT1: flange moment diagram.

0123456789

10

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Beam length (mm)

∆/2

(mm

)

F = 26.0 kN F = 42.0 kNF = 78.0 kN F = 85.6 kN

Fig. D.16 Specimen WT1: flange gap diagram.


242

-0.33-0.30-0.27-0.24-0.21-0.18-0.15-0.12-0.09-0.06-0.030.000.03

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Beam length (mm)

θz (m

m)

F = 26.0 kNF = 42.0 kNF = 78.0 kNF = 85.6 kN

Fig. D.17 Specimen WT1: flange rotation diagram.

-0.7

-0.5

-0.3

-0.1

0.1

0.3

0.5

0.7

0 5 10 15 20 25 30 35 40 45 50 55 60 65

Beam length (mm)

Plas

tic st

rain

F = 26.0 kNF = 42.0 kNF = 78.0 kNF = 85.6 kN

εp.u

εp.u

Fig. D.18 Specimen WT1: flange plastic strain diagram.

0

15

30

45

60

75

90

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

L/m

Loa

d, F

(kN

)

Fig. D.19 Specimen WT1: length of the equivalent cantilever.


243

Table D.16 Prediction of the failure modes.

Predicted poten-tial failure

mode.

Test ID Actual deter-mining frac-ture element

Mu Eq. (2.4)

Mu Eq. (2.5)

Determining fracture ele-ment in the beam model (nominal mech. Proper-

ties)

T1 Bolt 13 13 Flange, at (1*) P1 Bolt 13 13 Flange, at (1*) P2 Bolt 13 13 Bolt P3 Bolt 13 13 Flange, at (1*) P4 Bolt 13 13 Bolt P5 Bolt 23 23 Bolt P6 Bolt 13 13 Flange, at (1*) P7 Bolt 13 13 Flange, at (1*) P8 Bolt 13 13 Flange, at (1*) P9 Bolt 23 23 Bolt P10 Flange 11 11 Flange, at (1*) P11 Bolt 23 23 Bolt P12 Flange 11 11 Flange, at (1*) P13 Bolt 13 13 Flange, at (1*) P14 Flange 11 11 Flange, at (1*) P15 Flange 11 11 Flange, at (1*) P16 Bolt 23 23 Bolt P17 Bolt 13 13 Flange, at (1*) P18 Flange 11 11 Flange, at (1*) P19 Bolt 13 13 Flange, at (1*) P20 Bolt 23 23 Bolt P21 Bolt 13 13 Flange, at (1*) P22 Bolt 13 13 Flange, at (1*) P23 Bolt 23 23 Bolt Weld_T1(i) Bolt 11 11 Flange, at (1*) Weld_T1(ii) Bolt 13 13 Flange, at (1*) Weld_T1(iii) Bolt 13 13 Flange, at (1*)

WT1 Bolt and flange 11 13 Flange, at (1*)

WT2A Bolt and flange 11 13 Flange, at (1*)

WT2B Bolt and flange 13 13 Flange, at (1*)

WT4A Bolt 23 23 Bolt WT7_M12 Bolt 23 23 Bolt WT7_M16 Flange 11 13 Flange, at (1*) WT7_M20 Flange 11 11 Flange, at (1*)


244

predicted failure modes may slightly change (Tables 6.12 and D.16 – see T- stubs T1, P1, P6-P8, P22 and Weld_T1(iii)). The responses for this new ap-proach are traced in Figs. D.20-D.53. These graphs also trace the actual re-sponse as well as the prediction by using the actual material properties. For most specimens whose failure is determined by the bolt (black circle), the pre-dictions of deformation capacity improve. If the flange governs ultimate col-lapse (black square in the graphs), then the maximum deformation decreases since the nominal ultimate strain is lower than the actual value. In terms of strength, the ultimate resistance is lower now when compared to the actual properties. However, it can be seen from the graphs that the (nominal) ultimate resistance is identical to the value predicted by the actual mechanical properties if the bolt is determinant. These new predictions at ultimate conditions are summarized in Table D.17 for the various specimens. From a design point of view, and on average, the

0

15

30

45

60

75

90

105

120

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 16.5 18.0


Loa

d, F

(kN

)

Actual response

Simplified response (Simple beam model)

Simplified response with nominal properties

Fig. D.20 Specimen T1: force-deformation behaviour (nominal properties).

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual response



Fig. D.21 Specimen P1: force-deformation behaviour (nominal properties).


245

015

3045

6075

90105

120135

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15


Loa

d, F

(kN

)

Actual response




0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response




015

3045

6075

90105

120135

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0


Loa

d, F

(kN

)

Actual response





246

015

3045

6075

90105

120135

0 1 2 3 4 5 6 7 8 9 10 11 12


Loa

d, F

(kN

)

Actual response




0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual response




015

3045

6075

90105

120135

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response





247

0

1530

4560

7590

105120

135

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response




0

10

20

30

40

50

60

70

80

0 3 6 9 12 15 18 21 24 27 30 33 36


Loa

d, F

(kN

)

Actual responseSimplified response (Simple beam model)Simplified response with nominal properties



248

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response




0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Loa

d, F

(kN

)

Actual response




0

1530

4560

7590

105120

135

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response





249

0

1020

3040

5060

7080

90

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Loa

d, F

(kN

)

Actual response




020

4060

80100

120140

160180

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual responseSimplified response (Simple beam model)Simplified response with nominal properties


0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response





250

0

30

60

90

120

150

180

210

240

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response




0

40

80

120

160

200

240

280

0 3 6 9 12 15 18 21 24 27 30


Loa

d, F

(kN

)

Actual response




0

30

60

90

120

150

180

210

240

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response





251

030

6090

120150

180210

240270

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response




0

40

80

120

160

200

240

280

320

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response




0

50

100

150

200

250

300

350

400

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14


Loa

d, F

(kN

)

Actual response





252

0

50100

150200

250300

350400

450

0 1 2 3 4 5 6 7 8 9 10 11 12


Loa

d, F

(kN

)

Actual response




0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.44 Specimen Weld_T1(i): force-deformation behaviour (nominal prop-

erties).

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.45 Specimen Weld_T1(ii): force-deformation behaviour (nominal prop-

erties).


253

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.46 Specimen Weld_T1(iii): force-deformation behaviour (nominal

properties).

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.47 Specimen WT1: force-deformation behaviour (nominal properties).

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.48 Specimen WT2A: force-deformation behaviour (nominal properties).


254

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.49 Specimen WT2B: force-deformation behaviour (nominal properties).

015

3045

6075

90105

120135

0 1 2 3 4 5 6 7 8 9 10 11 12


Loa

d, F

(kN

)

Actual response



Fig. D.50 Specimen WT4A: force-deformation behaviour (nominal properties).

015

3045

6075

90105

120135

0 1 2 3 4 5 6 7 8 9 10 11 12


Loa

d, F

(kN

)

Actual response



Fig. D.51 Specimen WT7_M12: force-deformation behaviour (nominal prop-

erties).


255

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




erties).

0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




erties). predictions for resistance are very good (ratios of 0.96 approximately) but they overestimate the deformation capacity. In addition, the predicted failure type does not always correspond to the actual mode. The reason for the good resis-tance predictions derives from the mechanical σ-ε law, which differs signifi-cantly at yielding conditions from the nominal law but approximates it at ulti-mate conditions. D.6 Comparative graphs: simple beam model and sophisticated beam

model accounting for the bolt action Figs. D.54-D.75 compare the actual response of the several T-stubs with the


256

Table D.17 Prediction of deformation capacity and ultimate resistance (nomi-nal properties of steel).



T1 13 103.99 8.70 104.13 1.00 16.92 1.94 P1 13 91.76 10.77 87.94 0.96 19.30 1.79 P2 13 116.72 6.18 126.50 1.08 14.30 2.31 P3 13 95.41 10.17 91.54 0.96 16.34 1.61 P4 13 115.97 4.68 124.12 1.07 12.42 2.65 P5 23 130.20 3.63 129.04 0.99 9.19 2.53 P6 13 95.53 10.06 91.54 0.96 16.34 1.62 P7 13 111.34 7.56 117.31 1.05 18.03 2.38 P8 13 112.71 8.08 120.92 1.07 16.48 2.04 P9 23 131.43 3.31 144.71 1.10 8.04 2.43 P10 11 76.79 32.75 42.69 0.56 24.44 0.75 P11 23 121.15 2.94 138.24 1.14 6.87 2.34 P12 11 154.06 24.22 104.58 0.68 15.29 0.63 P13 13 93.71 11.38 104.13 1.11 16.92 1.49 P14 11 86.57 24.15 86.88 1.00 17.12 0.71 P15 11 171.08 18.02 130.79 0.76 13.75 0.76 P16 23 125.69 3.06 131.06 1.04 5.99 1.96 P17 13 192.01 9.29 183.01 0.95 17.06 1.84 P18 11 266.57 26.07 184.10 0.69 15.83 0.61 P19 13 186.52 9.29 183.01 0.98 17.06 1.84 P20 23 225.94 4.06 255.70 1.13 8.28 2.04 P21 13 281.33 17.67 243.84 0.87 16.60 0.94 P22 13 305.21 6.40 317.41 1.04 14.35 2.24 P23 23 346.01 5.22 427.41 1.24 10.59 2.03


Weld_T1(i) 11 92.02 10.85 70.00 0.76 9.84 0.91 Weld_T1(ii) 13 102.75 8.01 84.42 0.82 12.41 1.55 Weld_T1(iii) 13 113.10 6.22 101.77 0.90 17.43 2.80 WT1 11 91.91 14.32 84.39 0.92 10.74 0.75 WT2A 11 86.82 17.98 75.00 0.86 9.93 0.55 WT2B 13 97.88 13.09 93.44 0.95 12.63 0.96 WT4A 23 103.26 4.33 112.20 1.09 5.12 1.18 WT7_M12 23 100.34 4.60 111.31 1.11 5.13 1.11 WT7_M16 11 132.34 11.47 140.10 1.06 10.34 0.90 WT7_M20 11 145.72 9.12 140.33 0.96 10.25 1.12



257

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response


Simplified response accounting for the bolt action

Fig. D.54 Specimen T1.

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual response



Fig. D.55 Specimen P1.

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response





258

015

3045

6075

90105

120135

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0


Loa

d, F

(kN

)

Actual response

Simplified response (Simplebeam model)Simplified response accountingfor the bolt action


0

10

20

30

40

50

60

70

80

0 3 6 9 12 15 18 21 24 27 30 33 36


Loa

d, F

(kN

)

Actual responseSimplified response (Simple beam model)Simplified response accounting for the bolt action


0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Loa

d, F

(kN

)

Actual response





259

010

2030

4050

6070

8090

0 2 4 6 8 10 12 14 16 18 20 22 24 26


Loa

d, F

(kN

)

Actual response




0

30

60

90

120

150

180

210

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




0

50

100

150

200

250

300

0 3 6 9 12 15 18 21 24 27 30


Loa

d, F

(kN

)

Actual responseSimplified response (Simple beam model)Simplified response accounting for the bolt action



260

030

6090

120150

180210

240270

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0


Loa

d, F

(kN

)

Actual response



050

100150

200250

300350

400450

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0


Loa

d, F

(kN

)

Actual response


Fig. D.64 Specimen 23.

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.65 Specimen Weld_T1(i).


261

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.66 Specimen Weld_T1(ii).

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.67 Specimen Weld_T1(iii).

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.68 Specimen WT1.


262

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.69 Specimen WT2A.

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.70 Specimen WT2B.

0

15

30

45

60

75

90

105

120

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response




263

0

25

50

75

100

125

150

175

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.72 Specimen WT7_M16.

0

25

50

75

100

125

150

175

200

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




0

30

60

90

120

150

180

210

240

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response




264

030

6090

120150

180210

240270

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual response



Fig. D.75 Specimen WT57_M20. predictions from the simplified approach (bolt modelled as a single extensional spring) and with a sophistication of the beam model. This sophistication con-sists in assuming that the bolt effect can be reproduced with a set of extensional springs along a certain length here taken as the bolt diameter. The figures clearly show that such sophistication improved the agreement of the results with the actual predictions, in most cases. Table D.18 gives the actual results for resistance and deformation capacity and compare them with the aforemen-tioned two-dimensional approaches. Four alternative ultimate conditions are imposed for the sophisticated approach. The determinant conditions however are either fracture of the bolt at mid-section or cracking of the material at sec-tion (1*) – values in bold. D.7 Comparative graphs: influence of the distance m for the WP-T-stubs The influence of the geometrical parameter m is evident in the graphs from Figs. D.76-D.89. These graphs compare the actual response with the beam model predictions and highlight the effect of m: increase on resistance and stiffness (see also Table D.19) and decrease on ductility. Tables 6.13 and D.20 set out the predictions of deformation capacity and show that for the original distance there is an average ratio between experiments and analytical predic-tions of 1.81 (coefficient of variation of 0.34). If the “new” m is adopted, the average ratio drops to 1.02 with a coefficient of variation of 0.47. In general, for WP-T-stubs, the new value of m gives a better agreement with the experi-ments, particularly for specimens made up of S355.


265

Table D.18 Comparison of the predicted values for ultimate resistance and deformation capacity by applying the simple beam model and the sophisticated beam model accounting for the bolt action.

Num. or Exp. results

Simple beam model pre-

dictions

Beam model accounting for bolt action

Fmax ∆u.0 Fmax ∆u.0 Fmax Ra-tio ∆u.0

Ra-tio

Test ID

(kN) (mm) (kN) (mm) (kN) (mm)

Ultimate condi-tions

98.21 0.94 6.35 0.73 Bolt 1/4 104.04 1.00 8.54 0.98 Flange (1) 113.87 1.09 12.74 1.46 Bolt 1/2 T1

103.99

8.70

114.45

16.76

129.79 1.25 20.69 2.38 Flange (1*) 85.78 0.93 8.79 0.82 Bolt 1/4 91.55 1.00 12.10 1.12 Flange (1)

103.27 1.13 19.88 1.85 Bolt 1/2 P1 91.76

10.77

103.25

25.34

109.08 1.19 24.23 2.25 Flange (1*) 90.54 0.95 7.42 0.73 Bolt 1/4 94.37 0.99 9.11 0.90 Flange (1)

108.18 1.13 16.06 1.58 Bolt 1/2 P3 95.41

10.17

111.96

24.17

116.14 1.22 20.65 2.03 Flange (1*) 117.91 0.91 2.88 0.79 Bolt 1/4 121.71 0.93 3.37 0.93 Bolt 1/2 154.84 1.19 11.82 3.25 Flange (1) P5

130.20

3.63

123.76

4.63

188.42 1.45 23.75 6.54 Flange (1*) 48.36 0.63 12.87 0.39 Flange (1) 52.29 0.68 17.72 0.54 Bolt 1/4 59.17 0.77 27.79 0.85 Flange (1*) P1

0 76.79

32.75

50.25

32.40

112.97 1.47 134.63 4.11 Bolt 1/2 119.29 0.77 8.02 0.33 Flange (1) 125.30 0.81 10.00 0.41 Bolt 1/4 149.02 0.97 19.22 0.79 Flange (1*) P1

2 154.06

24.22

122.43

19.67


4 86.57

24.15

79.54

20.49


5 171.08

18.02

153.17

16.48

199.91 1.17 19.47 1.08 Bolt 1/2 217.87 0.82 7.42 0.28 Flange (1) 244.55 0.92 11.69 0.45 Bolt 1/4 P1

8 266.57

26.07

215.75

20.18

280.34 1.05 18.69 0.72 Flange (1*) 241.83 1.07 3.03 0.75 Bolt 1/4 P 20

225.94

4.06

246.14

4.03 247.76 1.10 3.51 0.87 Bolt 1/2

373.18 1.08 3.41 0.65 Bolt 1/4 P 23 346.01

5.22

408.62

5.24

399.89 1.16 4.00 0.77 Bolt 1/2


266

Table D.18 Comparison of the predicted values for ultimate resistance and deformation capacity by applying the simple beam model and the sophisticated beam model accounting for the bolt action (cont.).

Num. or Exp. results

Simple beam model pre-

dictions

Beam model accounting for bolt action

Fmax ∆u.0 Fmax ∆u.0 Fmax Ra-tio ∆u.0

Ra-tio

Test ID

(kN) (mm) (kN) (mm) (kN) (mm)

Ultimate condi-tions

80.57 0.88 11.11 1.02 Bolt 1/4 84.55 0.92 14.00 1.29 Flange (1) 88.57 0.96 17.17 1.58 Flange (1*) W

eld_

T1(i)

92.02

10.85

84.80

18.27

96.38 1.05 23.93 2.20 Bolt 1/2 88.34 0.86 8.39 1.05 Bolt 1/4 94.29 0.92 11.55 1.44 Flange (1)

104.18 1.01 17.61 2.20 Bolt 1/2 Wel

d_T1

(ii)

102.75

8.01

101.03

20.00

106.21 1.03 18.97 2.37 Flange (1*) 95.74 0.85 6.44 1.04 Bolt 1/4

103.47 0.91 9.57 1.54 Flange (1) 113.33 1.00 14.16 2.27 Bolt 1/2 W

eld_

T1(ii

i) 113.10

6.22

114.21

18.82

125.23 1.11 20.46 3.29 Flange (1*) 82.72 0.90 10.40 0.73 Flange (1) 84.56 0.92 11.53 0.81 Bolt 1/4 90.37 0.98 15.67 1.09 Flange (1*) W

T1

91.91

14.32

85.60

18.51

111.79 1.22 40.67 2.84 Bolt 1/2 77.64 0.89 12.88 0.72 Flange (1) 79.60 0.92 14.44 0.80 Bolt 1/4 81.50 0.94 16.11 0.90 Flange (1*) W

T2A

86.82

17.98

75.16

16.51

112.53 1.30 69.47 3.86 Bolt 1/2 87.31 0.89 9.42 0.52 Flange (1) 89.21 0.91 10.40 0.58 Bolt 1/4

102.67 1.05 19.83 1.10 Flange (1*) WT2

B

97.88

13.09

92.51

19.18

114.33 1.17 31.74 1.77 Bolt 1/2 104.11 1.07 3.80 0.96 Flange (1) 111.82 1.15 5.07 1.28 Bolt 1/4 119.08 1.23 9.00 2.27 Flange (1*) W

T51

97.08

3.96

112.86

9.43


T7

M16

132.34

11.47

140.79

17.78

176.09 1.33 25.98 2.26 Bolt 1/4

157.51 1.08 8.27 0.91 Flange (1)

WT7

M

20

145.72

9.12

141.17

17.87

177.93 1.22 14.71 1.61 Flange (1*) 183.04 1.05 3.49 0.59 Flange (1) 212.19 1.22 8.35 1.42 Flange (1*) W

T57_

M 16 173.64

5.88

196.62

9.13


T57_

M 20 241.71

15.98

196.77

8.36

310.09 1.28 45.68 2.86 Bolt 1/4


267

Table D.19 Prediction of axial stiffness by using the modified proposal for m.

Standard m Modified m Test ID Num./Exp. stiffness ke.0 Ratio ke.0 Ratio

Weld_T1(i) 73.77 52.36 0.71 59.00 0.80 Weld_T1(ii) 89.12 68.82 0.77 85.26 0.96 Weld_T1(iii) 107.29 87.68 0.82 119.32 1.11 WT1g/h (av.) 71.08 71.61 1.01 84.68 1.19 WT2Aa/b (av.) 61.83 58.97 0.95 66.00 1.07 WT2Ba/b (av.) 79.75 82.62 1.04 103.73 1.30 WT4Aa/b (av.) 86.96 115.86 1.33 136.22 1.57 WT7_M16 60.73 60.73 1.00 71.39 1.18 WT7_M20 64.23 62.12 0.97 72.98 1.14 WT51a/b (av.) 52.90 64.08 1.21 75.47 1.43 WT53C 116.09 117.27 1.01 138.40 1.19 WT53D 137.70 120.60 0.88 142.93 1.04 WT57_M12 85.78 99.95 1.17 116.81 1.36 WT57_M20 150.96 109.09 0.72 128.83 0.85


0.19

0.18

Table D.20 Prediction of ultimate resistance and deformation capacity (WP-T-

stubs) by using the modified proposal for m.

Num. or Exp. results Modified m Fmax ∆u.0 Fmax ∆u.0

Test ID


(mm) Ratio

Weld_T1(i) 92.02 10.85 83.42 0.91 14.57 1.34 Weld_T1(ii) 102.75 8.01 95.86 0.93 10.99 1.37 Weld_T1(iii) 113.10 6.22 108.05 0.96 8.46 1.36 WT1g/h (av.) 91.91 14.32 80.56 0.88 10.27 0.72 WT2Aa/b (av.) 86.82 17.98 74.12 0.85 13.23 0.74 WT2Ba/b (av.) 97.88 13.09 86.75 0.89 8.85 0.68 WT4Aa/b (av.) 103.26 4.33 127.24 1.23 9.36 2.16 WT7_M16 97.08 3.96 107.02 1.10 3.78 0.95 WT7_M20 98.90 4.24 108.12 1.09 3.49 0.82 WT51a/b (av.) 117.36 5.54 111.74 0.95 3.04 0.55 WT53C 132.34 11.47 134.63 1.02 10.74 0.94 WT53D 145.72 9.12 134.73 0.92 10.68 1.17 WT57_M12 121.87 4.33 162.92 1.34 5.38 1.24 WT57_M20 241.71 15.98 185.74 0.77 2.83 0.18


0.16

0.47


268

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response


Simplified response with new m

Fig. D.76 Specimen Weld_T1(i).

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.77 Specimen Weld_T1(ii).

0

15

30

45

60

75

90

105

120

0 2 4 6 8 10 12 14 16 18 20 22


Loa

d, F

(kN

)

Actual response



Fig. D.78 Specimen Weld_T1(iii).


269

0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




0

15

30

45

60

75

90

105

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response



Fig. D.81 Specimen WT2B.


270

015

3045

6075

90105

120135

0 1 2 3 4 5 6 7 8 9 10 11 12


Loa

d, F

(kN

)

Actual response




0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response




0

20

40

60

80

100

120

140

160

0 2 4 6 8 10 12 14 16 18 20


Loa

d, F

(kN

)

Actual response





271

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response




0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response



Fig. D.86 Specimen WT53C.

0

20

40

60

80

100

120

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response



Fig. D.87 Specimen WT53D.


272

0

30

60

90

120

150

180

210

0 1 2 3 4 5 6 7 8 9 10


Loa

d, F

(kN

)

Actual response




0

40

80

120

160

200

240

280

0 2 4 6 8 10 12 14 16 18 20 22 24


Loa

d, F

(kN

)

Actual response




273

PART III: MONOTONIC BEHAVIOUR OF BEAM-TO-COLUMN

BOLTED END PLATE CONNECTIONS

275

7 EXPERIMENTAL TESTS ON BOLTED END PLATE CONNEC-

TIONS 7.1 INTRODUCTION An experimental investigation of eight statically loaded extended end plate moment connections undertaken at the Delft University of Technology is de-scribed in this chapter. It provides a better understanding of the behaviour of this joint type up to collapse and complements the study on welded T-stubs re-ported in Chapter 3. The specimens were designed to confine failure to the end plate and/or bolts without development of the full plastic moment capacity of the beam (partial strength joint). The parameters investigated were the end plate thick-ness and steel grade. The main objective was the analysis of the ultimate be-haviour of the components end plate in bending and bolts and eventually the proposal of sound design rules for this elemental part within the framework of the so-called component method. The description of this test programme and results is given below. Comparisons with the code predictions [7.1] are also drawn. 7.2 DESCRIPTION OF THE TEST PROGRAMME 7.2.1 Test details The experimental programme essentially comprised four test details (two specimens for each testing type) on the above joint configuration. Two main parameters were varied in the four sets: the end plate thickness, tp and the end plate steel grade. The specimens were fabricated from one column/beam set, as detailed in Table 7.1. The steel grade specified for the beams was S355. Unfor-tunately, due to a laboratory misunderstanding, steel grade S235 was ordered instead. This brought a problem in terms of the beam resistance that was natu-rally lower than expected. Therefore, for the critical cases, the beam flanges were stiffened with continuous plates in order to increase the beam flange thickness and minimize the chance of premature failure. End plates were con-nected to the beam-ends by full strength 45º-continuous fillet welds. The fillet welds were done in the shop in a down-hand position. The procedure involved manual metal arc welding in which consumable electrodes were used. Basic, soft, low hydrogen electrodes were used in the process. Hand tightened full-threaded M20 grade 8.8 bolts in 22 mm diameter drilled holes were employed in all sets. Two different batches of bolts were employed. The first batch of

Monotonic behaviour of beam-to-column bolted end plate connections

276

bolts were employed in tests FS1a-b, FS2a-b and FS3a in both tension and compression zones. The second batch of bolts were used to fasten the end plate and the beam in the tension zone in the remaining tests. The geometry of the specimens is depicted in Figs. 7.1-7.2. The column had a section profile HE340M that was chosen so that it behaves almost as a rigid element. In addition, for the available column, the clearance above and below the end plate was less than 400 mm. However, since this is a rigid column, this limitation proved not to be severe. Regarding the joint geometry, the top bolt Table 7.1 Details of the test specimens.

# Column Beam End Plate Test ID Profile Steel

grade Profile Steel

grade tp

(mm) Steel grade

FS1 2 HE340M S355 IPE300 S235 10 S355 FS2 2 HE340M S355 IPE300 S235 15 S355 FS3 2 HE340M S355 IPE300 S235 20 S355 FS4 2 HE340M S355 IPE300 S235 10 S690

h b =

300

b c =

309

b b =

150

HE340M

h p =

400

H

c.up

= 4

00

1200

hc = 377

IPE300

5.5 ~ 6

3.5 ~ 4

ts = 10 tp = 10, 15, 20

Lbeam = 1200Lload = 1000 200

Hc.

low =

400

ts ~ 10

Lstiffened ~ 500

Fig. 7.1 Geometry of the specimens (dimensions in [mm]).

Experimental tests on bolted end plate connections

277

bp = 150w = 90e = 30 30

h p =

400

e X

=

30

L X =

69

.65

p =

90

e com

p =

75

p 2-3

= 2

05

aw = 3.5 ~ 4

aw = 5.5 ~ 6

d 0 =

22

L c

omp =

30

.35

1 2

3 4

5 6

150

300

aw = 5

aw = 5

(a) Detail of the end plate. (b) Detail of the stiffener. Fig. 7.2 Details of the specimens (dimensions in [mm]). row corresponds to specimen WT7_M20 (refer to Chapter 3) from the former test series on isolated T-stubs. All the end plate specimens were designed com-plying with the Eurocode 3 requirements [7.1] so that the components end plate and bolts in the tension zone were the determining factor of collapse. 7.2.2 Geometrical properties The actual geometry of the various connection elements was recorded before starting the test. For the various specimens the profiles and plates actual di-mensions and connection geometry are summarized in Table 7.2. These values are given as an average value of the several measurements from each series. Table 7.3 indicates the bolts measurements for each test. 7.2.3 Mechanical properties 7.2.3.1 Tension tests on the bolts Two different batches of bolts were used in the experiments. Having performed tests from series FS1 and FS2 and test FS3a, it was decided to use a different batch of bolts, from another manufacturer as explained later in the text. Three machined bolts from each group were tested in tension in order to determine the mechanical properties of the bolt material, in accordance with ISO 898-1:1999(E) [7.2]. The average properties are set out in Table 7.4.


278

Table 7.2 Actual geometry of the connection (averaged dimensions, [mm]).

Test ID Column profile Stif. hc bc tfc Hc.up Hc.low ts

FS1 175.00 219.00 10.76 FS2 174.50 219.50 10.50 FS3 177.50 216.50 10.46 FS4

376.00 307.50 40.21

174.50 219.50 10.42 Beam profile hb bb tfb twb Lbeam Lload

FS1 300.45 150.50 10.76 7.20 1200.00 1002.50 FS2 301.40 149.60 10.67 7.01 1200.38 1000.25 FS3 301.46 149.75 10.57 7.03 1191.50 992.63 FS4 300.66 149.54 11.86 7.03 1218.75 991.88

End plate and connection geometry hp bp tp e w

FS1 401.04 149.84 10.40 30.01 89.91 FS2 400.84 149.41 15.01 29.76 89.89 FS3 401.40 150.47 20.02 30.27 89.93 FS4 401.69 149.76 10.06 29.94 89.88

eX LX p p2-3 ecomp. FS1 29.90 69.35 90.03 205.90 76.45 FS2 30.10 69.30 89.98 205.04 75.44 FS3 29.74 68.90 90.14 204.84 76.82 FS4 29.83 69.86 89.95 205.28 76.13

7.2.3.2 Tension tests of the structural steel The test programme included two different steel grades for the end plate: S355 and S690. According to the European Standards EN 10025 [7.3] and EN 10204 [7.4], the steel qualities are S355J0 (ordinary steel) and N-A-XTRA M70 (high-strength steel for plates), respectively. For the beam profile, steel grade S235JR was ordered. Table 7.5 summarizes the chemical composition of the different steel grades. The coupon tension testing of the structural steel material was performed according to the RILEM procedures [7.5]. The average characteristics are set out in Table 7.6. In this table the values for the Young modulus, E, the strain hardening modulus, Est, the static yield and tensile stresses, fy and fu, the yield ratio, ρy, the strain at the strain hardening point, εst, the uniform strain, εuni, and the ultimate strain, εu are given. Note that for the 10 mm thickness end plates, the structural steel is the same that had been used in the testing of the isolated T-stub connections (cf. Chapter 3).


279

Table 7.3 Bolt hole clearance and length (dimensions in [mm]; H-tght: Hand-tightening; Aft. clps.: after collapse).

Test ID

#1 #2 #3 #4 #5 #6

d0 21.93 21.98 21.98 21.75 21.98 21.93 Initial 94.00 94.00 94.10 94.25 93.00 92.90 H-tght 94.00 94.00 94.10 94.25 93.10 93.00 FS1a Bolt

length Aft.

clps. 94.65 94.40 94.50 94.90 93.00 93.10

d0 22.05 22.00 22.03 22.05 22.10 21.90 Initial 94.00 94.25 94.40 94.05 93.15 93.20 H-tght 94.00 94.25 94.40 94.05 93.15 93.20 FS1b Bolt

length Aft.

clps. 94.95 96.00 95.40 94.85 93.00 93.15


length Aft.

clps. 95.70 96.18 102.06 96.62 93.24 93.78


length Aft.

clps. 95.16 97.02 101.30 96.52 93.28 93.04


length Aft.

clps. 95.56 95.10 96.04 96.12 93.48 93.44


length Aft.

clps. 95.30 95.00 95.25 99.22 93.24 93.24


length Aft.

clps. 94.40 93.94 99.62 102.62 93.06 93.10


length Aft.

clps. 94.16 94.82 100.94 100.26 93.26 93.38


280

Table 7.4 Average characteristic values for the bolts.

Batch E (MPa) fy (MPa) fu (MPa) εu 1 223166 857.33 913.78 0.184 2 222982 854.31 916.81 0.156

Table 7.5 Chemical composition of the structural steels according to the

European standards.

% max. C Mn Si P S N CEV S235JR 0.17 1.40 0.045 0.045 0.012 0.35 S355J0 0.20 1.60 0.55 0.040 0.045 0.009 0.40 N-A-XTRA M70 0.20 1.60 0.80 0.020 0.010 0.48 Table 7.6 Average characteristic values for the structural steels.

Specimen Steel grade

E (MPa)

Est (MPa)

fy (MPa)

fu (MPa)

ρy

tp = 10 S355 209856 2264 340.12 480.49 0.708 tp = 15 S355 208538 2901 342.82 507.85 0.675 tp = 20 S355 208622 2771 342.62 502.59 0.682

End plate

tp = 10 S690 204462 2495 698.55 741.28 0.940 Web S235 208332 1856 299.12 446.25 0.670 Beam Flange S235 209496 1933 316.24 462.28 0.684

Specimen Steel grade

εst εuni εu

tp = 10 S355 0.015 0.224 0.361 tp = 15 S355 0.020 0.198 0.475 tp = 20 S355 0.017 0.196 0.457

End plate

tp = 10 S690 0.014 0.075 0.174 Web S235 0.016 0.235 0.464 Beam Flange S235 0.016 0.235 0.299

7.2.4 Test arrangement and instrumentation The main features of the test apparatus are illustrated in Figs. 7.3a-b. Concern-ing the T arrangement depicted in Figs. 7.1-7.2, the actual connection was ro-tated 180º for practical reasons. The column was bolted to a reaction wall. The reader should bear in mind that the goal of these tests was the study of the end plate in the tension zone and therefore it had to be ensured that the column was not governing any failure mode. The load was applied by a 400 kN testing machine (hydraulic jack with maximum piston stroke of ±200 mm), through a purpose-built device (Fig.


281

(a) Test apparatus (illustra-tion with specimen FS2a).

(b) Detail of the beam and connection zone (il-lustration with specimen FS1a).

(c) Detail of the load application device. Fig. 7.3 Equipment and test specimen. 7.3c) that was clamped to the beam at 200 mm from the free end. A beam guidance device near the loading point was provided to prevent lateral torsional buckling of the beam with the course of loading. For that purpose, a special de-vice located at 250 mm from the load point was attached to the specimens (Figs. 7.3a-b). The length of the beam was chosen to ensure a realistic stress pattern de-veloped at the connection, on one hand, and to ensure that fracture of the sev-eral specimens, i.e. ultimate load, was attained with the specific testing ma-chine. The instrumentation plan is described in Figs. 7.4-7.6 below. The primary requirements of the instrumentation were the measurement of the applied load, the relevant displacements of the connection (e.g. vertical displacement of the beam, horizontal displacement of the assembly end plate-tensile beam flange) and bolt elongation. The record of all measurements was made automatically with intervals of 1 second. The displacements were measured by means of


282

LVDTs located as indicated in Fig. 7.4. These were attached to the elements with special glue. Four LVDTs with an accuracy of 0.5% were used to measure the beam vertical displacements (DT1-4). The range of these transducers is 480 mm for DT1, 425 mm for DT2 and 200 mm for both DT3 and DT4. The hori-zontal displacements of the assembly end plate-beam flanges were measured with 50 mm LVDTs with a precision of 0.5% (DT6-7, compression side, DT9-10, tension side). In order to measure the end plate vertical displacement due to elongation of the bolt holes, an additional 35 mm LVDT (DT5 – accuracy of 0.5%) was attached to the lower part of the end plate (tension side), as illus-trated in Fig. 7.4. To ensure that the displacements of the column could be ne-glected, two LVDTs (DT8,11) were attached to the back side of the column. These transducers could measure up to 1.5 mm displacement with an accuracy of 0.5% as well. This was the precision of the electrical components connected to the data logger. The bolts deformations were measured with special measuring brackets, MBs (horseshoe device), as common practice in the Stevin Laboratory of the Delft University of Technology. These devices were attached to the bolts only

IPE300

DT8

DT11

DT6/7

DT9/10

DT5 DT3 DT2 DT1

100 200

DT4

300 300300

HE340M

Load

1000

Top

Bottom

DT8/11

DT6/9

DT7/10 DT5 DT4 DT3 DT2 DT1

Front

Back

Bolts 1/3/5

Bolts 2/4/6 Fig. 7.4 Location of the displacement transducers.


283

Fig. 7.5 MBs 1,3 and LVDTs 6,9 (illustration with specimen FS4a).

SG3

SG1 SG2

19

10

30

90

30

250

4530 3045

4 5 6

7 8

9

10

1112

13

1415

45 30

4 5 6

7 8

9

10

11 12

13

14

15 30

90

30

5 19

19

19

Bac

k

Top

2

4

2 1

43

3

6 5

xy strain gauges Unidirectional strain gauges

(a) Sketch of the location of the strain gauges on the beam and end plate.

(b) Strain gauges located at the beam and end plate extension.

(c) Strain gauges located at the end plate.

Fig. 7.6 Location of the strain gauges.


284

on the tension side. They could only measure up to 2 mm of displacement. However, they were removed before collapse to prevent damage. Fig. 7.5 shows these devices for bolts 1 and 3. Finally, strain gauges, SGs, TML (maximum strain 21000 µm/m) were added to the end plate (back side) in the tension zone to provide insight into the strain distribution in this zone (Fig. 7.6). In addition, the specimens were pro-vided with strain gauges at the top of the tension beam flange. For good comparison of the results, all specimens used the same arrange-ment for the location of the strain gauges and measuring devices. 7.2.5 Testing procedure Before installation of the specimens into the testing rig, the dimensions of the plates were recorded and the bolts were hand-tightened and measured. The specimens were then placed into the machine and aligned. The bolts were fas-tened with an ordinary spanner (45º turn) and measured. In order to sketch the yield line patterns the specimens were painted with chalk. The measurement devices and strain gauges were then connected. Elec-tronic records started and all the equipment was verified. The specimens were subjected to monotonic tensile force, which was ap-plied to the beam as explained before. The tests were carried out under dis-placement control with a constant speed of 0.02 mm/s up to collapse of the specimens. The test itself then started with loading of the specimen up to 2/3Mj.Rd, which corresponds to the theoretical elastic limit. Mj.Rd is the full plas-tic resistance and is determined according to Eurocode 3. Complete unloading followed on and the specimen was then reloaded up to collapse. In this third phase, the test was interrupted at the load levels corresponding to 2/3Mj.Rd, Mj.Rd, at the knee-range and after this level each six minutes, equivalent to an actuator displacement of 7.2 mm. The hold on of the test lasted for three min-utes. The testing procedures adopted for the full-scale tests were identical to those described in Chapter 3 for the individual T-stubs. Four collapse failure modes or a combination of those were observed in the test: (i) weld cracking, (ii) plate cracking, (iii) bolt fracture and (iv) bolt nut stripping (see Table 7.7). After collapse, the bolts were measured again (Table 7.3). 7.3 TEST RESULTS The results presented in the following sections relate to the third phase of the tests, after elimination of slippery and after settlement of the connecting parts. The plotted graphs refer to the applied load, displacement and strain direct readings and to the corresponding bending moment and deformations. The bending moment, M, acting on the connection corresponds to the ap-


285

Table 7.7 Description of failure types.

Test ID Mode of failure FS1a Weld failure of the assembly beam-end plate, both at the flange and

web sides. FS1b Weld failure of the assembly beam-end plate, both at the flange and

web sides and plate cracking at opposite sides. FS2a Nut stripping of bolt #4 and weld failure along the whole end plate

extension width but not at the inner part. FS2b Nut stripping of bolts #1 and #4 with no plate cracking or weld fail-

ure. FS3a Nut stripping of bolts #3 and #4 and some weld failure close to bolt

#3 but without development of a crack. FS3b Nut stripping of bolt #3. FS4a Fracture of bolt #4 and some weld failure at the end plate extension

close to bolt #1 but without development of a complete crack. FS4b Fracture of bolt #3. plied load, “Load” multiplied by the distance between the load application point and the face of the end plate, Lload:

Load loadM L= × (7.1) The rotational deformation of the joint, Φ, is the sum of the shear deforma-tion of the column web panel zone, γ and the connection rotational deforma-tion, φ, that is defined as the change in angle between the centrelines of beam and column, θb and θc. In these tests, the column hardly deforms as it behaves as a rigid element. This statement will be validated later in the text. Then, both γ and θc are nought and so:

bφ θΦ = = (7.2) The beam rotation is approximately given by (Fig. 7.4):

DT3DT1 DT2. . .

DT4.

arctan arctan arctan900 600 300

arctan100

b b el b el b el

b el

δδ δθ θ θ θ

δθ

= − = − = − =

= − (7.3)

where δDTi are the vertical displacements at LVDT DTi and θb.el is the beam elastic rotation. The above expression disregards the effect of shear deforma-tion in the beam and assumes that the vertical displacements of the end plate are negligible, i.e. DT5 0δ ≈ . Some differences in the results from DT4 are ex-pected when compared to the remaining LVDTs since it is located closer to the end plate. In this region, the beam theory is not valid and the stress distribution is not smooth. By using the above relationships, the M-φ curve of the connection can be characterized. The main features of this curve are: resistance, stiffness and ro-


286

tation capacity. In particular, for the different tests the following characteristics are assessed [7.6]: the knee-range of the M-φ curve, the plastic flexural resis-tance, Mj.Rd, the maximum bending moment, Mmax, the initial stiffness, Sj.ini, the post-limit stiffness, Sj.p-l, the rotation corresponding to the maximum load level,

maxMφ and the rotation capacity, φCd (see Figs. 1.28 and 7.7). The stiffness values are computed by means of linear regression analysis of the quasi-elastic branches before and after the knee-range. A brief summary of the observed collapse failure modes is given in Table 7.7 and some illustrations are given in Fig. 7.8. Failure occurred due to a vari-ety of reasons, but the collapse modes always involved the components end plate and bolts in the tension zone.

0

40

80

120

160

200

0 10 20 30 40 50 60 70

Connection rotation φ (mrad)

Ben

ding

mom

ent,

M (k

Nm

)

Knee-range

Φ MRd Φ Xd

Mj.Rd

Mmax

Φ Cd

Sj.ini

Sj.p-l

Fig. 7.7 Moment-rotation characteristics from tests.

(i) General view. (ii) Detail: weld failure, front side.

(iii) Detail: bolts #2-#4 after failure (notice the bending of bolt #2).

(a) Specimen FS1b. Fig. 7.8 Illustration of the various failure types observed in the experiments.


287

(iv) Detail: end plate cracking (extension), back side.

(v) Elongation of the bolt holes in the tension zone.

(a) Specimen FS1b (cont.).

(i) General view of the end plate. (ii) Nut stripping of bolt #4

(column side).

(iii) Detail of the weld fracture in the tension zone.

(iv) Detail of tension bolts (bolt #3 nearly fractures).

(b) Specimen FS2a. Fig. 7.8 Illustration of the various failure types observed in the experiments

(cont.).


288

(i) Bolt #3.

(ii) Bolt #4.

(c) Specimen FS3a. (d) Specimen FS4b. Fig. 7.8 Illustration of the various failure types observed in the experiments

(cont.). 7.3.1 Moment-rotation curves As explained above, the M-φ curves for the different connections are obtained from the beam vertical displacement readings and the applied load. For illustra-tion, Fig. 7.9 plots the load vs. vertical displacement of the beam for specimen FS1a. This curve can be converted into a moment-“gross beam rotation” curve by application of Eqs. (7.1) and (7.3) excluding θel, as shown in Fig. 7.10a for the four LVDTs DT1-4. Examination of these four curves indicates a good agreement of the results obtained for DT1-3 and some deviation for DT4. These differences have already been explained earlier in the text. Therefore, the results from DT1 are kept for further analysis. If now the beam elastic de-formation is subtracted from the “gross rotation” (see Eq. (7.3)), the connection rotation can be completely characterized (Fig. 7.10b). This value is taken as equal to the beam rotation because the column rotation, θc, can be disregarded in comparison with θb (see Fig. 7.11) and also because the end plate vertical deformation due to the bolt hole elongation can be neglected when compared to the δDT1 (see Fig. 7.12). Note that for specimen FS1a the slippery at circa 110 kN has to be disregarded. The M-φ responses for the eight connection details are reported in Fig. 7.13. Almost identical responses are obtained for each set over the entire elastoplas-tic range. This proves that the test procedure and the instrumentation setup adopted for the programme were satisfactory. The main features of the eight curves are summarized in Table 7.8. All characteristic values are referred to the readings from LVDT DT1. In all cases, the knee-range domain of the curves is alike for the same connection detail. The maximum resistance is also similar,


289

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70

Vertical displacement of the beam (mm)

Tota

l app

lied

load

(kN

) DT1DT3DT4 DT2

Fig. 7.9 Beam vertical displacement readings of LVDTs DT1-4 for specimen

FS1a.

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100

Beam rotation includ. elastic def. (mrad)

Ben

ding

mom

ent (

kNm

)

DT4DT3DT1

DT2

(a) Beam rotation computed from the displacement readings of LVDTs DT1-4 [arctan(δDTi/LDTi)].

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100

Beam rotation θb (mrad)

Ben

ding

mom

ent (

kNm

)

Beam rotation including the beam elasticdeformationConnection rotation (equal to the beamrotation)

(b) Beam rotation computed by means of Eq. (7.3) from the readings of DT1. Fig. 7.10 Beam rotation for specimen FS1a.


290

0

20

40

60

80

100

120

140

160

-0.20 -0.16 -0.12 -0.08 -0.04 0.00 0.04 0.08 0.12 0.16 0.20

Horizontal displac. (column side) (mm)

Tot

al a

pplie

d lo

ad (k

N)

DT8 (compres-sion side)

DT11 (tensionside)

(a) Column horizontal displacements.

0

20

40

60

80

100

120

140

160

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Column rotation θc (mrad)

Ben

ding

mom

ent (

kNm

)

(b) Corresponding column rotations DT8 DT11arctancb fbh t

δ δθ

+ = −

.

0

20

40

60

80

100

120

140

160

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

θc/θb

Ben

ding

mom

ent (

kNm

)

(c) Ratio between column rotation and beam rotation. Fig.7.11 Column rotation for specimen FS1a.


291

0

30

60

90

120

150

180

210

-7 -6 -5 -4 -3 -2 -1 0 1

Vertical displac. of the end plate (δ DT5) (mm)

Tot

al a

pplie

d lo

ad (k

N)

FS1a

FS4b

Fig. 7.12 End plate vertical displacement for specimens FS1a and FS4b.

0

30

60

90

120

150

180

210

240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

) FS1a FS1b

(a) Series FS1.

0

30

60

90

120

150

180

210

240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

) FS2a FS2b

(b) Series FS2. Fig. 7.13 Moment-rotation curves for the four test series.


292

0

30

60

90

120

150

180

210

240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

) FS3a FS3b

(c) Series FS3.

0

30

60

90

120

150

180

210

240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

) FS4a FS4b

(d) Series FS4. Fig. 7.13 Moment-rotation curves for the four test series (cont.). though in series FS1 and FS3 some differences were observed. In series FS1, experimental observations show that the welding quality in set FS1a is poor, i.e. the welding procedure resulted in a glue weld instead of a burnt-in weld. This induced premature cracking of the specimen. Regarding series FS3, the discrepancies arise because different bolts were employed in the two sets and also because there was a disturbance in test FS3a at a load level of 190 kN that may have had some effect on the final results. In terms of rotational stiffness, some differences arise, particularly for Sj.ini in the case of series FS1 and Sj.p-l for series FS3. Identical values of the ratio Sj.ini/Sj.p-l are obtained for the four test types. Exception is made for joint FS3a, which shows some disturbance in the post-limit regime, and therefore the results are not reliable in this domain. Now, in terms of maximum rotation, the values at Mmax are close for all sets (again, the results for FS3a are not reliable in the post-limit domain), particu-larly for specimen FS2. Higher deviations appear for φCd, especially for series


293

Table 7.8 Main characteristics of the moment-rotation curves.

Resistance [kNm] Test ID Knee-range Mj.Rd Mmax

FS1a 65-112 105.60 ( )5.81 mradRdMφ = 142.76

FS1b 68-120 109.30 ( )6.49 mradRdMφ = 161.17

FS2a 120-174 165.65 ( )7.08 mradRdMφ = 193.06

FS2b 117-181 170.22 ( )7.74 mradRdMφ = 197.31

FS3a 112-186 172.27 ( )7.47 mradRdMφ = 202.91

FS3b 122-200 192.66 ( )8.94 mradRdMφ = 214.35

FS4a 110-170 165.60 ( )10.24 mradRdMφ = 185.32

FS4b 110-170 163.52 ( )9.53 mradRdMφ = 187.67

Stiffness [kNm/mrad]

Sj.ini Sj.p-l Ratio

Sj.ini/Sj.p-l FS1a 18.19 ( )2 0.9717R = 0.84 ( )2 0.9384R = 21.55 FS1b 16.84 ( )2 0.9921R = 0.74 ( )2 0.9681R = 22.78

FS2a 23.39 ( )2 0.9925R = 0.84 ( )2 0.8611R = 27.93 FS2b 22.00 ( )2 0.9968R = 0.92 ( )2 0.8405R = 23.91

FS3a 23.23 ( )2 0.9905R = 1.81 ( )2 0.8629R = 12.82 FS3b 21.56 ( )2 0.9972R = 1.03 ( )2 0.8003R = 20.96

FS4a 16.18 ( )2 0.9936R = 0.78 ( )2 0.8004R = 20.61 FS4b 17.15 ( )2 0.9956R = 0.74 ( )2 0.8681R = 23.29 Rotation [mrad] φXd φCd maxMφ

FS1a 18.23 68.91 ( )127.71 kNmCd

Mφ = 61.55 FS1b 20.00 111.22 ( )70.29 kNm

CdMφ = 77.05

FS2a 17.45 82.88 ( )66.00 kNmCd

Mφ = 41.72 FS2b 19.17 60.89 ( )147.93 kNm

CdMφ = 40.30

FS3a 13.75 42.76 ( )108.16 kNmCd

Mφ = 25.00 FS3b 18.25 48.74 ( )153.10 kNm

CdMφ = 29.99

FS4a 19.25 61.69 ( )150.25 kNmCd

Mφ = 37.70 FS4b 18.33 64.24 ( )158.09 kNm

CdMφ = 43.85


294

FS1 and FS2. The differences that are observed in series FS1 have already been explained above. For series FS2 and FS3, φCd is not well defined since it corre-sponds to the beginning of final unloading of the test. No actual rupture was observed in this case. The test was stopped because the deformations were al-ready too high and there was fear of damaging the equipment if the test went on any further. One connection from each set is now chosen for the purpose of a compara-tive study. In all the cases, the assembly end plate-bolts is the main source of connection deformability. Fig. 7.14 compares the rotational behaviour of the four test types and shows an increase in resistance and rotational stiffness and a loss of rotation capacity with the end plate thickness (FS1, FS2 and FS3). The effect of the steel grade is identical (FS1 and FS4).

0

30

60

90

120

150

180

210

240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS1b

FS2a

FS4b

FS3b

Fig. 7.14 Comparison of the moment-rotation curves for the four test series. 7.3.2 Behaviour of the tension zone 7.3.2.1 End plate deformation behaviour The most significant characteristic describing the overall end plate deformation behaviour in the tension zone is the F-∆ response. The test setup does not allow a direct measurement of the force at the component level but the information gathered from LVDTs DT9 and DT10 permits a full characterization of the end plate deformation behaviour. These transducers are attached to the beam flange and they measure the gap between the end plate and the column flange (see Fig. 7.4). As an example, Fig. 7.15a traces the moment-gap response obtained for DT9 and DT10 for specimens FS1b and FS4a and indicates a good agree-ment over the whole loading history. For comparison, Fig. 7.15b shows that these measurements are also identical for the two sets from one test type.


295

0

30

60

90

120

150

180

210

0 3 6 9 12 15 18 21 24 27 30 33

End plate (horiz.) deformation (δ DT9-10) (mm)

Ben

ding

mom

ent (

kNm

) FS1b: DT9 FS1b: DT10FS4a: DT9 FS4a: DT10

(a) Comparison of the responses for the two devices (DT9, DT10) for tests FS1b and FS4a.

0

20

40

60

80

100

120

140

160

180

0 3 6 9 12 15 18 21 24 27 30 33

Horizontal displac. (tension side) (mm)

Ben

ding

mom

ent (

kNm

)

FS1a FS1b

(b) Comparison of the responses for the two tests from series FS1 (deforma-tions from DT9). Fig. 7.15 End plate deformation in the tension zone for several specimens.

0

30

60

90

120

150

180

210

240

0 3 6 9 12 15 18 21 24 27 30 33

End plate (horiz.) deformation (δ DT9) (mm)

Ben

ding

mom

ent (

kNm

)

FS1b

FS2a

FS4b

FS3b

Fig. 7.16 Comparison of the moment-end plate deformation curves for the four test series.


296

Fig. 7.16 compares the end plate deformation behaviour for the four con-nection details. The deformability of the end plate increases for smaller values of tp and lower steel grades. This behaviour is identical to the connection rota-tion, as expected, since the components end plate and bolts are the main sources of connection deformability. Fig. 7.17 illustrates the evolution of the end plate deformation response with the applied load for the specific case of FS4b and Figs. 7.17d and 7.18 compare the collapse conditions for the four test types. A comparative analysis of the influence of the end plate deformability over the connection rotational behaviour is plotted in the graph of Fig. 7.19. For se-ries FS2, FS3 and FS4 where the bolts mainly determine failure, either by frac-ture or by stripping, the shape of the curves is identical. In series FS1 where end plate cracking and weld fracture are engaged in the collapse mode, the shape of the curve is slightly different. Even so, these curves clearly demon-strate that the ratio between end plate deformation behaviour is higher for lower end plate thickness values and lower steel grades.

(i) General view. (ii) Tension zone. (a) Load = 80 kN (theoretical elastic limit; elastic branch of the M-φ curve).

(i) General view. (ii) Tension zone. (b) Load = 120 kN (theoretical plastic resistance; knee-range branch of the M-φ curve). Fig. 7.17 Evolution of the end plate deformations until failure conditions for

test series FS4b.


297

(i) General view. (ii) Tension zone. (c) Load = 162 kN (post-limit branch of the M-φ curve).

(i) General view. (ii) Tension zone. (d) Load = 188 kN (maximum load attained during the test).

(i) General view. (ii) Tension zone. (e) Collapse conditions. Fig. 7.17 Evolution of the end plate deformations until failure conditions for

test series FS4b (cont.). Finally, Fig. 7.20 shows an alternative procedure for computation of the connection deformation from the readings of the horizontal LVDTs, in the compression and tension zone of the end plate (e.g. specimen FS1a). As ex-pected, the agreement between both procedures is excellent.


298

(a) Specimen FS1a. (b) Specimen FS2a. (c) Specimen FS3b. Fig. 7.18 Comparison of the end plate deformations at failure conditions for

test series FS1-3.

0

30

60

90

120

150

180

210

240

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

Ratio δ DT9/φ (mm/mrad)

Ben

ding

mom

ent (

kNm

)

FS1b

FS2a

FS4b

FS3b

Fig. 7.19 Comparison of the ratio end plate deformation vs. connection rota-

tion for the four test series.

0

20

40

60

80

100

120

140

160

0 10 20 30 40 50 60 70 80 90 100

Connection rotation (mrad)

Ben

ding

mom

ent (

kNm

)

Connection rotation as defined above

Connection rotation computed from DT9

Connection rotation computed from DT10

Fig. 7.20 Comparison of the moment-rotation curve for test FS1a by using al-

ternative definitions of connection rotation.


299

7.3.2.2 Yield line patterns Figs. 7.21 and 7.22 depict the yield line patterns of the inner tension bolt #3 for specimens FS1b and FS2b at collapse conditions. These patterns could be sketched because the specimens were painted with chalk. Clearly, for series FS1 the yielding of the end plate in this area spreads to the compression bolt, whilst for FS2, with a thicker plate, there is a small amount of plasticity in the end plate.

(a) Load = 93 kN. (b) Load = 151 kN. (c) Collapse

conditions. Fig. 7.21 Yield line patterns around the inner tension bolt for different load

levels (e.g. specimen FS1b).

(a) Load = 130 kN. (b) Load = 188 kN. (c) Near collapse conditions. Fig. 7.22 Yield line patterns around the inner tension bolt for different load

levels (e.g. specimen FS2b). 7.3.2.3 Bolt elongation behaviour The experimental results demonstrate that the two rows of tension bolts carry unequal forces (Fig. 7.23): the inner tension bolts carry a larger proportion of the load than the outer bolts. This conclusion is also supported by the graphs


300

shown in Fig. 7.24 that compare the ratio between the bolt elongation and the gap end plate-column flange. This ratio increases for the inner tension bolts. The graphs also highlight the influence of the bolt tension deformation on the overall behaviour with the increase of tp and steel grade. This conclusion is in line with the above observations.

0

30

60

90

120

150

180

210

0.00 0.15 0.30 0.45 0.60 0.75 0.90 1.05 1.20 1.35 1.50

(Tension) Bolt elongation (mm)

Tot

al a

pplie

d lo

ad (k

N)

MB1 MB2

MB3 MB4

Fig. 7.23 Bolt elongation behaviour (e.g. specimen FS4b).

0

2

4

6

8

10

12

14

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

δ b/δ DT9 (mm/mm)

End

pl. (

hor.)

def

. (δ

DT9

) (m

m)

FS1b

FS2b

FS3a

FS4b

(a) Bolt #1. Fig. 7.24 Comparison of the “nondimensional” bolt elongation behaviour for

the four specimen types. 7.3.2.4 Strain behaviour This section illustrates some of the experimental strain results. Unfortunately, the travel range of the gauges used for recording the strains was often exceeded before the connection failure occurred and in many specimens, the gauges were damaged in early stages of loading. In some cases, the strain gauges were not


301

0

2

4

6

8

10

12

14

0.00 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.30

δ b/δ DT9 (mm/mm)

End

pl. (

hor.)

def

. (δ

DT9

) (m

m)

FS1b

FS2b

FS3a

FS4b

(b) Bolt #2.

0

2

4

6

8

10

12

14

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

δ b/δ DT9 (mm/mm)

End

pl. (

hor.)

def

. (δ

DT9

) (m

m)

FS1b

FS2bFS3a

FS4b

(c) Bolt #3.

0

2

4

6

8

10

12

14

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

δ b/δ DT9 (mm/mm)

End

pl. (

hor.)

def

. (δ

DT9

) (m

m)

FS1b

FS2b

FS3a

FS4b

(d) Bolt #4. Fig. 7.24 Comparison of the “nondimensional” bolt elongation behaviour for

the four specimen types (cont.).


302

attached correctly and consequently their results are not trustworthy. Anyhow, some results can be retained for future comparisons. Fig. 7.25 shows the results obtained in different gauges (for their location, please refer to Fig. 7.6). These results also allow an assessment of the yield line patterns (hogging, SG11 and sagging yield lines, SG5, 7 and 9).

0

30

60

90

120

150

180

210

240

0 3000 6000 9000 12000 15000 18000 21000 24000

Strain (SG5) (µm/m)

Tota

l app

lied

load

(kN

)

FS1b

FS2a

FS3b

FS4b

Limit of the strain gauges

(a) Strains at SG5, located at the end plate extension.

0

30

60

90

120

150

180

210

240

0 3000 6000 9000 12000 15000 18000 21000 24000Strain (SG7) (µm/m)

Tota

l app

lied

load

(kN

)

FS1b

FS2aFS3b

FS4b

Limit of the strain gauges

(b) Strains at SG7, located at the inner end plate side, near the beam tension flange. Fig. 7.25 Comparison of some strain results obtained for the four specimen

types at different strain gauges. 7.4 DISCUSSION OF TEST RESULTS Eurocode 3 gives quantitative rules for the prediction of the joint flexural plas-tic resistance and initial rotational stiffness. These structural properties are evaluated below by using the actual geometrical characteristics from Table 7.2


303

0

30

60

90

120

150

180

210

240

0 3000 6000 9000 12000 15000 18000 21000 24000


Tota

l app

lied

load

(kN

)

FS1b

FS2aFS3b

FS4bLim

it of the strain gauges

(c) Strains at SG9, located at the inner end plate side, near the beam web.

0

30

60

90

120

150

180

210

240

-24000 -21000 -18000 -15000 -12000 -9000 -6000 -3000 0


Tota

l app

lied

load

(kN

)

FS1b

FS2a

FS3b

FS4b

Lim

it of

the

stra

in g

auge

s

(d) Strains at SG11, located at the bolt axis (end plate extension). Fig. 7.25 Comparison of some strain results obtained for the four specimen

types at different strain gauges (cont.). and the mechanical properties from Tables 7.4 and 7.6. The recommendations on rotation capacity are also verified to investigate if there is enough rotation capacity according to the code. The provisions are compared with the test re-sults. 7.4.1 Plastic flexural resistance According to Eurocode 3, the joint plastic flexural resistance is evaluated by means of Eq. (1.60). As the overall connection behaviour was dominated by the end plate and bolts, the computation of Fti.Rd relies on the T-stub idealiza-tion of the tension zone that can fail according to the three possible plastic col-lapse mechanisms.


304

Table 7.9 Evaluation of the resistance of the test specimens (the experimen-tal values correspond to the average of the two tests per connection detail; Ratio = [Theory/Experiments]).

Row 1 Row 2 h1 Ft1.Rd Plastic h2 Ft2.Rd Plastic

Mj.Rd Test ID

(mm) (kN) mode (mm) (kN) mode (kNm)

Ratio

FS1 334.52 83.86 Type-1 244.49 202.34 Type-1 77.52 0.72 FS2 335.26 176.07 Type-1 245.28 297.87 Type-2 132.09 0.79 FS3 335.34 274.06 Type-2 245.20 389.01 Type-2 187.29 1.03 FS4 334.76 161.16 Type-1 244.81 287.94 Type-2 124.44 0.76 Application of the procedure detailed in §1.6.1.2 provides the results pre-sented in Table 7.9. It is worth mentioning that the predicted yield line patterns (double curvature for the bolt row located at the end plate extension and side yielding near the beam flange) are in line with the experimental observations (cf. Figs. 7.22-7.23 for the inner bolt row, for instance). By comparing the code predictions with the experiments, they are within the knee-range bounds but below the experimental values of flexural resistance. 7.4.2 Initial rotational stiffness The initial rotational stiffness was evaluated according to the Eurocode 3 pro-cedure, as explained in §1.6.1.1. For simplicity, z was taken as equal to the dis-tance from the centre of compression to a mid point between the two bolt rows in tension [7.1]. Table 7.10 sets out the predicted values for the initial stiffness and com-pares them with the experiments. The ratio between the predicted values and the experiments shows that Eurocode overestimates this property. The differ-ences may derive from the fact that the expression as presented in the code was calibrated for a certain range of joints. The particular joints that were tested were not ‘balanced’, i.e. there was a much weaker component than the remain-ing ones. This situation is unlikely to occur in common joints for which the ex-pression was calibrated. 7.4.3 Rotation capacity The experimental values of rotation capacity and corresponding moment values for the various tests are set out in Table 7.8. It can be easily seen that test FS1, which employs a thinner end plate and steel grade S355, presents higher ductil-ity than the remaining tests. Application of the Eurocode 3 guidelines to the characterization of the rota-


305

Table 7.10 Evaluation of the initial rotational stiffness of the test specimens (the experimental values correspond to the average of the two tests per connection detail; Ratio = [Theory/Experiments]).

keff.1 keff.2 keq kcws kcwc Test ID (kN/mm)

FS1 225.023 333.57 541.95 2718.64 4867.62 FS2 375.03 453.37 816.22 2717.60 4983.72 FS3 2453.12 496.04 942.50 2711.50 5052.06 FS4 202.18 315.65 500.25 2710.23 4857.34

Test ID z Sj.ini Ratio (mm) (kNm/mrad)

FS1 289.51 34.66 1.98 FS2 289.62 46.76 2.06 FS3 290.27 51.76 2.31 FS4 290.41 32.77 1.97

Table 7.11 Verification of the recommendations for rotation capacity.

Test ID tp Maximum tp (mm) (mm)

FS1 10.40 11.80 Yes. FS2 15.01 11.75 No. FS3 20.02 11.76 No. FS4 10.06 8.25 No.

tion capacity [7.1] – cf. §1.6.2 – shows that the first condition is guaranteed for all specimens (the joint moment resistance is governed by the resistance of the end plate in bending), whilst the second condition (Eq. (1.68)) is only fulfilled for specimens FS1 (Table 7.11). Though these recommendations are only valid for steel grades up to S460, they were also applied to series FS4 that includes end plates from grade S690. 7.5 CONCLUDING REMARKS Tests on eight extended end plate moment connections were conducted under static loading. All specimens were designed to trigger failure in the end plate rather than in the beam or the column. The following conclusions can be drawn from the test programme: 1. The joint moment resistance increases with the increase of end plate thick-ness and with the yield stress of the plate;


306

2. The joint initial rotational stiffness also increases with the end plate thick-ness, but the sensitivity to the thickness variation is not as noticeable as for re-sistance. The steel grade has little influence if any on this property; 3. The joint post-limit rotational stiffness is identical for all specimens, i.e. the variation with end plate thickness or steel grade is not significant; 4. The Eurocode 3 proposals give safe approaches for the prediction of the joint resistance but overestimate the joint initial stiffness in this particular case; 5. The available rotation capacity and hence the joint ductility decreases with the plate thickness (series FS1, FS2 and FS3) and with the plate steel grade (FS1 and FS4); 6. In terms of the verification of sufficient rotation capacity, Eurocode 3 gives safe criteria but perhaps too conservative [7.7]. For instance, in terms of overall rotation capacity, specimens from series FS2 and FS4 exhibit rotation values of 40 mrad. 7.6 REFERENCES [7.1] European Committee for Standardization (CEN). PrEN 1993-1-8:2003,

Eurocode 3: Design of steel structures, Part 1.8: Design of joints, Stage 49 draft, May 2003, Brussels, 2003.

[7.2] International Standard ISO 898-1:1999(E). Mechanical properties of fasteners made of carbon steel and alloy steel – Part 1: Bolts, screws and studs, August 1999, Switzerland, 1999.

[7.3] European Committee for Standardization (CEN). PrEN 10025:2000E: Hot rolled products of structural steels, September 2000, Brussels, 2000.

[7.4] European Committee for Standardization (CEN). EN 10204:1995E: Me-tallic products, October 1995, Brussels, 1995.

[7.5] RILEM draft recommendation. Tension testing of metallic structural materials for determining stress-strain relations under monotonic and uniaxial tensile loading. Materials and Structures; 23:35-46, 1990.

[7.6] Weynand K. Sicherheits-und Wirtsschaftlichkeitsuntersuchungen zur anwendung nachgiebiger anschlüsse im stahlbau. PhD thesis (in Ger-man). University of Aachen, Aachen, Germany, 1996.

[7.7] Girão Coelho AM, Bijlaard F, Simões da Silva L. Experimental assess-ment of the ductility of extended end plate connections. Engineering Structures (in print), 2004.

307

8 DUCTILITY OF BOLTED END PLATE CONNECTIONS 8.1 INTRODUCTION The methodology developed in this chapter provides a characterization of the full nonlinear M-Φ behaviour of bolted end plate connections. The assessment of the available joint rotation is addressed in particular. Kuhlmann and Küh-nemund [8.1] assume that the available joint rotation should be taken as the to-tal joint rotation or rotation capacity, ΦCd. In the context of the component method, for a direct computation of the joint rotation capacity the following steps have to be fulfilled [8.2-8.3]: (i) the F-∆ curve of each joint component up to failure is modelled, (ii) the weakest joint component, i.e. the component with lower resistance, is identified, (iii) the plastic engagement of the remain-ing components is determined, (iv) the global displacements of the individual components at the level of maximum resistance are evaluated to finally (v) de-termine ΦCd. Literature suggests that most of the joint rotation in thin end plates comes from the end plate deformation [8.4-8.5]. The tests described in Chapter 7 con-firm this statement. For really thin end plates, the end plate deformation would be sufficient to characterize the M-Φ curve since it becomes the weakest joint component. In general, extended end plate connections are characterized by the participation of two or more components to the joint plastic deformation, as highlighted by Faella et al. [8.2]. In the framework of the component method, in this joint configuration the following sources of deformability for characteri-zation of the rotation capacity are identified (Fig. 8.1): column web in shear (cws), column web in compression (cwc), column web in tension (cwt), col-umn flange in bending (cfb), end plate in bending (epb) and bolts in tension (bt). Components beam web and flange in compression (bfc) and beam web in tension (bwt) are not taken into account in the model since they basically pro-vide a resistance limitation [8.6]. Components column flange in bending, end plate in bending and bolts in tension are modelled as equivalent T-stubs, as al-ready explained. The full M-Φ response is characterized from the F-∆ curve of the joint components, which are assembled into an appropriate mechanical model. Chapter 1 (§1.6) discusses alternative component models that are illus-trated in Fig. 8.2. If most of the joint rotation comes in fact from the sub-assembly end plate-bolts, the several models yield identical solutions since the only deformable components are the end plate in bending and the bolts in ten-sion (Fig. 8.3). Consequently, the component models illustrated above are equivalent, as shown in Fig. 8.3. This is the case of the tested joints that were reported in Chapter 7.


308

T-stub idealization

M

cwc

cwt

bfc

bwt z cws

Fig. 8.1 Basic joint components of an extended end plate connection with

two bolt rows in tension.

Φ(cwc) (cws)

(cw

t.1)

(cfb

.1)

(epb

.1)

(bt.1

)

(cw

t.2)

(cfb

.2)

(epb

.2)

(bt.2

)

Φ z

T-stubs (row 1)

T-stubs (row 2)

M

Φ

(cw

t.1)

(cfb

.1)

(epb

.1)

(bt.1

)

(cw

t.2)

(cfb

.2)

(epb

.2)

(bt.2

)

Φ

T-stubs (row 1)

(cwc) (cws)

T-stubs (row 2)

M

(a) Mechanical model adopted in Eurocode 3. (b) UC component model.

Φ(cwc) (cws)

(cfb

.1)

(epb

.1)

(bt.1

)

(cfb

.2)

(epb

.2)

(bt.2

)

Φ

T-stubs (row 1)

T-stubs (row 2)

(cw

t.1)

(cw

t.2)

M (c) Innsbruck mechanical model. Fig. 8.2 Alternative component models for analysis of the rotational behav-

iour of an extended end plate connection (two bolt rows in tension).

Ductility of bolted end plate connections

309

The following sections address the characterization of the rotational behav-iour of extended end plate connections up to failure. The full M-Φ curve is de-rived by using a computational tool, NASCon [8.7]. This software allows for a multilinear definition of the deformation behaviour of components and uses the spring model illustrated in Fig. 8.2b. Since ductility is such an important prop-erty in a partial strength scenario, particular attention is given to this issue. The available experimental tests (Chapter 7) were basically aimed at the in-vestigation of the end plate behaviour. Therefore, the proposed methodology is illustrated and validated only for this connection type. However, from a theo-retical point of view, the procedure can be applied to any beam-to-column joint configuration, as long as the F-∆ response of each component can be predicted with sufficient accuracy. This research work focuses on those components

(epb

.1)

Φ

(epb.1) (bt.1)

Φ

T-stub (row 1)

(epb.2) (bt.2)

T-stub (row 2)

M

Φ(cwc) (cws)

(cw

t.1)

(cfb

.1)

(epb

.1)

(bt.1

)

(cw

t.2)

(cfb

.2)

(epb

.2)

(bt.2

)

Φ

M

(cw

t.1)

Φ

(cfb

.1)

(bt.1

)

(cw

t.2)

(cfb

.2)

(epb

.2)

(bt.2

)

Φ

(cwc) (cws) M

Φ(cwc) (cws)

(cfb

.1)

(epb

.1)

(bt.1

)

(cfb

.2)

(epb

.2)

(bt.2

)

Φ

(cw

t.1)

(cw

t.2)

M Fig. 8.3 Equivalence of component models for analysis of the rotational be-

haviour of tested joints.


310

modelled by equivalent T-stubs. Chapter 3 describes some experiments per-formed on WP-T-stubs. A three-dimensional FE model has been proposed in Chapter 4 for the assessment of the F-∆ behaviour of this component. Chapter 6 proposes a simplified beam model for the assessment of the overall deforma-tion behaviour of individual T-stubs and also describes alternative simplified methods recommended by other authors. These methodologies are used below for characterization of the end plate behaviour. 8.2 MODELLING OF BOLT ROW BEHAVIOUR THROUGH EQUIVALENT T-

STUBS The T-stub idealization of the tension zone of a connection consists is substitut-ing this zone for T-stub sections of appropriate effective length (Fig. 8.4). These T-stub sections are connected by their flange to a rigid foundation (half-model) and subjected to a uniformly distributed force acting in the web plate [8.6]. The extension of the end plate and the portion between the beam flanges are modelled as two separate equivalent T-stubs (Fig. 8.4). On the column side, two situations have to be analysed: (i) the bolt rows act individually or (ii) the bolt rows act in combination (Fig. 8.4). To define the effective length, the complex pattern of yield lines that occurs around the bolts is converted into a simple equivalent T-stub. This effective length does not represent any actual length of the connection. The typical ob-served yield-line pattern in thin end plates is shown in Figs. 8.5 and 8.6, for two different cases: (i) end plate with one bolt row below the tension beam flange and (ii) end plate with two bolt rows below the flush line, respectively. For thicker end plates, the patterns may not develop fully as the bolt elongation behaviour may govern the overall behaviour. For end plates with more than one bolt row below the flush line, the cases of individual and combined bolt row behaviour have to be taken into consideration, as illustrated in Fig. 8.6. 8.3 APPLICATION TO BOLTED EXTENDED END PLATE CONNECTIONS The above procedures are applied to the extended end plate connections from Chapter 7 that were tested monotonically up to failure. In these examples, the tension zone on the end plate side that is idealized as a T-stub was always criti-cal. The remaining joint components behaved elastically until collapse. 8.3.1 Component characterization The four joint configurations FS1-FS4 comprise two bolt rows in tension. For each test detail, on the end plate side, two equivalent T-stubs are identified (Figs. 8.4-8.5). For further reference, these two T-stubs are designated by “T-


311

Bolt row 1

beff.ep.r2

Bolt row 2

End plate side

beff.fc.r1

beff.fc.r2

Bolt row 1, indivi-dually

beff.cf.r(1+2)

Bolt rows 1 and 2

Column side

Bolt row 2, indivi-dually

beff.ep.r1

Fig. 8.4 T-stub idealization of an extended end plate bolted connection with

two bolt rows in tension. stub top” and “T-stub bottom”, for bolt rows 1 and 2, respectively (cf. Figs. 8.4-8.5). The characterization of these components in terms of F-∆ behaviour is performed by means of four alternative procedures (Table 8.1): (i) experimen-tally, (ii) numerically (three-dimensional FE model), (iii) analytically (simple beam model) and (iv) simplified bilinear approximation proposed by Jaspart [8.8]. The experimental results are not available for all equivalent T-stubs and the numerical model is not implemented for all T-stubs, as shown in Table 8.1. For the equivalent T-stubs top from joints FS1 and FS4, the tests on WT7_M20 and WT57_M20, respectively, provide an experimental F-∆ curve that can be


312

Hogging yield-line Sagging yield-line

Bolt row 1 . 1 2eff r epb b=

Bolt row 2 . 2eff r epb mα=

(a) Plot. (b) Illustration: spec. FS1b. Fig. 8.5 Typical yield-line pattern in thin extended end plates with two bolt

rows in tension.

Hogging yield-line Sagging yield-line

Bolt row 1

Bolt row 2

acting in combination

p2-3

beff.r(2+3)p2-3

Influence of bolt row 2

(0.5beff.r2 + 0.5p)

Influence of bolt row 3

(0.5beff.r3 + 0.5p)

beff.r1

Fig. 8.6 T-stub idealization of an extended end plate with three bolt rows in ten-

sion with the bolts below the tension beam flange acting in combination.


313

Table 8.1 Alternative procedures for characterization of the T-stub response.

Characterization procedures Test ID Equivalent T-stub Experimental Numerical Beam

model Jaspart

approxi-mation

Top (WT7_M20)

FS1

Bot. Top FS2 Bot. Top FS3 Bot. Top

(WT57_M20)

FS4 Bot.

used for component characterization. The individual T-stub specimens do not correspond exactly to the equivalent T-stubs top as the bolt properties are dif-ferent. However, no major differences are expected. For application of the rec-ommendations of Jaspart [8.8], the bolt deformability is associated to that of the end plate. The effective length of the different components is defined according to Eurocode 3 [8.6] and is summarized in Table 8.2. The actual geometric proper-ties of the joints are used (Table 7.2). Figs. 8.7-8.10 illustrate the T-stub re-sponses for the various configurations and with the alternative methodologies. Table 8.3 sets out the predictions of ultimate resistance and deformation capac-ity of the above equivalent T-stubs, as ascertained by the different procedures. The experimental results correspond in fact to experimental failure (see Figs. 8.7a and 8.10a). Concerning the numerical predictions for the equivalent T-stub top for joint FS1, the values that are indicated in the table do not account for any reduction of the failure ductility of the HAZ (see also §5.4). The graphs in Figs. 8.7-8.10 also plot the experimental end plate deforma-tion behaviour, which is obtained directly from the measurement of the dis-placement of the tension beam flange with the course of loading. The corre-sponding force level is evaluated indirectly, Ft = M/z, whereby z is the lever arm determined from Eq. (1.59). In these graphs, this force Ft acting at the level of the tension beam flange was divided equally by the two bolt rows. This procedure gives a reasonable agreement with the predictions for the T-stub top but deviates from the predicted behaviour for the T-stub bottom in the same case (e.g. specimen FS2, Fig. 8.8). In fact, the division of the tensile force by the two bolt rows modelled as two equivalent T-stubs seems more appropriate for the top T-stub, rather than the bottom T-stub. The equivalent T-stub top shares the tensile beam flange whereas the T-stub bottom shares the beam web.


314

Therefore, the force acting at the web of the T-stub top is directly related to the tensile force Ft. This is not true for the bottom T-stub. Furthermore, the as-sumption of an equal division of Ft by the two bolt rows is questionable. Con-sequently, the graphs that were traced are merely illustrative and should be re-garded as such. The experimental deformation of the end plate at the tensile beam flange level is obtained from the readings of the LVDTs. Table 8.3 indicates these values at failure (in bold). They are directly related to the equivalent T-stub top Table 8.2 Effective length of the equivalent T-stubs.

beff Test ID (mm) FS1 FS2 FS3 FS4

T-stub top 74.92 74.71 75.24 74.88 T-stub bot. 205.77 202.67 202.73 206.42

0

45

90

135

180

225

270

0 2 4 6 8 10 12 14 16 18 20

∆ep.r1 (mm)

F ep.

r1 (k

N)

End plate deformation (exp.)Exp. results WT7_M20Numerical FE resultsBeam modelJaspart approximation

(a) T-stub top.

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10 12 14 16 18 20

∆ep.r2 (mm)

F ep.

r2 (k

N)

End plate deformation (exp.)

Beam model

Jaspart approximation

.

(b) T-stub bottom. Fig. 8.7 Equivalent T-stubs for joint FS1.


315

0

50

100

150

200

250

300

350

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0

∆ep.r1 (mm)

F ep.

r1 (k

N)


Beam model


(a) T-stub top.

0

100

200

300

400

500

600

700

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0

∆ep.r2 (mm)

F ep.

r2 (k

N)

End plate deformation (exp.)Beam modelJaspart approximation

.


0

100

200

300

400

500

600

0 1 2 3 4 5 6 7 8 9 10 11 12

∆ep.r1 (mm)

F ep.

r1 (k

N)


Beam model


(a) T-stub top. Fig. 8.9 Equivalent T-stubs for joint FS3.


316

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6 7 8 9 10 11 12

∆ep.r2 (mm)

F ep.

r2 (k

N)

End plate deformation (exp.)Beam modelJaspart approximation

.

(b) T-stub bottom. Fig. 8.9 Equivalent T-stubs for joint FS3 (cont.).

0

50

100

150

200

250

300

350

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 16.5

∆ep.r1 (mm)

F ep.

r1 (k

N)


Exp. results WT57_M20

Beam model


(a) T-stub top.

0

80

160

240

320

400

480

0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 16.5

∆ep.r2 (mm)

F ep.

r2 (k

N)


Beam model


.



317

Table 8.3 Assessment of the ultimate conditions of the equivalent T-stubs by means of the proposed alternative characterization procedures.

Characterization procedures Test ID Equiva-lent

T-stub Experimental Numeri-

cal Beam model

Jaspart approx.

Evaluation of Fep.ri.u (kN) Top 105.29 177.53 137.80 137.68 FS1 Bot. 360.18 275.70 Top 316.00 273.52 FS2 Bot. 375.79 366.47 Top 526.93 324.28 FS3 Bot. 865.40 448.49 Top 207.97 182.99 195.17 FS4 Bot. 439.82 310.90

Evaluation of ∆ep.ri.u (kN) Top 27.42 9.35 12.68 9.35 7.04 FS1 Bot. 10.20 4.62 Top 14.55 8.87 5.87 FS2 Bot. 10.83 4.83 Top 11.79 11.21 3.77 FS3 Bot. 10.69 3.77 Top 16.03 11.76 4.02 4.39 FS4 Bot. 7.53 5.34

deformation capacity. The predictions do not compare well to the experiments as they clearly underestimate the deformation capacity, particularly for the thinner end plates (ratios between the beam model predictions and the actual results from the LVDTs range from 3 to 4 for FS1 and FS4). For the thicker end plates the agreement improves considerably. In fact, for specimen FS3 the beam model predictions are quite accurate. The nondimensional analysis of these equivalent T-stubs, at the top bolt row, at failure, i.e. in terms of the component ductility index, ϕep.r1, is summa-rized in Table 8.4 (BM: beam model; JBA: Jaspart approximation; Num: Nu-merical results for T-stub top and beam model for T-stub bottom; Exp: Ex-perimental results for T-stub top and beam model for T-stub bottom). These indexes are evaluated from Eq. (1.39). The values in italic correspond to the ra-tios to the experimental results for the end plate deformation, at the beam flange level. Generally speaking, the predictions given by the beam model are good, showing a pronounced underestimation for FS4, which uses S690, and a clear overestimation for FS3. The average error is 14% but the coefficient of variation is significant (0.75). Jaspart [8.8], on the other hand, gives estima-tions with an average error of 25% but the scatter of results is lower, with a co-


318

efficient of variation of 0.26. The experimental results for the single T-stub are available for specimens FS1 (WT7_M20) and FS4 (WT57_M20). The corre-sponding indexes show some deviations from the actual joint results. The results just described are again analysed in the following sections in order to establish some criteria regarding the ductility requirements for the overall joint behaviour. Table 8.4 Evaluation of the equivalent “T-stub top” component ductility in-

dex (characterization of the T-stubs for evaluation of the analytical response: BM – beam model, JBA – Jaspart bilinear approxima-tion, Num – numerical FE model, Exp – T-stubs top characterized experimentally).

Test ID ϕep.r1 FS1 FS2 FS3 FS4

Average Coeff. var.

Experimental 27.98 15.99 16.15 11.06

BM 18.33 (0.66)

13.86 (0.87)

25.48 (1.58)

3.72 (0.34) 0.86 0.75

JBA 20.71 (0.74)

16.31 (1.02)

11.42 (0.71)

6.01 (0.54) 0.75 0.26

Num 23.92 (0.86)

A

naly

tical

Exp 18.33 (0.66)

15.08 (1.36) 1.01 0.50

8.3.2 Evaluation of the nonlinear moment-rotation response The full M-Φ joint response is evaluated using the software NASCon [8.7]. This software is a computational implementation of the component method. The model file is written by means of the user-friendly “Connection Assistant” tool. All the details of the joint and joint components are specified in this file (see Fig. 8.11 for illustration). The multilinear component behaviour is input in this file. The model is then imported by NASCon to generate the M-Φ curve (Fig. 8.12). A displacement control-based strategy was selected (Fig. 8.13). Fi-nally, the overall M-Φ curve can be visualized (Fig. 8.14). The various curves are shown in the graphs from Figs. 8.15-8.18 and are compared with the experiments. The graphs trace the responses obtained in NASCon for the different characterization processes described in the previous section. The critical component is also indicated in the graphs as well as the governing part (flange or bolt). Whenever the components are characterized with the bilinear approximation proposed by Jaspart [8.8], the critical failure mode at ultimate conditions (1, 2 or 3) is indicated. Note that for different characterization processes, the determinant T-stub for rotation capacity can change (e.g. joint FS1 and the beam model or the bilinear approximation pro-


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Fig. 8.11 Modelling of the connection and component behaviour (e.g. FS1).

Fig. 8.12 Model loading (e.g. FS1).


320

Fig. 8.13 NASCon strategy selection and prescribed loading (e.g. FS1).

Table 8.5 Evaluation of initial stiffness (experimental results correspond to the average results of the two tests).

Sj.ini Test ID (kNm/mrad) FS1 FS2 FS3 FS4

Average Coeff. var.

Experimental 17.52 22.69 22.39 16.67

Eurocode 3 34.66 (1.98)

46.76 (2.06)

51.76 (2.31)

32.77 (1.97) 2.08 0.08

BM 30.78 (1.76)

45.26 (1.99)

54.78 (2.45) 1.96 1.96 0.18

JBA 34.45 (1.97)

47.65 (2.10)

53.23 (2.38) 2.10 2.10 0.10

Num 30.29 (1.73)

A

naly

tical

Exp 29.51 (1.68)

42.47 (2.55) 2.12 0.29

posed by Jaspart for characterization of the T-stubs – Figs. 8.15a-b). Table 8.5 summarizes the characteristics of the curves in terms of initial stiffness. Again, the values in italic correspond to the ratio to the experiments.


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(a) Behaviour of component T-stub top (which determines ultimate conditions).

(b) Behaviour of component T-stub bottom. Fig. 8.14 Moment-rotation curve (e.g. FS1).


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0

20

40

60

80

100

120

140

160

180

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS1a

FS1b

NASCon prediction (T-stubtop critical - flange)

(a) Equivalent T-stubs characterized by means of the beam model.

0

20

40

60

80

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120

140

160

180

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS1a

FS1b

NASCon prediction (T-stubbottom critical - mode 2U)

(b) Equivalent T-stubs characterized by means of the Jaspart bilinear model.

0

20

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60

80

100

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160

180

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

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)

FS1a

FS1b


(c) Equivalent T-stub top characterized numerically (three-dimensional model). Fig. 8.15 Moment-rotation curve for joint FS1.


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0

20

40

60

80100

120

140

160

180

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS1a

FS1b


(d) Equivalent T-stub top characterized experimentally. Fig. 8.15 Moment-rotation curve for joint FS1 (cont.).

0

30

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210

240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS2a

FS2b



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30

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0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS2a

FS2b

NASCon prediction (T-stubtop critical - mode 2U)

(b) Equivalent T-stubs characterized by means of the Jaspart bilinear model. Fig. 8.16 Moment-rotation curve for joint FS2.


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0

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300

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400

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ding

mom

ent (

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)

FS3a

FS3b

NASCon prediction (T-stubbottom critical - bolt)


0

30

60

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240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS3a

FS3b


(b) Equivalent T-stubs characterized by means of the Jaspart bilinear model. Fig. 8.17 Moment-rotation curve for joint FS3.

0

30

60

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150

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210

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS4a

FS4b


(a) Equivalent T-stubs characterized by means of the beam model. Fig. 8.18 Moment-rotation curve for joint FS4.


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0

30

60

90

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150

180

210

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS4a

FS4b


(b) Equivalent T-stubs characterized by means of the Jaspart bilinear model.

0

30

60

90

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150

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0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

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)

FS4a

FS4b

NASCon prediction (T-stubtop critical - flange and bolt)

(c) Equivalent T-stub top characterized experimentally and T-stub bottom characterized by means of the beam model. Fig. 8.18 Moment-rotation curve for joint FS4 (cont.). In general, the analytical predictions overestimate the initial stiffness in com-parison with the experiments, Sj.ini, (e.g. specimen FS1 – Sj.ini.Exp = 17.52 kNm/mrad, Sj.ini.BM = 30.78 kNm/mrad = 1.76 Sj.ini.Exp). This is quite straightfor-ward from the statistical analysis of the ratios to the experiments also presented in Table 8.5 in italic. The examination of the curves also shows that the analytical predictions for resistance can also be slightly overestimated for some specimens, particularly in the plastic domain (e.g. FS3, Fig. 8.17) though for thinner end plates the pre-dictions are good (e.g. FS1, FS4, Figs. 8.15 and 8.18). The rotation capacity is clearly underestimated by the analytical methods, even for the cases of FS1 and FS4 with the experimental component charac-terization. Table 8.6 sets out the rotation predictions (experimental and analyti-cal; values in italic represent the ratio to the experimental values). Experimen-


326

tally, two rotation values were evaluated: the rotation corresponding to maxi-mum load level,

maxMΦ , and the rotation capacity, ΦCd (see also Table 7.8). Analytically, the rotation capacity is attained when the first component reaches failure. The experimental values in Table 8.6 are the averaged values between the tests for each configuration, except for FS1 and FS3 for which the value of tests “b” are adopted. This table also indicates the critical component for each methodology (EPX: cracking at the extension of the end plate; BNSo+i: bolt nut stripping of the outer and inner bolt; BNSi: bolt nut stripping of the inner bolt; BTi: inner bolt in tension; Tt-fl: T-stub top, flange; Tb-b: T-stub bottom, bolt; Tt-fl+b: T-stub top, flange and bolt; 1U: mode 1 critical at ultimate condi-tions; 2U: mode 2 critical at ultimate conditions). The statistical investigation of the results shows that the application of the beam model for the characterization of the individual T-stubs provides an aver-age ratio to the experiments of 0.40 with a coefficient of variation of 0.58. The predictions obtained from application of Jaspart’s approximation [8.8] yield a lower value for the average ratio but also a lower coefficient of variation. Never-theless, when both approaches are compared in terms of failure predictions, the beam model gives a better agreement with the experimental observations. The joint ductility properties are further analysed in the following sections. Table 8.6 Comparison of the predictions of rotation capacity of the various

joints and failure modes.

φCd Test ID (mrad) FS1 FS2 FS3 FS4

Average Coeff. var.

111.22 71.89 48.74 62.97 Experimental EPX BNSo+i BNSi BTi 29.20 (0.26)

28.80 (0.40)

35.00 (0.72)

13.50 (0.21) 0.40 0.58 BM

Tt-fl Tt-fl Tb-b Tt-fl 20.96 (0.19)

19.44 (0.27)

13.52 (0.28)

14.56 (0.23) 0.24 0.17 JBA

2U 2U 2U 1U 39.84 (0.36)

Num

Tt-fl 29.52 (0.27)

36.80 (0.58) 0.43 0.52

Ana

lytic

al

Exp Tt-fl Tt-fl+b

8.3.3 Evaluation of the rotation capacity according to other authors Having discussed the results obtained from the author’s methodology in terms


327

of predictions of rotation capacity, the proposals from other researchers are now analysed. The verifications on ductility requirements for these specimens according to Eurocode 3 have already been carried out in Chapter 7. The main conclusions are summarized in Table 8.7. Three alternative procedures for evaluation of the rotation capacity are il-lustrated. These procedures were proposed by Adegoke and Kemp [8.5], for thin end plates, Beg et al. [8.3] and Zoetemeijer [8.9]. This latter method is re-stricted to those cases where type-2 plastic failure mode is critical and conse-quently it is only validated by specimen FS3, for which the plastic failure mode of both equivalent T-stubs is of type-2. The three methodologies have been de-scribed in Chapter 1, §1.6.2. Table 8.8 sets out the main results for the above procedures. In general, the rotation capacity is underestimated. The application of the methodology proposed by Adegoke and Kemp [8.5] requires the definition of the location of the neutral axis of the connection at plastic and ultimate conditions (cf. §1.6.2 and references [8.5,8.10]). This loca-tion was defined from the results obtained through application of the UC me-chanical model. This method reflects the tendency observed in the experi-ments: the rotation capacity decreases with the plate thickness, for identical plate steel grades. For the specimen with steel S690, the rotation capacity is overestimated. However, the scope of the method is restricted to current steel grades and consequently the latter results are just illustrative. The method pro-posed by these authors yields an average ratio to the experiments of 0.53 with a Table 8.7 Verification of the recommendations for rotation capacity accord-

ing to Eurocode 3 (values in [mm]).

Test ID

tp Maximum tp

Critical component governing the joint resistance

Verification?

FS1 10.40 11.80 End plate in bending Yes. FS2 15.01 11.75 End plate in bending No. FS3 20.02 11.76 End plate in bending No. FS4 10.06 8.25 End plate in bending No.

Table 8.8 Analytical evaluation of the rotation capacity according to other

authors.

φCd Test ID (mrad) FS1 FS2 FS3 FS4

Average Coeff. var.

Experimental 111.22 71.89 48.74 62.97 Adegoke and

Kemp 31.66 (0.28)

22.71 (0.32)

17.47 (0.36)

72.88 (1.16)

0.53

0.79

Beg et al. 48.40 (0.44)

47.88 (0.67)

105.40 (2.16)

49.34 (0.78)

1.01

0.77

Zoetemeijer 17.53 (0.36)


328

(rather high) coefficient of variation of 0.79. Beg and co-authors’ proposals [8.3] do not reproduce well the actual behav-iour. In fact, for specimen FS3 that employs a thicker end plate, the predictions are the highest. Though the averaged ratio in this case is unitary, the high coef-ficient of variation indicates that the methodology is not sufficiently accurate. Finally, the predictions for FS3 by applying the Zoetemeijer’s proposals [8.9] underestimate the experimental results. 8.3.4 Characterization of the joint ductility A joint ductility index has been proposed in Chapter 1 and it has been defined as follows:

Rd

Cdj

M

ϑΦ

=Φ

(8.1)

Essentially, it relates the rotation value at ultimate conditions with a rotation value attained in a plastic situation. In Eq. (8.1), the values of ΦCd and

RdMΦ were adopted. However, in this work other distinct values of rotation have been defined: ΦXd, corresponding to the rotation at which the moment first reaches Mj.Rd, and

maxMφ , the rotation value at maximum load (see Figs. 1.28 and 7.7). Tables 8.9 and 8.10 evaluate the experimental joint ductility indexes and ex-plore the above differences. In these examples, the joint rotation and the con-nection rotation are equal and so the latter values are indicated. As expected, if the index is related to

RdMφ (ϑj.Rd), usually lower than φXd, its value is greater

than if related to φXd (ϑj.Xd). The differences between the two indexes vary be-tween 47% in FS4a and 68% in FS1b. Two possibilities are considered in terms of the rotation at ultimate conditions: φCd (failure) and

maxMφ (at Mmax). The ductility indexes are naturally bigger in the first case, with deviations that vary between 31% in FS1b and 50% in FS2a. Again, in these comparisons, the test results corresponding to specimens FS1a and FS3a should be excluded. Another relevant aspect that warrants some attention relates to the indexes values within the same test series. This aspect is restricted to series FS2 and FS4. The differences between the two tests can diverge 8% for ϑj.Xd (at failure) and 33% for the same index, for specimens FS4 and FS2, respectively. These differences, however, are not consistent for the alternative definitions and within the same test series. Analytically, the joint ductility indexes are also evaluated. Table 8.6 sets out the analytical predictions for rotation capacity. From the analytical point of view, these are also the values at Mmax except when the experimental charac-terization of the T-stub top is input (e.g. Fig. 8.15 for specimens FS1). These values are again repeated in Table 8.11 along with the rotation values for

RdMφ .


329

Table 8.9 Experimental evaluation of the joint ductility indexes at RdMφ .

Test ID Rotation values at the KR [mrad] Ductility indexes at Mmax φKR.inf

RdMφ φKR.sup maxMφ ϑj.inf ϑj.Rd ϑj.sup

FS1a 3.0 5.8 17.5 61.6 20.53 10.62 3.52 FS1b 4.2 6.5 25.0 77.1 18.36 11.86 3.08 FS2a 6.5 7.1 20.0 41.7 6.42 5.87 2.09 FS2b 6.3 7.7 20.5 40.3 6.40 5.23 1.97 FS3a 5.5 7.5 15.0 25.0 4.55 3.33 1.67 FS3b 5.5 8.9 18.0 30.0 5.45 3.37 1.67 FS4a 6.9 10.2 21.0 37.7 5.46 3.70 1.80 FS4b 6.9 9.5 21.6 43.8 6.35 4.61 2.03

Test ID Rotation values at the KR [mrad] Ductility indexes at failure φKR.inf

RdMφ φKR.sup φCd ϑj.inf ϑj.Rd ϑj.sup


Table 8.10 Experimental evaluation of the joint ductility indexes at φXd.

Test ID Rotation values at the KR [mrad] Ductility indexes at Mmax φKR.inf φXd φKR.sup

maxMφ ϑj.inf ϑj.Xd ϑj.sup


Test ID Rotation values at the KR [mrad] Ductility indexes at failure φKR.inf φXd φKR.sup φCd ϑj.inf ϑj.Xd ϑj.sup



330

For further comparisons, this is the relevant rotation at plastic conditions. Table 8.12 evaluates the joint ductility index for the various joints (cf. rotation values in Table 8.11). For the analytical procedures, the ductility index was evaluated at the analytical value for rotation capacity but with respect to the analytical and experimental values of

RdMφ (ϑj.Rd.anl and ϑj.Rd.exp, respectively), as shown in Table 8.12 (cf. rotation values in Tables 8.9 and 8.11). For the various proc-esses the index ϑj.Rd.anl is bigger than ϑj.Rd.exp. The analytical predictions of the ductility index are quite severe, particu-larly for the thinner end plate specimens, FS1 and FS4 (2nd-4th columns, Table 8.12 and 7th column on Table 8.9). This situation also results from the underes-timation of the T-stub component ductility itself (e.g. FS4, Fig. 8.10 and Table 8.3). The analytical predictions for deformation capacity of the individual T-stubs are rather conservative, as seen above. Consequently, the rotation capac-ity of the overall joint, which is calculated from the individual components contribution, is also underestimated. On the contrary, for specimen FS3 that uses a 20 mm thick end plate, the ductility index is overestimated (2nd on Table 8.12 and 7th column on Table 8.9). Table 8.11 Analytical predictions of rotation of the various joints (in [mrad]).

Analytical predictions BM JA Num Exp

Test ID

RdMφ φCd RdMφ φCd RdMφ φCd RdMφ φCd FS1 3.0 29.2 2.4 21.0 4.6 39.8 3.3 29.5 FS2 3.4 28.8 3.0 19.4 FS3 4.4 35.0 3.4 13.5 FS4 5.0 13.5 4.0 14.6 4.4 36.8

Table 8.12 Analytical evaluation of the joint ductility indexes.

Analytical predictions BM JA Num Exp

Test ID

ϑj.Rd.anl ϑj.Rd.exp ϑj.Rd.anl ϑj.Rd.exp ϑj.Rd.anl ϑj.Rd.exp ϑj.Rd.anl ϑj.Rd.exp FS1 9.73 4.49 8.75 3.23 8.65 6.12 8.94 4.54 FS2 8.47 3.89 6.47 2.62 FS3 7.95 3.93 3.97 1.52 FS4 2.70 1.36 3.65 1.47 8.36 3.72 8.4 DISCUSSION The rotational behaviour of bolted extended end plate beam-to-column connec-tions was evaluated in the context of the component method. The methodology


331

was restricted to joints whose behaviour was governed by the end plate mod-elled as equivalent T-stubs in tension. It has been shown that the overall M-Φ response can be modelled fairly accurately provided that the T-stub component F-∆ behaviour is well characterized. Because ductility is such an important characteristic in connection performance, the evaluation of the joint rotation capacity, i.e. the available joint rotation, was addressed with greater depth. In order to meet the ductility requirements, the required joint rotation, Φj.req must be less or equal to the available joint rotation, Φj.avail:

. .j req j availΦ ≤ Φ (8.2) It is generally accepted that a minimum of 40-50 mrad ensures “sufficient rota-tion capacity” of a bolted joint in a partial strength scenario [8.11]. Tables 8.9 and 8.10 show that joints FS2 and FS4 also guarantee this condition at maxi-mum load. Therefore, the Eurocode 3 current provisions seem too conservative as far as rotational capacity is concerned (Table 8.7). This study affords some basis for the proposal of some additional criteria on this topic. From the analysis of the ductility indexes in Table 8.9 (top half of the ta-ble), computed at maximum load, a minimum joint ductility index of 4.0 seems appropriate in order to ensure “sufficient rotation capacity”. This limitation should be set in conjunction with an absolute minimum value of 40 mrad and is valid for steel grade S355. For steel grade S690 similar criteria might be estab-lished. However, the T-stub component in isolation has to be further explored for higher steel grades because of the inherent specificities. In addition, the analytical procedure has to be calibrated with other joint specimens since the joint ductility indexes are not yet accurate enough (cf. Tables 8.9 and 8.12). Naturally, as the above mentioned joints were designed to confine failure to the end plate and bolts, the deformation behaviour is exclusively dependent on these two components that form an equivalent T-stub. Therefore, the conclu-sions are only valid if the T-stub determines collapse. In this case it would be preferable to set a criterion in terms of the component ductility index, rather than the joint ductility index. However, it is found out that the information con-tained in Table 8.4 is not sufficient and can even be contradictory. For instance, take specimen FS3 as an example. In terms of joint ductility index (Tables 8.9 and 8.12), it is quite lower than the remaining cases. As for the single T-stub, the ductility index is higher than for specimen FS2 or FS4. This situation may arise in the definition of the equivalent T-stub itself. For specimens FS1 or FS4, corresponding to thin end plates, the predictions for rotation capacity are underestimating but the ratio to the experiments is consistent (see Table 8.6 and the BM characterization). In both specimens, where the equivalent T-stubs are governed by a type-1 plastic mode, the whole yield line pattern is likely to develop. For the other two cases, type-2 “plastic” failure mode is also present and therefore the complete pattern may not develop fully. This means that the actual T-stub effective width may be different from beff in Table 8.2. Conse-quently, the T-stub response for assessment of the joint rotational behaviour would also be different.


332

Finally, although it has been shown that deemed-to-satisfy criteria for suffi-cient rotation capacity stated in Eurocode 3 are overconservative, the estab-lishment of more accurate criteria still requires further work. 8.6 REFERENCES [8.1] Kuhlmann U, Kühnemund F. Ductility of semi-rigid steel joints. In: Pro-

ceedings of the International Colloquium on Stability and Ductility of Steel Structures (SDSS 2002) (Ed.: M. Ivanyi), Budapest, Hungary; 363-370, 2002.

[8.2] Faella C, Piluso V, Rizzano G. Structural Semi-Rigid Connections – Theory, Design and Software. CRC Press, USA, 2000.

[8.3] Beg D, Zupančič E, Vayas I. On the rotation capacity of moment connec-tions. Journal of Constructional Steel Research; 60:601-620, 2004.

[8.4] Zandonini R, Zanon P. Experimental analysis of end plate connections. In: Proceedings of the First International Workshop on Connections in Steel Structures, Behaviour, Strength and Design (Eds.: R. Bjorhovde, J. Brozzetti and A. Colson), Cachan, France; 40-51, 1988.

[8.5] Adegoke IO, Kemp AR. Moment-rotation relationships of thin end plate connections in steel beams. In: Proceedings of the International Confer-ence on Advances in Structures, ASSCCA’03 (Eds.: G.J. Hancock, M.A. Bradford, T.J. Wilkinson, B. Uy and K.J.R. Rasmussen), Sydney, Australia; 119-124, 2003.

[8.6] European Committee for Standardization (CEN). prEN 1993-1-8:2003, Part 1.8: Design of joints, Eurocode 3: Design of steel structures. Stage 49 draft, May 2003, Brussels, 2003.

[8.7] Borges LAC. Probabilistic evaluation of the rotation capacity of steel joints. MsC thesis. University of Coimbra, Coimbra, Portugal, 2003.


[8.9] Zoetemeijer P. Summary of the research on bolted beam-to-column connections. Report 25-6-90-2. Faculty of Civil Engineering, Stevin Laboratory – Steel Structures, Delft University of Technology, 1990.

[8.10] Kemp AR, Nethercot DA. Required and available rotations in continu-ous composite beams with semi-rigid connections. Journal of Construc-tional Steel Research; 57:375-400, 2001.

[8.11] Grecea D, Statan A, Ciutina A, Dubina D. Rotation capacity of MR beam-to-column joints under cyclic loading. In: Proceedings of the Fifth International ECCS/AISC Workshop on Connections in Steel Structures, Innovative steel connections, Amsterdam, The Netherlands; 2004 (to be published).

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9 CONCLUSIONS AND RECOMMENDATIONS 9.1 CONCLUSIONS The primary goal of this dissertation was to develop a methodology for the characterization of the full nonlinear rotational behaviour of bolted end plate beam-to-column steel connections based on the component method. Because of the emphasis recently placed on the design of joints within the partial strength/semi-rigid approach, special attention was addressed to the characteri-zation of the ductility of this joint type. The scope of the research was re-stricted to end plate connections for which the collapse was governed by the tension zone idealized by means of T-stubs. This goal was achieved by firstly conducting a comprehensive experimental test programme of thirty-two individual T-stubs that were supplemented by ro-bust FE analyses. The research on T-stubs constitutes a reliable database for validation of a simplified analytical (beam) model for characterization of the F-∆ behaviour of T-stub connections. This investigation drew particular attention to the assemblies made up of welded plates that model the end plate behaviour in the context of the T-stub idealization. Additionally, eight monotonic full-scale tests on end plate connections were conducted to analyse the ultimate re-sponse of this joint type and assess their behaviour from a ductility point of view. The tests showed that end plate connections can achieve rotation capacity provided that the end plate is a “weak link” relative to the bolts. There are some original contributions in this research work: 1. A detailed review on the state-of-art of the characterization of the M-Φ be-haviour of bolted end plate beam-to-column steel connections which high-lighted the current methodologies and Eurocode 3 provisions [9.1]; 2. A comprehensive test programme on WP-T-stub connections that constitutes a database of experimental results on this simple connection. Previous research work on this assembly type is not documented in technical literature. Piluso et al. [9.2] refer a single test on a WP-T-stub to validate an analytical methodol-ogy. This test programme provided insight into the actual behaviour of this simple connection, failure modes and deformation capacity. The main parame-ters affecting the deformation response of WP-T-stubs were identified and their influence on the overall behaviour of the connection was qualitatively and quantitatively assessed. The role of the welding and the presence of transverse stiffeners were also tackled. Additionally, the behaviour of WP-T-stubs and HR-T-stubs was confronted in order to clarify the main differences between both assembly-types; 3. Documentation of the problems with the welding consumables and the pro-

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cedures is made. During the experiments on WP-T-stubs, some of the speci-mens showed early damage of the plate material near the weld toe due to the effect of the welding consumable that induced premature cracking and reduced the overall deformation capacity. A solution to this problem was given by set-ting requirements to the weld metal to be used; 4. Advanced FE modelling was conducted on HR-T-stubs and WP-T-stubs. A robust three-dimensional model that encompasses material and geometrical nonlinearities and contact friction phenomenon was developed. The model provided qualitative and quantitative understanding about the T-stub behav-iour. It may also be used as a benchmark for FE modelling of bolted end plate steel connections; 5. Although no new models were developed in this work, some problems with existing models were identified and some modifications were tested (e.g. the modification on the definition of the distance m for WP-T-stubs on Chapter 6); 6. The completion and documentation of monotonic tests on bolted end plate connections in bending, up to failure; 7. Tests on bolted connection employing high-strength steel grade S690 were carried out. There is growing demand for high-strength steels in construction and insufficient knowledge on these steel grades. In this work some results on bolted connections that use S690 are given to provide additional information on this subject; 8. A methodology for characterization of the rotational response of a joint based on the component method was implemented and calibrated against ex-perimental results. The methodology was restricted to joints whose behaviour was governed by the end plate modelled as equivalent T-stubs in tension. The results of this particular study along with the conclusions drawn from the analysis of individual T-stubs afforded some basis for the proposal of some cri-teria for the verification of sufficient rotation capacity. The proposal was made in terms of a non-dimensional parameter, the joint ductility index. Naturally, this limitation was set in conjunction with an absolute minimum value of 40 mrad. This proposal was restricted to S355 as it was recognized the data were insufficient for higher steel grades. Several conclusions are drawn from this research work: 1. The prediction of failure should be based upon a deformation-based criterion rather than a resistance-based parameter. However, for consistency with Euro-code 3 that uses the β-ratio at design conditions to predict the critical “plastic” collapse mode, in this work a similar ratio βu (at ultimate conditions) was brought in, to identify the potential fracture mode. Naturally, this brings some inconsistencies with the observed and the predicted failure type; 2. The experimental-numerical work on the T-stub behaviour (both assembly types) identifies the major contributions of the overall T-stub deformation: the flange flexural deformation and the tension bolt elongation. Usually, a higher deformation capacity of the T-stub is expected if the flange cracking governs the collapse instead of bolt fracture. The cracking associated to the flange mechanism, in the case of the welded plates assembly, also depends on struc-

Conclusions and recommendations

335

tural constraint conditions and modifications in the mechanical properties in the HAZ, particularly those linked to the presence of residual stresses; 3. During the experiments, the importance of the correct selection of electrodes and welding procedures in the case of the testing of WP-T-stubs was high-lighted. It has been shown that the use of evenmatch soft low hydrogen elec-trodes ensures a ductile behaviour; 4. In general, bolts fail in tension before stripping. The stripping of the bolt threads and/or nut is not likely to occur in most cases. In the full-scale tests, the nut stripping phenomenon occurred in four tests. In the experimental investiga-tion on individual T-stubs the same problem was observed in one test. In fact, this phenomenon is rather frequent in practice. Research indicates that when the nut hardness is below a certain level (89 Rockwell B or 180 Brinell) there is a risk of stripping. This phenomenon limits the ductility performance of the whole joint and therefore it should be avoided. A solution to this problem can be given by setting requirements to the hardness and strength properties of the nut; 5. A two-dimensional beam model for assessment of the T-stub behaviour was developed. It retains all the relevant behavioural characteristics. To obtain the F-∆ curve, a numerical incremental procedure is required and, consequently, the model is not suitable for hand calculations. However, it clearly simplifies the proc-ess of behaviour characterization when compared to the three-dimensional FE ap-proach or the experimental technique. The applicability of the model was well demonstrated within the range of examples shown in the text. The behaviour predicted by this model is rather good in terms of resistance. With respect to ductility, it reflects an overestimation of test results that is within an acceptable error. These differences may derive from a great sensitivity of the model to strain hardening parameters and bolt ductility. Additionally, the model encom-passes a major simplification regarding the T-stub width, which is kept constant with the course of loading. It is well known that as the load increases, the flange width tributary to load transmission also increases. The implementation of such a variation is not straightforward. Ideally, the T-stub breadth should vary with the loading and this variation should be dependent on the failure mode as well; 6. Concerning the evaluation of the F-∆ response of T-stubs by means of other simplified methodologies, the bilinear approximation proposed by Jaspart [9.3] is accurate in terms of curve mimicry. However, the prediction of the potential failure mode is sometimes incorrect; 7. The T-stub idealization of end plate behaviour is reliable in the elastic-yielding domain. When strain hardening is present such idealization should be re-evaluated, especially in terms of effective width that is clearly different from the initial elastic behaviour; 8. The methodology developed for the evaluation of the M-Φ curve of joints and its ductility, in particular, mainly depends on the T-stub idealization of end plate behaviour as the joints were designed to confine failure to the end plate and bolts. As a result, the conclusions drawn in Chapter 8 are only valid if the

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collapse is determined by the T-stub component. The characterization of the T-stub behaviour and failure modes is therefore crucial. Two simplified method-ologies were implemented for that purpose: (i) the proposed beam model and (ii) the bilinear approximation proposed by Jaspart. The outcomes were quite good, in general. However, the prediction of the failure modes was more accu-rate in the first case, as already explained; 9. This methodology provided satisfactory results in terms of joint ductility, perhaps too conservative. However, a correcting factor can be defined to im-prove the results. This work does not permit the establishment of such a correc-tion due to lack of data. Furthermore, the conclusions for series FS2 and FS3 are quite limiting as the governing failure mode was the nut stripping of the in-ner bolts. This phenomenon should be avoided as explained and thus further investigation on the behaviour of these specimens is required; 10. As already mentioned above, a minimum joint ductility index of 4.0 was proposed in order to ensure “sufficient rotation capacity”. Additionally, an ab-solute minimum rotation value of 40 mrad should also be guaranteed. It would have been preferable to set a criterion in terms of the T-stub component ductil-ity index, rather than the joint ductility index. However, there was not enough data to make such a proposal; 11. For steel grade S690, similar criteria for rotation capacity should be estab-lished. However, the T-stub component in isolation has to be further explored for higher steel grades because of the inherent specificities; 12. With reference to the end plate behaviour modelled as equivalent T-stubs (Chapter 8), the results for specimens FS2 and FS3 could be further improved if the effective width of the T-stub bottom was reduced. The suggestion for this reduction is based on experimental observations of the yielded portions of the end plate below the tension beam flange. If the following T-stub breadth:

. . 20.5eff red bot ep ep hb m e d m= + + + (9.1)

is implemented, then for the above specimens, ( 2). . 122.37FS

eff red botb = mm and ( 3)

. . 124.07FSeff red botb = mm (0.60 and 0.61 times the original value obtained from

Eurocode 3, respectively – cf. Table 8.2). If the equivalent T-stub response is re-evaluated with these changes (beam model characterization), the corre-sponding joints M-Φ curves will fit the experiments better, as shown in Figs. 9.1 and 9.2. From a resistance point of view, the results are clearly improved. Also, the failure mode is compliant with experimental evidence. In terms of ductility, the results do not vary significantly, though. This problem is probably linked to the T-stub idealization itself and so additional research should be car-ried out. 9.2 FUTURE RESEARCH Some relevant issues were exposed during this investigation that warrant fur-

Conclusions and recommendations

337

0

30

60

90

120

150

180

210

240

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS2a

FS2b

NASCon original prediction (T-stub top critical - flange)NASCon prediction (T-stubbottom critical - bolt)

Fig. 9.1 Moment-rotation curve for joint FS2 (T-stub characterization by means

of the beam model and reduced effective length of the T-stub bottom).

0

50

100

150

200

250

300

350

400

0 10 20 30 40 50 60 70 80 90 100 110 120


Ben

ding

mom

ent (

kNm

)

FS3a

FS3b

NASCon original prediction(T-stub bottom critical - bolt)

NASCon prediction (T-stubstop - flange & bottom critical- bolt)

Fig. 9.2 Moment-rotation curve for joint FS3 (T-stub characterization by means

of the beam model and reduced effective length of the T-stub bottom). ther consideration. They are listed below and are proposed as future research: 1. The bolt force-elongation curve that was proposed in Chapter 6 for the bolt response simplified modelling requires further investigation as far as full-threaded bolts are concerned. This curve was derived for short-threaded bolts. This work clearly shows that for full-threaded bolts the predictions of bolt frac-ture overestimate the overall results. The formula for evaluation of the bolt fracture should include explicitly the ratio between the bolt shank and threaded lengths. Additionally, there should be a resistance limitation as it was observed that the bolt force at fracture could be as high as 1.30Bu, whereby Bu is the bolt tensile strength, evaluated in engineering stresses; 2. A clarification of the definition of the distance m is needed. Chapters 3-5

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gave experimental and numerical results for the stress and strain results on WP-T-stubs and showed that the yield lines near the flange-to-web connection would potentially develop at the end of the fillet weld. This would change the expression for computation of the distance m. According to Eurocode 3, m in these cases is defined as follows:

0.8 2 wm d a= − (9.2) Chapter 6 compared the beam model results obtained when this distance was employed with those obtained from:

2 wm d a= − (9.3) which are further improved. The latter definition is more compliant with the observations and should be regarded as a possible modification. Additional work on this subject is essential; 3. Further research on the T-stub idealization of the end plate behaviour is re-quired. Three-dimensional FE analysis may be helpful for investigating this specific topic. The numerical results presented in this research work can be used as benchmarks for validation of the global joint model. Naturally, the ex-perimental results are also essential for the calibration procedures. The estab-lishment of more appropriate rules for the definition of the effective equivalent T-stub width, particularly in the post-limit regime, are fundamental. The ex-periments can not provide enough results for this analysis. Advanced FE mod-elling provides all the necessary data and will be carried out by the author as a follow up study to this investigation. 9.3 REFERENCES [9.1] European Committee for Standardization (CEN). prEN 1993-1-8:2003,


[9.2] Piluso V, Faella C, Rizzano G. Ultimate behavior of bolted T-stubs. II: Model validation. Journal of Structural Engineering ASCE; 127(6):694-704, 2001.


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