Nonlinear Analysis of Steel-concrete Composite Beams Curved in Plan

7/29/2019 Nonlinear Analysis of Steel-concrete Composite Beams Curved in Plan

1/15

*Corresponding author. Tel.: 0065-874-2158.

E-mail address: [email protected] (V. Thevendran)

Finite Elements in Analysis and Design 32 (1999) 125}139

Nonlinear analysis of steel}concrete composite beams curvedin plan

V. Thevendran*, S. Chen, N.E. Shanmugam, J.Y. Richard Liew

Department of Civil Engineering, National University of Singapore 10 Kent Ridge Crescent, Singapore 119260, Singapore

Abstract

This paper deals with the behavior of structural steel}concrete composite beams curved in plan. The "nite

element package ABAQUS has been used to study the nonlinear behavior and ultimate load-carrying

capacity of such beams. A three-dimensional "nite element model has been adopted. Shell elements have

been used to simulate the behavior of concrete slab and steel girder, and rigid beam elements to simulate the

behavior of shear studs. The proposed "nite element model has been validated by comparing the computed

values with available experimental results. An acceptable correlation has been observed between the

computed and experimental results obtained for beams of realistic proportion. 1999 Elsevier Science B.V.

All rights reserved.

Keywords: Steel}concrete composite beams; Beams curved in plan; Ultimate load behavior; Nonlinear analysis; Finiteelement method

1. Introduction

I-girders curved in plan are frequently employed in structures such as highway bridges, inter-changes in large urban areas and balconies of buildings. Despite the advantages of composite

construction, engineers are reluctant to use curved composite girders in construction because ofmathematical complexities associated with geometry and material. Under gravity loading, beamscurved in plan are subjected to twisting moments in addition to #exural moments. In a highlycurved beam, the interaction between #exural and torsional stresses along the span length is rathercomplex. Using conventional analytical methods to analyze a structure with both geometric and

0168-874X/99/$- see front matter 1999 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 1 0 - 4


2/15

MSC-NASTRAN is a trademark of MacNeal-Schendler Corporation.

Nomenclature

The following symbols are used in this paper:c

" crack widthE

" Young's modulus of concreteE

" Young's modulus of steelf

" stress of concretef

" cube strength of concretef

" yield stress of steelf

" ultimate stress of steelf

" ultimate tensile stress of concreteG " shear modulusG

" elastic shear modulus of the uncracked concretel

" typical distance between cracksP

" ultimate loading

"direct strain across the crack

"strain of concrete

" strain of concrete at which the maximum compressive stress is reached

" strain of concrete at which the concrete crushes

" cU

/lA"0.005

" Poisson's ratio of concrete

" Poisson's ratio of steel

material nonlinearities might be di$cult, if not impossible. However, the availability of high-speed

digital computers makes it somewhat possible to study the complex nonlinear behavior of suchstructural elements and to account for in designs.In the past, researchers have used "nite element method to analyze the inelastic large-displace-

ment behavior of straight composite beams. They have used di!erent "nite element models tosimulate the behavior of concrete, steel and stud connectors in their studies of straight compositebeams. Hirst and Yeo [1] set up a two-dimensional model for use with standard "nite elementprograms. Four-noded plane elements were used to simulate the concrete slab and steel beam whilestandard quadrilateral elements were used in connecting the nodes on concrete part and steel part.The material properties of these quadrilateral elements were adjusted to make them equivalent inboth strength and sti!ness to the actual stud connector. In addition, since the main function of

shear connectors is to transfer shear force across the steel concrete interface, pin jointed barelements of e!ective in"nite sti!ness have been added to prevent transfer of direct stress across theinterface of the elements.

A three-dimensional "nite element model was proposed by Brockenbrough [2] to study curvedmultiple I-girder bridge using the software MSC/NASTRAN. In this three-dimensional model,the concrete deck had been modeled with QUAD4 shell elements, the girder #anges with BAR

126 V. Thevendran et al. /Finite Elements in Analysis and Design 32 (1999) 125}139


3/15

ABAQUS is a trademark of HKS.

elements (including axial and bending strains in two directions, and torsional e!ects), the girderweb with QUAD4 shell elements (four elements through depth of girder), and the connectorsbetween steel #ange to concrete deck with RBAR elements (rigid links connecting all degrees offreedom to simulate composite action with the slab). The concrete deck was treated uncrackedthroughout the bridge.

Razaqpur and Nafal [3] investigated the behavior of straight composite beam using the "nite

element software NONLACS [4] by adopting a three-dimensional "nite element model. Theyapplied facet shell elements on modeling concrete slab and steel beam. Shear stud connectorelements used in NONLACS permitted the modeling of full, partial, and no interaction at theinterface of the concrete slab and the steel beam. Tan et al. [5] and Liew et al. [6] adopteda three-dimensional "nite element model to study the behavior of steel I-girder curved in plan usingthe "nite element software ABAQUS [7]. The curved beam was discretized into small "bersconsisting of triangular and quadrilateral shell elements.

However, there is a lack of research "ndings about the elastic and ultimate load behavior ofcomposite beams curved in plan. The present study is concerned with the ultimate load behavior ofsuch beams. Five curved composite beams, which were tested earlier by the authors, have been

analyzed using ABAQUS software. The ultimate strength values, load-de#ection curves and stressdistribution across the section obtained using ABAQUS are compared with the correspondingexperimental results to verify the accuracy of the proposed "nite element model.

2. Experimental investigation

As a part of the present study, experiments were carried out on steel}concrete composite beamscurved in plan to investigate the behavior and to determine the ultimate failure load. A series of"ve

large-scale composite beams (SP1}

SP5) with span-length to radius of curvature (/R) ratiosranging from 0 to 0.5 were tested to failure under a concentrated load applied at midspan. Eachspecimen was 6.2 m long simply supported over a span of 6 m and consisted of a main girder andthree secondary beams. The main girder and secondary girders were made of UB356;171;57 kg/m. The concrete slab of all specimens was a normal weight concrete slab with overallthickness of 100 mm. The width of the slab was 1500 mm. The test setup was basically same forboth straight and curved specimens. The tests were carried out after the concrete had achieved itsdesign strength. A rest rig built on a strong #oor and capable of applying up to a maximum load of2000 kN was used to test the specimens. The rollers were inserted at one end of the specimen andthe supporting beam to ensure the simply supported condition. The transverse beams at the ends of

specimen were to simulate the intersection brace at support in practice. The transverse beam at themiddle was for transmitting concentrated load to the specimen more smoothly. The load wasapplied by a 1000 kN Shimadzu actuator as a monatonic concentrated load through a steel boxsection welded at the midspan of steel beam. The straight beam was instrumented for measurementof vertical de#ections along the length of the specimen, steel and concrete strains at midspan andthe slip between concrete slab and steel beam. For curved beams, transducers were mounted to

V. Thevendran et al. /Finite Elements in Analysis and Design 32 (1999) 125}139 127


4/15

Fig. 1. Typical "nite element mesh for composite beams curved in plan.

measure both vertical and lateral de#ections. Furthermore, concrete strains at quarter span werealso measured for SP3, SP4, and SP5. During testing, the steel beam was examined for yielding andthe surface of concrete was carefully inspected for cracks developing on the concrete surface.Testing was terminated when crushing of the concrete occurred or the de#ection became excessive-ly large and crack width excessively widened with the loading. The load}displacement curves,ultimate load and mode of failure were recorded for each specimen.

3. Finite element analysis

3.1. Finite element model

A three-dimensional "nite element model with the following characteristics had been used in thestudy:

1. Concrete slab } modeled by four-node isoparametric thick shell elements with the coupling ofbending and membrane sti!nesses.

2. Steel #ange and web } modeled by four-node isoparametric thin shell element with the couplingof bending and membrane sti!nesses.

3. Shear connectors between concrete slab and steel #ange } modeled by rigid beam elements.

Full composite action between steel beam and concrete slab was assumed. The beams were simplysupported at their ends. A typical three-dimensional model with 1257 elements used in the study isshown in Fig. 1.



5/15

Table 1Summary of coupon test results

Specimen Couponlocation

Young's modulus E

(GPa) Yield stress (MPa) Ultimate stress (MPa)

(E)

(E)

()

()

(

)

(

)

SP1 Flange 206.5 } 365.0 } 505.0 }Web 210.0 } 390.0 } 542.0 }

SP2 Flange 220.0 225.0 371.0 385.0 522.0 537.0Web 205.0 210.0 380.0 409.0 495.0 542.0

SP3 Flange 200.0 210.0 370.0 388.0 491.0 529.0Web 210.0 220.0 401.0 401.0 502.0 530.0

SP4 Flange 212.5 217.5 352.0 391.0 535.0 549.0Web 203.0 210.0 363.0 429.0 531.0 540.0

SP5 Flange 190.0 210.0 345.0 383.0 518.0 520.0Web 199.0 220.0 338.0 400.0 520.0 525.0

(E)"Modulus before bending; (E)

"Modulus after bending; (

)"Yield stress before bending; (

W)?@"Yield

stress after bending; (

)"Ultimate stress before bending; (

)"Ultimate stress after bending.

The rigid connection beam elements were used to model the shear studs in the "nite elementanalysis of the beams. This is based on the assumption that no slip occurs between the concrete slaband steel girder. In the experimental study, the interfacial slip at both ends between the slab and thetop #ange of steel girder was measured. The relative displacements at failure were found to benegligibly small for all specimens and the maximum value recorded was 0.09 mm and hence theslips could be ignored. The assumption of perfect bonding between the concrete slab and steel beam

in the analysis is, therefore, justi"ed.Coupon test results for steel beams in all test specimens are summarized in Table 1. For each

specimen, average values of Young's modulus and yield stress given in the table have been used inthe "nite element analyses of the test specimens.

3.2. Material modeling

Steel was assumed to behave as an elastic}plastic material with strain hardening in both tensionand compression. Strain hardening had been modeled based on incremental plasticity theory. Theidealized stress}strain curve used in the numerical analysis is shown in Fig. 2(a). The constitutive

relation curve for concrete both in compression and tension is shown in Fig. 2(b). The materialmodel of concrete has the following characteristics:

1. Compressive behavior. Concrete in compression is considered to be elasto-plastic and strain-hardening material. Its uniaxial compressive stress}strain curve is assumed to follow the expressiongiven below [8]:

f"f

2

!M

, (1)where f is the cylinder compressive strength of concrete in MPa. The strain,

, at which the

maximum compressive stress is attained, is taken as 0.002; the strain, , at which the concretereaches crushing, is taken as 0.0038.



6/15

Fig. 2. (a) Idealized uniaxial stress}strain relationships for steel. (b) Idealized uniaxial stress}strain relationships for concrete.

2. Tensile behavior. The constitutive relation for the tensile behavior of concrete is approximatedby two linear parts. The "rst linear part joins the origin (zero stress at zero strain) to the maximumuniaxial tensile stress, f

, at the strain at which the concrete cracks. A linear softening model is

assumed with the tensile stress decreasing with increasing tensile strain. Beyond the strain value at



7/15

which f

is attained, the tensile stress is assumed to decrease linearly from f

to zero in order tore#ect the softening of the concrete due to crack.

3. Shear retention. The loss of shear modulus due to cracks is taken into account by usinga multiplying factor, which de"nes the modulus for shearing of cracks as G"G

, where G

is the

elastic shear modulus of the uncracked concrete. The shear retention model assumes that the shearsti!ness of open cracks reduces linearly to zero as the crack opening widens. The multiplying factor

is

"1!

for 0)(, (2)"0 for *

, (3)

where is the direct strain across the crack and

is the value given on the data card of theoption. Cedolin and Poli [9] proposed that the slope of the shear stress}shear displacement curvedecreases with crack width. The crack width for which the slope becomes zero is given asc"0.75 mm. Referring this magnitude to a typical distance between cracks in the real structures,

l"

150 mm, "

c/lA"

0.005. The model also assumes that cracks which subsequently closehave a reduced shear modulus where is assumed as 0.95.

3.3. Analysis parameters

The material properties of steel are speci"ed using the elastic}plastic with strain hardeningoptions. ABAQUS requires for this purpose the input of the Young's modulus of steel, E

; Poisson's

ratio, ; the yield stress of steel, f

; and the ultimate stress of steel f

. For concrete, the elastic

properties are de"ned by the elastic option and its compressive stress}strain relationship outsidethe elastic range is speci"ed using the concrete option. The values of Young's modulus, E

;

Poisson's ratio, ; and the values off for several values of and the corresponding values given by(!f

/E

) are required as input. To de"ne the shape of the failure surface of concrete, the ultimate

stress and strain values in uniaxial and biaxial stress states are speci"ed using the failure ratiosoption. The ratio of the ultimate biaxial compressive stress to the uniaxial compressive ultimatestress is taken as 1.16. The absolute value of the ratio of uniaxial tensile stress to the uniaxialcompressive stress at failure is taken as 0.1. The ratio of the magnitude of a principal component ofplastic strain at ultimate stress in biaxial compression to the plastic strain at ultimate stress inuniaxial compression is taken as 1.28.

The nonlinear response of a beam under loading has been analyzed using the Newton's iterativetechnique. The initial and "nal load increments as well as the allowable minimum and maximum

load increments are required in input "le. The program iterates to obtain the equilibrium conditionfor the initial load increment and the subsequent increment amplitudes are automatically adjustedby using the modi"ed Riks method [10] in conjunction with the modi"ed Newton}Raphsonmethod.

3.4. Convergence study

Convergence studies have been carried out separately on a straight composite beam and ona curved composite beam in order to determine a suitable "nite element model for the analysis. The



8/15

Fig. 4. Convergence study } load}vertical displacement curves of SP4.

Fig. 3. Comparison of"nite element and experimental load displacement curves for the straight composite beam [11].

convergence study for straight composite beam has been carried out on a beam that was tested byChapman and Balakrishnan [11]. The beam 5.5 m long consisted of concrete slab 1220 mm wideand 152 mm deep and a steel I-beam section 12 in;6 in;44 lb. B.S.B. The material propertieswere: (a) concrete: compressive strength, f

"50 MPa; tensile strength, f

"5 MPa; Young's

modulus, E"26.7 GPa; compressive strain under maximum stress,

"0.003; ultimate compres-

sive strain, "0.0045 and (b) steel: yield stress, f

"240 MPa; Young's modulus, E

"184 GPa;

Poisson's ratio, "0.3. Three independent convergence studies had been carried out on the meshsizes for concrete slab, steel web, and along the beam span, respectively. Based on the results fromthese convergence studies, a mesh with 8 elements along concrete slab width, 4 elements along steelweb depth and 35 elements along the span was adopted for "nite element analysis. The di!erencebetween the values of ultimate load obtained by analysis and experiment is about 8% as shown inFig. 3. To analyze the straight composite beam tested in the present study, however, a mesh with 10elements along the width of the concrete slab, 4 elements along the steel web and 46 elements alongthe span (total 1257 elements including connection rigid beam elements) was chosen.

In the convergence study of SP4 which is a composite beam curved in plan, the resultscorresponding to three di!erent meshes involving a total of 837, 1257, 1677 elements were

compared. The results of comparison are illustrated in Fig. 4. The di!erence between the ultimate



9/15

Table 2

Comparison of ultimate load predicted by ABAQUS with experiment

Specimen

R

Ultimate load P

(kN) (P

)/31

(P

)#6.2(P

)/31

(P

)#6.2

SP1 0.00 450 490 0.92

SP2 0.05 422 448 0.94

SP3 0.10 430 460 0.93

SP4 0.25 378 438 0.86

SP5 0.50 205 235 0.87

strength corresponding to 837 elements and 1257 elements is about 6%, and that between thevalues corresponding to 1257 elements and 1677 elements is less than 1%. The two curvescorresponding to the modeling with 1257 elements and 1677 elements lie very close throughout theloading cycle. Therefore, "nite element analysis based on 1257 elements seems to be adequate inpredicting the elastic as well as ultimate load behavior of curved composite beam. Such a mesh hasbeen adopted in the "nite element modeling for all the composite beams curved in plan.

4. Results and discussion

The "nite element analyses give detailed picture of the complete behavior of the beams fromelastic to ultimate load. The stress distribution across the cross sections and along the span,de#ected pro"les of the beam and ultimate load behavior can be obtained from the analysis.However, distribution at selected locations, de#ection pro"le and failure load are chosen fordiscussion herein.

The analytical values of the ultimate loads of "ve beams are summarized along with the

corresponding experimental values in Table 2. The comparisons between experimental and "niteelement values are also presented in the table. It can be seen that the "nite element predictions forall beams are in relatively close agreement with the corresponding experimental results. Themaximum deviation is about 14%. The "nite element modeling underestimates the ultimate load inmost cases. It appears, therefore, that the "nite element model used in the analysis is reliable and itis conservative in predicting the ultimate strength of composite curved beams.

Load}de#ection curves for the specimens SP3, SP4 and SP5 are shown in Fig. 5 in which thecorresponding experimental curves are superimposed. The theoretical and experimental curves lievery close to each other at initial stages for all the three specimens. However, there seems to be

some deviation between the results near the failure. The discrepancy may be due to the inadequacyin concrete modeling. The concrete is not a homogeneous material and the concrete material modelused in the analysis signi"cantly simpli"es the actual behavior. Furthermore, the smeared &crack'concept is used for numerical modeling of crack initiation and crack propagation. This smeared&crack' model does not track individual `macroa cracks. Instead, constitutive calculations areperformed independently at each integration point of the "nite element model. However, when



10/15

Table 3

Comparison initial crack load and yielding load predicted by ABAQUS with experiment

Specimen Initial yield load P

(kN) Initial crack load P

(kN)

(P)#6.2

(P)/31

(P)/31

(P)#6.2

(P)#6.2

(P)/31

(P)/31

(P)#6.2

SP1 325 350 1.07 440 412 0.93

SP2 315 340 1.08 405 410 1.01

SP3 310 320 1.03 325 400 1.23

SP4 305 300 0.98 280 250 0.89

SP5 220 200 0.91 140 100 0.71

Fig. 5. Load}vertical displacement curves of composite curved beams SP3, SP4 and SP5.

a crack developed in the concrete during the experiment, the moment of inertia of the whole sectiondecreased and the ratio of the deformation to loading increased considerably.

Loads corresponding to initial crack in concrete and initial yield in steel beam determinedanalytically are presented in Table 3. The corresponding experimental values are also given in thetable for comparison. It can be seen that the yield loads predicted by analysis are somewhat largerthan the values obtained from the experiment and the maximum deviation is about 10%. The

deviation for initial cracking load is between!29% and 23%. The results show that ABAQUScan predict yield loads well but cannot predict crack loads accurately.

Even though the "nite element analysis provides a detailed picture of the de#ection pro"le alongthe span and tangential stress distribution at a number of locations for di!erent stages of loadings,only selected sets of results are presented for brevity. Results for the beams SP3 and SP4 arepresented for further consideration. The variation of in-plane de#ected pro"le along the curved



11/15

Fig. 7. Variation of vertical de#ection along the curved length for SP4.

Fig. 6. Variation of vertical de#ection along the curved length for SP3.

length at three di!erent loading stage (viz. 150, 200 and 250 kN, respectively) is shown in Figs.6 and 7. The corresponding experimental results are also given in the "gures. Generally, goodagreement is observed between the experimental and analytical values for all three loading stages.

Fig. 8(a) and Fig. 9(a) show the analytical variations of tangential stresses across the slab width atmidspan and at quarter span of SP3 and SP4, respectively, corresponding to an applied load of200 kN and the experimental values are superimposed in the "gures. Fig. 8(b) and 9(b) show

tangential stresses across steel section at midspan and at quarter span of SP3 and SP4, respectively;the results obtained from the "nite element modeling for an applied load of 200 kN are given alongwith the corresponding experimental values. Generally good agreement between analytical andexperimental values is observed. The predicted stress values for steel are somewhat smallercompared to experimental values in some cases. In concrete part, discrepancies are observedbetween the predicted values and the experimental values. The predicted values are smaller thanthose obtained from experiment. It may be due to the following two reasons: (i) since the concrete isnot a homogeneous material, the concrete material model in the analysis is a simpli"ed model andcannot re#ect the true behavior; (ii) the strength of concrete at top surface is less than that at theinside since the concrete slab is cast vertically and thus the aggregates and sand may not be evenly

distributed across the depth of the slab.



12/15

Fig. 8. (a) Tangential stress distribution across the width of concrete slab at midspan and quarterspan of SP3 for an

applied load of 200 kN. (b) Tangential stress distributions in steel beam section at midspan SP3 for an applied load of

200 kN.



13/15

Fig. 9. (a) Tangential stress distribution across the width of concrete slab at midspan and quarterspan of SP4 for an

applied load of 200 kN. (b) Tangential stress distributions in steel beam section at midspan of SP4 for an applied load of

200 kN.



14/15

Although some discrepancies exist between stress values predicted by analysis and thoseobtained experimentally at the concrete surface, the trend in stress variation is same. It can be seenthat the tangential stresses at the top of concrete slab in outer curvature side at midspan are greaterthan those at inner curvature side. At quarter span, the stress distribution along slab width isreversed and the tangential stresses at outer curvature side are less than those at inner curvatureside.

5. Concluding remarks

Finite element modeling of structural steel}concrete composite beams curved in plan is present-ed in this paper. The nonlinear behavior of composite beams has been studied with reference tothose beams tested earlier by the authors. The software package ABAQUS was employed in theanalysis. Load}de#ection curves, de#ection pro"le, ultimate strength values and tangential stressdistribution across the cross section were obtained from the "nite element analysis. These results

have been compared with the corresponding results obtained from the experiments. The closeagreement between the "nite element and experimental results has been observed. The maximumdeviation in the prediction of ultimate strength has been found to be 14% and the resultsestablished the validity of the proposed "nite element model. In addition, "nite element methodprovides extensive information on the behavior of these beams up to failure.

As discussed in Section 4, concrete model in ABAQUS is quite simpli"ed and results indiscrepancies between "nite element and experimental values. Correct modeling especially inconcrete is, therefore, essential in order to ensure more accuracy in theoretical prediction. Thedevelopment of such an analytical model will be useful to designers as it will save from the need ofcarrying out expensive and time consuming full-scale tests to predict the behavior of composite

curved beams.

Acknowledgements

The authors gratefully acknowledge the research grant (RP940660) provided by the NationalUniversity of Singapore towards this study.

References

[1] M.J.S. Hirst, M.F. Yeo, The analysis of composite beams using standard "nite element programs, Comput. Struct.

11 (3) (1980) 233}237.

[2] R.L. Brockenbrough, Distribution factors for curved I-girder bridges, J. Struct. Eng. ASCE 112 (10) (1986)

2200}2215.

[3] A.G. Razaqpur, M. Nofal, A "nite element for modeling the nonlinear behaviour of shear connectors in composite

structures, Comput. Struct. 32 (1) (1989) 169}174.

[4] M. Nofal, Inelastic load distribution of composite concrete-steel slab-on girder bridges: an analytical and

experimental study. MEng Thesis, Department of Civil Engineering, Carlton University, Ottawa, 1988.



15/15

[5] L.O. Tan, V. Thevendran, J.Y.R. Liew, N.E. Shanmugam, Analysis and design of I-beams curved in plan, J.

Singapore Struct. Steel Soc. Steel Struct. 3 (1) (1992) 39}45.

[6] J.Y.R. Liew, V. Thevendran, N.E. Shanmugam, L.O. Tan, Behaviour and design of horizontally curved steel beams,

J. Construct. Steel Res. 32 (1995) 37}67.

[7] ABAQUS Theory Manual Version 5.4, ABAQUS User's Manual Version 5.4, Hibbit, Karlsson and Sorensen, Inc.,

USA, 1994.

[8] R. Park, T. Paulay, Reinforced Concrete Structure, Wiley, Canada, 1975.

[9] L. Cedolin, S.D. Poli, Finite element studies of shear-critical R/C beams, ASCE J. Eng. Mech. Div. 103 (3) (1977)395}410.

[10] G. Powell, J. Simons, Improved iterative strategy for nonlinear structures, Int. J. Numer. Methods Eng. 17 (1981)

1455}1467.

[11] J.C. Chapman, S. Balakrishnan, Experiments on composite beams, Struct. Eng. 42 (11) (1964) 369}383.


Documents

Nonlinear Analysis of Steel-concrete Composite Beams Curved in Plan