Experimental and Numerical Study on the Impact Strength of Beams and Plates

EXPERIMENTAL AND NUMERICAL STUDY ON THE IMPACT

STRENGTH OF BEAMS AND PLATES

Bin Liu

Thesis for obtaining the degree of Master in

Naval Architecture and Marine Engineering

Jury

President : Prof. Yordan Ivanov Garbatov

Supervisor : Prof. Carlos Antonio Pancada Guedes Soares

Member : Dr. Leigh Stuart Sutherland

December 2011

Abstract

i

Abstract

The structural design of ships concerning collision requires an accurate prediction of the energy

absorption and damage of plates and stiffeners under impact loading. Thus, in this thesis experimental

and numerical analyses are conducted in order to study the impact plastic response and failure of

structural components, such as beams, plates and stiffeners with attached plate. The experimental

program includes drop weight impact tests in a fully instrumented falling weight machine using different

types of indenters. The numerical simulations are carried out using the finite element package

LS-DYNA Version 971 which is appropriate for nonlinear explicit dynamic simulations with large

deformations.

The assumptions adopted in the numerical model are validated by means of drop weight impact

tests. The impact behavior of the structural elements is validated by comparison of their experimental

and numerical force-displacement responses. Special attention is paid in the definition of the true

material properties and the representation of the experimental boundary conditions. Thus, the plastic

response and failure of the material is calibrated by numerical simulation of tensile tests used to obtain

the mechanical properties of the material, and the experimental boundary conditions are partially

represented by coarse mesh of rigid shell elements.

Keywords: Impact; Experiment; Numerical simulation; Beam; Plate; Stiffener with attached plate;

Boundary condition; Force-displacement response; Axial displacement.

Abstract

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Resumo

iii

Resumo

O projeto estrutural de navios resistentes a colisão precisa previsão do absorção de energia e dano

das placas e reforços sujeitos a impacto. Nesta tese análises experimentais e numéricas são

realizadas para estudar a resposta plástica ao impacto e fractura dos componentes estruturais, tais

como vigas, placas e placas reforçadas. O programa experimental inclui testes de impacto

instrumentado usando diferentes tipos de projectil. As simulações numéricas são realizadas no

software de elementos finitos LS-DYNA versão 971 que é apropriado para simulações dinâmicas com

grandes deformações.

As premissas adotadas no modelo numérico são validada por meio de testes de impacto. O

comportamento dos elementos estruturais ao impacto é validado por comparação da resposta de

força-deslocamento entre ensaios e simulações numéricas. Atenção é dada na definição do material e

na representação das condições de fronteira. A resposta plástica e a fratura do material são calibrados

com simulações numéricas dos ensaios de tração utilizados para obter as propriedades mecânicas do

material, e as condições de contorno experimentais são representadas com elementos de placa rígida.

Palabra chave: Impacto; Experimento; Simulação numérica; Viga; Placa; Placa reforçada;

Condição de fronteira; Resposta de força-deslocamento; Deslocamento longitudinal.

Resumo

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Acknowledgements

v

Acknowledgements

The author would like to thank Professor Carlos Guedes Soares and my friend Richard Villavicencio for

all the guidance and unlimited help during the research and completion of this thesis. The author would

also like to thank his family and friends for all their patience, understanding and support.

Acknowledgements

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Table of contents

vii

Table of contents

Abstract......................................................................................................................................................i

Resumo ................................................................................................................................................... iii

Acknowledgements ..................................................................................................................................v

Table of contents .................................................................................................................................... vii

List of figures ........................................................................................................................................... xi

List of tables ........................................................................................................................................... xv

Nomenclature ....................................................................................................................................... xvii

CHAPTER 1 Introduction ......................................................................................................................... 1

1.1 Overview and background ......................................................................................................... 1

1.2 Objectives and scope of the work .............................................................................................. 5

CHAPTER 2 Nonlinear finite element simulation .................................................................................... 7

2.1 Material stress-strain relationship .............................................................................................. 7

2.1.1 Engineering stress-strain curve ....................................................................................... 7

2.1.2 True stress-strain curve ................................................................................................... 8

2.1.3 Mathematical expressions for the true material curve ..................................................... 9

2.2 Dynamic yield strength ............................................................................................................. 10

2.3 Dynamic fracture strain ............................................................................................................ 12

2.3.1 Failure criteria ................................................................................................................ 12

2.4 Contact-impact algorithm ......................................................................................................... 13

2.4.1 Kinematic constraint method .......................................................................................... 14

2.4.2 Penalty method .............................................................................................................. 14

2.4.3 Initial contact Interpenetrations ...................................................................................... 15

2.4.4 Friction definition ............................................................................................................ 15

Table of contents

viii

2.4.5 Contact automatic surface to surface ............................................................................ 15

2.5 Mesh size ................................................................................................................................. 15

2.6 Simulation of tensile tests ........................................................................................................ 16

2.7 Representation of experimental supports ................................................................................ 19

CHAPTER 3 Plastic response of beams subjected to lateral impact .................................................... 21

3.1 Impact test machine ................................................................................................................. 21

3.2 Impact tester software .............................................................................................................. 22

3.3 Experimental details ................................................................................................................. 23

3.4 Experimental results ................................................................................................................. 25

3.5 Numerical model ...................................................................................................................... 27

3.6 Numerical results...................................................................................................................... 28

3.7 Concluding remarks ................................................................................................................. 31

CHAPTER 4 Failure prediction of pre-notched beams subjected to lateral impact .............................. 33



4.3 Numerical model ...................................................................................................................... 37

4.3.1 Boundary conditions ...................................................................................................... 38

4.3.2 Material definition using tensile test simulation .............................................................. 38


4.5 Comparison with a theoretical analyses .................................................................................. 42


CHAPTER 5 Plastic response of rectangular plates subjected to lateral impact .................................. 45



Table of contents

ix

5.3 Numerical model ...................................................................................................................... 49

5.3.1 Boundary conditions ...................................................................................................... 49


5.4.1 Thin plates ...................................................................................................................... 49

5.4.2 Thick plates .................................................................................................................... 53


CHAPTER 6 Failure prediction of rectangular plates subjected to lateral impact ................................. 57

6.1 Experimental details and results .............................................................................................. 57

6.2 Numerical model and results .................................................................................................... 58


CHAPTER 7 Plastic response of stiffeners with attached plate subjected to lateral impact ................. 63



7.3 Numerical model ...................................................................................................................... 65



CHAPTER 8 Conclusions and further work ........................................................................................... 69

8.1 Conclusions .............................................................................................................................. 69

8.2 Future work .............................................................................................................................. 70

References ............................................................................................................................................ 71

Table of contents

x

List of figures

xi

List of figures

Figure 2.1 Engineering stress-strain curve. Intersection of the dashed line with the curve

determines the offset yield strength. (cf. Dieter 1986) .............................................................. 7

Figure 2.2 Engineering and true stress-true strain curves. (cf. Dieter 1986) ................................... 8

Figure 2.3 Log-log plot of true stress-strain curve n is the strain-hardening exponent; K is the

strength coefficient. (cf. Dieter 1986) ...................................................................................... 10

Figure 2.4 Dynamic yield strength, σYd (normalized by the static yield strength, σY), plotted versus

strain rate, εɺ , for mild and high-tensile steels. (cf. Paik 2007). ............................................ 11

Figure 2.5 Dynamic fracture strain (normalized by the static fracture strain) versus strain rate for

mild steels. (cf. Paik 2007) ...................................................................................................... 12

Figure 2.6 Nodes of the master slide surface designated with an “x” are treated as free surface

nodes in the nodal constraint method. (cf. Hallquist 2010) ..................................................... 14

Figure 2.7 Undetected interpenetration. Such interpenetrations are frequently due to the use of

coarse meshes. (cf. Hallquist 2010) ........................................................................................ 15

Figure 2.8 Finite element model. (cf. Villavicencio and Guedes Soares 2011c) ........................... 17

Figure 2.9 Time steps of a typical tensile test simulation. (cf. Villavicencio and Guedes Soares

2011c) ..................................................................................................................................... 18

Figure 2.10 Fracture propagation. (cf. Villavicencio and Guedes Soares 2011c) ......................... 18

Figure 2.11 Sketch of end view of rectangular test piece after fracture showing constraint at

corners indicating the difficulty of determining reduced area. (cf. Villavicencio and Guedes

Soares 2011c) ......................................................................................................................... 18

Figure 2.12 Models of boundary conditions. (cf. Villavicencio and Guedes Soares in press a) .... 19

Figure 3.1 Fully instrumented Rosand IFW5 falling weight machine. (Drawing provided by R.

Villavicencio) ........................................................................................................................... 21

Figure 3.2 Experimental results of force-time, displacement-time, absorbed energy-time and

force-displacement. (cf. Villavicencio and Guedes Soares 2011a) ........................................ 23

Figure 3.3 Beam struck transversely by a mass. ........................................................................... 23

Figure 3.4 Experimental set-up. ..................................................................................................... 24

Figure 3.5 Deformed shape of beams after the impact.................................................................. 26

Figure 3.6 Deformation of the beam impacted at velocity 2.0 m/s. ................................................ 26

Figure 3.7 Force-displacement responses using different torque on the bolts. Impact velocity 1.0

m/s........................................................................................................................................... 27

Figure 3.8 Numerical model. .......................................................................................................... 27

List of figures

xii

Figure 3.9 Engineering and true stress-strain curve of material. ................................................... 28

Figure 3.10 Force-displacement responses with different friction coefficient at the supports. ...... 29

Figure 3.11 Maximum force and displacements at different impact velocities............................... 29

Figure 3.12 Shape of the deformation and von Mises stress distribution at maximum force. ....... 29

Figure 3.13 Axial displacements at the supports at different velocity. ........................................... 30

Figure 3.14 Experimental and numerical force-displacement responses using different torque at

the bolts. Experimental results: dashed lines. Numerical results: continuous lines. Impact

velocity 1.0 m/s. ...................................................................................................................... 30

Figure 3.15 Axial displacements at the supports using different torque at the bolts. Impact velocity

1.0 m/s. ................................................................................................................................... 30

Figure 4.1 Pre-notched beam stuck transversely by a mass. ........................................................ 33


Figure 4.3 Measure the depth of notch. ......................................................................................... 34

Figure 4.4 Position of notch and striking mass. ............................................................................. 35

Figure 4.5 Tension failure. (a): Specimen N115_4mm_Q; (b) Specimen N15_4mm_M. .............. 36

Figure 4.6 Shear failure. (a): Specimen N75_4mm_Q; (b) Specimen N115_4mm_M. ................. 36

Figure 4.7 Shape of the deformation: (a) Specimen N15_2mm_M; (b) Specimen N15_4mm_M. 36

Figure 4.8 Experimental results of force-time, displacement-time, absorbed energy-time and

force-displacement. P: Specimen N15_2mm_M (plastic deformation); F: Specimen

N15_4mm_M (fracture). .......................................................................................................... 37

Figure 4.9 Maximum forces of beams with different notch. ........................................................... 37

Figure 4.10 Mesh sizes of beam and striking mass. ...................................................................... 38

Figure 4.11 Boundary conditions of pre-notched beam. ................................................................ 38

Figure 4.12 True and engineering material curves. ....................................................................... 39

Figure 4.13 Results of numerical simulations. ............................................................................... 39

Figure 4.14 Comparison of experimental and numerical force-displacement responses: (a)

Specimen N15_2mm_M; (b) Specimen N15_4mm_M. .......................................................... 40

Figure 4.15 Comparisons of experimental and numerical force-displacement curves with different

boundaries (Specimen N15_4mm_M). ................................................................................... 40

Figure 4.16 Failure modes (Specimen N15_4mm_M). .................................................................. 40

Figure 4.17 Time steps of a typical simulation of the pre-notched beams (Specimen N15_4mm_M).

................................................................................................................................................ 41

List of figures

xiii

Figure 4.18 The failure of the notch (Specimen N15_4mm_M). (a): Numerical; (b): Experimental.

................................................................................................................................................ 41

Figure 4.19 Comparison of the numerical triaxiality (Specimen N15_4mm_M)............................. 42

Figure 4.20 Variation of dimensionless maximum permanent transverse deformation /fW H with

dimensionless external dynamic energy λ . —— Equation (4.1) with static yield stress σ0; - - -

- Equation (4.1) with dynamic flow stress σ0׳ given by equation (4.1); (1) circumscribing yield

curve (2) inscribing yield curve; Experimental results: ▲ Simulation results: ■ ................. 43


Figure 5.2 Different types of indenters. .......................................................................................... 46

Figure 5.3 Engineering stress-strain curve of material with thickness 1.4 and 4.0 mm. ................ 46

Figure 5.4 Force-displacement responses of thin plates at different velocity. ............................... 47

Figure 5.5 Deformation of thin plate at velocity of 2.7 m/s. ............................................................ 47

Figure 5.6 Experimental force-displacement responses using different diameters of indenters. .. 48

Figure 5.7 Experimental force-displacement responses using different types of indenters with the

same diameter. ....................................................................................................................... 48

Figure 5.8 Boundary conditions of rectangular plate. .................................................................... 49

Figure 5.9 Comparison of different materials. ................................................................................ 50

Figure 5.10 Comparison of the shape of deformation. ................................................................... 50

Figure 5.11 Comparison of different mesh sizes. .......................................................................... 51

Figure 5.12 Comparison of shell model and solid model. .............................................................. 51

Figure 5.13 Comparison of different support. ................................................................................ 51

Figure 5.14 Comparison of different dynamic yield strength. C40.4q5: mild steel coefficients;

C3200q5: high tensile steel coefficients. ................................................................................. 52

Figure 5.15 Position of selected element. ...................................................................................... 52

Figure 5.16 Strain rate of selected elements from numerical simulation. ...................................... 53

Figure 5.17 Force-displacement responses of the clamped and supported models. (Impact velocity

1.94 m/s) ................................................................................................................................. 53

Figure 5.18 Force-displacement responses of experimental results and numerical results. E:

experimental, S: Simulation .................................................................................................... 54

Figure 5.19 Different type of indenter............................................................................................. 54

Figure 5.20 Comparison of numerical results using different type of indenter. (Impact velocity 1.94

m/s) ......................................................................................................................................... 54

List of figures

xiv

Figure 6.1 Experimental force-displacement responses of thin plates with different indenter. ...... 57

Figure 6.2 Failure modes of the plates. (a) indenter 20 mm; (b) indenter 30 mm ......................... 58

Figure 6.3 Force-displacement responses of experimental and numerical results with different true

material curves. (Indenter 30 mm) .......................................................................................... 58

Figure 6.4 Experimental and numerical failure modes. (Indenter 30 mm) ..................................... 59

Figure 6.5 Force-displacement responses of shell model and solid model. (Indenter 30 mm) ..... 59

Figure 6.6 Force-displacement responses of different support. (Indenter 30 mm) ........................ 60

Figure 6.7 Force-displacement responses with different failure strain. (a): indenter 10 mm; (b):

indenter 16 mm; (c): indenter 20 mm; (d): indenter 30 mm; ................................................... 61

Figure 6.8 Force-displacement responses with different mesh size and corresponding failure strain.

(a) indenter 10 mm; (a) indenter 16 mm; (a) indenter 20 mm; (a) indenter 30 mm. ............... 61

Figure 7.1 Experimental set-up ...................................................................................................... 63

Figure 7.2 Specimens: stiffeners with attached plate .................................................................... 63

Figure 7.3 Force-displacement responses ..................................................................................... 65

Figure 7.4 Force-displacement responses. Panel A2 and Panel A3 impacted at 2.7 m/s ............. 65

Figure 7.5 Details of finite element model ...................................................................................... 65

Figure 7.6 Previous and new finite element models ...................................................................... 66

Figure 7.7 Force-displacement response, Specimen A2V2.7. (E): Experimental. (1): Shell. (2):

Solid. (3): Shell Weld. (4): Solid Weld. .................................................................................... 66

Figure 7.8 Force-displacement response, Panel A2. Experimental results: dashed lines. Numerical

results: continuous lines (Solid Weld model). ......................................................................... 67

Figure 7.9 Shape of deformation and von mises stress distribution. Panel A2. (a) Transversal view;

(b) Longitudinal view. .............................................................................................................. 67

List of tables

xv

List of tables

Table 2.1 Sample coefficients for the Cowper--Symonds constitutive equation (cf. Paik 2007) ... 11

Table 2.2 Static and sliding coefficient of friction (cf. Hallquist 2010) ............................................ 15

Table 3.1 Mechanical properties of material .................................................................................. 25

Table 3.2 Summary of experimental results at different impact velocity ........................................ 25

Table 3.3 Axial displacements at the supports .............................................................................. 26

Table 3.4 Experimental results using different torques on the bolts. (Impact velocity 1.0 m/s) ..... 26

Table 3.5 Axial displacements at the supports using different torque on the bolts. (Impact velocity

1.0 m/s) ................................................................................................................................... 27

Table 4.1 Experimental results ....................................................................................................... 35

Table 5.1 Mechanical properties of material .................................................................................. 46

Table 5.2 Summary of experimental results of thin plates at different velocity .............................. 47

Table 5.3 Summary of experimental results using different indenters (Impact velocity 1.94 m/s). 48

Table 6.1 Summary of experimental results of thin plates with different indenter ......................... 58

Table 6.2 Mesh size with corresponding failure strain ................................................................... 60

Table 7.1 Mechanical properties of the material. ........................................................................... 64

Table 7.2 Results of impact tests ................................................................................................... 64

List of tables

xvi

Nomenclature

xvii

Nomenclature

A True area

A0 Original area

Ag Maximal uniform strain

B Width

D Diameter

E Yong’s modulus

e Natural logarithmic constant

G Mass

H Thickness

K Torque coefficient

L True length

L0 Original length

P Load

Rm Ultimate tensile stress

T Torque

V0 Initial velocity

Wffff Maximum permanent transverse deformation

λ Dimensionless external dynamic energy

σeng Engineering stress

σt True stress

σY Static yield stress

σYd Dynamic yield stress

εeng Engineering strain

εt True strain

εɺ Strain rate

Nomenclature

xviii

CHAPTER 1 – Introduction

1

CHAPTER 1 Introduction

1.1 Overview and background

Ship grounding and collision could cause loss of human lives and severe environmental damage. In

order to minimize these consequences, it is necessary to design crashworthy marine structures. The

structural design of ships concerning collision requires an accurate prediction of the damage of ships

under impact loading. Finite element analysis is a useful tool to predict the extent of ship collision and

consequent damage to structural components. However, the nonlinear dynamic analysis should be

compared with experimental tests before being used for structural design. Unfortunately, experimental

tests on full scale ship collision are rare and very expensive. One approach is to perform scaled

collision test on typical ship structural members to validate the numerical methods for impact analysis.

Theoretical and experimental analyses of individual ship structural components under lateral

impact loads, such as beams and plates, have been widely analyzed. However, comparison between

experiments and numerical simulations still require investigation especially when fracture occurs. Thus,

this thesis aims at summarizing definitions adopted in numerical models that use lateral impact load to

reproduce the experimental impact response until maximum load and fracture.

Studies of the energy absorption of different structural elements have been made for a long time in

order to understand the basic mechanisms associated with large plastic deformations. Initially studies

were made comparing rigid plastic theory with experiments and more recently comparisons have been

made with finite element results, which allow obtaining detailed information of the structural response of

the specimens.

Concerning to the behavior of beams, Parkes (1955) studied the deformation of a cantilever

modeled as a rigid-plastic material struck transversely at its tip by a moving mass. The theoretical and

experimental results were compared, finding good agreement at points remote from the impact, and

concluding that the prediction of local damage depends on accurate definition of the boundary

conditions at the striking point. It was found that the motion producing the deformation can be divided

into three phases, which were represented by curves that define the plastic behavior in each phase.

Menkes and Opat (1973) conducted an experimental study on the dynamic plastic response and

failure of fully clamped beams subjected to different velocities over the entire span, proposing three

basic failure modes for fully clamped beams: large inelastic deformation, tensile tearing and transverse

shear failure at the supports.

Jones (1973) conducted an approximate theoretical study in order to examine the influence of axial

displacement at the supports of rotationally fixed and rotationally free rigid perfectly plastic beams.

Jones found that small in-plane displacements at the boundaries can change the response from a

completely restrained beam to one with complete axial freedom and no increase in strength beyond the

limit load.

Based on the theoretical analysis of Jones, Hodge (1974) re-examined those loaded beams with a

more physical basis, proposing that a simply supported beam can undergo a relative axial


2

displacement proportional to the induced axial force. Hodge proposed complete formulae and curves

for transverse displacements of the order of the beam thickness.

Complete load-deflection relationships of a rigid-plastic beam loaded through a rigid circular

indenter at mid-span were derived by Low (1981). In that analysis the axial displacements at the

supports and the geometrical effects of large rotations were considered, concluding that the elasticity of

the beam material has a similar effect to the elastic displacement at the supports.

Gürkök (1981) studied the influence of finite deflections on the behavior of rigid perfectly-plastic

beams with support conditions specified by an axial and a rotational constraint factor. He first proposed

a complete solution for axially-restrained rectangular beams with a certain degree of rotational

end-fixity constraint. Then, the restriction of complete axial restraint was removed and a separate

analysis was performed considering the effect of horizontal displacements at the supports. It was found

that the load-carrying capacity is considerably increased showing a strong dependence on the axial

restraint provided at the supports.

Similar studies were conducted by Tin-Loi (1990) proposing load-deflection relationships of a

beam subjected to a non-symmetrically placed point load. In the analysis the supports were considered

capable of combine axial and rotational restraints concluding that the increase in load-carrying capacity

with deflection can be quite large for certain support conditions.

Numerical studies on the dynamic failure of beams have been conducted, proposing theoretical

predictions on the initial impact energy for the different failure modes of beams using various failure

criteria. However, experimental studies are still very important in order to depth study the failure modes

of structures, to propose the more accurate theoretical analysis predictions on the dynamic failure of

structure and to verify the finite element simulations.

One of the first numerical simulations using beam elements was performed by Symonds and

Fleming (1984). They examined the problem of rigid-plastic structural dynamics, finding the

deformations of a beam carrying a mass at its tip which was subjected to a short pulse loading.

A finite element analysis of a clamped aluminum beam struck transversely by a mass was

presented by Yu and Jones (1989). As in previous experiments, two specimen types were modeled,

one with enlarged ends and the other with flat ends. The numerical predictions agreed with the

experimental results, however significant differences were found between the behavior of flat end

beams and beams with enlarged ends.

Experimental investigations on the failure of clamped steel beams under impact loads were

simulated numerically by Yu and Jones (1997). They observed that the large deformations of the

beams in the experimental study caused the upper surfaces of the beams to lose contact with the

supports.

Chen and Yu (2004) conducted experiments aiming to systematically investigate the failure

behavior of clamped beams with one pre-notch or two pre-notches under impact loading from a

projectile strike.


3

Dimas and Guedes Soares (2006) performed numerical studies of the absorbed energy in

clamped steel beams under transverse impact at the mid-span and along their length by relatively

heavy masses and compared them with experiments.

Harsoor and Ramachandra (2009) reported the experimental and numerical results on deformation

and failure of clamped mild steel beams with and without notches subjected to low velocity impact.

Analysis of the response of clamped beams to impact loading along their length, for various

thicknesses, was performed by Villavicencio and Guedes Soares (2009). They compared the

experimental tests with the theoretical rigid plastic analysis proposed by Liu and Jones (1987) and with

a finite element analysis. Good agreement was found between them, although in the numerical

simulations the material was defined as a rigid perfectly plastic material, which does not give a true

indication of the deformation characteristics of the metal.

Villavicencio and Guedes Soares (2011a) presented a numerical model to simulate the

experimental boundary conditions of beam impacted along its span, proposing rigid shell support plate

simulating the pressure on the supported length of the beam and the axial displacement between the

supports.

Plates are one of the principal structural components of ship structure. The history of the dynamic

plastic response of plates goes back to the fifties. Cox and Morland (1959) gave the theoretical solution

for a simply supported square plate subjected to a uniformly distributed rectangular pressure pulse.

The most important results concerned the maximum displacement and the total time of motion.

Jones (1971) proposed an approximate theoretical procedure to estimate the permanent

transverse deflections of rectangular plates under uniformly distributed loading. The influence of

finite-deflections or geometry changes was retained in the analysis but elastic effects were

disregarded.

Yu and Chen (1992) completed a theoretical investigation to trace the large deflection dynamic

plastic response of simply-supported or fully-clamped rectangular plates, assuming a kinematically

admissible time-dependent velocity field and considering the global equilibrium of all the forces acting

on each rigid segment during large deflection of a plate.

Zhu and Faulkner (1994a) reported results on the dynamic response of plates under impact load in

minor ship collision using a simplified model. Their work gave a better understanding of the collision

process.

Zhu et al. (1994b) presented an experimental investigation on clamped metal plates struck by a

rigid wedge mass. A simple theoretical procedure based on the rigid perfectly plastic method was used

to study the dynamic behavior of locally impacted plates, using a strain-rate sensitivity factor on the

average dynamic yield stress.

Caridis et al. (1994) summarized the response of thin plates subjected to dynamic loads, obtaining

good predictions of the permanent deflections.

Shen (1997) presented a theoretical analysis to examine the dynamic response of thin rectangular


4

plates struck transversely by a wedge. The analysis employed a pure membrane model with two

traveling hinge phases. Good agreement between the theoretical predictions and the experimental

results obtained on the maximum permanent deflections for various impact energies.

Shen et al. (2002a) presented a series of tests to examine the dynamic response and petalling

failure of thin circular plates struck transversely at the centre by a mass. He observed that a necking

circle was initiated approximately in the central part of plates along a small circle, which was directly

under the transition circle from the spherical surface to conical surface of the drop mass. This was due

to the excessive in-plane tensile strain. A through-thickness crack, then, was formed at one point on the

circle, which was recognized as in-plane tearing failure.

Shen (2002b) proposed a theoretical analysis of the petalling failure of thin circular plates. A failure

criterion of plastic work density was employed for predicting the onset of petalling failure. Good

agreement between the theoretical predictions and the experimental results was obtained for the

critical impact energy required to cause failure of the plates with various diameters struck transversely

by a mass.

Shen et al. (2003) presented a series of experimental results to examine the dynamic response

and failure of thin rectangular plates struck transversely at the centre by wedges. He found that the

critical impact energy required for the onset of failure varied significantly with the thickness of plates.

Jones et al. (2008) studied the perforation of mild steel square and rectangular plates struck

normally by cylindrical projectiles having blunt, hemispherical, and conical impact faces. The plates

were struck at the center and at several positions near the fully clamped supports. The effect of the

aspect ratio on the perforation energies of rectangular plates was examined, and comparisons were

made with the perforation behavior of fully clamped circular plates. The predictions of several empirical

equations were compared with the corresponding experimental values of the perforation energies.

Simple design equations were presented for predicting the maximum permanent transverse

displacements of square plates prior to any cracking or perforation.

The structural design of ships concerning collision also requires an accurate prediction of the

damage of stiffened plates under impact loading. Thus, experimental studies on laterally loaded panels

have been conducted in order to derive analytical expressions. Hagiwara et al. (1983) proposed a

method for predicting low-energy ship collision damage based on combined experiments, which

determined the initiation of plate fracture, the effects of structural details and the deformation of a

typical ship panel.

Manolakos and Mamalis (1985) used a rigid plastic analysis for predicting the structural behavior

of longitudinally framed shell plating of struck vessel during a minor oblique collision.

Cho and Lee (2009) developed a simplified method for the prediction of the extent of damage on

stiffened plates due to lateral collisions.

Lehmann and Peschmann (2002) presented a large-scale collision experiment to validate

numerical calculations of the collision process. The results obtained with respect to the failure strains


5

were used in the calculations and other parameters numerical calculations were performed of a

double-skin structure with austenitic inside wall and austenitic shell and inside wall.

Wu et al. (2004) presented results of a scaled double hull structure representing ship to ship

collision, obtaining good results in terms of general structural response.

Ehlers et al. (2008) performed numerical simulations of the collision response of ship side

structures, finding a strong sensitivity of the failure criteria.

Alsos and Amdahl (2009a) dealt with hull damage in ships which were subjected to grounding

actions. Various configurations of stiffened panels were loaded laterally by a cone shaped indenter until

fracture occurs.

Alsos and Amdahl (2009b) investigated two failure criteria, which were implemented into the

impact analysis. The influence of the element size with respect to onset of failure was studied.

Villavicencio and Guedes Soares (2011b) studied numerically the deflection and failure of small

panels subjected to lateral impact using different stiffener distributions and impact locations. The

analysis of the sensitivity of different parameters, such as the impact velocity, type and distribution of

stiffeners, width of panel, and impact along the width is summarized.

1.2 Objectives and scope of the work

The thesis aims at studying small-scale ship structural elements subjected to lateral impact.

Experimental and numerical methods are used to study the impact strength of structural components.

Detailed information of the impact response of structural components is obtained through drop weight

impact tests and nonlinear finite element simulations. The study considers the following ship structural

components: beams, plates and stiffeners with attached plate. The impact behavior of the structural

elements is validated by comparison of their experimental and numerical force-displacement responses.

Special attention is paid to the definition of the true material properties and the representation of the

experimental boundary conditions.

The thesis is organized as follows:

Chapter 2 introduces the definitions for nonlinear finite element simulations: material stress-strain

relationship, dynamic yield strength, dynamic fracture strain, failure criteria, contact-impact definition,

and mesh size. The simulation of tensile tests and the representation of the boundary conditions are

also summarized.

Chapter 3 studies the plastic response of beams using the definition of the coefficient of friction at

the modeled experimental supports. The true material curve and boundary conditions are studied in the

definition of the numerical simulations, comparing the numerical results with the experimental results.

Chapter 4 studies the plastic behavior and fracture propagation of clamped pre-notched beams

subjected to lateral impact by a mass. The true material and critical failure strain in the numerical

simulation of impact test is obtained from numerical simulation of tensile test of the material.

Chapter 5 presents experimental and numerical analyses of laterally loaded rectangular plate to


6

study the impact plastic response. The impact tests use different impact velocities and different types of

indenters.

Chapter 6 presents the initiation and propagation of fracture on laterally loaded rectangular plate

through experiments and numerical simulations, studying the influences of indenter type on the failure

of rectangular plates. The sensitivity of mesh size and critical failure strain are reviewed using the

force-displacement response of plates.

Chapter 7 summarizes results from experiments and numerical simulations of stiffeners with

attached plate subjected to lateral loads, predicting the absorption of energy during the impact event.

The sensitivity of the incident velocity and the stiffener type is reviewed using the force-displacement

response of the tested specimens.

Chapter 8 contains the conclusions and the further work.

CHAPTER 2 – Nonlinear finite element simulation

7

CHAPTER 2 Nonlinear finite element simulation

The nonlinear finite element method is one of the most powerful approaches to simulate the impact

response of structural elements subjected to large deformation. However, the results of finite element

simulations are significantly affected by the modeling technique used. There are different parameters

that must be defined in numerical models in order to represent a realistic impact event. This chapter

summarizes the definitions in nonlinear finite element simulations, such as: material stress-strain

relationship, dynamic yield strength, dynamic fracture strain, failure criteria, contact-impact definition

and mesh size. The simulation of tensile tests and the representation of the boundary conditions are

also summarized in this chapter.

2.1 Material stress-strain relationship

Material stress-strain relationship is an important definition in nonlinear finite element analysis. The

mechanical behavior of materials is described by their deformation and fracture characteristics under

applied tensile, compressive or multi-axial stresses. The engineering tension test is widely used to

provide basic design information on the strength of materials. In this section, the interpretation of the

tension test results and the determination of the true stress-strain curve are briefly summarized (Dieter

1986). The procedures for conducting tensile tests can be found in ASTM (1989).

2.1.1 Engineering stress-strain curve

The engineering stress-strain curve (Figure 2.1) is a graphical representation of the result of metal

tensile test which is from the load-elongation measurements of the test specimen. The engineering

stress (σeng) is the average longitudinal stress in the tensile test specimen. It is obtained by dividing the

load (P) by the original area of the cross section of the specimen (A0).

0eng

P

Aσ = (2.1)

Figure 2.1 Engineering stress-strain curve. Intersection of the dashed line with the curve determines the offset yield

strength. (cf. Dieter 1986)


8

The engineering strain (εeng) is the average linear strain, which is obtained by dividing the

elongation of the gauge length of the specimen (δ) by its original length (L0).

0

0 0 0eng

L LL

L L L

δε −∆= = = (2.2)

There are many influence factors to the shape and magnitude of the stress-strain curve, such as

composition, heat treatment, prior history of plastic deformation, strain rate, temperature, and state of

stress imposed during the testing. The parameters that are used to describe the stress-strain curve of a

metal are the tensile strength, yield strength or yield point, percent elongation, and reduction in area.

The first two are strength parameters; the last two indicate ductility (Dieter 1986). In the elastic region,

stress is linearly proportional to strain. When the stress exceeds the yield strength, the specimen

undergoes gross plastic deformation.

After the yield stress, the engineering stress continues to rise with increasing strain until ultimate

tensile stress. A point is reached where the decrease in specimen cross-sectional area is greater than

the increase in deformation load arising from strain hardening (Dieter 1986). This condition will be

reached first at some point in the specimen that is slightly weaker than the rest. All further plastic

deformation is concentrated in this region, and the specimen begins to neck or thin down locally.

Because the cross-sectional area now is decreasing far more rapidly than the deformation load is

increased by strain hardening, the actual loading required to deform the specimen falls off, and the

engineering stress defined in Equation (2.1) continues to decrease until fracture occurs. More details of

the engineering stress-strain curve can be found in Dieter (1986).

2.1.2 True stress-strain curve

The engineering stress-strain curve is only based on the original dimensions of the test specimen

(A0 and L0), not indicating the true deformation characteristics of a metal. The engineering stress based

on the original area decreases after necking. If the true stress and true strain based on the actual

cross-sectional area of the specimen is used, the stress-strain curve increases continuously until

fracture, as shown in Figure 2.2.

Figure 2.2 Engineering and true stress-true strain curves. (cf. Dieter 1986)


9

The true stress (σt) expressed in terms of engineering stress and strain by:

0

( 1) ( 1)t eng eng eng

P

Aσ ε σ ε= + = + (2.3)

Equation (2.3) can be used only until the onset of necking, because its assumption is a

homogeneous distribution of strain along the gauge length of the test specimen, which is far from the

truth after necking. Thus, beyond the ultimate tensile stress, the true stress should be determined from

actual cross-sectional area.

t

P

Aσ = (2.4)

The true strain (εt) can be determined from the engineering strain (εeng) by:

0

ln( 1) lnt eng

L

Lε ε= + = (2.5)

Equation (2.5) also can be used only until the onset of necking. Beyond maximum load, the true

strain should be determined from actual cross-sectional area.

0lnt

A

Aε = (2.6)

However, highly accurate optical measuring systems are needed to measure the actual

cross-sectional area. Ehlers and Varsta (2009) obtained experimentally true stress-strain material

curves from tensile test specimens using highly accurate optical measuring systems. Unfortunately, in

most of the cases the material property information is just reduced to the engineering stress-strain

curve. More details of the true stress-strain curve can be found in Dieter (1986).

2.1.3 Mathematical expressions for the true material curve

Since a dynamic calculation involves extreme structural behavior with both geometrical and material

nonlinear effects, the input of material properties up to the ultimate tensile stress has a significant

influence on the extent of critical deformation energy. The information of the engineering stress-strain

curve can only be used to obtain true stress-strain curve only until necking. Thus, mathematical

approximations should be used to define the true material curve beyond the maximum load. Two true

material curves are introduced in this section: the power law curve and the true material curve

proposed by Zhang et al. (2004).

The true curve of many metals in the region of uniform plastic deformation can be expressed by

the simple power curve relation:

nt tKσ ε= (2.7)

where n is the strain-hardening exponent, and K is the strength coefficient. A log-log plot of true stress

and true strain up to maximum load will results in a straight line (Figure 2.3). The linear slope of this line

is n, and K is the true stress at εt = 1.0 (correspond to A/A0 = 0.63). For most metals, n has values

between 0.10 and 0.50.


10

Figure 2.3 Log-log plot of true stress-strain curve n is the strain-hardening exponent; K is the strength coefficient.

(cf. Dieter 1986)

Another mathematical expression defining the true material curve is the one proposed by Zhang et

al. (2004). They recommended to use the following true stress-strain relationship:

pt tCσ ε= (2.8)

where

ln(1 )gp A= + (2.9)

and

( / ) pmC R e p= (2.10)

Ag is the maximal uniform strain related to the ultimate tensile stress Rm, and e is the natural

logarithmic constant. Both values can be measured from a specimen tensile test. If only the ultimate

stress Rm (MPa) is available, the following approximation can be used to obtain the proper Ag value:

1/(0.24 0.01395 )g mA R= + (2.11)

The material was defined by Villavicencio and Guedes Soares (2011c, in press b) to represent the

true stress and strain relationship finding good agreement when compared with the experimental

engineering stress and strain curve. This material combine two of the previous defined material, i.e.

before necking the material is defined by Equations (2.3) and (2.5) and beyond localization is

represented by Equation (2.8). The two material definitions are well combined by the material model of

Villavicencio and Guedes Soares (2011c) to express the true stress-stain relationship.

In the present work, the material curve defined by Zhang et al. (2004) is denoted by “GL”, and the

true stress-strain curve until the onset of necking is denoted by “UN”, and the power law curve is

denoted by “PL”. The true material defined with “UN” curve and beyond continued with “GL” curve is

denoted by “UN+GL”. Thus, Equations (2.3) and (2.5) define the process before and Equation (2.8)

after the necking is localized.

2.2 Dynamic yield strength

The material yield stress is dependent on both the rate of deformation (strain rate) and the temperature

at which the deformation occurs. The relationship between yield stress and strain rate at constant


11

temperature can be well described (Paik 2007). The dynamic yield strength of the material may be

expressed as follows (Jones 1989):

( ) ( )Yd

Y

f gσ ε εσ

= ɺ (2.12)

where Yσ , Ydσ = static and dynamic yield stresses, ( )f εɺ = strain rate sensitivity effect function,

( )g ε = a material strain-hardening function, and εɺ = strain rate.

Table 2.1 Sample coefficients for the Cowper--Symonds constitutive equation (cf. Paik 2007)

Material C (s−1) q Reference

Mild steel 40.4 5 Cowper and Symonds (1957)

High-tensile steel 3,200 5 Paik and Chung (1999)

Aluminum alloy 6,500 4 Bodner and Symonds (1962)

α-Titanium (Ti 50A) 120 9 Symonds and Chon (1974)

Stainless steel 304 100 10 Forrestal and Sagartz (1978)

If the strain-hardening effect is negligible, one can take that ( )g ε = 1. The strain rate sensitivity

function ( )f εɺ is often given using the Cowper–Symonds equation (Cowper and Symonds 1957) as

follows:

1/1.0 ( ) qYd

Y C

σ εσ

= +ɺ

(2.13)

where C and q are coefficients determined on the basis of test data, see Table 2.1. It is evident that

these coefficients depend on the material. Figure 2.4 plots the Cowper–Symonds equation together

with the relevant coefficients for mild or high-tensile steels when ( )g ε = 1.

Figure 2.4 Dynamic yield strength, σYd (normalized by the static yield strength, σY), plotted versus strain rate, εɺ ,

for mild and high-tensile steels. (cf. Paik 2007).


12

2.3 Dynamic fracture strain

Both crushing effects and yield strength increase as the loading speed gets faster, whereas any

fracture or tearing of steel (and the welded regions) of a structure tends to occur earlier (Paik 2007).

The following approximate formula, which is the inverse of the Cowper–Symonds constitutive equation

for the dynamic yield stress, is then used for the estimation of the dynamic fracture strain as a function

of the strain rate (Jones 1989), namely,

1/ 1[1.0 ( ) ]fd q

f C

ε εξε

−= +ɺ

(2.14)

where fε , fdε = static and dynamic fracture strains, ξ = ratio of the total energies to rupture for

dynamic and static uniaxial loadings. The dynamic fracture strain fdε will then be set for the finite

element simulations in place of static fracture strain fε .

If the energy to failure is assumed to be invariant, i.e., independent of εɺ , then it may be taken that

ξ = 1. Figure 2.5 plots Equation (2.14) with three sets of the coefficients together with experimental

results for mild steels when ξ = 1. The expression in Equation (2.14) represents the decrease of the

dynamic fracture strain with increase in the strain rate, but the coefficients for the dynamic fracture

strain differ from those for the dynamic yield strength. It is again evident that the strain rate is a primary

parameter affecting the impact mechanics and the structural crashworthiness. Also, it is seen from the

figure that Equation (2.14) with C = 40.4 and q = 5 for mild steel gives a very small value for the fracture

strain. Rather, it is recommended to adopt C in the range from 7,000 to 10,000 and q in the range from

2 to 4.

Figure 2.5 Dynamic fracture strain (normalized by the static fracture strain) versus strain rate for mild steels. (cf.

Paik 2007)

2.3.1 Failure criteria

The failure predicted by a finite element analysis can be represented by the initial fracture of a finite

element which has an extreme large plastic strain (Zhang 2004).Usually the first rupture of an element


13

will be defined with a failure strain value. If the calculated strain, such as plastic effective strain,

principal strain or for a shell element strain in the thickness direction exceeds its defined failure strain

value, the element will be fractured and deleted from the finite element model. The deformation energy

in this element will keep in a constant value in the further calculation steps.

Numerical calculations have shown that the deformation energy is very sensitive to the defined

failure criteria. The definition of the failure strain value is a most important key point for a correct

prediction of realistic critical deformation energy and it can result in an incorrect assessment of the

energy absorption, if an improper failure criterion is defined.

In fact the development of rupture of a structural component is a very complicated process and is

influenced from many factors. Firstly it is directly related to material characteristics such as yield stress,

the maximal uniform strain and the fracture strain. Secondly it is well known from numerous practical

experiences and theoretical investigations that an initiation of a fracture depends also on the stress

states resulting under complicated loads on the structures. In addition, it is also influenced by the

production process, manufacture quality as well as environmental and operational conditions. For a

mesh size, element shape as well as selected element types plays also very important roles because in

reality a fracture process is developed from a uniform deformation state over the whole component to a

very local necking in a very small area with extreme large strain values. To obtain practical failure strain

definitions under consideration of element size, stress state and manufacture influence many thickness

measurements from prototype damaged structure components such as shell plating and stiffeners etc.

have been carried out and the uniform strain, the necking as well as the necking length have been

determined.

There are two failure criteria for the definition of failure strain of a shell element. The criterion

proposed by Peschmann (2001) is based on quasi-static uniaxial tension. The ratio of uniform and

necking strain must be scaled by the length of necking:

( )ecrit g m

x t

t lε ε ε= + ⋅ ⋅ (2.15)

gε is the uniform strain, mε is the necking strain, ex the length of neck, t the plate thickness and l

the element length. Thus, based on tests, Peschmann proposed the following values for the critical

failure strain for plate thickness smaller than 12 mm:

0.1 0.8crit

t

lε = + ⋅ (2.16)

The values of uniform and necking strain achieved from thickness measurements related to the

calculated stress states proposed by Zhang et al. (2004) of 0.056 for the uniform strain and 0.54 for the

necking strain in the case of shell elements.

0.056 0.54crit

t

lε = + ⋅ (2.17)

2.4 Contact-impact algorithm

The treatment of sliding and impact along interfaces are important definitions in nonlinear analysis


14

when two bodies interact (Hallquist 2010). Interfaces can be defined in three dimensions by triangular

and quadrilateral segments. One side of the interface is designated as the slave side, and the other as

the master side. Nodes lying in those surfaces are referred to as slave and master nodes, respectively.

Thus, the slave nodes are constrained to slide on the master surface and must remain on the master

surface until a tensile force develops between the node and the surface. Automatic contact definitions

are commonly used. In this approach the slave and master surfaces are generated internally from the

part given for each surface. Two distinct methods for defining contact are implemented in LS-DYNA:

the kinematic constraint method and the penalty method.

2.4.1 Kinematic constraint method

In this method, the constraints are imposed on the global equations by a transformation of the nodal

displacement components of the slave nodes along the contact interface. This transformation has the

effect of eliminating the normal degree of freedom of nodes. To preserve the efficiency of the explicit

time integration, the mass penetrates into the specimen allowing that the global degrees of freedom of

each master node are coupled. The impact is imposed to the conservation of moment.

Problems arise with this method when the master surface mesh is finer than the slave surface

mesh as shown in two dimensions in Figure 2.6. Here, certain master nodes can penetrate through the

slave surface without resistance and create a twist node in the slide line. Such twist nodes are relatively

common with this formulation, and, when interface pressures are high, these twist nodes occur whether

one or more quadrature points are used in the element integration.

Slave surfaceMaster surface

Indicates nodes treated as free surface nodes

Figure 2.6 Nodes of the master slide surface designated with an “x” are treated as free surface nodes in the nodal

constraint method. (cf. Hallquist 2010)

2.4.2 Penalty method

The penalty method consists of placing normal interface springs between all penetrating nodes and the

contact surface. Quite in contrast to the nodal constraint method, the method excites little if any mesh

hourglassing is used. Three implementations of the penalty algorithm are available in LS-DYNA:

(1) Standard penalty formulation, where the interface stiffness is chosen to be approximately the same

order of magnitude as the stiffness of the interface element.

(2) Soft constraint penalty formulation, which treat contact between bodies with dissimilar material

properties (e.g. steel-foam).

(3) Segment-based penalty formulation, which uses a slave segment-master segment approach


15

instead of a traditional slave node-master.

2.4.3 Initial contact Interpenetrations

The offset to account the thickness of the shell elements contributes to initial contact interpenetrations,

which cannot be detected since the contact node interpenetrates completely through the surface at the

beginning of the calculation. This is illustrated in Figure 2.7.

Detected Penetration Undetected Penetration

Figure 2.7 Undetected interpenetration. Such interpenetrations are frequently due to the use of coarse meshes. (cf.

Hallquist 2010)

2.4.4 Friction definition

Friction in LS-DYNA is based on a Coulomb Formulation. The frictional algorithm uses the equivalent of

an elastic plastic spring. The interface shear stress that develops as a result of Coulomb friction can be

very large and in some cases may exceed the ability of the material to carry such a stress. Typical

values of friction are indicated in Table 2.2.

Table 2.2 Static and sliding coefficient of friction (cf. Hallquist 2010)

Materials Static Sliding

Hard steel on hard steel 0.78 (dry) 0.08 (greasy), 0.42 (dry)

Mild steel on mild steel 0.74 (dry) 0.10 (greasy), 0.57 (dry)

Aluminium on mild steel 0.61 (dry) 0.47 (dry)

Aluminium on aluminium 1.05 (dry) 1.4 (dry)

Tires on pavement (40 psi) 0.90 (dry) 0.69 (wet), 0.85 (dry)

2.4.5 Contact automatic surface to surface

Automatic contact formulations are recommended for most explicit simulations. Contact automatic

surface to surface can handle situations such as shell edge to surface, and beam to shell surface. The

contact search algorithms employed by automatic contacts is well suited to handling disjoint meshes. In

the case of shell elements, automatic contact types determine the contact surfaces by projecting

normally from the shell mid-plane a distance equal to one-half the ‘contact thickness’. Further, at the

exterior edge of a shell surface, the contact surface wraps around the shell edge with a radius equal to

one-half the contact thickness thus forming a continuous contact surface.

2.5 Mesh size


16

In the finite element simulation, mesh size has important effects on the calculation results (Paik 2007).

To properly capture highly nonlinear characteristics of structures, a very fine finite element mesh is

required. For many reasons, however, it is not always the case that a very fine mesh modeling can be

adopted. For example, very large complex structures need a huge number of finite elements so that it is

not easy to execute the numerical simulations with such number of elements. Convergence studies

with varying mesh size and element number often need to be undertaken to define the relevant mesh

size, with large computational efforts still being required for that purpose. In this regard, it will be very

helpful for nonlinear finite element structural modeling if the relevant mesh size can be readily

determined, even without a convergence study. Actually, the mesh size is not very important in analysis

of plastic deformation, but plays an important role when fracture occurs. In collisions or grounding,

major failure modes are crushing, fracture (tearing or cutting), and plate tension. Among them, the

crushing mechanism requires very fine mesh size to reflect folded configuration.

2.6 Simulation of tensile tests

A tensile specimen is a standardized sample cross-section. It has two shoulders and a gauge section in

between. The shoulders are large so that they can be gripped. The gauge section has a smaller

cross-section so that the deformation and failure can occur in this area. The standard test specimen

dimensions and tolerances can be found in ASTM (1989).

Finite element simulations of structures subjected to impact loads require the input of the true

stress-strain relationship until failure. The adopted stress-strain relationship affects the necking and

rupture behavior obtained by the finite element simulations. The mesh size for simulating the internal

mechanics needs to be fine enough to capture the nonlinear structural response. Another important

parameter is the prediction and simulation of initiation and propagation of fracture. However, the failure

due to material rupture is still not well resolved numerically, because the fracture length is much smaller

than the side length of the elements in a finite element model. Thus, it is difficult to establish a

procedure suitable for prediction of failure in the engineering practice.

In practical terms, numerical simulations of tensile tests have been conducted to predict the critical

failure strains used in the finite element models of small-scale structures such as plates subjected to

impact loads. For example, Ehlers and Varsta (2009) obtained experimentally true stress-strain

material curves from tensile test specimens using highly accurate optical measuring systems.

Simonsen and Lauridsen (2000) approached the critical failure strain used in finite elements models

through numerical simulations of the tensile tests using different failure strains and mesh densities.

Villavicencio et al. (in press a) conducted tensile tests to calibrate the plastic response of aluminum

plates, as a result the true stress-strain relationship for thin and thick circular plates subjected to impact

was defined and the local out-of-plane deformation and the indentation into the thickness were

adequately predicted.

For a purely plastic response without necking or fracture, the plastic parameters of the material can

be determined by true stress-strain curve until necking which is from the results of a tensile test.

However, fracture and necking occur over a length which is much smaller than the side length of the


17

elements considered in a finite element model. Thus, these elements cannot capture such a local

phenomenon.

The mesh sensitivity can be approached with an engineering method at the level of advanced

industry practice (Simonsen and Lauridsen 2000) in which the critical failure strain required to give the

material fracture strain is found through numerical simulations of the tensile tests using different failure

strains and mesh densities. Here, failure strain denotes the strain value when fracture occurs. The

failure strain is defined as the average normal strain over the element.

The engineering tension tests are modeled using LS-DYNA Version 971 (Hallquist 2010) finite

element package. The tensile specimens are modeled with shell (4-nodes, 5-integration points) or solid

(8-nodes, 1-integration point) elements, depending on the type of structure in analysis.

In the tensile test simulations, only the length of the tensile specimen between the clamping edges

is modeled (Figure 2.8). For initiating necking, the width of the specimen at the centre is gradually

reduced by 0.5 %. The mesh is diagonally orientated to avoid hourglassing and all components of the

hourglass force vector are orthogonal to rigid body rotations. As the critical failure strain depends on

the mesh density, various mesh sizes are simulated.

Figure 2.8 Finite element model. (cf. Villavicencio and Guedes Soares 2011c)

The material selected from the library of LS-DYNA allows the definition of a true stress-strain curve

as an offset table. Also, failure based on a plastic strain and arbitrary strain rate dependency can be

defined (Mat.024-Piecewice linear plasticity).

The translational degrees of freedom are restricted at one end and a constant displacement of 100

times the experimental speed is prescribed at the other. The force of the displaced nodes at the free

end is obtained and then plotted versus the applied displacement, and these values used to give the

engineering stress-strain behavior.

To model failure, the solver deletes elements when their average strain reaches a critical value.

The numerical simulations are calibrated using the experimental data to give the critical strain value

that fit the experimental results using an iterative procedure.

Figure 2.9 shows the time steps of a typical tensile test simulation. It is observed that when the

stress exceeds the yield strength, the specimen undergoes gross plastic deformation and its

cross-sectional area decreases uniformly along the gauge length (step 1). At some time, the decrease

in specimen cross-sectional area is greater than the increase in axial deformation (step 2). All further

plastic deformation is concentrated in this region, and the specimen begins to neck locally (steps 3 and


18

4) until fracture occurs (step 5).

Figure 2.9 Time steps of a typical tensile test simulation. (cf. Villavicencio and Guedes Soares 2011c)

The beginning of a fracture is indicated by a crack in the center of the specimen (neutral axis),

which propagates through the surface perpendicularly to the applied tension (Figure 2.10).

Figure 2.10 Fracture propagation. (cf. Villavicencio and Guedes Soares 2011c)

Reduction of cross-sectional area is customarily measured only on test pieces with an initial circular

cross section because the shape of the reduced area remains circular or nearly circular throughout the

test for such test pieces. With rectangular tests pieces, in contrast, the corners prevent uniform flow

from occurring, and consequently, after fracture, the shape of the reduced area is not rectangular

(Figure 2.11).

Figure 2.11 Sketch of end view of rectangular test piece after fracture showing constraint at corners indicating the

difficulty of determining reduced area. (cf. Villavicencio and Guedes Soares 2011c)


19

2.7 Representation of experimental supports

In the experimental tests it is almost impossible to satisfy precisely the zero displacement condition at

the supports, because the load capacity of the structure is strongly dependent on the axial restraint

provided at the supports. Therefore, when developing numerical models that are to be compared with

experimental results, it is necessary that the real boundary conditions are represented in the numerical

model, instead of the ideal ones that are the intention in the experimental program.

The assumptions of simply supported or fully clamped boundaries in finite element analyses can

lead to errors in the load carrying capacities of the analyzed structure. On the other hand, a complete

representation of the experimental supports simulating the boundary conditions requires extra time in

modeling and computations.

In most of the numerical simulations, the support plates are represented using shell undeformable

elements compressing the specimen as occurred in the experiments (Figure 2.12).

Figure 2.12 Models of boundary conditions. (cf. Villavicencio and Guedes Soares in press a)

In this boundary condition representation, all the support plate length is modeled and no gap

between the support plates and the supported area of the specimen is considered. The lower support

plate is constrained in all degrees of freedom. The upper support plate is constrained in all degrees of

freedom, except for vertical translation, because a prescribed vertical motion is imposed to compress

the supported surface of the panel simulating the clamped condition. The value of the prescribed

displacement is equal to εyt/3 (Ehlers 2010), where εy is the yield strain of the material and t is the

thickness of the supported stiffened plate. Using this boundary condition representation, some small

longitudinal displacements of the supported portion of the specimen between the support plates are

observed, which are due to the high incident energy applied. More details of the representation of

experimental supports may be found in Villavicencio and Guedes Soares (2011a).


20

CHAPTER 3 – Plastic response of beams subjected to lateral impact

21

CHAPTER 3 Plastic response of beams subjected to lateral impact

Experiments and numerical simulations were conducted in order to study the plastic response of

beams laterally impacted by a mass. Most of researches use standard boundary conditions, such as

simple supported and fully clamped, and studies on the influence of the coefficient of friction at the

supports are rare. Thus, this chapter studies the plastic response of beams using the definition of the

coefficient of friction at the modeled experimental supports and proposes some useful conclusions.

Also, as is difficult to define all the parameters in a finite element model, such as the true material curve

and boundary conditions, in the numerical simulations these important parameters were obtained

experimentally, for instance measuring the displacements of the supported length of the beams.

3.1 Impact test machine

The basic requirement of the instrumented falling weight impact test (Figure 3.1), is to gain an

understanding of the mechanism by which materials (e.g. plastics, metals, ceramics etc.) or structures

(e.g. fabricated articles, composites, adhesive joints), fail in impact situations, i.e. at high rates of strain.

ROSAND

Optical gate

Indenter

Load cell (50kN)

Guide railsInstrumented Falling Weight Impact Tester, Type 5

ROSAND

ROSAND

SpecimenSupport

plates

Structural support

Flag

Striking mass

Figure 3.1 Fully instrumented Rosand IFW5 falling weight machine. (Drawing provided by R. Villavicencio)

Details of the impact machine are summarized in Rosand Precision Impact Tester User manual

(version 1.3). The impact machine is composed by two main parts:

(1) A frame to hold the sample, and guide the falling weight onto it.

(2) The electronics, which:

i. Control the impact process, in a reproducible and safe manner.

ii. Acquire, display and analyze the high-speed data.


22

iii. Communicate with the computer system, which is used for further analysis, and for data storage

and retrieval and for presentation of results.

The experiments were conducted using an instrumented falling weight impact test machine. A

small, light hemispherical ended cylindrical projectile is dropped from a known, variable height between

guide rails onto a clamped horizontally supported plate target. A much larger, variable mass is attached

to the projectile and a load cell between the two gives the variation of impact force with time. An optical

gate gives the incident velocity, and hence the displacement and velocity and the energy it imparts are

calculated from the force–time data by successive numerical integrations. Since the projectile is

assumed to remain in contact with the specimen throughout the impact event, the indenter

displacement is used to give the displacement and velocity of the top face of the specimen, under the

indenter. By assuming that frictional and heating effects are negligible, the energy imparted by the

indenter is that absorbed by the specimen. Thus, this energy value at the end of the test is that

irreversibly absorbed by the specimen.

3.2 Impact tester software

Details of the software are summarized in Rosand Precision Impact Tester Software manual (version

1.3). This section describes the capabilities of the operating software of the Rosand impact machine.

The software can be used to control the machine, and collect and analyze the test data. Two distinct

functions of the software are to control the apparatus and to present the results of the test.

The software provides direct control over the following aspects:

� Data capture speed and sample length

� Transducer amplifier gain, and filtering

� Catcher position and winch speed

� Chamber and anvil temperature

� Drop height, velocity and energy

� Operation of optional equipment, such as sample strippers and second strike preventers

� Drop cycle initiation, and optionally direct control over the drop itself

� Calibration of all transducers (impact force, catcher height and bungee force)

� Second data collection channel control

After a drop test, data is automatically transmitted to the computer. The software allows great

flexibility in the manipulation of this data, including the following functionality:

� Calculation, from the basic force-time (or acceleration-time) information, of velocity, distance and

energy

� Plotting of force, and derived quantities, against time and distance

� Multiple, simultaneous, on-screen graphs. Flexible control over axis scaling, including fully

automatic scaling. Simple graph zooming using the mouse to investigate features in detail.

� Saving and recalling of data to disk

� Recalling of multiple data sets for comparison and averaging

� Tabular test results


23

� Moveable markers, used to define the start of event, peak force and end of event. Initial values set

automatically, may be manually adjusted.

� Printing of graphs and tables

� Crack analysis calculations

Data of force-time, displacement-time, absorbed energy-time and force-displacement can be

obtained from the impact tester software (Figure 3.2), which are used to analyze the experimental

results and compare with numerical simulation of these impact tests.

Figure 3.2 Experimental results of force-time, displacement-time, absorbed energy-time and force-displacement.

(cf. Villavicencio and Guedes Soares 2011a)

3.3 Experimental details

The impact tests represent a situation in which it is considered that a partially clamped beam is struck

at the mid-span by a mass travelling with an initial impact velocity, as shown in Figure 3.3. The beam is

partially clamped between two support plates, i.e. the beam experiences some axial displacement at

the supports during the impact event.

V

BEAM

FRICTION STRIKING MASS

SUPPORT PLATE

Figure 3.3 Beam struck transversely by a mass.

The experimental set-up can be seen in Figure 3.4. The specimen beams were 125 mm span

length, 20 mm width and 3 mm thickness. Tests were carried out by using a striking mass of 24.42 kg


24

and impact velocities from 0.5 to 2.5 m/s. The length of the beams is divided in three: span length and

two support lengths, where the span length is the distance between the supports and the support

length is the length of the upper support plate (Figure 3.4).

Figure 3.4 Experimental set-up.

The specimen beams were supported between two thick rectangular steel plates (upper and lower

support plates) and were compressed by two M16 bolts at each support. The lower support plates

(thickness 16 mm) were stiffened by two relatively thick plates (12 mm) one of which was located below

the supported length of the beam, and were fixed to a strong structural base to prevent their movement.

Although the structural supports were made of mild steel and they could experience some deformation

during the impact, it was assumed that they were stiff enough and did not suffer any important

deformations.

The torque applied to screw the bolts and to compress the supported length of the specimen

beams was 60 N·m. This torque (T) gives the preload force (F) to the upper support plate. The

relationship between the torque and preload force can be expressed by:

F=T/KD (3.1)

where K is the dimensionless torque coefficient, ranging from 0.10 for a well lubricated or waxed

assembly to over 0.30 for one that is dirty or rusty, and D is the diameter of the bolt. In this case, the


25

bolts are plain and slightly oily, so the torque coefficient takes 0.2. Therefore, the preload force of one

bolt was estimated as 18750 N.

The material is high-tensile steel and its mechanical properties were obtained from in-house

tensile test which results are summarized in Table 3.1.

Table 3.1 Mechanical properties of material

Property Units Steel 3.0 mm

Density kg/m3 7850

Young’s modulus GPa 206

Poisson’s ratio -- 0.3

Yield stress MPa 403

Ultimate stress MPa 492

3.4 Experimental results

The experimental results of the tested beams are summarized in Table 3.2. It is observed that the force

and the displacement are in direct proportion to the impact velocity. When the impact velocity increases

0.5 m/s, the peak force increases about 1.6 kN and the maximum displacement about 4 mm. The

rebound displacement is the difference between the maximum and the end displacement. This rebound

displacement is larger at low impact velocities. As is a drop weight impact experiment, the maximum

absorbed energy during the impact is larger than the initial input kinetic energy of the striking mass.

Hence, the total absorbed energy must include the potential energy of striking mass caused by the

vertical deformation of the beam.

Table 3.2 Summary of experimental results at different impact velocity

Impact Velocity

(m/s)

Values at Peak Force Values at End

Force (kN) Displ (mm) Energy (J) Displ (mm) Energy (J)

0.5 1.60 3.65 3.89 2.21 2.98

1.0 2.86 7.37 13.67 6.38 12.72

1.5 4.27 12.64 29.57 11.81 28.31

2.0 5.85 16.64 52.73 16.12 52.14

2.5 8.35 21.99 81.70 21.60 80.63

The axial displacements at the supports were measured at the end of the impact event in order to

analyze the coefficient of friction between the supported portion of the beam and the support plates. An

average axial displacement was used, because of the asymmetrical displacement at each supports.

The axial displacements at the supports are shown in Table 3.3. The axial displacement would be

doubled when the impact velocity increases 0.5 m/s.

The deformed shape of the beams at different impact velocities was shown in Figure 3.5. The

maximum lateral displacement is in direct proportion to the impact velocity, which is different from the


26

axial displacement. Figure 3.6 shows a typical beam after the impact. The V-shape deformation was

formed with plastic hinges at the impact position and the supports.

Table 3.3 Axial displacements at the supports

Velocity (m/s) Displacement (mm)

0.5 0.10

1.0 0.20

1.5 0.42

2.0 0.94

2.5 2.22

2.5m/s1.5m/s

0.5m/s2.0m/s

1.0m/s

Figure 3.5 Deformed shape of beams after the impact.

Figure 3.6 Deformation of the beam impacted at velocity 2.0 m/s.

Table 3.4 Experimental results using different torques on the bolts. (Impact velocity 1.0 m/s)

Torque

(Nm)



60 2.83 7.10 13.80 6.74 13.08

70 3.01 6.93 13.73 6.11 12.57

80 3.24 6.85 13.68 5.45 11.84

The friction between the supported length of the beam and the support plates depends on the

preload force. Thus, the beams were tested using different torque on the bolts in order to observe

different axial displacements at the supports. Three beams were tested using an impact velocity of

1.0 m/s and using torques of 60, 70 and 80 Nm. The experimental results are summarized in Table 3.4,


27

and the force-displacement responses are shown in Figure 3.7. It is noted that the torque on the bolt

has important influence on the response of beam, increasing the impact forces and, consequently,

decreasing the lateral displacements. Also, the axial displacement at supports decreased in

proportional magnitudes using larger torque (Table 3.5).

Figure 3.7 Force-displacement responses using different torque on the bolts. Impact velocity 1.0 m/s.

Table 3.5 Axial displacements at the supports using different torque on the bolts. (Impact velocity 1.0 m/s)

Torque (Nm) Displacement (mm)

60 0.20

70 0.15

80 0.10

3.5 Numerical model

The numerical model is illustrated in Figure 3.8. The specimen beam and the striking mass were

modeled with solid (8-nodes, 1-integration point) elements. The mesh size of the beam and the striking

mass was 2×2×1.5 mm and 1×1×1 mm, respectively. The striking mass was defined as a rigid material

to ensure no deformation. The ‘Mat.020-Rigid’ was selected from the material library of LS-DYNA,

assigning mild steel mechanical properties (Young’s modulus 206GPa and Poisson’s ratio 0.3). As the

falling weight assembly was modeled as a simple box, an artificially large density was used to give the

same mass as used in the experiments. The contact between the striking mass and the beam was

defined as “Automatic Surface to Surface”. The “Load Body Z” was chosen to define the acceleration of

gravity in the simulation. In the constraint of striking mass, only the vertical translation was free, in

which direction the initial impact velocity was assigned.

V

Tx,Ty,TzRx,Ry,Rz

Tx,Ty,TzRx,Ry,Rz

F FTx,TyRx,Ry,Rz

Tx,TyRx,Ry,Rz

Figure 3.8 Numerical model.


28

The definition of the beam material is the most important one, and thus the mechanical properties

of the material used in the finite element model were obtained from in-house tensile tests carried out on

the same beams from which the impact specimens were done. The material selected from the library of

LS-DYNA was ‘Mat.024-Piecewice lineal plasticity’, which allows the definition of a true stress-strain

curve as an offset table. The true material was defined using the exact true stress-strain relationship

until maximum load and beyond necking was used the approximate relationship proposed by Zhang et

al. (2004), (UN+GL, Section 2.1.3), see Figure 3.9.

Figure 3.9 Engineering and true stress-strain curve of material.

Although the experiments allow studying the plastic response of the beams, the aim of the

experiments was to measure the displacements at the supports in order to observe the influence of the

coefficient of friction. The contact between the support plates and the beam was defined as “Automatic

Surface to Surface”. Thus, the support plates modeled with 4-node shell elements were defined as a

rigid material. All the degrees of freedom of the lower support plates were constrained, whereas only

the vertical translation was free on the upper support plates. The “Boundary Prescribed Motion Node”

was selected to define preload force of bolts on the upper support plates.

3.6 Numerical results

The tested beam with an impact velocity of 1.0 m/s was selected to compare with the numerical

simulations. The sensitivity of the coefficient of friction was evaluated by modifying its magnitude in the

contact definition. The results of the simulations are shown in Figure 3.10. The importance of the

coefficient of friction is evident. Several coefficients of friction were chosen to analyze the influence on

the plastic responses of beam. If the coefficient of friction is big enough, the beam does not experience

any axial displacement at the support, i.e. represents a fully clamped condition. If the coefficient of

friction is zero, the impact force remains constant after the elastic limit, whose characteristics is due to

the high axial displacement at the support. The result of numerical simulation with coefficient of friction

of 0.21 gives better agreement with the experimental result.

The maximum force and displacement are shown in Figure 3.11. The numerical results show a

straight line, predicting well the maximum displacement in the full range of impact velocities. However,

the force is under predicted at higher impact velocities reaching a maximum difference of 14.9%.


29

Figure 3.10 Force-displacement responses with different friction coefficient at the supports.

Figure 3.11 Maximum force and displacements at different impact velocities.

The maximum stress and deformation are mainly concentrated at the impact location and at the

supports, as shown in Figure 3.12. The remaining length of the beams has low stresses and small axial

deformations.

Figure 3.12 Shape of the deformation and von Mises stress distribution at maximum force.

The comparison of the axial displacements at the supports is shown in Figure 3.13. The axial

displacements of the simulation are in agreement with the experimental results. The axial displacement

has a relation with the maximum force and displacement shown in Figure 3.11. A good results would be

obtained if the maximum force and displacement having agreement with the experimental results.


30

Figure 3.13 Axial displacements at the supports at different velocity.

The comparison of experimental and numerical force-displacement responses using different

torque at the bolts are shown in Figure 3.14. Higher forces and smaller displacements are obtained

increasing the magnitude of the torque. It is noted that the three force-displacement responses are

similar at the beginning. The numerical results agree with the experimental results, only having the

error of maximum force and displacement of 2.8% and 3.6%, respectively.

Figure 3.14 Experimental and numerical force-displacement responses using different torque at the bolts.

Experimental results: dashed lines. Numerical results: continuous lines. Impact velocity 1.0 m/s.

Figure 3.15 Axial displacements at the supports using different torque at the bolts. Impact velocity 1.0 m/s.


31

The axial displacements at the supports using different torque at the bolts are shown in Figure 3.15.

The tendency of the axial displacement is the same with the experiments, decreasing with the torque at

the bolts. Although, are seen big differences in the axial displacement at the supports, the magnitudes

of these differences only reach 0.1 mm which is a magnitude difficult to measure in experiments.

3.7 Concluding remarks

(1) The force-displacement response is in direct proportion with the impact velocity. Also, the rebound

displacement decreases with the impact velocity.

(2) Moreover, the axial displacement at the supports increases with the impact velocity. The lateral

displacement increases with the axial displacement using the same velocity.

(3) The good agreements obtained between experiments and simulations obey a correct

representation of the boundary condition. Thus, the preload force and the coefficient of friction are the

most important parameters in the definition of the representation of the experimental supports.


32

CHAPTER 4 – Failure prediction of pre-notched beams subjected to lateral impact

33

CHAPTER 4 Failure prediction of pre-notched beams subjected to

lateral impact

Various studies on uniform and perfect structural members under impact loads have been made in

order to understand their dynamic plastic behavior. However, real structures have imperfections in their

geometry, such as notches or cracks. In engineering practice, the imperfections in the cross section are

easier to fracture. Thus, it is imperative for the study of structural components including initial damage

in their geometry.

This chapter describes a series of impact tests conducted to study the plastic behavior and fracture

propagation of clamped pre-notched beams subjected lateral impact by a mass. Also, numerical

simulations were carried out using the LS-DYNA to predict the force-displacement response and the

initiation of fracture in the pre-notched beams. The material definition in the numerical model was

obtained from numerical simulation of tensile test, and the experimental supports were represented in

order to obtain the stresses on the elements at the supports.


The descriptions of impact test machine and its software were given in Section 3.1-3.2. The impact test

represents a situation in which it is considered that a clamped pre-notched beam is struck by a mass

travelling with an initial impact velocity, as shown in Figure 4.1. The experimental set-up can be seen in

Figure 4.2. The specimen beams were 250 mm span length, 20 mm width and 8.0 mm thickness. Tests

were carried out using a striking mass of 74.8 kg and an impact velocity of 4.2 m/s. The indenter was a

hemispherically ended cylindrical projectile of diameter 30 mm. The impact point was at the mid-span

and at one-quarter of the support. Beams with and without notch were tested. Two notches were

manually mechanized with a saw. Two depth of notch were considered: 2.0 mm and 4.0 mm (Figure

4.3). The width of the notch is 1.0 mm. The positions of the notch were 15, 75 and 115 mm from the

support, as shown in Figure 4.4.

V

notch

striking mass

Figure 4.1 Pre-notched beam stuck transversely by a mass.

The length of the beams is divided in three: span length and two support lengths, where the span

length is the distance between the supports and the support length is the length of the upper plate. Two

short beams of length 100 mm were welded at the ends of the specimen beams (Figures 4.2). These

short beams named stoppers were designed to prevent the axial displacement of the beams between

the support plates. No gap between the stopper and the upper support plate was considered. The

specimen beams were supported between two thick rectangular steel plates (upper and lower support

plates) and were compressed by two bolts at each support. The lower support plates (thickness 16 mm)


34

were stiffened by two relatively thick plates (12 mm) one of which located below the supported length of

the beam, and were fixed to a strong structural base to prevent their movement. Although the structural

supports were made of mild steel and they could experience some deformation during the impact, it

was assumed that they were stiff enough and did not suffer any important deformations.


Figure 4.3 Measure the depth of notch.


35

notch1

notch2

notch3

15 mm

75 mm

115 mm

10 mm

10 mm

Figure 4.4 Position of notch and striking mass.


The experimental results are summarized in Table 4.1. Three failure modes were observed in the

experiments: large inelastic deformation (failure mode I), tension failure (failure mode II) and transverse

shear failure (failure mode III). The notation of the failure modes is only valid in this thesis. The failure

mode II and III are shown in Figures 4.5 and 4.6, respectively. Although both II and III failure modes are

under a combination of tension and shear force, the tension failure denotes the specimen in which the

cut area is smaller than the area seen in the shear failure, and also in the tension failure it is thought

that the development of membrane force is more evident.

Table 4.1 Experimental results

Specimen Values at Peak Values at End

Failure Mode Force [kN] Defln [mm] Energy [J] Defln [mm] Energy [J]

WN_Q 44.6 33.4 669.8 29.1 617.6 I

WN_M 38.9 43.2 683.3 38.7 637.3 I

N15_2mm_Q 41.6 34.0 678.3 29.3 622.3 I

N15_4mm_Q 37.9 36.4 646.4 36.4 646.4 III

N15_2mm_M 38.2 44.2 677.3 39.2 620.4 I

N15_4mm_M 29.7 40.6 542.0 40.6 542.0 II

N75_2mm_Q 43.3 33.6 678.8 27.6 618.6 I

N75_4mm_Q 32.9 33.8 480.2 33.8 480.2 III

N75_2mm_M 38.5 44.3 685.5 39.0 628.5 I

N75_4mm_M 28.3 39.5 468.3 39.5 468.3 II

N115_2mm_Q 43.1 33.8 686.7 27.6 627.1 I

N115_4mm_Q 41.7 37.5 663.3 37.5 663.3 II

N115_2mm_M 38.5 44.3 685.5 37.5 620.6 I

N115_4mm_M 24.6 34.7 347.8 34.7 347.8 III

*WN: Without notch; Q and M: Impact at one-quarter of the support and mid-span, respectively; N15, N75 and

N115: Notch position 15, 75 and 115 mm from the support, respectively; 2mm and 4mm: The depth of notch

2.0 mm and 4.0 mm, respectively.


36

(a) (b)

Figure 4.5 Tension failure. (a): Specimen N115_4mm_Q; (b) Specimen N15_4mm_M.

(a) (b)

Figure 4.6 Shear failure. (a): Specimen N75_4mm_Q; (b) Specimen N115_4mm_M.

In all cases, the beams with 2.0 mm notch experienced large plastic deformation, whereas the

beams with 4.0 mm notch fractured at the notch position. Two of the tested beams were selected to

compare their impact response. The selected beams were specimens N15_2mm_M and N15_4mm_M,

and their deformed shapes are shown in Figure 4.7.

(a) (b)

Figure 4.7 Shape of the deformation: (a) Specimen N15_2mm_M; (b) Specimen N15_4mm_M.

The time dependant curves of force, displacement and absorbed energy and the

force-displacement response are shown in Figure 4.8. The absorbed energy can be calculated by

integrating the force-displacement curve. Both curves are similar at the beginning of the response,

indicating that the depth of notch has small influence on the plastic response of pre-notched beams.

The beam with 4.0 mm notch fractured, whereas the beam with 2.0 mm notch only reproduced plastic

deformation. The absorbed energy until fracture is the most important parameter of the impact

response of pre-notched beam.

The maximum forces of the beams with different notches are shown in Figure 4.9. The results of

the beams with 2.0 mm notches are similar with the results of the beams without notch. But the results

with 4.0 mm notch differ from the results of the beams without notch, especially when the notch position


37

near the impact position, because the absorbed energy of fractured beam is smaller than the incident

energy, and the beam is much easier to fracture when the notch near the impact position.

Figure 4.8 Experimental results of force-time, displacement-time, absorbed energy-time and force-displacement.

P: Specimen N15_2mm_M (plastic deformation); F: Specimen N15_4mm_M (fracture).

notch 1 notch 2 notch 320

25

30

35

40

For

ce [

kN]

without notch

notch 2mm

notch 4mm

Impact at Mid

notch 1 notch 2 notch 330

35

40

45

50

For

ce [

kN]

without notch

notch 2mm

notch 4mm

Impact at Quarter

Figure 4.9 Maximum forces of beams with different notch.

4.3 Numerical model

The computations were carried out using the finite element package LS-DYNA Version 971 (Hallquist

2010) which is appropriate for non-linear explicit dynamic simulations with large deformations. The

finite element model was designed with the specimen beam and striking mass. The specimen beam

was modeled in 8-nodes solid elements with 1-integration point using constant stress solid element

formulation. The striking mass was modeled with shell elements. As no material response information

was obtained from the striking mass, zero-integration points were defined. The mesh size of the beam


38

and the striking mass was 1×1×1 mm, as shown in Figure 4.10. The material of beam and the striking

mass were defined as ‘Mat.024-Piecewice lineal plasticity’ and ‘Mat.020-Rigid’ selected from the

material library of LS-DYNA, respectively, assigning mild steel mechanical properties (Young’s

modulus 206GPa and Poisson’s ratio 0.3).

Figure 4.10 Mesh sizes of beam and striking mass.

4.3.1 Boundary conditions

The experimental restraints are similar to fully clamped boundary conditions. In order to check the

accuracy, two definitions of the boundary condition were evaluated in the numerical model. The

clamped and supported boundary conditions are shown in Figure 4.11. In the constraint of striking

mass, only the vertical translation was free, in which direction the initial impact velocity was assigned.

In the clamped model, the beam length was the span length, and all the degrees of freedom of the ends

of beam were constrained. But in the supported model, the beam length was the total of the span

length and the two support lengths, and all the degrees of freedom of the ends of beam were

constrained. The support plates modeled with 4-node shell elements were defined as a rigid material

and constrained all the degrees of freedom.

V Clamped model

Tx,Ty,TzRx,Ry,Rz

Tx,Ty,TzRx,Ry,Rz

Tx,Ty,TzRx,Ry,Rz

Tx,Ty,TzRx,Ry,Rz

Tx,Ty,TzRx,Ry,Rz

Tx,Ty,TzRx,Ry,Rz

V

Supported model

Figure 4.11 Boundary conditions of pre-notched beam.

4.3.2 Material definition using tensile test simulation

Three true material curves were evaluated in order to select the one that best fit the plastic response of

the material and also to predict the point of fracture. The engineering and true material curves (GL,

UN+GL and PL, Section 2.1.3) are plotted in Figure 4.12.


39

Figure 4.12 True and engineering material curves.

The size of model is 100×20×8 mm, and the mesh size is 1.0 mm. The critical failure strain was

obtained by successive numerical simulation (εf = 1.2). Figure 4.13 shows the results of the numerical

simulations using the three material curves (GL, UN+GL, PL). The best approximation is given by the

power law (PL) material curve, which follows the experimental curve quite precisely.

Figure 4.13 Results of numerical simulations.


First, the clamped model is used to analyze the numerical simulation of the impact test. The true

stress-strain curve (PL), mesh size (1.0 mm) and failure strain (1.2) were determined by the numerical

simulation of the tensile test. A series of numerical simulations are presented in this section to compare

with the experimental results. The beams shown in Figure 4.7 (N15_2mm_M and N15_4mm_M) were

selected to predict the experimental results. The experimental and numerical force-displacement

curves are plotted in Figure 4.14. The results are similar to the experiments. The cause of the

difference is that the boundary condition in the experiments is not absolutely fully clamped, but the

experimental result is still reasonable. The appropriate true stress-strain curve and critical failure strain

obtained from the simulation of the tensile test, helped to reproduce the experimental response in the

impact model.


40

(a) (b)

Figure 4.14 Comparison of experimental and numerical force-displacement responses: (a) Specimen

N15_2mm_M; (b) Specimen N15_4mm_M.

The beam with notch of 4.0 mm and impact near the support (N15_4mm_M) was selected to study

the difference between clamped and supported model. The force-displacement responses are shown

in Figure 4.15, which are very similar between them. The displacement of the supported model is larger

than the one of the clamped model, because the former experienced axial displacement at the support.

The failure modes of both models are similar (Figure 4.16). Although both responses are similar, the

supported model is selected as the best option because allows obtaining stress and strain information

at the supports.

Figure 4.15 Comparisons of experimental and numerical force-displacement curves with different boundaries

(Specimen N15_4mm_M).

Figure 4.16 Failure modes (Specimen N15_4mm_M).

Figure 4.17 shows the time steps of a typical simulation of the impact specimen. The original


41

model of the pre-notched beam before impact (step 1). The specimen is at the first stages of impact

and the notch remains undeformable (step 2). The notch starts to undergo gross plastic deformation

and its cross-section elongates when the stresses exceed the yield strength (step 3). All further plastic

deformation is concentrated in this region (step 4). The notch begins to neck locally (steps 5). Fracture

occurs (step 6).

The fracture at the notch is shown in Figure 4.18. The fractured section is smaller than the original

section, because at this location the beam necked. It is observed that the impact event is described

sufficiently well by the numerical model. The failure strain predicted by tensile test simulations is

considered satisfactory when used in the material definition of the impact model. The reacting force and

the displacement at the point of fracture are well predicted. The shape of the failure mode observed in

the experiment is captured accurately by the numerical simulation (Figure 4.18). The complete failure

of the impact specimen starts at the notch corner and extends upwards to the neutral axis of the beam

resulting in strength failure given as a combination of tension and shear. However, as an element in the

original cross-section of the beam is deformed severely and displaced from the initial position, the most

important effect of this change is the development of membrane force.

Figure 4.17 Time steps of a typical simulation of the pre-notched beams (Specimen N15_4mm_M).

(a) (b)

Figure 4.18 The failure of the notch (Specimen N15_4mm_M). (a): Numerical; (b): Experimental.

The numerical triaxiality at failure for the tensile and impact specimens is compared in Figure 4.19

in order to verify the calibration of the critical failure strain by tensile test simulations. The presented


42

triaxiality is calculated by the ratio between the hydrostatic stress and the effective stress and is plotted

versus the normalized axial displacement of the tensile model, on one hand, and versus the normalized

transverse displacement of the impact model, on the other hand. The chosen critical failure strain is

justified due to the close range of observed triaxiality values at the point of failure.

Figure 4.19 Comparison of the numerical triaxiality (Specimen N15_4mm_M).

4.5 Comparison with a theoretical analyses

Liu and Jones (1987) proposed a theoretical analysis to examine the transverse shear and bending

response of clamped beams struck transversely by a mass at any point on the span. The maximum

permanent transverse deformation at the impact point of a rigid-perfectly plastic clamped beam may be

estimated from Equation (4.1), which includes the influence of finite transverse displacements, axial

restraints and bending moments, but disregards material elasticity and the influence of the transverse

shear force in the yield condition. Wf is the maximum permanent transverse deformation; 1l , 2l is

the length of beam from the impact point to the right-hand and left-hand supports, respectively ( 1 2l l≤ );

H is the thickness of beam; B is the width of beam; G is the mass of striker; 0V is the initial impact

velocity; σ0, σ0׳ is the static and dynamic yield stresses, respectively.

8[ 1 1 ]

2 (1 )f

HW

r

λ= − + ++

, 1 2/r l l= (4.1)

where

20 13

02

GV l

BHλ

σ= (4.2)

The lower and upper bound solutions of Equation (4.1) are obtained using circumscribing and

inscribing square yield conditions, respectively. The strain rate sensitivity of the material is considered

using the Cowper-Symonds empirical expression:

1/0 0= [1+( / ) ]pDσ σ ε′ ɺ (4.3)

where σ0 and σ0׳ are the static and dynamic flow stresses, respectively, and D = 40.4 s-1 and p = 5 for

mild steel. The yield stress of material σ0 is equal to 318 MPa.

Material strain rate sensitivity is a highly non-linear phenomenon. Therefore, in order to make an


43

estimate of its importance in the present problem, it is assumed that the strain rate remains constant

throughout the entire response of the beam and equals 45 s-1 (Liu and Jones 1987). Therefore, the

dynamic flow stress for the steel beams is

0 0 0=[1+1.0218] =2.0218σ σ σ′ (4.4)

It is possible that Equation (4.4) might overestimate the influence of material strain rate effects

because the strain rate is less than 45 s-1 during the later stages of a beam response. The theoretical

predictions corresponding to an inscribing square yield condition use σ0.6180.618=׳σ0׳. The static yield

stress σ0 is replaced by 0.618σ0 in Equation (4.4):

0 0 02.0218 0.618 1.249σ σ σ′ = × = (4.5)

The maximum permanent transverse deformation of test beams is calculated by Equations 4.1-4.5

(Figure 4.20). λ is equal to 25.3 when the impact is at mid-span; λ is equal to 12.7 when the impact is at

one-quarter of the support. The experimental and numerical results are between the upper and lower

bound solution of Equation (4.1) without inclusion of strain rate effect.

Figure 4.20 Variation of dimensionless maximum permanent transverse deformation /fW H with dimensionless

external dynamic energy λ . —— Equation (4.1) with static yield stress σ0; - - - - Equation (4.1) with dynamic flow

stress σ0׳ given by equation (4.1); (1) circumscribing yield curve (2) inscribing yield curve; Experimental results: ▲

Simulation results: ■


(1) The selection of the mesh size and critical failure strain by numerical simulations of the tensile tests

is valid for these particular experiments, because some elements in the original cross-section have

deformed severely and the most important effect is the development of membrane forces.

(2) The two basic assumptions used in the finite element simulations: model of the supports to

represent the experimental boundary conditions and prediction of the critical failure strain by tensile

tests are the most important parameters obtained from this simple failure analysis. These parameters

are valid in small-scale structures subjected to impact where such a small mesh size can be defined

and full modeling of the boundary conditions can be performed. Future work should extend the

applicability of these definitions in complex structures and to investigate the effects of all the

parameters involved in this type of impact analysis.


44

(3) The experimental results show three failure modes: large inelastic deformation, tension failure and

transverse shear failure.

CHAPTER 5 – Plastic response of rectangular plates subjected to lateral impact

45

CHAPTER 5 Plastic response of rectangular plates subjected to

lateral impact

In this chapter, experimental and numerical analyses of laterally loaded rectangular plate are described.

The impact tests use different types of indenters. Then, the numerical simulations were carried out

using the LS-DYNA to predict the experimental plastic behavior of the plates. Discussion of the results

and useful conclusions are given.


The descriptions of impact test machine and its software were given in Section 3.1-3.2. The impact

tests represent a situation in which it is considered that a fully clamped rectangular plate is struck at the

center by a mass travelling with an initial impact velocity. The experimental set-up can be seen in

Figure 5.1. The plate thicknesses are 1.4 and 4.0 mm (henceforth referred to as ‘thin’ and ‘thick’,

respectively) in the experiments. Specimen plates were fully clamped by four bolts between two thick

rectangular steel plates with internal cut-out of 127 × 76.2 mm. The bolts passed through holes in the

specimen plate, providing the clamping force. The striking mass is 54.34 kg. A hemispherical indenter

of Ø 30 mm was used to impact two thin plates at different velocity to study the plastic response of plate.

In order to investigate the effects of global deformation and local indentation, tests were carried out on

thick plates using of 6 different types of indenters with a impact velocity of 1.94 m/s. The indenters were

hemispherically ended projectiles of diameter 10, 20 and 30 mm, and cylindrically ended projectiles of

diameter 10, 20 and 30 mm. The different types of indenters are shown in Figure 5.2.


The material of the plate is structural carbon steel and its mechanical properties were obtained by


46

in-house tensile tests using standard tensile specimens and procedures (ASTM 1989). The results of

the tensile tests of the thin and thick plates are presented in Table 5.1, and the engineering

stress-strain curves are shown in Figure 5.3.

Figure 5.2 Different types of indenters.

Table 5.1 Mechanical properties of material

Properties Unit Thickness 1.4 mm Thickness 4 mm

Density kg/m3 7850 7850

Young’s modulus GPa 206 206

Poisson’s ratio -- 0.3 0.3

Yield stress MPa 228 347

Ultimate stress MPa 364 466

Figure 5.3 Engineering stress-strain curve of material with thickness 1.4 and 4.0 mm.


Two thin plates were impacted at velocity 2.7 and 3.3 m/s and using the hemispherical indenter of

Ø 30 mm. The maximum and end results of force, displacement and energy are summarized in Table

5.2. The end of the test is determined by zero contact force. It occurs when the indenter leaves the

surface of the specimen and the specimen acquires its permanent deformation (‘Values at End’ in

Table 5.2). The force-displacement responses of thin plates at different velocity are shown in Figure 5.4.


47

The shape of the deformation at velocity of 2.7 m/s is shown in Figure 5.5. The impacted plates

experienced large plastic deformation, and the force-displacement responses are similar at the

beginning of the response. The differences just depend on the incident energy.

Table 5.2 Summary of experimental results of thin plates at different velocity

Impact Velocity

(m/s)

Input Energy

(J)

Maximum Values Value at End


2.7 200 21.5 17.3 202.9 16.7 202.1

3.3 300 26.8 20.7 303.2 20.7 303.2

Figure 5.4 Force-displacement responses of thin plates at different velocity.

Figure 5.5 Deformation of thin plate at velocity of 2.7 m/s.

In respect to the thick plate it was observed that the shape of the indenter has strong influence on

the impact response. The results of the thick plates are summarized in Table 5.3, and Figure 5.6 shows

the force-displacement responses using different types of indenters with impact velocity of 1.94 m/s. All

the plates experience large plastic deformation, and the maximum energies are basically the same.

The displacement increases and the force decreases using the smaller diameter of indenter. This effect

is more evident with the cylindrical indenters. Figure 5.7 shows the responses using indenters with

same diameter but different ends (sphere and cylinder). The force is larger using the cylindrical

indenter and consequently reproduced smaller displacements. This effect is produced because the

cylindrical indenter has more contact area and thus indentation out of the plane of the plate is less


48

evident.

Table 5.3 Summary of experimental results using different indenters (Impact velocity 1.94 m/s)

Indenter

Maximum Values Values at End


Sphere 10 mm 23.2 6.0 102.8 5.7 101.5

Sphere 20 mm 24.2 5.9 99.5 5.5 97.6

Sphere 30 mm 25.8 5.8 103.4 5.5 101.8

Cylinder 10 mm 26.1 5.6 107.3 5.2 105.3

Cylinder 20 mm 31.6 5.0 104.7 4.5 100.4

Cylinder 30 mm 36.2 4.3 104.5 3.7 98.2

Figure 5.6 Experimental force-displacement responses using different diameters of indenters.

Figure 5.7 Experimental force-displacement responses using different types of indenters with the same diameter.


49

5.3 Numerical model

The plate was modeled with 4-node shell elements with 5-integration points through the thickness. The

contact mass-specimen uses nodal normal projections resulting in a continuous contact surface. The

selected material provides a definition of the true stress-strain relationship as an offset table. The

striking mass was modeled as a rigid material, and since the falling weight assembly was modeled as a

simple hemisphere, an artificially large density was used to give the same mass as the one to use in the

experiments.

5.3.1 Boundary conditions

The experimental restraints are similar to fully clamped boundary conditions. In order to check the

accuracy, two definitions of the boundary condition were evaluated in the numerical model. The

clamped and supported boundary conditions are shown in Figure 5.8. For the striking mass, only the

vertical translation was free, in which direction the initial impact velocity was assigned. In the clamped

model, the edges of the plate were constrained in all degrees of freedom. But in the supported model,

the nodes of the plate at the bolts position were fully clamped. All the degrees of freedom of the lower

support plates were constrained in numerical model, whereas in the upper support plates all the

degrees of freedom except vertical translation were constrained. The “Boundary Prescribed Motion

Node” was selected to define preload force to the upper support plates, loading an initial imposed

displacement to clamp the plate. The contact between the support plates and the plate was defined as

“Automatic Surface to Surface”. The support plates modeled with shell elements were defined as a rigid

material.

Figure 5.8 Boundary conditions of rectangular plate.


5.4.1 Thin plates

The simulations were evaluated in terms of the true stress-strain curve, mesh size, element type and

boundary conditions. The experimental results of the thin plates at impact velocity of 2.7 m/s were used


50

to compare the different parameters studied in the numerical model.

Firstly, three true stress-strain material curves (PL, GL and UN+GL, Section 2.1.3) were used to

compare their differences, as shown in Figure 5.9. The mesh size is 2.0 mm. As the three true material

curves were similar, the impact responses are almost the same. The comparison of the shape of the

deformation is shown in Figure 5.10, showing good agreement between them.

Figure 5.9 Comparison of different materials.

Figure 5.10 Comparison of the shape of deformation.

Three mesh sizes (4.0, 2.0 and 1.0 mm) were evaluated (Figure 5.11). The influence of the mesh

size is very small, and thus the impact model reproduces similar plastic response. Smaller mesh sizes

help to increase the displacement, because the indention out of the plane of plate is better defined and

then the plate takes the same shape of the indenter as seen in the experiments.


51

Figure 5.11 Comparison of different mesh sizes.

The influence of the element type is illustrated in Figure 5.12. Better results were obtained with the

shell model, because this model has 5-integration points through the thickness whereas the solid

model only has one. This implies that a higher number of solid elements are necessary to reproduce

the experimental plastic response.

Figure 5.12 Comparison of shell model and solid model.

A mesh size 2.0 mm was used for the simulations. The comparison between the clamped model

and the supported model is shown in Figure 5.13. The model with supported plates reproduced higher

displacement, but smaller force in the event, because of the sliding between the supports.

Figure 5.13 Comparison of different support.


52

The material strain rate sensitivity is evaluated in the support model using the coefficient of the

Cowper and Symonds constitutive equation (Equation 2.13) for mild and high tensile steel (Figure 5.14).

Using mild steel coefficients (C=40.4 and q=5), the force-displacement response shows a stiffer

behavior. However, using high tensile steel coefficients (C=3200 and q=5), good agreement with the

experiments is noticed at the first stage of impact although the results deviate at the end, leading in

higher forces and consequently smaller displacement.

Figure 5.14 Comparison of different dynamic yield strength. C40.4q5: mild steel coefficients; C3200q5: high tensile

steel coefficients.

Two elements near the impact point were selected to analyze the strain rate. The two elements are

the typical elements under impact, having the maximum stresses in the impact process, and the

positions of these elements (1 and 2) are shown in Figure 5.15. The strain-time and strain rate-time

curve are shown in Figure 5.16. At the beginning of the impact, the average strain rate goes up to 80 s-1,

but then, decreases to about 40 s-1. Actually, when time is 0.004 s, the displacement is 10 mm. At this

point, the strain rate decreases to about 40 s-1. At the same time, the force-displacement response of

high tensile steel coefficient starts to deviate with the experimental results.

impact point

element 1

element 2

2 mm

Figure 5.15 Position of selected element.


53

Figure 5.16 Strain rate of selected elements from numerical simulation.

5.4.2 Thick plates

As mentioned the different indenters influenced the impact response of plate. The UN+GL true material

curve was selected for the numerical simulation of the thick plate. First, the clamped model and the

supported model mentioned above were compared. The numerical simulations using a spherical

indenter of Ø30 mm with impact velocity of 1.94 m/s were selected to compare with the experimental

result. The comparison of the clamped and supported models is shown in Figure 5.17. As similar

results between them were obtained, the clamped model was selected for the remaining numerical

simulations.

Figure 5.17 Force-displacement responses of the clamped and supported models. (Impact velocity 1.94 m/s)

The cylindrical and spherical indenters of Ø30 mm were selected to compare the experimental

force-displacement responses (Figure 5.18). Good agreement between them was found, being the

maximum forces and displacement well predicted. The obtained results were used to compare

intermediate responses using a combination of spherical and cylindrical indenter as shown in Figure

5.19. Two transitive types of indenter between sphere and cylinder were defined cutting out the end of

the sphere (Figure 5.19). The numerical results are shown in Figure 5.20. These two

force-displacement curves are between the results of the spherical and cylindrical indenter,

representing the transition between the responses. When the cut out at the bottom is small, the force

decrease and the displacement increase.


54

Figure 5.18 Force-displacement responses of experimental results and numerical results. E: experimental, S:

Simulation

Figure 5.19 Different type of indenter.

Figure 5.20 Comparison of numerical results using different type of indenter. (Impact velocity 1.94 m/s)


(1) The impact response of the plates is similar at the beginning when using the same indenter at

different velocity.

(2) The numerical analysis demonstrated that the results are insensitive to the mesh size and the type

of element.

(3) The material strain rate has strong effect on the plastic response of plate. Using the high tensile

steel Cowper-Symonds coefficient gives good agreement in most of the response, especially the


55

magnitude of the strain rate is higher than the one that can be obtained for mild steel.

(4) The types of indenter have strong influence on the impact response of plate. If the contact area is

small, such as a sphere, the force decreases and the displacement increases.


56

CHAPTER 6 – Failure prediction of rectangular plates subjected to lateral impact

57

CHAPTER 6 Failure prediction of rectangular plates subjected to

lateral impact

In this chapter, experiments and numerical simulation of laterally loaded plates were conducted in order

to predict the initiation and propagation of fracture. The influence of the indenter on the initiation of

failure plates was studied. The sensitivity of the mesh size and the critical failure strain are reviewed

using the force-displacement response of plates. Discussion of the results and conclusions are given.

6.1 Experimental details and results

The descriptions of impact test machine and its software were given in Section 3.1-3.2, and the

experimental details are the same given in Chapter 5, except for the impact velocity which in this case

is 3.8 m/s. The plate thickness is 1.4 mm (thin plate). The indenters were hemi-spherically ended

projectiles of diameter 10, 16, 20 and 30 mm. In all cases the initial incident energy is 400 J, which is

large enough to provoke fracture.

The results at peak force are summarized in Table 6.1. The maximum force and the displacement

differ for each diameter of indenter. The absorbed energy increases with the diameter of indenter. The

force-displacement responses are shown in Figure 6.1. The slope of force-displacement curve

decreases for the smaller indenters. The failure of the plates by smaller indenters occurred at lower

displacement and force, and thus smaller ratio of the input energy was absorbed for these specimens.

The energy at fracture using the indenter of Ø 10 mm is only one third of the energy at fracture using

the indenter of Ø 30 mm. Thus, it is needed more energy to yield fracture using the other indenters. All

plates initiated fracture at peak force, being rapidly propagated.

The failure modes of plates using the indenters of Ø 20 and 30 mm are shown in Figure 6.2. There

are large plastic deformations in the impact region and the plates start to fracture at the side of impact

region.

Figure 6.1 Experimental force-displacement responses of thin plates with different indenter.


58

Table 6.1 Summary of experimental results of thin plates with different indenter

Diameter of Indenter

(mm)

Values at Peak Force

Force (kN) Displ (mm) Energy (J)

10 11.8 12.9 76.9

16 18.9 16.0 149.4

20 21.8 17.8 209.6

30 27.5 18.7 257.1

(a) (b)

Figure 6.2 Failure modes of the plates. (a) indenter 20 mm; (b) indenter 30 mm

6.2 Numerical model and results

The numerical models are the same as given in Section 5.3. The failure of the plate is predicted using

different true stress-strain curves, mesh sizes, element types and boundary conditions. The specimen

using an indenter of Ø 30 mm was selected to evaluate the numerical simulation.

First, the three true stress-strain curves (PL, GL and UN+GL curve, Section 2.1.3) using a critical

failure strain of 0.9 were used to predict the experimental response (Figure 6.3). All the predicted

force-displacement responses were similar to the experimental response, selecting the good true

material curve and critical failure strain.

Figure 6.3 Force-displacement responses of experimental and numerical results with different true material curves.

(Indenter 30 mm)

The predicted deformation shape of the plate was similar to the experimental one (Figure 6.4). The

plate underwent a large plastic deformation when the stresses exceed the yield strength, and all further


59

plastic deformation was concentrated in the impact region. The plate began to failure locally at the side

of impact region, and then the crack propagated. The shape of the failure mode observed in the

experiment was captured accurately by the numerical simulation.

Figure 6.4 Experimental and numerical failure modes. (Indenter 30 mm)

Two models, one designed in shell and the other in solid elements, were simulated and their

comparison is shown in Figure 6.5. Here the material curve was defined by the UN+GL approximation.

The response of shell model is better than the one of solid model, because of its higher number of

integration points. The shell element has 5-integration points through the thickness and the solid

element only has one, so the shell element is much more suit to analyze the large deformation and

failure in the numerical simulation.

Figure 6.5 Force-displacement responses of shell model and solid model. (Indenter 30 mm)

The clamped model and the supported model mentioned in Section 5.3 were also used to predict the

failure using a mesh size of 2.0 mm (Figure 6.6). It is noted that the sliding between the supports has


60

important influence on the response of plate, decreasing the impact forces and, consequently,

increasing the displacements. As the numerical result of clamped model is more similar with the

experimental result, it is selected for the remaining numerical simulations.

Figure 6.6 Force-displacement responses of different support. (Indenter 30 mm)

Two failure criteria were mentioned in Section 2.3. As in the actual study t equals 1.4 mm and l

equals 2 mm in this case, the failure strain of Peschmann (2001) and Zhang et al. (2004) is 0.660 and

0.434, respectively. These failure strains are used in the numerical model. The previous failure strain of

0.9 which was obtained by numerical simulation of tensile test is also included in the results. The

comparisons of the different failure strain using different indenter are shown in Figure 6.7. The

numerical simulations using the critical failure strain of 0.9 predict better the experimental results. The

failure strain of 0.660 (Peschmann 2001) is in good agreement with the indentation of 30 mm. But all

the numerical results using failure strain of 0.660 (Peschmann 2001) are smaller than the experimental

results. Moreover, the numerical results using failure strain of 0.434 (Zhang et al. 2004) have big

differences from the experimental results.

As the critical failure strain depends on the mesh size (Paik 2007), the numerical models were

calculated using smaller mesh sizes and their corresponding failure strains. The failure strains were

calibrated by succesive numerical simulation of the tensile tests. The mesh size and their

corresponding failure strain are shown in Table 6.2. The material curve proposed by Zhang (2004) was

selected to represent the true material curve. The comparisons of force-displacement responses with

different mesh size are shown in Figures 6.8. Similar impact responses were obtained using different

mesh size and failure strain. The slope of force-displacement response with coarse mesh size is larger

before fracture.

Table 6.2 Mesh size with corresponding failure strain

Mesh size (mm) Failure strain

4 0.6

2 0.9

1 1.3


61

(a) (b)

(c) (d)

Figure 6.7 Force-displacement responses with different failure strain. (a): indenter 10 mm; (b): indenter 16 mm; (c):

indenter 20 mm; (d): indenter 30 mm;

(a) (b)

(c) (d)

Figure 6.8 Force-displacement responses with different mesh size and corresponding failure strain. (a) indenter 10

mm; (a) indenter 16 mm; (a) indenter 20 mm; (a) indenter 30 mm.


62

The numerical models can just predict the point of fracture. Fracture propagation is still not

predicted in numerical analysis because the length of the element is much larger than the one seen in

experimental results. The element would be deleted from the finite element model if its strain exceeds

the defined failure strain, which is different from the fracture propagation. But the crack appears at the

point of fracture. So, if only considering the crack and water inflow, the force-displacement response

before fracture is enough to predict the impact response.


(1) The critical failure strain is the most important parameter in the numerical simulation, which

depends on the mesh size of the finite elements.

(2) The deformed shape is well predicted by the numerical model. The plate undergoes a large plastic

deformation when the stresses exceed the yield strength, and all further plastic deformation is

concentrated in the impact region. The plate begins to failure locally at the side of impact region, and

then the crack propagates.

(3) The indenter type has strong effects on the failure of plate. The small indenter is easier to break

through the plate, absorbing less energy until fracture.

CHAPTER 7 – Plastic response of stiffeners with attached plate subjected to lateral impact

63

CHAPTER 7 Plastic response of stiffeners with attached plate

subjected to lateral impact

This chapter summarizes results from experiments and numerical simulations of stiffeners with

attached plate subjected to lateral loads, thus allowing for predicting the absorption of energy during

the impact event. The sensitivity of the incident velocity and the stiffener type is reviewed using the

force-displacement response of the tested specimens. This chapter is summarized from

Villavicencio et.al (2011d).


The descriptions of impact test machine and its software were given in Section 3.1-3.2. The

experimental tests represent a situation in which a partially supported stiffener with attached plate is

struck at the mid-span by a mass travelling with an initial impact velocity. After the impact, the striker is

assumed to remain in contact with the specimen having an initial velocity at the instant of contact and a

common velocity throughout the entire response.

The experimental set up can be seen in Figure 7.1. The design of the specimens (denoted by

Panel A2 and Panel A3) is shown in Figure 7.2. The indenter is a hemispherically ended projectile of

diameter 30 mm which uses a striking mass of 54.0 kg.

Figure 7.1 Experimental set-up

Figure 7.2 Specimens: stiffeners with attached plate


64

The specimens were partially supported, i.e. the edges in the length direction were fully clamped

whereas the edges in the width direction were free (Figures 7.1 and 7.2). The restrained edges were

supported between two thick rectangular steel plates and were compressed by two bolts at each

support. The lower support plates were stiffened by two relatively thick plates, and were fixed to a

strong structural base to prevent their movement. The torque applied to screw the bolts and compress

specimens was measured providing a known clamping force.

The material of the plate and stiffeners is structural carbon steel and its mechanical properties

were obtained by in-house tensile tests using standard tensile specimens and procedures (ASTM

1989). The results of the tensile tests are presented in Table 7.1.

Table 7.1 Mechanical properties of the material.

Property Units PL. 4.0 FB 4x25

L 50x50x5

Yield stress MPa 286 367

Ultimate tensile strength MPa 426 488

Rupture stress MPa 322 384

Rupture strain (in 100 mm) - 0.21 0.18


The impact test results are summarized in Table 7.2. The resulting force-displacement responses

are shown in Figure 7.3. It is observed that when the impact velocity increases, larger transverse

displacements and impact forces are developed in both Panel A2 and Panel A3. The initial reacting

forces increase with the impact velocity. The magnitudes of the maximum force, maximum deflection

and permanent deflection are similar between both panels when impacted at the same velocity. The

shapes of the force-displacement responses of Panel A3 show more oscillations than the ones of Panel

A2, especially at the first moment of the impact. The instant change of slope, respectively transition

from plate-stiffener bending to membrane behavior, is less profound in Panel A2, as shown in Figure

7.4.

Table 7.2 Results of impact tests


Specimen* Force Defln Energy Defln Energy

(kN) (mm) (J) (mm) (J)

A2V2.0 35.7 4.51 110.9 2.34 80.7

A2V2.7 40.4 6.73 200.9 4.43 165.9

A2V3.3 46.6 9.83 350.9 8.08 326.2

A3V2.0 36.3 4.43 110.9 2.75 86.1

A3V2.7 40.5 6.44 200.1 4.30 168.1

A3V3.3 46.9 8.92 326.1 7.10 295.9

*A2 denotes Panel A2 and A3 denotes Panel A3. V denotes the impact velocity (m/s).


65

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8 9 10Displacement [mm]

For

ce [k

N] 2.0 m/s

2.7 m/s

3.3 m/sPanel A2

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8 9 10Displacement [mm]

For

ce [k

N]

Panel A3

2.0 m/s

2.7 m/s

3.3 m/s

Figure 7.3 Force-displacement responses

0

10

20

30

40

50

0 1 2 3 4 5 6 7Displacement [mm]

For

ce [k

N]

Panel A3

Panel A2

2.7 m/s

Figure 7.4 Force-displacement responses. Panel A2 and Panel A3 impacted at 2.7 m/s

7.3 Numerical model

The model, sketched in Figure 7.5, was computed in LS-DYNA. The plate and stiffener were modeled

in 4-node shell elements with 5-integration points through the thickness. The mesh size was 2.0 mm.

The supported perimeter was constrained in all degrees of freedom. For the striking mass, only the

vertical translation was free, in which direction the initial impact velocity was assigned. The contact

mass-specimen uses nodal normal projections resulting in a continuous contact surface. The selected

material provides a definition of the true stress-strain relationship as an offset table. The striking mass

was modeled as a rigid undeformable material, and since the falling weight assembly was modeled as

a simple hemisphere, an artificially large density was used to give the same mass as the one to use in

the experiments.

Figure 7.5 Details of finite element model


66

This basic finite element model was improved in order to reproduce the experimental plastic response.

Three new models were used (Figure 7.6): One was designed in solid elements and the other two

represented the weld joint using shell and solid elements. The fillet weld cross-section takes the shape

of a triangle and the measured leg length is 4.0 mm.

Figure 7.6 Previous and new finite element models


Specimen A2V2.7 (Table 7.2) was selected to compare the force-displacement responses (Figure 7.7).

The maximum reacting forces are similar in the four numerical models. The ‘Shell’ and ‘Solid’ model

reproduce almost the same response, overestimating the maximum and the permanent deflections.

Although ‘Shell Weld’ model improves the results, the maximum and end displacements are still

overestimated. The ‘Solid Weld’ model has been favorably validated against maximum force and

maximum deflection. However, the permanent deflection is not accurate. Certainly, the real

representation of the weld joint helps to reproduce the experimental response in this model. As a

validation of the ‘Solid Weld’ model, Figure 7.8 shows a good agreement for the whole range of impact

velocities.

0

10

20

30

40

50

0 1 2 3 4 5 6 7 8Displacement [mm]

For

ce [k

N]

(E)

(1) (2)

(3)(4)

Figure 7.7 Force-displacement response, Specimen A2V2.7. (E): Experimental. (1): Shell. (2): Solid. (3): Shell

Weld. (4): Solid Weld.


67

0

10

20

30

40

50

60

0 1 2 3 4 5 6 7 8 9 10Displacement [mm]

For

ce [k

N]

2.0 m/s 2.7 m/s 3.3 m/s

Figure 7.8 Force-displacement response, Panel A2. Experimental results: dashed lines. Numerical results:

continuous lines (Solid Weld model).

The deformed shape and Von Mises stress distribution are shown in Figure 7.9. The three

specimens suffer mainly global deformation. The observed local indentation in the plate thickness is

very small. The maximum stresses occur on the lower surface of the plate opposite the impact point

and on the lower edge of the stiffener. It is noted that the stresses are distributed in the modeled weld

even for low incident energies.

(a) (b)

Figure 7.9 Shape of deformation and von mises stress distribution. Panel A2. (a) Transversal view; (b) Longitudinal

view.


(1) Detailed information of the impact response of stiffeners with attached plate has been obtained

through drop weight impact tests and nonlinear explicit dynamic simulations. The discrepancies

between numerical and experimental results were due to overestimation of the permanent deformation,

whereas the maximum force and maximum deflection were generally very well predicted.

(2) In minor impact events, the stiffener type does not play an important role in the absorption of energy

and in the global deformation. The main influence of the stiffener is observed at the very beginning

impact where the specimen reproduces its stiffness and initial bending.

(3) The numerical simulations of small-scale structural elements require to include the weld joint in


68

order to increase the stiffener resistance and represent a smoother cross-section transition between

the stiffener and the plate.

CHAPTER 8 – Conclusions and further work

69

CHAPTER 8 Conclusions and further work

8.1 Conclusions

This thesis studied the impact strength of structural components, in order to provide the basis for the

studying ship collisions. Experimental and numerical methods were used to analyze the structural

components of ship subjected to lateral impact. Thus, detailed information of the impact response of

structural components was obtained through drop weight impact tests and nonlinear explicit dynamic

simulations.

In the analysis of plastic response of beams, the good agreements obtained between experiments

and simulations result from a correct representation of the boundary condition. The preload force and

the coefficient of friction are the most important parameters in the definition of the representation of the

experimental supports. The force-displacement response is in direct proportion with the impact velocity.

Also, the rebound displacement decreases with the impact velocity. Moreover, the axial displacement

at the supports increases with the impact velocity, and the lateral displacement increases with the axial

displacement using the same velocity.

In the analysis of failure of pre-notched beams, the selection of the mesh size and critical failure

strain by numerical simulation of the tensile tests is valid for numerical simulation of impact tests. Two

basic definitions used in the finite element simulations: model of the supports to represent the

experimental boundary conditions and prediction of the critical failure strain by tensile tests are the

most important parameters obtained from this simple crushing analysis. These parameters are valid in

small-scale structures subjected to impact where such a small mesh size can be defined and full

modeling of the boundary conditions can be performed. These definitions can be applied in complex

structures and to investigate the effects of all the parameters involved in this type of impact analysis.

In the analysis of plastic response of plates, the numerical analysis demonstrated that the results

are insensitive to the mesh size and the type of element. But, the material strain rate has strong effect

on the plastic response of plate. Using the high tensile steel Cowper-Symonds coefficient gives good

agreement in most of the response, especially are the magnitude of the strain rate is higher than the

one that can be obtained for mild steel. The impact response of the plates is similar at the beginning

when using the same indenter at different velocity. But, the indenter type has strong influence on the

impact response of plate. If the contact area is small, such as a sphere, the force decrease and the

displacement increase.

In the analysis of failure of plates, the critical failure strain is the most important parameter in the

numerical simulation, which depends on the mesh size of the finite elements. The deformed shape can

be well predicted by the numerical model. The plate undergoes a large plastic deformation when the

stresses exceed the yield strength, and all further plastic deformation is concentrated in the impact

region. The plate begins to failure locally at the side of impact region, and then the crack propagates.

The indenter type has strong effects on the failure of plate. The small indenter is easier to break

through the plate, absorbing less energy until fracture.

CHAPTER 8 – Conclusions and further work

70

In the analysis of plastic response of stiffeners with attached plate, the differences between

numerical and experimental results were due to overestimation of the permanent deformation, whereas

the maximum force and maximum deflection were generally very well predicted. The main influence of

the stiffener is observed at the very beginning impact where the specimen reproduces its stiffness and

initial bending. The numerical simulations of small-scale structural elements require including the weld

joint in order to increase the stiffener resistance and represent a smoother cross-section transition

between the stiffener and the plate.

All the above analysis studied the plastic response and failure of structural components,

comparing the numerical results with the experimental results. The good results could be obtained if the

material properties are correctly defined and the boundary conditions are correctly simulated. The

simple impact test is a good way to define the material properties which can be used in the analysis of

complex structures.

8.2 Future work

The object of this work is only structural components subjected to lateral impact, and all of them only

have one type of boundary conditions, which are the most important factor in the analysis of impact

strength. There are many works need to do in this research area, and the boundary conditions are also

very hard to simulate in the ship grounding or collision. The analysis of beams and plates subjected to

lateral impact has a big distance with the analysis of ship under grounding or collision. The boundary

conditions of actual collision should be more deeply studied in the future.

The dynamic yield strength is very hard to define the relation with the strain rate in the numerical

simulation and many existing definitions are still not well suit for the experimental results. It need more

work to study the relationship between dynamic yield strength and strain rate, and apply it into the

impact analysis.

Since the impact strength of complex structures requires more attention than the one of stiffened

plates, it is necessary to conduct further experiments and numerical simulations to study their plastic

response and failure propagation.

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Documents

Experimental and Numerical Study on the Impact Strength of Beams and Plates