High velocity impact under axial crushing and bending collapse on thin walled prismatic structures

i ABSTRACT Master’s Thesis HIGH VELOCITY IMPACT UNDER AXIAL CRUSHING AND BENDING COLLAPSE ON THIN-WALLED PRISMATIC STRUCTURES By TRAN HAI NIM: 23613011 Thin-walled prismatic structure

HIGH VELOCITY IMPACT UNDER AXIAL CRUSHING AND BENDING COLLAPSE ON THIN-WALLED PRISMATIC

STRUCTURES

THESIS

Submitted in partial fulfillment of the requirements

for the degree of Master of Engineering from the Institut Teknologi Bandung

by TRAN HAI NIM : 23613011 Aeronautics and Astronautics Study Program

Faculty of Mechanical and Aerospace Engineering

INSTITUT TEKNOLOGI BANDUNG

2015

HIGH VELOCITY IMPACT UNDER AXIAL CRUSHING AND BENDING COLLAPSE ON THIN-WALLED PRISMATIC

STRUCTURES

By

TRAN HAI NIM : 23613011 Aeronautics and Astronautics Study Program Faculty of Mechanical and Aerospace Engineering

Institut Teknologi Bandung

Approval of Supervisory Committee

i

ABSTRACT Master’s Thesis HIGH VELOCITY IMPACT UNDER AXIAL CRUSHING AND BENDING COLLAPSE ON THIN-WALLED

PRISMATIC STRUCTURES

By TRAN HAI NIM: 23613011

Thin-walled prismatic structures are one of the most efficient energy absorbing components in various engineering systems such as components of automobiles, aircrafts, ships and trains They are widely used in buses, coaches, special purpose vehicles, and other areas subject to safety requirement Especially, in modern society, transportation vehicles have higher speed and lager size Therefore, thin-walled structures become more important to absorb impact energy and prevent high deceleration on the passengers during high velocity impact This research investigates the behaviors as well as assesses the effect of strain rate to energy absorption capability of prismatic thin-walled structures subjected to high velocity impact under axial crush loading In addition, this research will also study the bending collapse behavior of thin-walled beams In particular, the deep plastic collapses of the thin-walled beams so that one can predict the ultimate strength and energy absorption capability

For high velocity impact under axial crush loading, the dynamic behavior of thin-walled square columns is examined by analytical and numerical analysis

An explicit-nonlinear commercial finite element code LS-DYNA is used to predict the response of thin-walled square columns subjected to high velocity under axial crushing conditions It revealed the effect of the impact velocity

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and strain rate of material during dynamic plastic buckling progression The strain rate sensitive material- mild steel St-37 is utilized in this study since its stress-strain relation depends on the strain rates Two groups of material models are used, one group has strain rates up to 100 s-1 and the other group has strain rates up to 4500 s-1 The study is conducted by using analytical solution and finite element analysis for impact speeds of 10 m/s, 20 m/s, 30 m/s, 40 m/s and 50 m/s Results indicate that for the impact speed less than 20 m/s, there are no significant differences on the crushing forces However, for impact speed higher than 20 m/s, the crushing forces of the columns with material model covering strain rates up to 4500 s-1 are higher than those obtained using material model covering strain rates up to 100 s-1 The results suggest that analysis of high velocity impact should be performed with material model covering high strain rates Furthermore, the compared results between analytical and numerical analysis have a good agreement in term of mean crushing forces and energy absorption capability of thin-walled structures under high velocity impact loading The axial impact under high velocity conditions for strain rate sensitive material, the peak crushing force, the energy absorption capability and mean crushing force are higher than that

of lower velocity

The bending response of thin-walled beams is studied through numerical and experimental studies in quasi-static and dynamic cases A combination of analytical and numerical results is used to predict the initial and post collapse response of the thin-walled beams The experimental validation was used to confirm analytical and numerical prediction of bending crush behavior of the thin-walled beams, such as peak moment, moment-rotation and energy absorption

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Acknowledgements

First and foremost, I would like to express my deepest gratitude to my advisors, Professor Ichsan Setya Putra, Dr Ir Leonardo Gunawan, Dr.ir Sigit Santosa, Dr.Ir Tatacipta Dirgantara and Dr Ly Hung Anh, for their dedicated guidance and conscientious helping of this research I also indebted a grateful thank to Dr Annisa Jusuf, Dr Hery Setiawan who have support me in many and various ways during this study

My appreciation goes out to Technical staff Mr Eddi Satriyo Wibowo and Engineer from General Motor, Indonesia Mr Didi Nuryadi Arifin who did not hesitate in helping to procure the specimens for the experiments I greatly appreciate helping from Metal Industry Development Center (MIDC), Bandung, Indonesia where many experiments in my research were executed

I gratefully acknowledge AUN/SEED-Net, JICA for the financial support All of that supports help me to have a peace of mind and spent the entire time to complete this study in the best way Thank are due to many my friends, my colleagues at Lightweight Structure Laboratory, Institut Teknologi Bandung, who have always been willing to offer advices Many grateful thanks to my best friend

Mr Vu Tien Dat who always shares and helps me in my research

This last group of people has the greatest influence on my life My father Tran Thanh and my mother Nguyen Thi Mung have support me emotionally throughout my academic career To them I love with all of my life I also would like to express my deep gratitude to my sisters Tran Thi Huong, Tran Thi Anh Nga and Tran Thi Hien, and my brothers Tran Hoa, Tran Khoa, who were always supportive

Finally, I want to thank my girl friend Ms Nguyen Kim Ngoc Hien, who has been waiting and always supported me in any circumstances Within her love, her patience, I have worked and tried to finish well the master program in ITB

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Contents

Abstract i

Acknowledgements iii

Table of Contents iv

List of Figures vii

List of Tables xii

List of Symbols xiii

Chapter 1: Introduction 1

1.1 Research background 1

1.2 Research objectives and methodology 5

1.2.1 Research objectives 5

1.2.2 Research methodology 6

1.3 Research contributions 8

1.4 Research outline 8

Chapter 2: Literature Review and Theoretical Background 11

2.1 Impact mechanics 11

2.2 Axial crushing collapse 11

2.2.1 Inertia effects 11

2.2.2 Strain rate sensitive material 14

2.3 Bending collapse on thin-walled prismatic beams 19

2.3.1 Theoretical bending collapse mechanism of prismatic thin-walled beams 22

2.3.2 Effect of strain rate in bending collapse 25

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2.4 Idealized material model 26

2.4.1 Small plastic deformation 26

2.4.2 Large plastic deformation 28

2.5 Crashworthiness parameters in axial crushing 30

2.5.1 Initial peak force 30

2.5.2 Mean crushing force 31

2.5.3 Total energy 32

2.5.4 Energy absorbed per unit length 32

2.5.5 Crushing force efficiency 33

2.5.6 Crushing length 33

2.6 Theoretical prediction of static and dynamic progressive buckling in axial crushing of thin-walled square columns 34

2.6.1 Static axial crushing of thin-walled square columns 34

2.6.2 Dynamic axial crushing of thin-walled square columns 39

Chapter 3: Axial Crushing of Thin-Walled Prismatic Columns under High Velocity Impact 41

3.1 Finite element analysis of thin-walled square columns under axial impact 41 3.1.1 Explicit finite element solver 42

3.1.2 Dynamic simulation of thin-walled square columns under axial impact 43

3.1.3 Material description 44

3.2 Finite element modeling 48

3.2.1 Effect of element size 49

3.2.2 Effect of initial impact velocity 51

3.2.3 Effect of thickness of thin-walled columns 55

3.2.4 Effect of trigger mechanism 57

3.2.5 Effect of strain rates 59

Chapter 4: Bending Crush Behavior of Prismatic Thin-Walled Beams 68

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4.1 Bending crush resistance of prismatic thin-walled beams 69

4.2 Finite element modeling 70

4.3 Material characterization 71

4.4 Experiment set up 75

4.5 Numerical and experimental results 76

4.5.1 Effect of frictionless coefficient 76

4.5.2 Effect of geometrical imperfection 82

4.5.3 Effect of thickness of thin-walled beams 85

4.6 Dynamic bending response of thin-walled beams 87

4.6.1 Effect of strain rates 87

4.6.2 Drop bending 90

4.6.2.1 Numerical simulation model and experimental set up 90

4.6.2.2 Numerical and experimental results 94

Chapter 5: Summary, Conclusions and Future Works 98

5.1 Summary and conclusions 98

5.2 Future works 101

Bibliography 102

Appendix 107

i Three point bending quasi-static test certificate from MIDC 107

ii Drawings of apparatus of three point bending dynamic drop test 110

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List of Figures

1.1 Distribution of type of collision of passenger car accidents

1.2 Types of crashworthiness application in space frame of car

1.3 Large deformation of passenger compartment during axial crash

1.4 Steel crash box application inspace frame of car

1.5 Large deformation of passenger compartment in side impact

1.6 Frame concept of passenger car in bending crash

1.7 Methodology of thesis

2.1 Dynamic progressive buckling

2.2 Two classes of structures

2.3 Stress-strain curves for mild steel at various uniaxial compressive strain

rates

2.4 Dynamic uniaxial tensile tests on smile steel at various plastic strain rates 2.5 Variation of strength with strain rate for dynamic uniaxial tensile tests on

smile steel

2.6 A typical hinge collapse mechanism

2.7 Schematic showing the bending collapse of aluminum hat profiles

2.8 Deformation pattern of fully filled beam with highly dense foam

2.9 Collapse mechanisms of box section beam

2.10 Hinge mechanism at various stages of development

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2.11 Characteristic circles for measuring the deformation pattern

2.12 Comparison of dynamic bending moment (M)- rotation (θ) curves

2.13 Idealized stress–strain curves of elastic materials

2.14 Idealized stress–strain curves of rigid materials

2.15 Load-deflection curve for axial crushing of thin-walled column

2.16 Mean-load defection response of thin-walled column

2.17 Total energy absorption of thin-walled column

2.18 The asymmetric collapse mode of square box and basic collapse element

(Type I)

2.19 Basic element type symmetric collapse element and development of part

of the basic collapse element

3.1 Description of the dynamic loading case: initial and constant velocity 3.2 INSTRON 4484/ Split Hopkinson bar for tensile test

3.3 True stress– plastic strains of mild steel-St37

3.4 Typical finite element models of the thin-walled square tubes

3.5 Effect of shell elemet on the crushing respone of square tube without

trigger mechanism in symmetric collapse (extensional mode)

3.6 Comparison on the crushing respone of square tube in case I and case II

crashing

3.7 Effect of initial velocity on the crushing respone of square tube without

trigger mechanism in asymmetric collapse (extensional mode)

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3.8 Effect of initial velocity on the crushing respone of square tube without

trigger mechanism in symmetric collapse (in-extensional mode)

3.9 Effect of wall thickness on the mean crushing force-Pm of square tube

without trigger mechanism in symmetric collapse

3.10 Thin-walled square tubes has pure asymmetric as wall thickness increase 3.11 Local folding of square tubes in within and without trigger

3.12 Folding pattern Comparison between without and within trigger

3.13 Effect of trigger mechnism on the crushing respone of square tube with

wall thickness t =1.5 mm

3.14 Yield stress at various strain rate of mild steel

3.15 Effect of strain rate on the peak force of square tube with wall thickness t

=1.5 mm at V = 20 m/s (up to 100/s and 4500/s)

3.16 Effect of strain rate on the mean crushing force- Pm of square tube with

wall thickness t =1.5 mm at V = 20 m/s

3.17 Peak force-Ppeak with strain rate = 4500/s: (a) without trigger (extensional

mode) and (b) with trigger (in-extensional mode), t =1.5 mm

3.18 Mean crushing force- Pm between extensional and in-extensional mode of

square tube with wall thickness t =1.5 mm at strain rate =4500/s

3.19 Mean crushing force- Pm of square tube with wall thickness t =1.5 mm at

V = 20 m/s, 30 m/s, 40 m/s and 50 m/s with strain rate up to 100/s and

4500/s (extensional mode)

3.20 Mean crushing force- Pm comparision between numerical anad analytical

method with wall thickness t =1.5 mm and strain rate up to 4500/s

(extensional mode)

x

3.21 Mean crushing force- Pm comparision between numerical anad analytical

method with wall thickness t =1.5 mm and strain rate up to 4500/s

(in-extensional mode)

3.22 Progressive buckling comparison between asymmetric and symmetric

collapse mode

3.23 Specific energy absorption with strain rate up to 100/s and 4500/s

4.1 Bending collapse mode of a thin-walled beam

4.2 Finite element model of a thin-walled square beam

4.3 Velocity profile of the punch during static bending process

4.4 Tensile test speciment to define stress- strain behavior

4.5 Tensile test specimen before and after do tensilte test

4.6 True stress- Plastic strain of high strength steel CR-CHSP420Y

4.7 Experimental set up for three poin bending test at MIDC (static)

4.8 Comparison between max, mean and minimum propeties of high strength

steel CR-CHSP420Y to test data

4.9 Deformation geometry and friction of the beam

4.10 Punch force- displacement response for various coefficient friction

4.11 Moment-rotation response at the center length for various coefficient

friction

4.12 Deformation pattern of thin-walled beam in three point bending

comparision between numerical and experimental results

4.13 Cross sectional crush of empty thin-walled beam

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4.14 Effect of radius - r to the punch forceof thin-walled beam

4.15 Effect of radius - r to the bending moment of thin-walled beam

4.16 Effect of thickness at the corner of (a) Punch force (b) bending moment 4.17 Punch force- displacement and Moment- rotation response varying

4.25 Load- displacement curves of dynamic and quasi-static three point

bending of thin-walled tubes high strength steel CR-CHSP420Y

4.26 Total energy absorption of dynamic and quasi-static of thin-walled tubes-

high strength steel CR-CHSP420Y

4.27 Calibrated data for load cell from MIDC

4.28 Load- displacement curves of dynamic three point bending of thin-walled

tubes high strength steel CR-CHSP420Y in numerical and experimental solution

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List of Tables

3.1 Material properties of mild steel material used in this study

3.2 Square tube geometry, mass, velocity of impactor

3.3 Pm with different element size (t =1 mm, V =30 m/s)

3.4 Pm comparison between theoretical and numerical analysis with

asymmetric collapse mode which has strain rate up to 100/s

3.5 Pm comparison between theoretical and numerical analysis with symmetric

collapse mode which has strain rate up to 100/s

3.6 Pm comparison between two models have strain rate up to 100/s and

4500/s with asymmetric extensional collapse mode

3.7 Pm comparison between theoretical and numerical analysis with

asymmetric extensional collapse mode and in-extensional mode has strain

rate up to 4500/s

4.1 Material properties of high strength steel CR-CHSP420Y

4.2 Comparison peak force of the punch- Ppeak between numerical and

experimental analysis

4.3 Comparison ultimate bending moment-Mu

4.4 Comparision peak force- Ppeak and ultimate bending moment-Mu with

varying thickness at corners of thin-walled beam

4.5 Varying ratio of t/b investigated in this work

4.6 Peak force comparison between numerical and experiment

b Width of square tubes

c Speed of stress wave propagation

E Energy absorbed in collapse mode Type II

EA Total energy absorption

H Haft folding length

SEA Specific energy absorption

t Thickness of thin-walled prismatic columns

t

 Time step

v Impact velocity of impacting mass

 The angle of bending rotation

c

 Critical angle of bending rotation

Figure 1.1 Distribution of type of collision of passenger car accidents [1]

Considering the impact energy management by absorbing and dissipating the kinetic energy to reduce the extreme acceleration or impulsive forces transfer to the occupants, crashworthiness is as a feature of safety structure commonly used

to build vehicle (Fig.1.2)

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Figure 1.2 Types of crashworthiness application in space frame of car [2]

Figure 1.3 Large deformation of passenger compartment during axial crash [3]

In order to avoid the catastrophic damage when the accident happens, the cars or vehicles have to be designed very well The design of vehicles structure must

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consider the amount of kinetic energy during the impact, which should be absorbed by structural components and the deceleration of the passenger must be limited In other words, the passenger compartment should not be deformed and intrusion must be minimized to protect the occupants [31-39]

In high velocity impact, steels have been intensively used to improve vehicle crashworthiness due to its higher energy absorption capability With high rate of loading, the energy absorbing capability of steel thin-walled structures is higher because the steel material has higher flow stress at high strain rates Therefore, clarifying the effect of various impact velocities on the energy absorption characteristics of thin-walled structures is significant to lead to comprehensive results The deformation mode of a single component, as well as the overall crash behavior of a vehicle may change due to different strain rates in high velocity axial impact

Figure 1.4 Steel crash box application inspace frame of car [4]

As shown, in Fig.1.1, front impact accounted for 52% while side impact accounted around 27% of automotive accidents When a car impacts the side of another vehicles, it is known as a side impact collision According to one study,

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approximately 8500 individuals are killed each year as a result of a side impact car crash [1] In side impact, there are limited structures and space between the occupant and the side structure of the impacted vehicle Therefore, side impact events expose passengers and driver to serious injuries

Contrary to front impact event, the structures of vehicles will be subjected to bending load in side impact collision The deformation mode of components in side impact is not considered as progressive buckling as in front impact events The front structures of car in front impact can absorb much higher impact energy than the side structures

Figure 1.5 Large deformation of passenger compartment in side impact [1]

To reduce injury severity in side impact events, many methods are applied on cars Newer vehicles are often equipped with side airbags to protect passengers and drivers from the risk of serious injury from a side impact event Depending on the location of the impact, passengers or drivers are at great risk for injury even when they are using seatbelts or the vehicle is equipped with airbags Therefore, this research studies the bending collapse behavior of thin-walled beams with respect to ultimate bending moment and energy absorption capability

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Figure 1.6 Frame concept of passenger car in bending crash [5]

1.2 Research objectives and methodology

1.2.1 Research objectives

The primary objective of this thesis is as follow:

“To study the crush behavior of thin-walled prismatic structures under high velocity in axial crushing and bending collapse load and to facilitate their application in energy absorbing structures or systems to manufacture safer vehicles”

The subsequent objectives of this thesis are:

(i) To predict the characteristics of thin-walled prismatic structures subjected

to high velocity in axial and bending load, and assess the effect of strain rate on energy absorption capability under varying impact velocities (ii) To compare the energy absorption of thin-walled prismatic structures

response for various impact velocities

(iii) To predict the bending collapse characteristics of thin-walled beams in

quasi-static and dynamic loading conditions

(iv) To develop finite element models of thin-walled structures in high velocity

impact under axial crushing load and bending collapse

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(v) To do bending collapse experiment with three point bending to validate

the numerical and theoretical results

to ensure they simulate the response of thin-walled structures with sufficient accuracy The FE model of quasi-static and impact response of rectangular structures under axial loading are verified using existing theoretical and numerical models Once verified, the FE model could be used to predict the deformation and energy absorption response of thin-walled square structures under high velocity impact conditions

In second part, the bending resistance of thin-walled beams after reaching the ultimate value at a small rotation is predicted and examined In other words, the bending strength and moment-rotation response of thin-walled prismatic beams are studied by using numerical and experimental analysis in static and dynamic bending collapse

All modeling in this thesis are conducted using the explicit commercial finite element code LS-DYNA for their static and dynamic crash behavior

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1.3 Research contribution

The contributions of this research are achieving good understanding of axial crush and bending collapse of thin-walled structures under quasi static and dynamic loading Two types of impact problems (axial crushing and bending collapse) that accounts for majority of vehicle accidents were studied These contributions are briefly summarized below:

(1) Using the finite element models to understand the effect of strain rate

of material when the impact velocity changes and to predict the energy absorption characteristics of steel square columns in axial and bending crash This knowledge is also useful for designed purposes Additionally, the knowledge generated in this study shows how energy absorption capability of such steel structures is influenced by various high velocities Based on the parametric studies, designed guidelines have been developed which can be used for manufacturing the energy absorbing systems especially in front impact accidents It is showed in Chapter 3 and Chapter 4 of this research

(2) Validated finite element models have been developed and they can be utilized to predict characteristic response of thin-walled square beams under static and dynamic bending crush loading conditions This study

is conducted by using theoretical, numerical and experimental analysis with high strength material that is used widely in modern cars Finally, the analysis of this research will provide designers with a guideline to reach acceptable safety level in vehicular manufacturing

1.4 Thesis outline

This thesis is divided in Five Chapters Following this introduction, Chapter 2

provides a review of literature and theoretical background related to the research objectives The literature review includes impact mechanics under axial crushing and in bending collapse In axial crushing, the survey of inertial effects (2.2.1), strain rate sensitive material (2.2.2) will be carried out In bending collapse of

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thin-walled prismatic beams in quasi static (2.3) with first theoretical analysis is presented The idealized material model is introduced in small and large deformation (2.4) In addition, the theoretical analysis of single walled prismatic column under axial crushing is reviewed in section (2.5) In this section, crashworthiness parameters: Initial peak crushing force (2.5.1), Mean crushing force (2.5.2), Total energy absorption (2.5.3), Energy absorption per unit length (2.5.4), Crushing force efficiency (2.5.5) and Crushing length (2.5.6) are defined After that, theoretical predictions in static and dynamic crushing resistance of square columns are presented in section (2.6.1), and (2.6.2)

Chapter 3 describes and focuses on the numerical analysis for axial crushing

behaviors of thin-walled square columns under high velocity conditions This includes geometry of structures and several factors that influence energy absorption capability of thin-walled square structures Much information about the finite element modeling are reviewed and developed: Mesh size effect in high velocity, Boundary condition, Effect of triggering imperfection and Effect of strain rate Furthermore, the resultant comparisons between numerical and theoretical analysis are also investigated under dynamic loading with various high velocities impact

Chapter 4 derives the theoretical method for predicting the bending collapse

behavior of thin-walled empty beams This Chapter devotes on the energy absorption in the hinges and ultimate bending collapse characteristics of thin-

walled empty beams Similar to Chapter 3, many details related to crushing

resistance in bending will be carried out In other words, the energy absorbing characteristics of thin-walled empty beams under various velocities and strain rate effect will be discussed extensively The numerical analysis for bending collapse under various effective factors: Triggering imperfection, friction coefficient, initial velocity, strain rate effect, effect of thickness are also emphasized and evaluated To validate theoretical and numerical analysis, three point bending testes are conducted to compare and verify the displacements, punch forces and

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ultimate bending moment during bending collapse event The three point bending experiments were completed not only in quasi-static but also in dynamic drop test

Finally, chapter Five summarizes the main conclusions of the research and their

implications Many experiences during numerical and experimental studies are summarized and shared Moreover, the suggestions for future research topics are given on the basic of this research

to the kinetic energy ( 1 2

2mv ) The collapse response of a structure subjected to dynamic load differs from the response under a quasi-static load due to two physical phenomena known as inertial effects and strain rate effects The effect depends critically on the relative velocity of the vehicle bodies to one another Furthermore, the value of force and displacement during collision is function of time In other words, the displacement and force are time dependent The energy absorption response of thin-walled energy absorbers is particularly influenced by these phenomena They were referred to Norman Jones [6], W Johnson [7], Tam

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known as progressive buckling Additionally, if the thin-walled structures subjected to a dynamic axial load, then the inertia effect will influence the structure‟s response It is called dynamic progressive buckling Thus, the deformed profile of structures is similar in case of both static buckling and dynamic progressive buckling

Figure 2.1 a) Dynamic progressive buckling Permanent profile of mild steel

structures impacted axially at University of Liverpool [6]

b) Dynamic plastic buckling Permanent profile of an aluminum 6061-T6 structures subjected to an axial impulse at the University

of Liverpool [6]

Inertia can influence impact response in different ways, depending on the nature

of the structures Calladine and English (1984) [10] defined two types of structures which plastically deform to absorb impact energy Type I structure is modeled as a laterally compressed circular structure, while the type II structure is modeled as a column with two separate bars fixed together at clamped supports,

in which deformation is assumed to take place at plastic hinges Furthermore, according to Calladine and English‟s experiments [10], the deformation of type II structure was more sensitive to impact velocity than type I structure by keeping

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the kinetic energy of impactor mass constant, higher impact velocities cause smaller final deflections, and this phenomenon is much more significant for type

II structure than type I structure

Figure 2.2 Two classes of structures: a) Type I b) Type II [9]

In type I structures, the load-displacement curve is flat-topped in the plastic range, for example in the arches and rings loaded radically The response is different in type II structure, there is an initial peak force followed by a rapidly descending portion in the load- displacement curve This difference can be attributed that in type II structure, the shape of the curve indicates that a disproportionately large amount of energy is absorbed in the first small increment of displacement, which

is direct consequence of the geometric effect [8] That reason leads on more energy will be absorbed in the first short time of deformation duration whereas high deceleration and hence significant inertia loading

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According to L L Tam [8], the deformation of type II structure has an initial of collapse initiated by the effect of lateral inertial force on the plates A considerable portion of the impactor‟s kinetic energy was absorbed during this phase via axial compression of the plates

Tam and Calladine [8] showed that the deformation of the Type II structure specimen involves two separate phases of behavior In phase 1, a high initial axial force is set up in the specimen This phase ends when the specimen has accelerated laterally, which is sufficient to enable the motion of the striker to be accommodated without further axial shortening of the specimen During phase 1, energy is absorbed in much the same way as it is during the collision of two compact masses, which adhere to each other after impact, and the fraction of energy „lost‟ depends strongly on the ratio between the mass of the striker and the mass of the specimen Phase 2 involves the kinetic energy, which remains in the striker and the specimen at the end of phase 1 The kinetic energy is absorbed in phase 2 by the rotation of plastic hinges The relevant yield stress in this process corresponds to the material of the specimen at the appropriate strain-rate

Overall, this section shows that the response of thin-walled tubes under dynamic impact loading conditions is sensitive to impact mass and velocity due to inertia effects

2.2.2 Strain rate sensitive material

High velocity impact problems involve large plastic deformation and high strain rate In particular, the stress- strain curves of the material, such as steel will change if the rate of subjected load changes (change impact velocity) Nevertheless, the properties of many materials subjected to dynamic loading are not same to the corresponding quasi-static property values [6], [11] and [12] For strain rate sensitive material such as steel, the stress-strain relation is sensitive to the testing velocity This phenomenon, which is known as strain rate sensitivity or viscoplasticity, can significantly influence the dynamic response of energy absorbing structures, and as such it is considered in this section

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According to Norman Jones and Abramowicz [6], [11], [12], when the strain rates increase, the yield and ultimate stress of the material increases due to the yield criteria or plastic flow of their material is sensitive to strain rate To evaluate the dependence of plastic flow on various strain rates, Marsh and Campbell‟s

experiment was reviewed by N.Jones [6] A vertical 1 m long mild steel bar (L= 1

m) which is struck at one end with a large mass dropped from a height 5 m and having an axial velocity of 10 m/s on impact It is demonstrated in Fig.2.3 under unaxial compressive strain rate The average axial strain rate of the bar will be

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Figure 2.3 Stress (ϭ)-strain (Ɛ) curves for mild steel at various uniaxial

compressive strain rates according to Marsh and Campbell 1 unit of ordinate is

103 lbf/in2 or 6.895 MN/ m2 [6]

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In case of tension, results were conducted by Manjoine [6] His experiment indicated that the lower yield stress and the ultimate tensile stress increased with increase in strain rate, the increase being more significant for the lower yield stress However according to Campbell and Cooper [13], the ultimate tensile stresses increase but more slowly His results showed that the fracture strain decreases with increase in strain rate (Fig.2.4) In other words, the material becomes more brittle at higher strain rates

Figure 2.4 Dynamic uniaxial tensile tests on smile steel at various plastic strain

rates, A: =106 s-1; B: =55 s-1; C: =2 s-1, D: =0.22 s-1; E: =0.001 s-1 1 unit of ordinate is 103 lbf/in2 or 6.895 MN/m2 [6]

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Figure 2.5 Variation of strength with strain rate for dynamic uniaxial tensile tests

on smile steel A: upper yield stress; B: lower yield stress; C: ultimate tensile stress; 1 unit of ordinate is 1kg/mm2 or 9.807 MN/m2 [6]

Cowper-Symonds constitutive equation for axial crushing

A constitutive relation shows reasonable agreement with available experimental data for various metals is the Cowper-Symonds constitutive equation, defined as [6]

1 ; :

q d

where d is the dynamic flow stress at a uniaxial plastic strain rate  , and p sis

the associated static flow stress C and q are material parameter According to

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Reid and Reddy (1986) [14], the Cowper-Symonds relationship is widely used to account for strain rate effects on dynamic structural plasticity problems The dynamic flow stress agrees reasonably well with the dynamic uniaxial tension and compression test results on several materials Eq.(2.1) can also be written as

log log d 1 log

plotted against logep

The parameter q is the slope of this straight line, while the intercept on the

ordinate is loge C The Eq.(2.2) may be rewritten in the form:

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q p d

In other words, these values for C and q were found suitable for mild steel

structures, which deform with strain close to those associated with the static ultimate tensile stress A number of studies on the crushing of mild steel structures have used these values (Abramowicz & Jones 1984 [11]; Reid & Reddy 1986; Reid et al 1986 [14])

2.3 Bending collapse on thin-walled prismatic beams

In this section, researcher has recently turned attention to the bending collapse mode of thin-walled members The application of bending collapse is in the rollover or side impact event where the thin-walled structures and the frame are subjected to bending collapse load The associated impact energy absorption of the structures is localized at plastic hinges

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Figure 2.6 A typical hinge collapse mechanism a) and cross section b) [15]

The bending mechanism firstly involves bulging of the side webs followed by development of the collapse mechanism with traveling of hinge lines These hinge lines define the growth of the inward and outward forming buckles As the rotation angle of the structures increases, the traveling hinge lines eventually stop and additional hinge lines develop The bending mechanism stops when jamming occurs between the two buckled halves of the compression flange

Kecman (1983) [15] provided the first theoretical analysis for bending collapse behaviors of thin-walled prismatic beams In this theorem, Kecman utilized the hinge moment to angle rotation for the bending collapse behaviors of rectangular and square section structures Chen (2001) [18] also investigated the bending collapse behaviors of aluminum hat profile in numerical and experimental analysis

Thin-walled beams with hat section have been used as energy absorbing elements

in vehicle body structures [19] Under bending load, such structures behave similarly to square and rectangular structures while the load –deflection curves largely resemble those for other sections

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Figure 2.7 Schematic showing the bending collapse of aluminum hat profiles [16]

Using thin-walled structures with foam filler will provide a higher bending resistance by retarding inward fold formation at the compression flange The presence of the foam filler changes the crushing modes from single stationary fold

to a multiple propagation fold In addition, many results of filled thin-walled beams were conducted in numerical and experimental methods by S Santosa, J Banhart and Wierzbicki (2001) [20]

Figure 2.8 Deformation pattern of fully filled beam with highly dense foam [20]

Hodes and Harvey [16] have examined the maximum bending strength for lipped channel section After that, Dawson and Walker [17] developed a similar method and they gave the formulae that could also be used for uniaxial bending of

rectangular section structures The maximum moment-Mm of rectangular structures was calculated by Dusan Kecman (1983) [15]:

Figure 2.9 Collapse mechanisms of box section beam [15]

According to Dusan Kecman (1983) [15], the theoretical model of an actual hinge collapse mechanism is based on the second phase of collapse, which includes hinge angles normally allowed and displayed in vehicle safety structures Theoretical mechanism is based on the assumptions that the walls of a section

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deform along the concentrated yield lines only, that the walls are incompressible and inextensible and that structural continuity is maintained in the two characteristic sections (Fig 2.10)

Figure 2.10 Hinge mechanisms (a) at various stages of development (b) [15]

a) Most of the plastic deformation is concentrated along the stationary yield

lines EF, GH, EB, GB, FC, HC, BC, BA, CJ, GK, NH, EL and FM