numerical modelling of reinforced concrete members under …
TRANSCRIPT
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NUMERICAL MODELLING OF REINFORCED CONCRETE MEMBERS UNDER IMPACT LOAD
by
MD. SHAHARIAR FEROJ HOSSAIN
MASTER OF SCIENCE IN CIVIL ENGINEERING (STRUCTURE)
DEPARTMENT OF CIVIL ENGINEERING
BANGLADESH UNIVERSITY OF ENGINEERING AND TECHNOLOGY
DHAKA, BANGLADESH
FEBRUARY, 2015
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The thesis titled “NUMERICAL MODELLING OF REINFORCED CONCRETE
MEMBERS UNDER IMPACT LOAD” submitted by MD. SHAHARIAR FEROJ
HOSSAIN, Roll no: 1009042322P, Session: October 2009; has been accepted as
satisfactory in partial fulfillment of the requirement for the degree of Master of Science
in Civil Engineering (Structure) on 18 February 2015.
BOARD OF EXAMINERS
Dr. Tahsin Reza Hossain Professor Department of Civil Engineering BUET, Dhaka-1000
Chairman
Dr. A.M.M. Taufiqul Anwar Professor and Head Department of Civil Engineering BUET, Dhaka-1000
Member (Ex-Officio)
Dr. Bashir Ahmed Professor Department of Civil Engineering BUET, Dhaka-1000
Member
Dr. Sharmin Reza Chowdhury Associate Professor Department of Civil Engineering Ahsanullah University of Science and Technology, Dhaka
Member (External)
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DECLARATION
It is hereby declared that except for the contents where specific references have been
made to the work of others, the study contained in this thesis is the result of
investigation carried out by the author. No part of this thesis has been submitted to any
other university for a Degree, Diploma or other qualification (except for publication)
Signature of the Candidate
(Md. Shahariar Feroj Hossain)
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ACKNOWLEDGEMENT
First of all, I praise and thank Almighty Allah for giving me strength and ability to
complete this research work.
I would like to gratefully and sincerely thank Dr. Tahsin Reza Hossain, Professor,
Department of Civil Engineering, Bangladesh University of Engineering and
Technology, BUET, Dhaka, for his guidance, understanding, patience, and most
importantly, his friendship during my research work. He helped me by providing
necessary references, books and valuable advices.
I would also like to thank the Head of the Department of Civil Engineering, BUET for
providing all the facilities of the Department in materializing this work. Additionally, I
am very gratefully acknowledges the cooperation of all concerned persons and offices
of BUET for their helps and advices.
Finally, it is also a good opportunity to express my sincere respect and gratitude to my
father and mother for their continuous encouragement and blessings in completing the
research work.
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ABSTRACT
Response due to impact load is different from that caused due to static load. Broadly,
the impact load can be classified into i) low velocity large mass impact and ii) high
velocity small mass impact. The first category involves collision of vehicle into crash
barriers, piers of bridges, drop of an object on slab etc. whereas the second one includes
bullet or missile hitting structures, birds hitting airplane etc. Reinforced concrete (RC)
members are often subjected to extreme dynamic loading condition due to direct
impact. In context of Civil Engineering problem, an investigation into the impact
behaviour of RC members subjected to low velocity high mass is very important. The
current work deals with low velocity large mass impact on RC structures.
Full-scale test of RC member under impact load is very expensive and time-consuming
work. The numerical finite element (FE) analysis of RC member has become an
effective and reliable solution to overcome this problem. Before carrying out numerical
simulation of RC member under impact load, some existing literatures on the relevant
field based on experimental, analytical and numerical approaches are thoroughly
reviewed. RC members have been modeled by nonlinear FE software ABAQUS
(2012). The nonlinearity of RC member has been achieved by incorporating nonlinear
effects due to cracking and crushing of concrete and yielding of steel reinforcement.
The Concrete Damage Plasticity (CDP) model has been used with appropriate
parameters to model the nonlinear behaviour of concrete material and elastic-plastic
material has been selected for steel reinforcement. Performance of this numerical
simulation has been validated against experimental as well as analytical results for
static loads.
The numerical simulation is then extended to impact loading. A number of beams and
slabs tested by Chan and May (2009) under impact load has been modeled and
observed responses have been found to be comparable. The transient impact force
histories and crack patterns obtained from FE analysis of these beams and slabs match
reasonably well with the test results but a time lag has been observed between peak
impact forces for FE analysis. So, CDP model provides consistent results for static as
well as impact analysis of RC members. A series of RC beams subjected to low speed
high mass impact, tested by Tachibana et al. (2010), has been numerically modeled and
analyzed. From the observation of these analyzed beams it is noted that, if only global
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damage is under consideration and analyzed RC beam is failed completely due to high
mass low velocity impactor‟s load then impulse, total area under time-force curve, only
depend upon the momentum of impactor. The duration of impact load varies
proportionally with the ratio of momentum of impactor to ultimate bending capacity of
beam. The beam fails completely, if the mean impact force exceeds 1.37 times of its
ultimate bending capacity. The bending capacity of RC column is also be increased by
1.37 times of its actual capacity, if the failure is governed by tension. But axial capacity
will be reduced by 0.91 times when failure is by crushing of concrete before tension
yielding.
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CONTENTS
DECLARATION .......................................................................................................... iii
ACKNOWLEDGEMENT ............................................................................................ iv
ABSTRACT .................................................................................................................... v
CONTENTS .................................................................................................................. vii
LIST OF FIGURES ..................................................................................................... xii
LIST OF TABLES ..................................................................................................... xvii
NOTATIONS ............................................................................................................... xix
Chapter 1: Introduction ................................................................................................. 1
1.1 Background and Present State of the Problem ........................................................ 1
1.2 Objectives with Specific Aims and Possible Outcome .......................................... 2
1.3 Methodology of Work ............................................................................................ 3
1.4 Outlines of the Thesis ............................................................................................. 4
Chapter 2: Literature Review ....................................................................................... 5
2.1 Introduction ............................................................................................................ 5
2.2 Classification of Impact Events Considering Response of Target Structure .......... 5
2.3 Investigation Technique of Impact Events ............................................................. 7
2.3.1 Impact Behaviour of Concrete Beams ............................................................. 7
2.3.2 Failure mode of RC beam under impact load .................................................. 9
2.4 Experimental Investigation of RC Bram under Impact load ................................ 10
2.4.1 Experimental investigation of RC beam carried out by Mylrea (1940) ......... 10
2.4.2 Experimental investigation of RC beam carried out by Feldman et al. (1956, 1958, 1962) ............................................................................................................. 11
2.4.3 Experimental investigation of RC beam carried out by Hughes and Speirs (1982) ...................................................................................................................... 11
2.4.4 Experimental investigation of RC beam carried out by Ando et al. (1999)... 11
2.4.5 Experimental investigation of RC beam carried out by Kishi et al. (2001) ... 12
2.4.6 Experimental investigation of RC beam carried out by Magnusson et al. (2000) ...................................................................................................................... 14
2.4.7 Experimental investigation of RC beam carried out by May et al. (2005, 2006) and Chen and May (2009) ............................................................................ 16
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2.4.8 Experimental investigation of RC beam carried out by Saatci and Vecchio (2009a) .................................................................................................................... 17
2.4.9 Experimental investigation of RC beam carried out by Fujikake et al. (2009) ................................................................................................................................. 18
2.5 Analytical Models for Impact Loading on RC Member ....................................... 25
2.5.1 Proposed analytical models carried out by Hughes and Speirs (1982) and Hughes and Beeby (1982) ....................................................................................... 26
2.5.2 Proposed idealizing single degree of freedom system carried out by Comite Euro-International Du Beton (CEB, 1988) ............................................................. 29
2.6 Numerical Model of RC Beam under Impact Load.............................................. 31
2.6.1 Numerical Model of RC Beam under Impact Load carried out by Sangi, A. J. (2011) ...................................................................................................................... 31
2.7 Impact Behaviour of RC Slabs ............................................................................. 33
2.7.1 Impact tests on RC slabs carried out by Sawan and Abdel-Rohman (1986) . 35
2.7.2 Impact tests on RC slabs carried out by Kishi et al. (1997) ........................... 36
2.7.3 Impact tests on RC slabs carried out by Zineddin and Krauthammer (2007) 36
2.7.4 Impact tests on RC slabs carried out by Chen and May (2009) ..................... 37
2.8 Impact Behaviour of Concrete Column ................................................................ 37
2.8.1 Impact tests on RC Column carried out by Leodolft (1989) ......................... 38
2.8.2 Impact tests on RC Column carried out by Feyerabend (1988) ..................... 38
2.8.3 Impact tests on RC Column carried out by Gebbeken et al. (2007) .............. 40
2.9 FE Modelling ........................................................................................................ 40
2.9.1 FE package ..................................................................................................... 41
2.9.2 An overview of ABAQUS (2012) ................................................................. 41
2.9.3 FE modelling of RC ....................................................................................... 44
2.9.4 Contact algorithms ......................................................................................... 47
2.10 Constitutive Concrete Material Models .............................................................. 47
2.10.1 Concrete Damage Plasticity model .............................................................. 48
2.10.2 Material model for reinforcing steel ............................................................ 50
2.11 Damping Coefficients ......................................................................................... 50
2.12 Summary ............................................................................................................. 50
Chapter 3: Nonlinear FE modelling Validation ........................................................ 52
3.1 Introduction .......................................................................................................... 52
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3.2 FE Modelling in ABAQUS (2012) ....................................................................... 52
3.3 Validation of FE Model of RC Beam under Static Load with Test Result .......... 54
3.3.1 Dimension of tested RC beam by Saatci (2007) ............................................ 54
3.3.2. Modelling of tested beam by Saatci (2007) .................................................. 56
3.3.3 Response of MS0 beam ................................................................................. 57
3.3.4 Response of MS1 beam ................................................................................. 58
3.3.5 Response of MS2 beam ................................................................................. 58
3.4 Validation of FE Model of RC Slab under Static Load Tested by McNeice (1967) .................................................................................................................................... 60
3.5 Validation of RC Beam Modelling with Theoretical Result ................................ 62
3.5.1 Response of beam .......................................................................................... 65
3.6 Linear FE Analysis of SDOF System under Dynamic Load ................................ 74
3.6.1 Lateral stiffness of the structure ..................................................................... 75
3.6.2 Response to free vibration.............................................................................. 76
3.6.3 Response to step load ..................................................................................... 80
3.6.4 Validation of dynamic equation of equilibrium ............................................. 82
3.7 Response of Nonlinear SDOF System to Free Vibration ..................................... 83
3.8 FE Analysis of SDOF Beam under Impact Load ................................................. 88
3.9 Summary ............................................................................................................... 90
Chapter 4: FE modelling of RC Beam and Slab under Impact ............................... 92
4.1 Introduction .......................................................................................................... 92
4.2 FE Analysis of RC Beam under Impact Load ...................................................... 92
4.3 Simulations of Beam with Plywood Pad at Interface of Beam and Impactor ...... 93
4.3.1 Element‟s modelling ...................................................................................... 93
4.3.2 Parts interaction ............................................................................................. 94
4.3.3 Material property ........................................................................................... 94
4.4 Mesh Sensitivity of Beam with Plywood Pad at Interface of Beam and Impactor .................................................................................................................................... 96
4.4.1 Sensitivity analysis for linear material properties of beam ............................ 97
4.4.2 Sensitivity analysis for nonlinear material properties of beam ...................... 98
4.5 Validation of FE Analysis Results ...................................................................... 101
4.5.1 Transient impact force ................................................................................. 101
4.5.2 Crack patterns and damage .......................................................................... 103
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4.5.3 Correlation between transient load and crack development ........................ 104
4.6 Computational Nonlinear Simulations of Beam without Plywood Pad ............. 104
4.7 Validation of FE Analysis Results ...................................................................... 106
4.7.1 Transient impact force ................................................................................. 106
4.7.2 Crack patterns and damage .......................................................................... 107
4.8 FE Analysis of RC Slab under Impact Load ...................................................... 108
4.8.1 Description of slabs tested by Chen and May (2009) .................................. 108
4.8.2 Experimental result of tested slab ................................................................ 109
4.9 Computational Nonlinear FE Analysis of Slab-2 ............................................... 110
4.9.1 Element‟s modelling of RC slab .................................................................. 111
4.9.2 Parts interaction ........................................................................................... 112
4.9.3 Material property ......................................................................................... 112
4.10 Mesh Sensitivity Analysis of Slab .................................................................... 113
4.10.1 Sensitivity analysis for linear material properties of slab .......................... 115
4.10.2 Sensitivity analysis for nonlinear material properties of slab .................... 116
4.11 Comparison of FE Analysis Results of Slab with Test Results ........................ 118
4.11.1 Transient impact force ............................................................................... 118
4.11.2 Crack patterns and damage ........................................................................ 119
4.12 Summary ........................................................................................................... 120
Chapter 5: Behaviour of RC Structure under Impact Load .................................. 121
5.1 Introduction ........................................................................................................ 121
5.2 RC Beam under Impact Load ............................................................................. 121
5.3 Description of Beam Used in Analysis ............................................................... 122
5.3.1 Dimension of beams .................................................................................... 122
5.3.2 Material of beams ........................................................................................ 124
5.3.3 Overview of impactor and respective beam ................................................. 125
5.4 Numerical Modelling of Beam ........................................................................... 126
5.5 Result of Beam Analysis .................................................................................... 129
5.6 Evaluation of Damage Level for RC Beam ........................................................ 136
5.6.1 Result of beam analysis ............................................................................... 136
5.7 RC Column under Impact Load .......................................................................... 140
5.7.1 Dimension and material properties of column ............................................. 141
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5.7.2 Overview of impactor .................................................................................. 143
5.8 Numerical Modelling of Column ....................................................................... 144
5.9 Result of Column ................................................................................................ 146
5.10 Impact Load on Flyover Pier ............................................................................ 148
5.12 Summary ........................................................................................................... 155
Chapter 6: Conclusion and Recommendation ......................................................... 157
6.1 Introduction ........................................................................................................ 157
6.2 Findings of Work ................................................................................................ 157
6.3 Summary ............................................................................................................. 159
6.4 Recommendation for Future Studies .................................................................. 160
Reference ..................................................................................................................... 161
Appendix-A ................................................................................................................. 168
Appendix-B ................................................................................................................. 174
Appendix-C ................................................................................................................. 180
Appendix-D ................................................................................................................. 186
Appendix-E ................................................................................................................. 188
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LIST OF FIGURES
Figure 2.1: Missile impact effects on concrete target, (a) Penetration, (b) Cone cracking, (c) Spalling, (d) Cracks on (i) proximal face and (ii) distal face, (e) Scabbing, (f) Perforation, and (g) Overall target response (Li et al., 2005) ..................................... 6 Figure 2.2: Contact-impact problem involving a concrete beam (Thabet, 1994) ........... 9 Figure 2.3: Dimensions of RC beams tested by Kishi et al. (2001) .............................. 12 Figure 2.4: Simplified model for the reaction force verses displacement loop (Kishi et al., 2002) ......................................................................................................................... 14 Figure 2.5: FE model of RC girder (Kishi and Bhatti, 2010) ....................................... 15 Figure 2.6: Stress-strain relations of materials (a) concrete (b) reinforcement (Kishi and Bhatti, 2010) ............................................................................................................ 15 Figure 2.7: Drop-weight test set up by Magnusson et al. (2000) .................................. 16 Figure 2.8: Drop-weight test set up at Heriot-Watt University (Chen and May, 2009) 18 Figure 2.9: Details of beams tested by Saatci and Vecchio (2009a) ............................. 19 Figure 2.10: FE model by Saatci and Vecchio (2009b) ................................................ 19 Figure 2.11: Comparison of midspan displacements for first impacts on undamaged specimens by Saatci and Vecchio (2009b) ..................................................................... 20 Figure 2.12: Observed and calculated crack profiles for SS1b-1 (Saatci and Vecchio, 2009b) ............................................................................................................................. 21 Figure 2.13: Impact tests by Fujikake et al. (2009): Specimen details ......................... 21 Figure 2.14: Impact tests by Fujikake et al. (2009): Test setup .................................... 22 Figure 2.15: Failure modes: (a) S1616 series; (b) S1322 series; and (c) S2222 series (Fujikake et al., 2009) .................................................................................................... 23 Figure 2.16: Impact response for S1616: (a) drop height = 0.15 m; (b) drop height = 0.3 m; (c) drop height = 0.6 m; and (d) drop height = 1.2 m (Fujikake et al., 2009) ..... 24 Figure 2.17: Impact responses: (a) maximum impact load; (b) impulse; (c) duration of impact load; (d) maximum midspan deflection; and (e) time taken for maximum midspan deflection (Fujikake et al., 2009) ..................................................................... 25 Figure 2.18: Impulse and duration of impact load (Fujikake et al., 2009) .................... 25 Figure 2.19: Midspan impact of a pin-ended beam (Hughes and Beeby, 1982) ........... 27 Figure 2.20: First three symmetrical vibration modes of a pin-ended beam (Hughes and Beeby, 1982) .................................................................................................................. 28 Figure 2.21: Measured and theoretical force histories (Hughes and Beeby, 1982) ...... 28 Figure 2.22: Mass-spring model for impact (CEB, 1988) ............................................ 29 Figure 2.23: Schematic Diagram of the beam simulated by Sangi, A. J. (2011) .......... 32 Figure 2.24: Comparison of final crack patterns and damage on beam simulated by Sangi, A. J. (2011) .......................................................................................................... 32 Figure 2.25: Comparison of concrete penetration depths calculated by various formulae for the case of a typical missile (Yankelevsky, 1997) .................................... 34 Figure 2.26: Comparison of concrete perforation thickness calculated by various formulae for the case of a typical missile (Yankelevsky, 1997) .................................... 35 Figure 2.27: Load-time histories of slabs under 610 mm drop (Zineddin and Krauthammer, 2007) ...................................................................................................... 37
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Figure 2.28: The test set-up by Feyerabend (1988) ...................................................... 39 Figure 2.29: C3D20, C3D8, T3D2 elements used to model by ABAQUS (2012) ....... 44 Figure 2.30: Smeared formulation for RC .................................................................... 45 Figure 2.31: Embedded formulation for RC ................................................................. 46 Figure 2.32: Discrete model of RC ............................................................................... 46 Figure 2.33: Penalty method for contact algorithm ...................................................... 47 Figure 2.34: Response of concrete to uniaxial loading in (a) tension and (b) compression .................................................................................................................... 49 Figure 2.35: Stress-strain relation for steel reinforcement ............................................ 50 Figure 3.1: Details of RC beam tested by Saatci (2007) ............................................... 54 Figure 3.2: FE model of tested RC beam by Saatci (2007) .......................................... 57 Figure 3.3: Comparison of reaction force Vs. midspan displacement diagram of numerically analyzed beam MS0 with test result found by Saatci (2007) ..................... 58 Figure 3.4: Comparison of reaction force Vs. midspan displacement diagram of numerically analyzed beam MS1 with test result found by Saatci (2007) ..................... 59 Figure 3.5: Comparison of reaction force Vs. midspan displacement diagram of numerically analyzed beam MS2 with test result found by Saatci (2007) ..................... 59 Figure 3.6: Geometry of RC slab tested by McNeice (1967) ....................................... 61 Figure 3.7: FE model of one-quarter of RC slab tested by McNeice (1967) ................ 61 Figure 3.8: Comparison of Load-deflection diagram of numerically analyzed slab with test result observed by McNeice (1967) ......................................................................... 62 Figure 3.9: Dimensional view of RC beam ................................................................... 63 Figure 3.10: Concrete modelling with C3D8R Brick elements .................................... 64 Figure 3.11: Reinforcement modelling with T3D2 truss elements ............................... 65 Figure 3.12: RC beam (a) cross section (b) transformed section .................................. 66 Figure 3.13: The stress distribution in psi at midspan of RC beam at 5.78 kip load .... 68 Figure 3.14: The stress distribution in psi at midspan of RC beam at 6.84 kip load (elastic, cracked) ............................................................................................................. 69 Figure 3.15: Crack pattern of RC beam at 6.84 kip load (elastic, cracked) .................. 69 Figure 3.16: The stress distribution in psi at midspan of RC beam at 16.90kip load (elastic, cracked) ............................................................................................................. 70 Figure 3.17: The stress distribution in psi at midspan of RC beam at 16.90kip load (elastic, cracked) ............................................................................................................. 70 Figure 3.18: Flexural crack pattern of under reinforced beam at 16.90kip load (elastic, cracked) .......................................................................................................................... 71 Figure 3.19: The stress distribution in psi at midspan of under RC beam at 20.44kip load (plastic, cracked) .................................................................................................... 71 Figure 3.20: The stress distribution in psi at midspan of under RC beam at 20.44kip load (plastic, cracked) .................................................................................................... 72 Figure 3.21: Flexural crack pattern of under RC beam at 20.44kip load (plastic, cracked) .......................................................................................................................... 72 Figure 3.22: Load-deflection diagram of analyzed RC beam ....................................... 73 Figure 3.23: Failure deflected shape of analyzed RC beam .......................................... 74 Figure 3.24: Single degree of freedom system (SDOF) ................................................ 75
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Figure 3.25: Response of SDOF system under static load ............................................ 76 Figure 3.26: Comparison of analytical and numerical Undamped free vibration response .......................................................................................................................... 77 Figure 3.27: Comparison of analytical and numerical 2% critically-damped free vibration response .......................................................................................................... 79 Figure 3.28: Comparison of analytical and numerical 100% critically-damped free vibration response .......................................................................................................... 80 Figure 3.29: Step loading acting on SDOF System ...................................................... 81 Figure 3.30: Comparison of analytical and numerical response to step loading response for 3% critically-damped system .................................................................................... 81 Figure 3.31: Dynamic equilibrium of SDOF system to step load ................................. 82 Figure 3.32: Dimensional view of RC column ............................................................. 84 Figure 3.33: FE Model of (a) concrete and (b) reinforcement of column ..................... 85 Figure 3.34: Comparison of analytical and numerical undercritically-damped free vibration response (Nonlinearity not triggered due to low stress level) ........................ 86 Figure 3.35: Comparison of analytical and numerical undercritically-damped free vibration response (involving nonlinearity) ................................................................... 87 Figure 3.36: Tensile crack develop at RC column ........................................................ 87 Figure 3.37: Schematic diagram of the beam ................................................................ 88 Figure 3.38: Impact load generated between two surfaces of beam and impactor ....... 89 Figure 3.39: Equilibrium of motion for SDOF beam to impact load ............................ 89 Figure 4.1: Detail of beam tested by Chen and May (2009) ......................................... 92 Figure 4.2: Pin-ended support used by Chen and May, (2009) .................................... 93 Figure 4.3: FE Modelling of pin-ended support used by Chan and May (2009) .......... 94 Figure 4.4: Schematic diagram of tested beam by Chan and May (2009) .................... 95 Figure 4.5: FE model of beam (a) complete beam mesh (b) reinforcement mesh ........ 97 Figure 4.6: Comparison of transient displacement histories of linear RC beam for different mesh size ......................................................................................................... 98 Figure 4.7: Comparison of transient impact force histories of linear RC beam for different mesh size ......................................................................................................... 99 Figure 4.8: Comparison of transient displacement histories of nonlinear RC beam for different mesh size ....................................................................................................... 100 Figure 4.9: Comparison of transient impact force histories of nonlinear RC beam for different mesh size ....................................................................................................... 101 Figure 4.10: Comparison of impact force history of numerically simulated beam with tested response by Chan and May (2009) .................................................................... 102 Figure 4.11: Comparison of crack and damage patterns of numerically simulated beam with test result observed by Chan and May (2009): (a) tested beam, (b) tension damage and (c) compression damage of analyzed beam. .......................................................... 103 Figure 4.12: Correlation between impact load and crack propagation for beam from tested by Chan and May (2009) and numerical analysis .............................................. 105 Figure 4.13: Schematic diagram of tested beam by Chan and May (2009) ................ 106 Figure 4.14: Comparison of impact force history of numerically simulated beam with tested response by Chan and May (2009) .................................................................... 107
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Figure 4.15: Comparison of crack and damage patterns of numerically simulated beam with test result observed by Chan and May (2009): (a) tested beam, (b) tension damage and (c) compression damage of analyzed beam. .......................................................... 108 Figure 4.16: Dimension of slab tested by Chen and May (2009) ............................... 109 Figure 4.17: Impact force histories of slabs tested by Chen and May (2009)............. 110 Figure 4.18: Damage at top and bottom faces of slabs tested by Chen and May (2009) ...................................................................................................................................... 111 Figure 4.19: Schematic diagram of tested Slab-2 by Chan and May (2009) .............. 112 Figure 4.20: FE model of slab-2 (a) complete slab with impactor and support mesh (b) reinforcement mesh ...................................................................................................... 114 Figure 4.21: Comparison of transient displacement histories of linear RC slab for different mesh size ....................................................................................................... 115 Figure 4.22: Comparison of transient impact force histories of linear RC slab for different mesh size ....................................................................................................... 116 Figure 4.23: Comparison of transient displacement histories of nonlinear RC slab for different mesh size ....................................................................................................... 117 Figure 4.24: Comparison of transient impact force histories of nonlinear RC slab for different mesh size ....................................................................................................... 117 Figure 4.25: Comparison of impact force history of numerically simulated slab with tested response by Chan and May (2009) .................................................................... 118 Figure 4.26: Comparison of crack and damage patterns of numerically simulated slab with test result observed by Chan and May (2009) ...................................................... 119 Figure 4.27: Stresses distribution of reinforcement of slab in psi ............................... 120 Figure 5.1: Details of RC beams tested by Tachibana et al. (2010) ............................ 123 Figure 5.2: Schematic view of RC beams with impactor ............................................ 128 Figure 5.3: Concrete beams with C3D8R brick element mesh ................................... 129 Figure 5.4: Reinforcement represented by T3D2 truss elements ................................ 129 Figure 5.5: Time response of stiffness force (reaction) .............................................. 130 Figure 5.6: Comparison of the displacement history at midspan for beam A-2(13) with test result conducted by Tachibana (2012). .................................................................. 131 Figure 5.7: Relationship between static bending capacity and impulse for numerically analyzed beam number A-2(13), A-1(14), A-4(15), B(16), C(17), D(18), E(19) and F(20) ............................................................................................................................. 133 Figure 5.8: Relationship between momentum of impactor and impulse observed from FE analysis of all beams ............................................................................................... 134 Figure 5.9: Relationship between static bending capacity and duration for numerically analyzed beam number A-2(13), A-1(14), A-4(15), B(16), C(17), D(18), E(19) and F(20) ............................................................................................................................. 134 Figure 5.10: Relationship between Mcol/Pu and duration of impact force observed from FE analysis of all beams ...................................................................................... 135 Figure 5.11: Relationship between static bending capacity and mean impact force ... 135 Figure 5.12: Relationship between momentum of impactor and impulse .................. 139 Figure 5.13: Relationship between Mcol/Pu and duration of impact force ................... 140 Figure 5.14: Details of RC column ............................................................................. 142
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Figure 5.15: Schematic view of RC column with impactor for model no. 1 to 5 ....... 145 Figure 5.16: Schematic view of RC column with impactor for model no. 6 .............. 145 Figure 5.17: Compression of strength interaction diagram of column under static and impact load. .................................................................................................................. 148 Figure 5.18: Schematic view of analyzed flyover pier................................................ 150 Figure 5.19: Flyover pier collided by bus (a) Stress contours in psi unit and (b) deflection contours in inch unit .................................................................................... 151 Figure 5.20: Flyover pier collided by truck (a) Stress contours in psi unit and (b) deflection contours in inch unit .................................................................................... 152 Figure 5.21: Typical railway track map and possible direction of derailment ............ 153 Figure 5.22: RC flyover pier and railway locomotive mesh at just before of collision (a) top view and (b) side view ...................................................................................... 153 Figure 5.23: Flyover pier collided by train (a) Stress contours in psi unit and (b) deflection contours in inch unit .................................................................................... 154 Figure 5.24: Flyover pier collision with train (a) tension damage contours and (b) compression damage contours ................................................................................ 154 Figure A.1: Stress-strain relation for uniaxial compressive loading ........................... 170 Figure A.2: Stress-strain relation for uniaxial tension loading ................................... 172 Figure C.1: Dimensional view of over-reinforced concrete beam .............................. 180 Figure C.2: Load-deflection diagram of FE analyzed over-reinforced concrete beam ...................................................................................................................................... 181 Figure C.3: Dimensional view of reinforced concrete beam without shear bars ........ 182 Figure C.4: Load-deflection diagram of simulated reinforced concrete beam without shear bar ....................................................................................................................... 183 Figure C.5: Dimensional view of reinforced concrete beam with shear bars ............. 184 Figure C.6: Load-deflection diagram of simulated reinforced concrete beam with shear bar ................................................................................................................................. 184 Figure D.1: North-South component of horizontal ground acceleration of El Centro Earthquake of May 18, 1940 ........................................................................................ 186 Figure D.2: Displacement response of SDOF Systems to El Centro Earthquake (T = 2 Sec. and ξ = 0%) ........................................................................................................... 187 Figure D.3: Displacement response of SDOF Systems to El Centro Earthquake (T = 2 Sec. and ξ = 2%) ........................................................................................................... 187 Figure D.4: Displacement response of SDOF Systems to El Centro Earthquake (T = 2 Sec. and ξ = 5%) ........................................................................................................... 187 Figure E.1: Details of RC foundation ......................................................................... 189 Figure E.2: Complete FE model of Foundation .......................................................... 190 Figure E.3: Comparison of reaction force histories of fixed ended column with flexible foundation supported column. ...................................................................................... 191 Figure E.4: Tension damage pattern at (a) perspective view and (b) bottom face of analyzed foundation ..................................................................................................... 192
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LIST OF TABLES
Table 2.1: Characteristics of the Feyerabend‟s (1988) test specimens ......................... 39 Table 2.2: Parameters used for failure criteria .............................................................. 49 Table 3.1: Transverse reinforcement ratios and stirrup spacing for beams ................... 55 Table 3.2: Material property of transverse and stirrup for beams ................................. 55 Table 3.3: Material Property of Concrete ...................................................................... 56 Table 3.4: Material property of reinforcement .............................................................. 60 Table 3.5: Material Property of Concrete ...................................................................... 60 Table 3.6: Material property of reinforcement .............................................................. 64 Table 3.7: Material Property of Concrete ...................................................................... 65 Table 3.8: Fundamental natural frequency of the column ............................................. 78 Table 3.9: Material property of reinforcement .............................................................. 83 Table 3.10: Material Property of Concrete .................................................................... 83 Table 3.11: Fundamental natural frequency of the column ........................................... 85 Table 4.1: Material Property of Concrete ...................................................................... 95 Table 4.2: Material Property of high yield reinforcement ............................................. 95 Table 4.3: Material Property of mild reinforcement ..................................................... 96 Table 4.4: Basic property of Plywood ........................................................................... 96 Table 4.5: Mesh data for linear analysis of beam .......................................................... 99 Table 4.6: Mesh data for nonlinear analysis of beam .................................................. 100 Table 4.7: Details of slab tests ..................................................................................... 110 Table 4.8: Material Property of Concrete .................................................................... 112 Table 4.9: Material Property of high yield reinforcement ........................................... 113 Table 4.10: Mesh data for slab .................................................................................... 113 Table 5.1: Design values of RC beams ....................................................................... 124 Table 5.2: Material Property of Concrete .................................................................... 125 Table 5.3: Material Property of bending reinforcement .............................................. 125 Table 5.4: Material Property of shear reinforcement .................................................. 125 Table 5.5: Overview of impactors and beams ............................................................. 127 Table 5.6: Numerical result of all analyzed beams ..................................................... 132 Table 5.7: Overview of impactors and beams ............................................................. 137 Table 5.8: Numerical result of all analyzed beams ..................................................... 138 Table 5.9: Material Property of Concrete .................................................................... 142 Table 5.10: Material Property of longitudinal reinforcement ..................................... 143 Table 5.11: Material Property of tie bar ...................................................................... 143 Table 5.12: Overview of impactors and column ......................................................... 144 Table 5.13: Numerical result of all analyzed models .................................................. 146 Table 5.14: Comparison of load carrying capacity of analyzed RC column .............. 147 Table 5.15: Impact test conducted by Texas Transportation Institute (1980 to 1988) 149 Table 5.16: Material Property of Concrete .................................................................. 150 Table 5.17: Material Property of reinforcement .......................................................... 150 Table B.1: Material Property with damage parameter of Concrete ............................ 174 Table B.2: Material Property with damage parameter of Concrete ............................ 175
xviii
Table B.3: Material Property with damage parameter of Concrete ............................ 176 Table B.4: Material Property with damage parameter of Concrete ............................ 177 Table B.5: Material Property with damage parameter of Concrete ............................ 178 Table B.6: Material Property with damage parameter of Concrete ............................ 179 Table C.1: Summary of reinforced concrete beam under static load .......................... 185 Table C.2: Summary of responses for different type of reinforced concrete beam under static load ...................................................................................................................... 185
xix
NOTATIONS
Tensile equivalent plastic strain
Compressive equivalent plastic strain
Temperature
Compressive damage parameter
Tensile damage parameter
Undamaged elastic stiffness
Modulus of elasticity for steel
Modulus of elasticity for Concrete
Modulus of rupture for Concrete
Allowable elastic stress of concrete
ultimate stress of concrete
yield stress of steel
Ultimate stress of steel
Tensile stress
Compressive stress
Effective tensile stress
Effective compressive stress
Tensile strain at
Compressive strain at or
Ultimate strain
Yield strain
Poison‟s ratio
Density of material
Descent function
Crushing energy
Characteristic length of simulated specimen
Crack opening
Maximum crack opening
Free parameter determined experimentally
Dilation angle
xx
Ratio of biaxial to uniaxial compressive strength
Second stress invariant ratio
Eccentricity value in ABAQUS
Percentage of critical damping
Fundamental natural frequency of structure
√
Rayleigh Stiffness proportional damping coefficient
Rayleigh mass proportional damping coefficient
T Period of time
k Stiffness of structure
Static displacement
Initial displacement
Initial velocity
Initial acceleration
C damping
I Modulus of elasticity
Area of reinforcement
Elastic moment
Nominal moment
Ultimate bending capacity
Ultimate Shear capacity
m Mass of impactor
Impact velocity of impactor at collision
Kinetic energy of impactor at collision
Momentum of impactor at collision
ma Inertia force
cv Damping force
kd Damping force
Duration of impact
Impulse of impact event, area under time-force curve
Maximum impact force
Mean impact force
xxi
Maximum displacement of structure under impact load
RC Reinforced concrete
FE Finite element
CDP Concrete Damage Plasticity
1
Chapter 1: Introduction
Introduction
1.1 Background and Present State of the Problem
Reinforced concrete (RC) members are often subjected to extreme dynamic loading
conditions due to direct impact. Common example of these conditions include
transportation structure subjected to vehicle crash impact, marine structure subjected to
water vessels direct impact, wind and storm generated missiles, accidental collisions of
train, aircraft, motor vehicle, dropped objects on to RC structures etc.
Response of RC structures to impact is different from that caused due to static load.
Also, impact load is a particular type of dynamic loading which needs special attention.
Broadly, the impact load can be classified into i) low velocity large mass impact and ii)
high velocity small mass impact. The first category involves collision of vehicle into
crash barriers, piers of bridges, drop of an object on slab etc. whereas the second one
includes bullet or missile hitting structures, birds hitting airplane etc. The majority of
studies carried out in the military sector mainly focused on high velocity small mass
impacts on RC structure. In context of Civil Engineering problem, an investigation into
the impact behaviour of RC members subjected to low velocity large mass is very
important. The current work deals with low velocity large mass impact on RC
structures.
Several researchers have conducted reviews of impact problems. Kennedy (1976)
provides a detailed review for the effects of missile impacts on concrete structures.
Corbett et al. (1996) reviewed the impact effects of plates and shells subjected to free-
flying objects. More recently, an extensive review of concrete impact problems for
local effects has been provided by Li et al. (2005). Thabet and Haldane (2000) have
described the failure mechanisms of RC beam under impact loading.
The experimental research into the impact behaviour of RC beams was initiated by
Mylrea (1940). Further studies were conducted by Feldman et al. (1956, 1958, 1962),
Hughes and Speirs (1982). A series of low speed impact experiments of RC beams
were performed by Tachibana, et. al. (2010). The flexural behaviour of RC beams
2
under impact loading predicted analytically has been presented by Ando et al. (1999).
The basic relations and empirical formula for the design of RC structure under impacts
have been presented by Daudeville and Malecot (2011).
To investigate the shear behaviour of RC beams, Kishi et al. (2001) conducted impact
tests and carried out FE impact analysis. Further studies were carried out by Kishi et al.
(2002) to establish a rational design procedure for shear-failure-type RC beams under
impact loads.
A series of experiment of RC beam and slab subjected to high-mass, low-velocity
impact load was performed by Chan and May (2009) which provide high-quality input
data and results to verify numerical model.
Sawan and Abdel-Rohman (1986) carried out low-velocity impact tests on RC slabs
75 x 75 x 5 cm in dimensions. To investigate the dynamic behaviour of slabs, large
scale RC slabs were tested under impact loading by Kishi et al. (1997).
Fererabend (1988) conducted an experimental investigation on 300 x 300 x 4000 mm
RC columns subjected to lateral impact at midspan. Thilakarathna, et. al. (2010) have
investigated the vulnerability of columns to low elevation vehicular impacts.
Since not much study has so far been conducted in the country, it is essential to initiate
study on response of impact load on RC structure since this is also important in
Bangladesh context.
1.2 Objectives with Specific Aims and Possible Outcome
The main objective of this study is to numerically simulate the response of RC
members subjected to low velocity large mass impact by using three dimension
nonlinear finite element (FE) analysis.
The following investigations are required to achieve the principal objective of the
thesis:
1. To carry out an extensive literature review to identify the related works carried
out and approaches followed.
2. To select and validate a nonlinear FE method employing a suitable material
constitutive model under static load.
3
3. To extend and validate the nonlinear FE method such that it can simulate the
structural response under impact loading.
4. To carry out a limited parametric study to identify the effect of different
parameter i.e. velocity, mass, period of impact, impulse etc. on dynamic
response of structures under impact load.
With the successful completion of the above objectives, it would be possible to
understand the response of impact loads on RC structural members.
1.3 Methodology of Work
Before carrying out the systematic computational investigation, existing literature on
the relevant field has thoroughly been reviewed.
In numerical model of RC members using proper constitutive material model is actually
the most critical factor for accurate analysis. Two different types of constitutive model
and element types have been used to model concrete and steel reinforcement. In the FE
modelling of RC members, concrete has been modeled by three dimensional eight
nodded solid elements while the reinforcing steel is modeled by one dimensional two
nodded link/truss elements. The impactor is also modeled by three dimensional eight
nodded solid elements. The interaction between reinforcing steel and concrete element
could be achieved by sharing the same node i.e. no slippage occurs. Again the
interaction between the impactor and solid concrete elements has been achieved by
using surface to surface contact (Explicit) algorithms, which uses a penalty method to
model contact interface between the different parts. The nonlinear effect due to
cracking and crushing of concrete and yielding of steel reinforcement is included.
Damping property is counted as mass and stiffness proportional damping factors.
ABAQUS (2012) is a powerful engineering simulation programs, based on the FE
method. This program can solve wide range of problem from simple linear to relatively
more complex nonlinear for both static as well as dynamic loading. ABAQUS (2012)
material library is also very rich and it can provide different constitutive material for
different engineering materials.
4
Initially nonlinear FE modelling has been validated for statically applied load on RC
beam and slab by comparing with experimental results. Then, the modelling is extended
to impact load on RC structures. Once the FE modelling is validated against
experimental results, a limited parametric study has been carried out to identify
different parameters i.e. velocity, mass, period of impact, impulse etc. which are
directly related with impact force histories of any large mass low velocity impact event.
1.4 Outlines of the Thesis
The thesis consists of 6 chapters. The current chapter is Chapter 1, which introduces the
general background and present state of problem of this research work and summary of
aims, objectives and methodology. A comprehensive review of the literature pertaining
to the impact behaviour of RC members is presented in Chapter 2. A brief discussion
regarding local and global responses of RC members subjected to impact loading
through experimental as well as numerical investigation are presented in this chapter.
Chapter 3 describes the linear as well as nonlinear FE modelling of RC beam, slab and
column under static or dynamic loading by popular FE software ABAQUS (2012).
Performance of this model is verified against different experimental and analytical
results in this chapter too. In Chapter 4, the nonlinear FE modelling of beam and slab
under impact load, which was tested by Chan and May (2009), are carried out using
ABAQUS (2012) to investigate and verify the impact behaviour of these RC structural
members. Chapter 5 is dedicated to a through limited parametric study to identify
different parameters which are directly related with impact force histories of any large
mass low velocity impact event. The investigation and findings of this chapter leads to
developing a convenient numerical equation which is helpful for any designer to predict
the failure condition of any RC beam and column. The conclusions drawn from the
present study and recommendations for future work related with this research are
presented in Chapter 6.
5
Chapter 2: Literature Review
Literature Review
2.1 Introduction
The study of impact phenomena covers an extremely wide range of situations and is of
interest to researchers and engineers from a number of different fields. For example,
vehicle manufacturers use their understanding of the response of structures to improve
the safety of their products; military engineers need to understand the phenomenon in
order to design structure that are more efficient to withstanding projectile impact; and
civil engineers have to consider the effects of abnormal load like blasts, falling objects
onto the structures for the safe and efficient design.
In the present chapter, previous research regarding experimental, analytical and
numerical approach on RC structural members like, beam, column and slab under
impact load have been discussed. The numerical modelling technique of RC members
under impact loads have also been discussed in this chapter.
2.2 Classification of Impact Events Considering Response of Target Structure
The impact is classified as soft or hard, based on the way that impact energy absorbs
during an impact. Generally, in a soft impact the striker absorbs most of the kinetic
energy through plastic deformation, while the structure experiences minor
deformations.
When subjected to impact, the target structure may respond in several ways depending
on the nature of impact. The responses of target structure under impact are:
Local response: Local damage only, as majority of the impact energy is dissipated
around the impact zone.
Global response: Bending and deformation of the entire reinforced concrete (RC)
member.
Combined response: Combination of both local and global damage.
6
Figure 2.1: Missile impact effects on concrete target, (a) Penetration, (b) Cone
cracking, (c) Spalling, (d) Cracks on (i) proximal face and (ii) distal face,
(e) Scabbing, (f) Perforation, and (g) Overall target response (Li et al.,
2005)
More recently, an extensive review of concrete impact problems for local effects has
been provided by Li et al. (2005). The phenomena generally associated with missile
impact effects on concrete targets are shown in Fig. 2.1 and have been defined by Li et
al. (2005). The responses phenomena of target concrete structure under impact are:
Penetration: Crater developed in the target at the zone of impact.
Cone cracking and plugging: Formation of a cone-like crack under the projectile and
the possible subsequent punching-shear plug.
Spalling: Ejection of target material from the proximal face of the target.
Radial cracking: Global cracks radiating from the impact point and appearing on either
the proximal or distal face of the concrete slab or both, when cracks develop through
the target thickness.
7
Scabbing: Ejection of fragments from the distal face of the target.
Perforation: Complete passage of the projectile through the target with or without a
residual velocity.
Overall structural responses and failures: Global bending, shear and membrane
responses, and their induced failures throughout the target.
From the above impact effects, penetration, spalling, cone cracking, scabbing and
perforation are considered as local impact effects.
2.3 Investigation Technique of Impact Events
The local and global response of RC structures subjected to impact loads can be
investigated using a number of different approaches:
Experimental approach
Analytical approach
Numerical approach
The majority of the research on global response was carried out on RC beams, since
local damage becomes much more important in the case of columns, slabs and shells
subjected to impact loading.
Earlier studies done mostly by military engineers on the design of fortification
structures mainly focused on high velocity (150-1000 m/s) hard impacts that cause
extensive local damage without any significant global response. These studies were
mainly experimental resulting in the development of empirical formulae, which had
little theoretical basis. However, for civil applications, the empirical formulae had very
limited application as they were only applicable to the range of available test data
(Kennedy, 1976) to overcome the limitations of the experimental studies, several
theoretical studies have been carried out to date, but the major breakthrough has been
the development of numerical methods to analyze the structures under impact loads,
thus eliminating the need of expensive and time consuming experimental
investigations.
2.3.1 Impact Behaviour of Concrete Beams
Figure 2.2 shows the most dominant mechanisms when a RC beam is subjected to
impact loading. Thabet and Haldane (2000) have described the following mechanisms:
8
Surface crushing: As the impactor strikes the beam, stress waves are transmitted into
the contact area on the beam during the first few microseconds. Concrete in this area is
crushed and a crater on the surface of the member is formed, Figs. 2.2(b) and (c).
Concrete plug: As the stress waves travel into the member, they encounter a large
number of internal wave reflectors, such as aggregate particles, voids and cement paste.
The momentum is progressively accumulated within the concrete as the stress waves
are dissipated. If the momentum deposited under the impacted area is large, a local
punching shear failure can occur before the beam has time to respond in flexure. This is
commonly referred as a concrete plug and is usually accompanied by the development
of cracks, Fig. 2.2(e).
Scabbing: The reflection of the incident compressive stress waves results in tension
failure in the concrete normal to its free surface. This localized detachment of an area
of concrete normally along the flexural reinforcing bars at right angles to the direction
of the impact load is referred to as scabbing and occurs on the opposite face to the
impact area, Fig. 2.2(d).
Global flexural response: As the momentum is transferred away from the impact area
towards the supports region, the whole beam progressively responds in flexure, which
occurs over a long period compared to the formation of the concrete plug. A reinforced
beam when subjected to static loading will exhibit a ductile flexural response or brittle
shear-critical behaviour depending on the design parameters. The ductile flexural
failure is characterized by the initiation and development of vertical flexural cracks at
the centre. As the beam continues to deform, the flexural cracks widen and propagate
towards the top of the beam.
9
Figure 2.2: Contact-impact problem involving a concrete beam (Thabet, 1994)
2.3.2 Failure mode of RC beam under impact load
A shear-critical beam under static loading shows a brittle behaviour characterized by
the development of diagonal shear cracks near the supports. However, under impact
loading, formation of diagonal cracks, originating at the impact point and propagating
downward with an angle of approximately 45 degrees forming shear plug has been
reported by many researchers regardless of the static behaviour (Saatci and Vecchio,
2009a). Further diagonal cracks parallel to the major shear-plug cracks may also
develop. Flexural cracks at the midspan and at the supports develop and usually
propagate vertically. The vertical cracks at the midspan start from the bottom surface,
10
whereas those near the support start from the top surface. The vertical cracks starting
from the top in a region of the beam away from the impact zone are associated with the
traveling of the stress waves from the impact zone towards the supports in the beam.
The failure modes and crack patterns in a RC beam subjected to impact loading at the
midspan may be broadly generalized depending on their static behaviour.
Mode 1: Flexural failure with some crushing under the impactor and diagonal cracks in
the impact zone forming a shear plug. Vertical flexural cracks form in the centre and
near the supports. Flexure-critical beams may exhibit this mode of failure.
Mode 2: Shear failure with some crushing under the impactor and diagonal cracks in
the impact zone forming a shear plug. Additional diagonal cracks develop alongside the
shear-plug, which start near the supports, propagate at an angle of approximately 45
degrees upward and become horizontal close to the top of the beam. Shear-critical
beams may exhibit this mode of failure.
Mode 3: Localized failure at the impact zone with extensive concrete crushing below
the impactor and yielding of the tension reinforcement.
2.4 Experimental Investigation of RC Bram under Impact load
As mentioned earlier, the impact behaviour of a RC beam mainly depends upon its
global response to impact loading as compared to its local response. A great amount of
experimental work was carried out in order to develop analytical models for impact and
impulsive loadings on RC beams. Fundamental theory of vibration and SDOF systems
have also been used (CEB, 1988; Hughes and Beeby, 1982; Hughes and Speirs, 1982).
With the development of numerical methods, especially the FE Method, many
researchers have carried out extensive numerical studies supported by experiments.
2.4.1 Experimental investigation of RC beam carried out by Mylrea (1940)
The experimental research into the impact behaviour of RC beams was initiated by
Mylrea (1940), who tested 8 feet span beams without shear reinforcement, subjected to
falling weights. The beams developed severe diagonal cracking. He concluded that the
beams have significant impact resistance based on his failure criteria of the rupture of
the longitudinal reinforcement.
11
2.4.2 Experimental investigation of RC beam carried out by Feldman et al. (1956,
1958, 1962)
Feldman et al. (1956, 1958, 1962) carried out a comprehensive series of impact tests on
RC beams. A total of 43 beams were tested under midspan and two-point loading,
which was applied using a pneumatic loading system comprising a piston with loading
and unloading by pressurized gases. The tests were very well instrumented to record the
time-histories of impact load, reactions, deflections, accelerations and strains in the
concrete and the reinforcement. Based on their tests, they developed an analytical
model assuming a SDOF system, which later became the basis of CEB (1988)
formulations.
2.4.3 Experimental investigation of RC beam carried out by Hughes and Speirs
(1982)
An extensive program of impact tests on RC beams was conducted by Hughes and
Speirs (1982). There were 80 impact tests on pin-ended RC beams and 12 tests on
simply supported beams. Impact force histories and beam displacements were
measured. In most of the tests, the beam failed in a flexural mode, with flexural cracks
at both the bottom of the beam, concentrated towards mid-span, and at the top close to
the supports. There were shear cracks also at both 1/3 spans. No shear failure was
observed although diagonal cracks appeared in many of the beams. It was found that
stiffness of the impact zone, which was a function of the impactor, the plywood pad
and the local stiffness of the beam, had a more significant influence than the supports
on the response of the beams. The simple beam vibration model developed was shown
to be applicable over the test range and gave good correlation with the measured impact
force - time history, but was shown to be inadequate for impacts with very stiff impact
zones due to the likelihood of higher modes of vibration being excited.
2.4.4 Experimental investigation of RC beam carried out by Ando et al. (1999)
The flexural behaviour of RC beams under impact loading predicted analytically has
been presented by Ando et al. (1999). Three-dimensional finite element method (FEM)
analysis was conducted on simply supported rectangular RC beams using LS-DYNA
(Hallquist, 2007). Thirteen RC beams were analyzed of varying cross sectional
dimensions and area of main reinforcement. A 200 kg steel weight was used to impact
with a predetermined velocity onto the midspan of the RC beam. Material models for
concrete and reinforcement were bi-linear elastic plastic model and elasto-plastic model
12
with isotropic hardening, respectively. The computed time histories of impact force,
reaction force and displacement at midspan were compared with experimental results
and showed good agreement.
2.4.5 Experimental investigation of RC beam carried out by Kishi et al. (2001)
To investigate the shear behaviour of RC beams, Kishi et al. (2001) conducted impact
tests and carried out finite element (FE) impact analysis. They tested thirteen simply
supported rectangular RC beams, each with dimensions of 200 x 400 x 2,400 mm, Fig.
2.3. The impact was applied at the mid-span using a 400 kg steel weight. The impact
velocity (3.7 - 10.2 m/s) and the shear reinforcement ratio were taken as variables. The
impact force, reaction forces and mid-span deflections were recorded. These were later
used to validate numerical simulations carried out using LS-DYNA. A simple elasto-
plastic FE analysis was then used to predict the time histories of impact force, reaction
force, mid-span displacement and the crack pattern on the side surface of RC beams
and good agreement was found (Bhatti et al., 2009).
Figure 2.3: Dimensions of RC beams tested by Kishi et al. (2001)
Further studies were carried out by Kishi et al. (2002) to establish a rational design
procedure for shear-failure-type RC beams under impact loads. They tested twenty
seven simply supported rectangular RC beams, all without shear reinforcement. The
13
longitudinal reinforcement and shear-span ratios were taken as variables. A free-falling
weight of 300 kg was dropped at the midspan and recordings for impact force, reactions
and mid-span deflections were made. The experimental results were utilized to propose
a simplified impact-resistance design procedure for RC beams without shear
reinforcement. They suggested a simplified model for the reaction force vs.
displacement loop for shear failure as a triangle, Fig. 2.4. For their range of test results,
they assumed the maximum reaction force Rud to be 1.5 times the calculated static shear
capacity Vusc and the absorbed energy Ea given by the loop-area of the reaction force vs
mid-span displacement curve to be 0.6 times the input kinetic energy Ek. They
calculated the required static shear capacity Vusd against the dynamic loading in terms
of the maximum reaction force as
(2.1)
The absorbed energy Ea is calculated using the simplified reaction force vs
displacement curve (Figure 2.4) as
(2.2)
The design input kinetic energy Ekd and absorbed energy Ea are related as
(2.3)
Substituting Equations (2.1) and (2.2) into (2.3), the required shear capacity Vusd can be
found as
(2.4)
They concluded that RC beams without shear reinforcement, failing in shear could be
designed for impact loads with a certain margin of safety by assuming a dynamic
response ratio of 1.5 and absorbed input energy ratio of 0.6
The research has been further extended to large scale RC girders under falling weight
impact loads (Bhatti et al., 2006; Kishi and Bhatti, 2010). A RC girder having
rectangular cross-section of 500 mm x 850 mm, with a clear span of 8 m was subjected
to falling-weight impact of 2000 kg dropped freely from the height of 10 m. This type
of girders has been used in the roofs of RC rock-sheds, constructed over the highways
14
to ensure the safety of public and vehicles. Measurements of impact force, reaction
forces and mid-span deflections were made, which were later compared with analytical
model developed in LS-DYNA. Figures 2.5 and 2.6 show the FE model of the girder
and material models used for concrete and steel reinforcement. They have extended the
concept of simple elasto-plastic modelling of concrete used in their earlier studies on
beams by including an equivalent tensile fracture energy concept. The equivalent
tensile strength for a given mesh size is computed based on the fracture energy of a
control element, thus allowing larger mesh sizes. The introduction of a fictitious tensile
strength for a concrete element resulted in similar results for a coarser mesh in span
directions to those obtained for a control element size of 35 mm.
Figure 2.4: Simplified model for the reaction force verses displacement loop (Kishi et
al., 2002)
2.4.6 Experimental investigation of RC beam carried out by Magnusson et al.
(2000)
In 1999, the investigation of the impact behaviour of beams was carried out in Sweden
by Magnusson et al. (2000). They have tested high strength RC beams under impact
loads. The concrete had a compressive strength of 112 MPa. A total of eight beams
were tested, three of which were subjected to quasi static loading and five beams were
subjected to impact loading. For the dynamic tests, the striker with a mass of 718 kg
was dropped from a height of 2.68 m, and struck a steel pad placed at the mid-span of
the 4000 mm long beam.
15
Figure 2.5: FE model of RC girder (Kishi and Bhatti, 2010)
Figure 2.6: Stress-strain relations of materials (a) concrete (b) reinforcement (Kishi
and Bhatti, 2010)
The test set up is shown in Fig. 2.7. Various measuring devices were used including
accelerometers, strain gauges on the concrete and the reinforcement. The tests were
also recorded by a high speed film camera at 1000 frames per second, which was
16
increased to 1540 for the last test. Acceleration time histories obtained were used to
calculate velocity and displacement time histories, which were then compared with
photos and pulse transducer readings. Strain rates in reinforcement and concrete were
also computed.
Figure 2.7: Drop-weight test set up by Magnusson et al. (2000)
2.4.7 Experimental investigation of RC beam carried out by May et al. (2005,
2006) and Chen and May (2009)
May et al. (2005, 2006) and Chen and May (2009) have described the results of an
investigation into large mass-low velocity impact behaviour of RC beams. They
conducted tests on fifteen 2.7 m or 1.5 m span beams under drop-weight loads. A high-
speed video camera was also used which recorded the images at the rate of up to 4,500
frames per second. Durham strain gauges were also used in some tests to measure
reinforcement strain, because of the location of the gauges inside the bar without
affecting the bond between the concrete and the reinforcement (Scott and Marchand,
2000). Impact loads, strains, accelerations etc. were recorded to obtain time histories.
Figure 2.8 shows the test set up for the impact tests. A software package ELFEN (2004)
17
was tested using the data obtained from the tests. The results of these tests have been
used by the author to validate FE modelling and analysis will be described in
Chapters 4.
2.4.8 Experimental investigation of RC beam carried out by Saatci and Vecchio
(2009a)
To investigate the impact behaviour of RC beams in shear, Saatci and Vecchio (2009a)
carried out an experimental program at University of Toronto. Eight RC beams, four
pairs, were tested under free-falling drop weights, impacting the specimens at the
midspan. Longitudinal reinforcement was identical for all the specimens, but shear
reinforcement ratio was varied. A total of 20 impact tests were performed which
included multiple tests on some specimens. The specimens were 250 mm wide,
410 mm deep and 4880 mm long. All the beams were simply supported with a shear
span of 1500 mm, with a 940 mm overhang at each end, Fig. 2.9.
The test data was later used to verify a two-dimensional nonlinear FE analysis
procedure using the Disturbed Stress Field Model (Saatci and Vecchio, 2009b).
NLFEA procedure was implemented into a two-dimensional, nonlinear FE analysis
program based on rotating smeared-crack approach. The 2-D model of half of the test
beam comprised of a total of 992 rectangular elements to represent concrete and 124
truss bar elements for longitudinal steel, Fig. 2.10. midspan displacements, crack
profiles and longitudinal reinforcement strains were compared with the experiments.
Figures 2.11 and 2.12 show the comparison of midspan displacements and crack
profiles, respectively.
The analyses were also carried out for the multiple impact tests on the same specimens.
The second impacts were analyzed, starting from the results of the analyses performed
on the undamaged specimens and the results compared for midspan displacements,
crack profiles and reinforcement strains. The program failed to analyze the same
specimens for the third impact tests, because of accumulated errors from the previous
analyses and numerical problems. Analysis of the second impact tests for some
specimens was also omitted because of the specimens suffered extensive damage,
which was beyond the analysis capabilities of the software.
18
2.4.9 Experimental investigation of RC beam carried out by Fujikake et al. (2009)
An experimental study was performed by Fujikake et al. (2009), using drop hammer
impact tests on RC beams to investigate the influence of drop height and the effect of
the amount of longitudinal steel reinforcement. A total of twelve beams were tested.
Figures 2.13 and 2.14 show the specimen details and test setup, respectively. A drop
hammer with a mass of 400 kg was dropped freely onto the top surface of the beam at
midspan from four different heights for three series of beams.
The impact force was recorded using a dynamic load cell, which was rigidly connected
to the drop hammer. A laser displacement sensor was used to measure the midspan
deflections. For series S1616, overall flexural failure was recorded at all the drop
heights. For the other two series S1322 and S2222, the overall flexural failure was
observed only at a drop height of no more than 0.6 m. Local failure with heavy
crushing under the loading point was observed at a drop height of not less than 1.2 m.
Figure 2.15 shows the typical failure modes for the impact tests. Measured impact loads
and midspan deflections are shown in Fig. 2.16 for series S1616 beams.
Figure 2.8: Drop-weight test set up at Heriot-Watt University (Chen and May, 2009)
19
Figure 2.9: Details of beams tested by Saatci and Vecchio (2009a)
Figure 2.10: FE model by Saatci and Vecchio (2009b)
20
Figure 2.11: Comparison of midspan displacements for first impacts on undamaged
specimens by Saatci and Vecchio (2009b)
21
Figure 2.12: Observed and calculated crack profiles for SS1b-1 (Saatci and Vecchio,
2009b)
Figure 2.13: Impact tests by Fujikake et al. (2009): Specimen details
Figure 2.17 shows the maximum impact load, the impulse, and the duration of the
impact load, the maximum midspan deflection, and the time taken for the maximum
midspan deflection obtained at each drop height. The impulse and duration of impact
load were defined, as shown in Fig. 2.18.
22
Figure 2.14: Impact tests by Fujikake et al. (2009): Test setup
Based on the results of the impact tests, following conclusions were presented Fujikake
et al. (2009):
The amount of longitudinal reinforcement significantly affected the failure
modes of RC beams under impact loading. The RC beam with comparatively
lower amounts of longitudinal steel reinforcement exhibited only overall
flexural failure, while the RC beam with the comparatively higher amounts of
longitudinal reinforcement exhibited not only the overall flexural failure but
also local failure located near impact loading point.
The amount of longitudinal compression reinforcement affected the degree of
the local failure. Local failure was substantially reduced when heavy
longitudinal compression reinforcement was provided.
The maximum impact load, the impulse, the duration of impact load, the
maximum midspan deflection, and the time taken for the maximum midspan
deflection increased as the drop height was increased. The duration of impact
load, the maximum midspan deflection, and the time taken for the maximum
midspan deflection were affected by the flexural rigidity of the RC beams.
23
(a) (b)
(c)
Figure 2.15: Failure modes: (a) S1616 series; (b) S1322 series; and (c) S2222 series
(Fujikake et al., 2009)
24
Figure 2.16: Impact response for S1616: (a) drop height = 0.15 m; (b) drop height =
0.3 m; (c) drop height = 0.6 m; and (d) drop height = 1.2 m (Fujikake et
al., 2009)
25
Figure 2.17: Impact responses: (a) maximum impact load; (b) impulse; (c) duration of
impact load; (d) maximum midspan deflection; and (e) time taken for
maximum midspan deflection (Fujikake et al., 2009)
Figure 2.18: Impulse and duration of impact load (Fujikake et al., 2009)
2.5 Analytical Models for Impact Loading on RC Member
There have been a number of attempts to develop analytical models for impact and
impulsive loadings on RC beams. Most of the models are based on fundamental theory
of vibration using SDOF systems. In some of the models, two-degree of freedom
systems have also been employed (Fujikake et al., 2009).
26
2.5.1 Proposed analytical models carried out by Hughes and Speirs (1982) and
Hughes and Beeby (1982)
Hughes and Speirs (1982) and Hughes and Beeby (1982) have proposed a `simple'
beam vibration model to analyze midspan impact of pin-ended and simply supported
beams by a rigid striker. Figure 2.19 shows the impact of a rigid spherical striker of
mass ms with impact velocity vo on the midspan of a pin-ended beam of mass mb and
first fundamental frequency w1. The local deformation at the impact zone, a, is given by
net compression at the impact zone as
(2.5)
where ys and yb are the striker and beam midspan displacements, respectively. If each
term of the above equation is expressed as a function of the force F, it in-fact becomes
the impact equation. They used the Hertz contact law, F = Ka3/2 to relate the
deformation to the corresponding impact force F. Therefore
(2.6)
The displacement of the rigid striker was given by
∫ [∫
]
(2.7)
Assuming the overall flexural response of the beam remained elastic, they modeled the
beam displacement in terms of its free vibration modes by solving the simple beam
equation of free vibration
(2.8)
27
Figure 2.19: Midspan impact of a pin-ended beam (Hughes and Beeby, 1982)
They assumed that for a mid-span impact, only symmetrical modes were excited
(Figure 2.20) and calculated the midspan displacement as
(
) ∑ ∫ { }
√
Therefore, the impact equation could be expressed as
⁄ (
)∫ *∫
+
(
)
∑ ∫ { }
28
The above equation is an integral equation of impact force F and its solution in terms of
the known quantities ms, vo, K, mb and w1 defines the impact.
Figure 2.20: First three symmetrical vibration modes of a pin-ended beam (Hughes and
Beeby, 1982)
They suggested that the accurate solution could be obtained using a finite number of
modes i = 1; 3….N. The solution was also related to a limiting case of a massive beam
with movement limited to deformation at the impact zone. They treated this limiting
case essentially to be an input pulse, soluble for quantities ms, vo and K, which
depended on only two `input' parameters of mass ratio (α = mb/ms) and pulse ratio (β =
τ∞/T1), which is the ratio of pulse duration τ∞ and first natural period of vibration (T1 =
2π/ω1). The analysis was compared with the experimental tests, which were described
earlier. Figure 2.21 shows measured and theoretical force histories for one of the
beams.
Figure 2.21: Measured and theoretical force histories (Hughes and Beeby, 1982)
29
2.5.2 Proposed idealizing single degree of freedom system carried out by Comite
Euro-International Du Beton (CEB, 1988)
Idealizing the beam as a single degree of freedom system was also recommended by the
Comite Euro-International Du Beton (CEB, 1988). They classified the impact problem
into soft and hard impacts. A single spring-mass system was proposed for soft impacts,
whereas for hard impact problems, two-spring mass model was suggested, Fig. 2.22.
For the soft impact case, the force-time history P(t) is determined from the dynamics of
the crashing body with a one mass model for the subsequent SDOF analysis. In the two
mass model for hard impact on a beam, the distributed mass of the beam is replaced by
an approximate
(a) Single mass model for soft impact
(b) Two mass model for hard impact
Figure 2.22: Mass-spring model for impact (CEB, 1988)
single mass, m1, which is impacted by mass m2. Springs R1 and R2 represent the
stiffness of the beam and the contact resistance, respectively. In this approach, the
30
“participating mass" m1 is determined from an estimated displaced shape resulting from
energy considerations, given as
∫
where is the distributed mass, L is the span and ϕ(x) is the assumed displaced shape
normalized with respect to the mid-span deflection. Following equations of equilibrium
are required for hard impact case.
where R1 = R1(u1); R2 = R2(Δu); Δu = u2 - u1. Displacement and force time histories can
be determined easily using, for example a finite difference scheme, if R1(u1) and
R2(Δu) are known.
In cases where u2 ≫ u1 the relations is expressed as;
This situation is also called Soft Impact where the kinetic energy of the striking body is
completely transferred into deformation energy of the striking body, while the rigidly
assumed resisting structure remains undeformed.
The analytical methods based on simple mass-spring models using SDOF approaches
have severe limitations. Determination of the force-deformation relationships for R1(u1)
and R2(Δu2) would involve a number of idealizations. For impact loads, the impact
force history is required which would have to be either measured or estimated as an
impulse force. The force deformation relationship for the contact spring also requires
thorough investigation and may be affected by several factors including impacting
mass, its shape and the material properties, friction conditions etc. Moreover, SDOF
idealizations can be applied only to simple structures with predominant deflection
mode that can be justified by SDOF simplification. Loading patterns may also limit the
use of SDOF methods. In order to overcome these limitations and analyze more
31
complex structures, use of numerical methods such as the FE method has been
preferred over the analytical methods.
2.6 Numerical Model of RC Beam under Impact Load
There has been tremendous growth in the development of numerical methods,
especially the FE method. Recently, codes have also been developed combining
different numerical methods. For example, ELFEN (2004) combines FE and discrete
element technologies and has been used for the dynamic analysis of RC structures
(Bere, 2004). Most of these methods can analyze various types of structures under
typical loading conditions. There are several commercial packages with the ability to
solve complex problems including impact simulations, such as ABAQUS (2012),
AUTODYN, ELFEN, LS-DYNA, LUSAS etc. Many of these packages utilize different
techniques of mesh descriptions such as Lagrangian, Eulerian, arbitrary Lagrangian and
Eulerian, smooth particle hydrodynamics (SPH). The choice of time-integration
(explicit or implicit) is also available. Anderson (1987) and Hamouda and Hashmi
(1996) provide a detailed discussion of the numerical techniques for the simulation of
highly dynamic events.
Several researchers have employed the LS-DYNA (Hallquist, 2007) commercial FE
package for the numerical simulations involving highly transient loading including
impact and blast effects. In the present study ABAQUS (2012) has been used for its
various modelling capabilities, which will be described in further details in Section 2.9.
In the following sections, a selection of numerical studies involving low-velocity
impacts on RC beams is described.
2.6.1 Numerical Model of RC Beam under Impact Load carried out by Sangi, A. J.
(2011)
The data obtained from the experimental tests by Chan and May (2009) has been
utilized by Sangi, A. J. (2011) for numerical simulation using LS-DYNA together with
a mechanical constitutive model for concrete. The concrete beam and drop weight were
discretized in space with eight-node cube elements. One-point Gauss integration and
viscous hourglass control was used for the beam and the weight. The reinforcement
bars were discretized with beam elements and the stirrups with truss elements. The
contact surfaces, striker-steel pad and steel pad-beam, were modeled using a surface to
surface constraint algorithm. Figure 2.23 shows the schematic diagram of beam which
32
was simulated by Sangi, A. J. (2011). Comparison of final crack patterns and damage
on beam are shown in Figure 3.24.
Figure 2.23: Schematic Diagram of the beam simulated by Sangi, A. J. (2011)
Figure 2.24: Comparison of final crack patterns and damage on beam simulated by
Sangi, A. J. (2011)
33
2.7 Impact Behaviour of RC Slabs
As discussed in Section 2.3, RC slabs are mostly subjected to local damage due to
impact loads. There has been a number of experimental studies on the local effects of
hard projectiles on RC targets (mainly slabs), which have resulted in a large number of
empirical formulae (Kennedy, 1976; Li et al., 2005). These investigations mainly
suggested the formulae for the penetration depth of the missile, and minimum thickness
to prevent scabbing and perforation under impact loading. These formulae were mostly
developed based on curve fitting of the test data, thus are limited to an applicable range.
Petry's formula (Kennedy, 1976; Li et al., 2005) developed in 1910 was most
commonly used prior to the second world war, which was later modified to include the
effects of concrete strength. Several other formulae were later developed, mostly in the
military including the Army Corps of Engineers (ACE) formula, the National Defense
Research Committee (NDRC) formula, the UK Atomic Energy Agency (UKAEA)
formula and the Ballistic Research Laboratory (BRL) formula. Details of these and
many other formulae have been discussed by many researchers (Corbett et al., 1996;
Kennedy, 1976; Li et al., 2005), therefore are not included in this review for brevity.
As mentioned earlier, the empirical formulae are based on experimental tests and are
valid only for a specific range. Therefore, each formula can result in a different
prediction for each local effect. For example, Yankelevsky (1997) presents the
comparison between several empirical formulae for prediction of the penetration depth
and the perforation thickness in a thick target, Figs. 2.25 and 2.26. As seen in the
figures, the compared empirical formulae give different results for penetration and
perforation predictions. The modified NDRC formula provides the predictions for
penetration depth and perforation thickness which are consistent and in the middle of
the range. For this reason, it was widely used in the nuclear industry. However, for
scabbing thicknesses, the modified NDRC formula is reported to severely
underestimate the phenomenon especially for low-velocity impacts (William, 1994).
34
Figure 2.25: Comparison of concrete penetration depths calculated by various
formulae for the case of a typical missile (Yankelevsky, 1997)
There are severe limitations associated with the use of these empirical formulae due to
a large scatter in the underlying experimental tests. Moreover, the formulae predict the
local effects independent of the influence of global response of the member, i.e.
bending and shear which may affect the local response especially in the low-velocity
impacts. Most of these formulae also do not account for the influence of the
reinforcement, which may be significant for heavily RC members. Only the UKAEA
formula (Barr, 1990) and more recently the UMIST formula (Li et al., 2005) include
the effect of reinforcement. The formulae are also limited to the case of normal impact
only.
35
Figure 2.26: Comparison of concrete perforation thickness calculated by various
formulae for the case of a typical missile (Yankelevsky, 1997)
With the development of numerical methods, the study of local impact effects on
concrete targets using such methods has several advantages over empirical and
analytical methods. The use of the FE method along with other numerical methods, not
only determines the local effects, but also predicts the influence of global response. To
validate the numerical models, several experimental studies have been carried out. A
selection of experimental investigations on RC slabs is described followed by the
numerical simulations.
2.7.1 Impact tests on RC slabs carried out by Sawan and Abdel-Rohman (1986)
Sawan and Abdel-Rohman (1986) carried out low-velocity impact tests on RC slabs
75 x 75 x 5 cm in dimensions. The slabs were impacted by a 12 cm diameter steel ball
at the centre from heights of up to 120 cm and the deflection was measured. They
36
investigated the effect of the percentage of steel reinforcement and impact velocity on
the deflection response of the slabs.
2.7.2 Impact tests on RC slabs carried out by Kishi et al. (1997)
To investigate the dynamic behaviour of slabs, large scale RC slabs were tested under
impact loading by Kishi et al. (1997). Nine rectangular specimens measuring 4 m wide
and 5 m long were repeatedly loaded onto the centre by a steel weight falling freely.
Masses of 1000, 3000 and 5000 kg were used depending on slab thickness. The
collapse was assumed when the accumulated residual displacement reached 1=200th of
span length. Variations of the slab thickness (25, 50, 75 cm), reinforcement ratio (0.5,
1.0 %), reinforcement arrangement type (single and double arrangements) and
reinforcement material were considered. The impact behaviour was considered by
recording maximum impact force, reaction forces, residual displacements and crack
patterns. The experiments showed that the maximum impact force was more affected
by slab thickness than reinforcement ratio, material strength and the reinforcement
arrangement type. The failure under repeated impact loading was initiated by flexural
cracks, but final failure was due to punching failure. They estimated the punching shear
capacity assuming a conical shape shear failure neglecting the reinforcement.
2.7.3 Impact tests on RC slabs carried out by Zineddin and Krauthammer (2007)
Zineddin and Krauthammer (2007) tested 90 x 1524 x 3353 mm slabs with three
different reinforcement configurations. They used welded steel wire meshes on both
faces, No. 3 steel bars located at the middle of the slab and No.3 bars on both faces for
three configurations, respectively. A 2608 kg mass was dropped from three different
heights of 152, 305 and 610 mm at the centre of the slabs. The tests were instrumented
using load cell, accelerometers, deflection gauges, reinforcement strains and high-speed
videos. Figure 2.27 shows the load-time histories of slabs subjected to 610 mm drop.
37
Figure 2.27: Load-time histories of slabs under 610 mm drop (Zineddin and
Krauthammer, 2007)
2.7.4 Impact tests on RC slabs carried out by Chen and May (2009)
Chen and May (2009) carried out a series of experimental studies to investigate the
high-mass, low-velocity impact behaviour of RC beams and slabs. The tests were
conducted to generate high quality input data to validate numerical modelling. They
tested four 760 mm square, 76 mm thick and two 2320 mm square, 150 mm thick slabs.
A drop-weight system was used to drop a mass of up to 380 kg with velocities of up to
8.7 m/s. Supports were provided by clamping at the four corners to restrain horizontal
and vertical movement. The author has used the results of these tests to perform the
three-dimensional FE analysis, which will be described in Chapter 4.
After that lot of researchers were used experimental data of Chen and May (2009) to
verified numerical simulation of slab under impact load. Mokhatar and Abdullah (2012)
simulated an RC slabs to investigate failure mechanism when subjected to impact
loading by using ABAQUS (2012) software.
2.8 Impact Behaviour of Concrete Column
Previous research on columns has mainly focused on improving the axial load carrying
capacity and stiffness, while improvement of impact resistance has been largely
unexplored. The few investigations conducted on laterally impacted columns highly
emphasized the importance of the stain rate effects. Some of the test results indicated
that the increased structural resistance is somewhat greater than the commonly accepted
maximum increase of 30% of the static resistance, Louw et al. (1992). Strain rate
effects, as well as the behaviour of the vehicle during the impact, are of primary
importance as far as the structural response is concerned, Prasad (1990).
38
2.8.1 Impact tests on RC Column carried out by Leodolft (1989)
Leodolft (1989) tested thirty-nine 350 x 150 x 1600 mm RC columns under soft impact
conditions. The soft impact condition was achieved by inserting a pipe buffer system in
between the pendulum and the column. The columns were axially preloaded with
100 kN and 200 kN forces and subjected to an impact velocity of about 7 m/s. The
applied loads were sufficient to permanently damage the impacted columns.
In this experiment, the peak load occurred later and is more likely to influence the
flexural shear resistance of the element, than its pure shear and inertia stiffness. During
the first 10 ms, the buffer system was subjected to elastic-plastic deformations. By that
time, substantial energy had been transferred to the column, which deflected
significantly. The generated axial load from the impact was increased as the column
increases in length and subsequently decreases and remained compressive. In addition,
the strain rate of up to 10-2 was generated at the rear surface of the column. It was
observed that the partially damaged columns exhibited the same static lateral capacity
as the undamaged columns. Moreover, the impacted columns were subjected to a series
of peak shear and corresponding moments and peak moments and corresponding shear.
According to the test results, it was concluded that the dynamically loaded slender
columns are considerably stronger than the ultimate load predicted by the modified ACI
equation (ACI 318-02) for the slenderness ratio.
2.8.2 Impact tests on RC Column carried out by Feyerabend (1988)
However, during a hard impact, the kinetic energy of the striker is mostly absorbed by
the structure and the striker itself suffers small deformations. Feyerabend (1988),
conducted an experimental investigation on 300 x 300 x 4000 mm RC columns
subjected to lateral impact at midspan. The columns were tested in a horizontal
position, where one end was restrained using a 20t mass to simulate the inertial restraint
provided by a bridge deck. The axial load was applied by pulling the free sliding end
using external prestressing bars towards the stationary end. The impact load was
generated by dropping a 1.14 ton mass onto the column at midspan and the shear
reinforcements were provided to ensure a flexural failure of column, Table 2.1. The
arrangement of test which conducted by Feyerabend (1988) has been presented in
Fig 2.28.
39
Table 2.1: Characteristics of the Feyerabend‟s (1988) test specimens
Figure 2.28: The test set-up by Feyerabend (1988)
An important feature of the impact behaviour of that column was the initial increase in
axial force as the column lengthened along its centre line. The authors also observed
that the initial peak of the applied impact load depended on the inertial characteristics
of the column and the boundary conditions. Under the effects of the impact load, the
column experienced shear deformations while local deformations occurred at the point
of impact. Even though these deformations were relatively small, the initial impact
force had a high initial peak. The initial force was opposed primarily by the inertia
forces of the element. The shear stiffness of the column was the main parameter that
controlled its response. As the shock wave progressed through the cross sections of the
elements, they were subjected to fluctuating moments, shear forces and axial loads.
40
After observing these responses the author has emphasized the impotency of the all
these forces in determining the critical section.
By assuming a 10% increment of the material properties due to strain rate effects, the
dynamic moment capacity of the tested column exhibited 20% increment compared to
that of its static value. On the other hand, observed dynamic shear capacity of the
column was substantially greater than the ultimate static shear capacity of the column.
Therefore, it was concluded that the initial peak shear force generated during hard
impact, is not an indication of the ultimate structural resistance of the column when
adequately reinforced in shear. In addition, under the hard impact condition the
moment-shear combination moves from a low moment high shear value to a higher
moment much lower shear value. Therefore there is a possibility to generate initial
shear cracks in a section which probably diminishes the flexural resistance that follows.
2.8.3 Impact tests on RC Column carried out by Gebbeken et al. (2007)
There are several disadvantages associated with individual column tests, Gebbeken et
al. (2007). The main disadvantage being the idealized boundary conditions. The
flexibility of the realistic support conditions was not taken into account in these tests.
This factor can shift the location of the plastic hinge and consequently, the failure
mechanism would be different from the usual fixed assumption. In addition, the effects
due the wave reflection at the boundaries cannot be neglected. At free boundaries, the
compressive wave is reflected as a tensile wave, while at fixed boundaries, the reflected
wave becomes a compressive wave. The small models are the ones that suffer most due
to the boundary conditions, Gebbeken et al. (2007). Even though the shear cracks,
spalling of the concrete cover and confinement failure are the ideal failure modes for
the individual columns, the effects of the global structural configuration cannot be
neglected.
2.9 FE Modelling
The FE method is a numerical method for solving problems of engineering and
mathematical physics. Typical problem areas of interest in engineering and
mathematical physics that are solvable by use of the FE method include structural
analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential.
41
2.9.1 FE package
A number of FE analysis computer packages are available to solve simple as well as
complex problem in the field of civil engineering problem. Most popular FE packages
are:
ABAQUS
ANSYS
LS-DYNA
SAP
STAAD etc.
Almost all FE packages are suitable for any type of structural problem but some
packages have special advantage to analyze particular problem. Of these, the package
ABAQUS (2012) has been used in this study for its flexibility and vastness of
capability.
2.9.2 An overview of ABAQUS (2012)
ABAQUS (2012) is one of the most popular commercial FE package used for
numerical analysis involving impacts, blast and many other transient phenomena.
ABAQUS (2012) contains an extensive library of elements that can model virtually any
geometry. It has an equally extensive list of material models that can simulate the
behaviour of most typical engineering materials including metals, rubber, polymers,
composites, RC, crushable and resilient foams, and geotechnical materials such as soils
and rock. Designed as a general-purpose simulation tool, ABAQUS (2012) can be used
to study more than just structural (stress/displacement) problems. It can simulate
problem in such diverse areas as heat transfer, mass diffusion, thermal management of
electrical components, acoustics, soil mechanics and piezoelectric analysis.
ABAQUS/CAE
ABAQUS/CAE is a complete ABAQUS (2012) environment that provides a simple,
consistent interface for creating, submitting, monitoring, and evaluating results from
ABAQUS/Standard and ABAQUS/Explicit simulations. ABAQUS/CAE is divided into
modules, where each module defines a logical aspect of the modelling process; for
example, defining the geometry, defining material properties, and generating a mesh.
ABAQUS (2012) is capable to move any model from one module to another module,
42
i.e. from ABAQUS/Explicit to ABAQUS/Implicit module and vice versa. It offers to
build the model from which ABAQUS/CAE generates an input file that could be
submitted to the ABAQUS/Standard or ABAQUS/Explicit analysis product. The
analysis product performs the analysis, sends information to ABAQUS/CAE to allow
analyzer to monitor the progress of the job, and generates an output database. Finally,
the Visualization module of ABAQUS/CAE (also licensed separately as
ABAQUS/Viewer) is used to read the output database and view the results of analysis.
ABAQUS/Explicit
An Explicit FE analysis does the incremental procedure and at the end of each
increment updates the stiffness matrix based on geometry change (if applicable) and
material changes (if applicable). Then a new stiffness matrix is constructed and the next
increment of load or displacement is applicable to the system. In this type of analysis
the hope is that if the increments are small enough the results will be accurate. One
problem with this method is that you do need many small increments for good accuracy
and it is time consuming. If the number of increments are not sufficient the solution
tends to drift from the correct solution.
The advantages of ABAQUS/Explicit are:
It has been designed to solve highly discontinuous, high-speed dynamic
problems efficiently.
It has a very robust contact algorithm that does not add additional degrees of
freedom to the model.
It does not require as much disk space as ABAQUS/Implicit for large problems,
and it often provides a more efficient solution for very large problems.
It contains many capabilities that make it easy to simulate quasi-static problems.
ABAQUS/Implicit
An Implicit finite analysis is the same as Explicit with the addition that after each
increment the analysis does newton-raphson iteration to enforce equilibrium of the
internal structure forces with the externally applied loads. The equilibrium is usually
enforced to some user specified tolerance. So this is the primary difference between the
two types of analysis, implicit uses newton-raphson iterations to enforce equilibrium.
43
This type of analysis tends to be more accurate and can take somewhat bigger
increment steps. Also, this type of analysis can handle problems better such as cyclic
loading, snap through, and snap back so long as sophisticated control methods such as
arc length control or generalized displacement control are used.
The advantages of ABAQUS/Implicit are:
It can solve for true static equilibrium, P − I = 0, in structural simulations.
It provides a large number of element types for modelling many different types
of problems.
It provides analysis capabilities for studying a wide variety of nonstructural
problems.
It uses a very robust and proven contact algorithm.
It uses an integration method for transient problems that has no mathematical
limit (stability limit) on the size of the time increment (the time increment size
is limited only by the desired accuracy of the solution).
ABAQUS/Viewer
ABAQUS/Viewer provides graphical display of ABAQUS (2012) FE models and
results. ABAQUS/Viewer is incorporated into ABAQUS/CAE as the Visualization
module.
Element Type
Different type of 3D elements has been offered by nonlinear FE package, ABAQUS
(2012), to predict the complicated behavior of RC structure. However frequently used
3D elements for modelling of concrete material are:
C3D20
C3D8
C3D20 element is a general purpose quadratic brick element with 27 (3 x 3 x 3)
integration points whereas C3D8 element is simple linear continual solid brick element
with 8 (2 x 2 x 2) integration points. In the present study, C3D8 element has been
employed to model concrete material of RC structure. On the other hand ABAQUS
(2012) offer width range of two noded link elements such as truss element T3D2, Beam
44
element B31 etc. In the present study, T3D2 element has been employed to model
reinforcement of RC structure.
Figure 2.29: C3D20, C3D8, T3D2 elements used to model by ABAQUS (2012)
2.9.3 FE modelling of RC
The explicit FE method has proven to be an effective tool, especially for transient and
impact analyses.
The FEM for RC structures have generally been based on replacing the composite
continuum by an assembly of elements representing the concrete and the steel
reinforcement. From the literature, it has been observed that three techniques are
mainly employed for modelling reinforcement in a 3D FEM of a concrete structure:
smeared model, embedded model, and discrete model. The specific technique is chosen
depending on the application and the degree of detail in which the model needs to be
developed. However, most of the difficulties in modelling RC behaviour depend on the
development of an effective and realistic concrete material formulation and not in the
modelling of the reinforcement.
45
Smeared Model
In the smeared model, the reinforcement is assumed to be uniformly distributed over
the concrete elements, Fig. 2.30. As a result, the properties of the material model in the
element are constructed from individual properties of concrete and reinforcement using
composite theory. This technique is generally applied for large structural models, where
reinforcement details are not essential to capture the overall response of the structure.
Figure 2.30: Smeared formulation for RC
Embedded Model
To overcome mesh dependency in the discrete model, the embedded formulation allows
independent choice of concrete mesh, as shown in Fig. 2.31. In this approach, the
stiffness of the reinforcement elements is evaluated independently from the concrete
elements, but the element is built into the concrete mesh in such a way that its
displacements are compatible with those of surrounding concrete elements. The
concrete elements and their intersection points with each reinforcement segment are
identified and used to establish the nodal locations of the reinforcement elements. The
embedded formulation is generally used with higher-order elements. In concrete
structures where reinforcement modelling is complex, the embedded representation is
advantageous. However, the additional nodes required for the reinforcement increase
the number of degrees of freedom (DOFS), and hence the solution time. Further,
researchers have found that although analyses with the embedded representation are in
general more computationally efficient than those with the discrete representation.
46
Figure 2.31: Embedded formulation for RC
Discrete Model
In the discrete model, reinforcement is modeled using bar or beam elements connected
to the concrete nodes. As a result, there are shared nodes between the concrete element
and the reinforcement element, Fig. 2.32. Also, since the reinforcement is
superimposed in the concrete mesh, concrete exists in the same regions occupied by the
reinforcement. The drawback of using the discrete model is that the concrete mesh is
restricted by the location of the reinforcement. Full bond is generally assumed between
the reinforcement and the concrete. In cases where bond issues are of importance,
fictitious spring elements are used to model bond slip between the concrete and the
reinforcement elements. These linkage elements connect concrete nodes with
reinforcement nodes having the same coordinates. These types of elements have no
physical dimension at all and only their mechanical properties are of importance.
Figure 2.32: Discrete model of RC
47
2.9.4 Contact algorithms
Several contact algorithms are available in the literature, namely, frictional sliding,
single-surface contact, nodes impacting on a surface, tied interfaces, 1-D slide lines,
rigid walls, material failure along interfaces, penalty and Lagrangian projection options
for constraint enforcement and fully automatic contact. Details of a typical algorithm,
automatic-single-surface contact are enumerated:
This algorithm uses a penalty method to model the contact interface between the
different parts. In this approach, the slave and master surfaces are generated
automatically. The method consists of placing normal interface springs to resist
interpenetration between element surfaces. An example of this approach is illustrated in
Fig. 2.33. It shows that, when a slave node penetrates a master surface in a time step,
the algorithm automatically detects it, and applies an internal force to the node
(represented by the spring) to resist penetration and keep the node outside the surface.
The internal forces added to the slave nodes are a function of the penetrated distance
and a calculated stiffness for the master surface. The stiffness is computed as a function
of the bulk modulus, volume, and surface area of the elements in the master surface.
Friction less tangential and hard normal behaviour between different parts in contact
has been used in the present study.
Figure 2.33: Penalty method for contact algorithm
2.10 Constitutive Concrete Material Models
There are several material models to represent concrete, which have been implemented
in commercial software used for simulation of concrete structures subjected to impact
48
loads. For example, the Concrete Damage Plasticity (CDP) model, Concrete Smeared
Crack model, and Modified Drucker-Prager/Cap model are most popular models for
modelling of concrete material in ABAQUS (2012). In the present study Concrete
Damage Plasticity model has been employed to model concrete material of RC
structure.
2.10.1 Concrete Damage Plasticity model
The concrete damaged plasticity model in ABAQUS (2012) provides a general
capability for modelling concrete and other quasi-brittle (rock, brick etc.) materials in
all types of structures (beams, trusses, shells, and solids). CDP model also uses
concepts of isotropic damaged elasticity in combination with isotropic tensile and
compressive plasticity to represent the inelastic behavior of concrete. It can be used for
plain concrete, even though it is intended primarily for the analysis of RC structures as
well as it is designed for applications in which concrete is subjected to monotonic,
cyclic, and/or dynamic loading under low confining pressures.
The model is a continuum, plasticity-based, damage model for concrete. It assumes that
the main two failure mechanisms are tensile cracking and compressive crushing of the
concrete material. The evolution of the yield (or failure) surface is controlled by two
hardening variables, and , tensile and compressive equivalent plastic strains
respectively, inked to failure mechanisms under tension and compression loading,
respectively.
The model assumes that the uniaxial tensile and compressive response of concrete is
characterized by damaged plasticity, as shown in Fig. 2.34. The details of Concrete
Damage Plasticity model has been presented in Section A.2 of Appendix-A
49
Figure 2.34: Response of concrete to uniaxial loading in (a) tension and (b)
compression
Failure criteria for concrete in tension and compression was developed by Kupfer
(1973) with five parameters have been used in FE analysis of RC member. These
parameters are given in the following Table 2.2
Table 2.2: Parameters used for failure criteria
Parameter Denotation Reference
Ψ = 38o Dilation angle Jankowiak et. al (2005)
Ratio of biaxial to uniaxial
compressive strength
Kupfer (1973)
K =0.67 Second stress invarient ratio Kmiecik et al. (2011)
e = 0.1 Eccentricity ABAQUS (2012)
50
2.10.2 Material model for reinforcing steel
The uniaxial material behaviour of the reinforcing steel is modeled by a bilinear stress-
strain relation. Therefore, the reinforcing steel material model only requires the yield
strength , the modulus of elasticity as well as the ultimate strength and the
corresponding ultimate strain . Elastic-perfectly plastic material property for steel
reinforcement has been used to model RC member as shown in Fig. 2.35.
Figure 2.35: Stress-strain relation for steel reinforcement
2.11 Damping Coefficients
Mass and stiffness proportional damping, normally referred to as Rayleigh damping, is
commonly used in nonlinear-dynamic analysis. Suitability for an incremental approach
to numerical solution merits its use. The formation of damping matrix accociate with
Raleingh damping is assumed to be proportional to the mass and stiffness matrices. The
details of the damping matrix formation which has been used in the presect numerical
simulation of different structural members under dynamic load has been presented in
Section A.2 of Appendix-A.
2.12 Summary
In this chapter, the literature on impact load applied on RC structural member has been
reviewed. An impact event may be classified as soft or hard impact depending upon the
energy absorption capacity of target structure. The structural responses of RC structural
member under impact load have been discussed in the present chapter. Some existing
σy
ε
σ
εy
Es
51
literature on RC beam, slab and column subjected to impact load based on experimental
investigation, analytical methods, numerical methods have also been thoroughly
reviewed. Numerical modelling technique of RC structural member under static as well
as dynamic load has been discussed in last portion of current chapter. The numerical
modelling technique includes constitutive material modelling, selection of proper type
of finite element on the basis of characteristics of structure, modelling of interaction
property between reinforcement and concrete, contact property between impactor and
target structure etc. have been reviewed.
52
Chapter 3: Nonlinear FE modelling Validation
Nonlinear FE modelling Validation
3.1 Introduction
The finite element (FE) method is one of the most powerful numerical techniques
which is frequently used for the solution of engineering problems. It has provided the
necessary tools to model and analyze virtually any engineering structural system
involving reinforced concrete (RC) structure. ABAQUS (2012), one of the most
powerful commercial software of FE analysis, is used to model and analyze almost any
type of RC structural member. The constitutive modelling technique with respect to RC
element has been presented in the Section 2.9. In the present chapter, the modelling
techniques of RC structural elements like beam, slab etc. in FE software ABAQUS
(2012) have been verified with test as well as theoretical results of RC structures.
Five different type of RC beam under static load with nonlinear property, a simple
SDOF system with different damping property under different type of dynamic loading
such as El Centro earthquake have been modeled and analyzed by ABAQUS (2012).
The numerical results of these analyses have been compared with the experimental
results as well as theoretical results as per different literatures. After successful
execution of these analyses, it will verify that the ABAQUS (2012) software has the
ability to model actual structural behaviour of any RC members.
3.2 FE Modelling in ABAQUS (2012)
It is possible to model any type of RC structural elements, such as beam, column, and
slab in ABAQUS (2012) with help of different constitutive materials available in
software library. Concrete Damage Plasticity (CDP) and concrete smeared crack
models are most suitable material modelling techniques for concrete portion of RC
structure. The reinforcement of RC structure has been modeled with the elastic-plastic
material. In the present analysis CDP model has been used for numerical model of
concrete material. This type of model is designed for applications in which concrete is
subjected to monotonic, cyclic, and/or dynamic loading under low confining pressures.
The details of CDP model have been presented in Section 2.5.5.
The available analysis methods of ABAQUS (2012) are divided into two types, one is
explicit analysis method and another is implicit analysis method. In explicit analysis
53
method total analysis is divided into a number of increments and at the end of each
increment updates the stiffness matrix based on geometry and material changes. An
Implicit analysis method is the same as explicit with the addition that after each
increment the analysis does Newton-Raphson iterations to enforce equilibrium of the
internal structure forces with the externally applied load. In the present analysis,
explicit analysis method of ABAQUS (2012) has been used for all modelling of RC
structures.
Explicit and implicit analysis methods of ABAQUS (2012) are designed to model
dynamic response of any structure. Static analysis of same structure is also possible to
model in explicit or implicit analysis method of ABAQUS(2012) but some
modification is required to fulfill the static equilibrium of the structure. The basic
difference between static and dynamic analysis is equilibrium of external and internal
forces. In static analysis the external force is balanced with stiffness force of structure.
Whereas in dynamic analysis equilibrium is done by equating external force with
internal inertia, damping and stiffness forces of structure. So any dynamic analysis will
become static analysis if inertia and damping forces of structure become close to zero.
The inertia and damping force of any structure will become very small when the rate of
displacement of structure is very small. Again the rate of displacement will be very
small if mass of structure is very high but the mass of structure is a part of external
load. To avoid this extra self-weight problem, the structure should be modeled without
gravity action. In the present analysis, mass scaling technique of ABAQUS (2012) has
been used to model a static problem in explicit analysis method.
Different types of FE are available in ABAQUS (2012) library. In the present section
eight noded continuum solid element (C3D8R) is used for modelling of concrete
portion of RC structures and two noded truss element (T3D) is used for reinforcement.
The detail properties of different Finite elements have been presented in Section 2.9.2.
The size of element used to discretize any structure is one of most important part of FE
analysis. The analysis results of FE are directly influenced by the element sizes. The
smaller element size gives more accurate results i.e. deflection and force of the
analyzed structure. The technique to find most usable element size for FE analysis is
called mesh sensitivity analysis.
54
3.3 Validation of FE Model of RC Beam under Static Load with Test Result
The static analyses presented in this section simulate the tests carried out by Saatci
(2007) on RC beams. These static beams have been modeled by ABAQUS (2012) with
the FE method and the analyses results are compared with the test results.
3.3.1 Dimension of tested RC beam by Saatci (2007)
The test specimens used for the modelling of RC beam are taken from test performed
by Saatci (2007). The test specimens constructed and modeled were four simply
supported RC beams with identical longitudinal reinforcement and varying shear
reinforcement. The dimension of the test specimens were 16.4 inch in height and
10 inch in width and 195 inch in length. The specimens were tested under simply
supported conditions with a shear span of 60 in, leaving 37.5 in at each end. The details
of tested RC beam are presented in Fig. 3.1.
Figure 3.1: Details of RC beam tested by Saatci (2007)
The beams were reinforced with symmetric longitudinal reinforcement in height such
that it would have equal moment capacity in positive and negative flexure. All beams
had the same amount of longitudinal reinforcement: two No. 30 (area = 1.1 in2) steel
55
bars placed with 1.5 in clear cover at the bottom and top of the beam. Shear
reinforcement was varied for different beams, thus allowing a better understanding of
how shear reinforcing affects the failure behavior. The three levels of shear
reinforcement include no shear reinforcing steel, 0.1% shear reinforcement, and 0.2%
shear reinforcement as shown in Table 3.1. The type of shear reinforcement used in
these tests were D8 reinforcing bar (area = 0.085 in2). The beam specimen MS1 has
shear reinforcement with 12 in spacing, since MS0 has no shear reinforcement and
MS2 has shear reinforcements with 6in spacing.
Table 3.1: Transverse reinforcement ratios and stirrup spacing for beams
Specimen Transverse Reinforcement
Ratio
Stirrup Spacing
(in)
MS0 0.0% --
MS1 0.1% 12
MS2 0.2% 6
The concrete compressive strength of the beams at testing was approximately 50ksi.
The material properties of reinforcements and concrete are given in Tables 3.2 and 3.3.
Detail material properties with damage parameters as used in Concrete Damage
Plasticity model are presented in Table B.1 of Appendix-B.
Table 3.2: Material property of transverse and stirrup for beams
Bar size Area (in2)
Yield
Strain
x10-3
Yield
Stress
(ksi)
Ultimate
Strength
(ksi)
Young's
Modulus
(ksi)
Ultimate
Strain
x10-3
30mm 1 2.5 67.3 100 29000 80
8mm 0.078 4.9 83 90.36 16930 50
56
Table 3.3: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable elastic stress
(ksi)
Allowable elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
2.25x10-4 4974 0.15 2.17 4.4x10-4 7.25 3.43x10-3
3.3.2. Modelling of tested beam by Saatci (2007)
The schematic view of RC beam is shown in Fig. 3.2. The individual parts of the model
such as concrete and reinforcement should be connected properly to each other. The
bending and shear reinforcements are connected with surrounding concrete at each
intersection points of the concrete and the reinforcement elements by embedded model
technique of ABAQUS (2012). Full bond is always assumed between the reinforcement
and the concrete. The support and load have been applied on beam by steel plate with
1 in thickness. Surface to surface (Explicit) contact method has been used to model the
contact behaviour between RC beam and steel support/load plates. The penalty method
[Section 2.9.4] has been used for contact algorithm. The element size of 1in (25 mm)
for concrete beam, reinforcement and steel support plate has been proved most
reasonable element size from mesh sensibility analysis. Figure 3.2 shows the RC beam
after divided into Finite elements.
The constitutive material property of concrete has been modeled by CDP model of
ABAQUS (2012). The CDP model help to visualize the damage level of concrete
elements as tension and compression damage. Other material of RC beam i.e.
reinforcement has been modeled by simple elastic-plastic material of ABAQUS (2012).
The beam has been modeled with full scale to ensure the actual behaviour of test.
57
Figure 3.2: FE model of tested RC beam by Saatci (2007)
3.3.3 Response of MS0 beam
Figure 3.3 shows the load verses midspan deflection of the beam MS0. The load has
been applied to the beam at midspan by loading plate (bearing plate) and analysis
executed by explicit method of ABAQUS (2012). So, reaction force between contact
surface of the beam and loading plate has been considered as the stiffness force i.e.
static load of the beam. Undulation has been observed at load vs. deflection diagram of
this beam because the beam has been vibrated by tension crack during analysis. This
phenomenon is also observed during flexural test of any RC beam. The response of
MS0 by FE analysis has been found to be relatively similar to the actual response
observed in the test. ABAQUS (2012) modeled the beam to be slightly flexible than the
actual response. The main variation in behaviour between the predicted and the
observed results can be viewed after the peak load has reached, where ABAQUS
(2012) found that the beam lost its capacity at 41.30 kip as the peak load, whereas the
actual beam sustained upto 44.48 kip load.
58
Figure 3.3: Comparison of reaction force Vs. midspan displacement diagram of
numerically analyzed beam MS0 with test result found by Saatci (2007)
3.3.4 Response of MS1 beam
Figure 3.4 shows the load verses midspan deflection of the beam MS1. The response of
MS1 has been found by FE analysis to be very similar to the actual response viewed in
the test. ABAQUS modeled the beam to be almost similar of actual response, but the
load reached at peak earlier than the test values. ABAQUS found the beam would
sustain a larger force with a displacement smaller than actually observed.
3.3.5 Response of MS2 beam
Figure 3.5 shows the load vs. deflection behaviour at midspan of MS2 beam as
predicted by the FE analysis along with the test result. Up to the yielding of
longitudinal reinforcement, the predicted response of MS2 has found to be very similar
to the actual response observed in the test. The yielding point of the beam has been
estimated well by ABAQUS model. However, the ductility of the beam has been
severely underestimated by the failure pattern of beam after yielding.
0
5
10
15
20
25
30
35
40
45
50
0.00 0.05 0.10 0.15 0.20 0.25 0.30
App
lied
Forc
e, k
ip
Mid point deflection, in
ABAQUS Test
59
Figure 3.4: Comparison of reaction force Vs. midspan displacement diagram of
numerically analyzed beam MS1 with test result found by Saatci (2007)
Figure 3.5: Comparison of reaction force Vs. midspan displacement diagram of
numerically analyzed beam MS2 with test result found by Saatci (2007)
0
10
20
30
40
50
60
70
80
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70
App
lied
forc
e, k
ip
Mid point Deflection, in
ABAQUS Test
0
10
20
30
40
50
60
70
80
90
100
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00
App
lied
load
, kip
Mid span deflection, in
ABAQUS Test
60
3.4 Validation of FE Model of RC Slab under Static Load Tested by McNeice
(1967)
A set of specimens were tested by McNeice (1967) consisting of corner supported slabs
subjected to a point load applied at the centre. The McNeice slab was often used as a
benchmark for calibrating nonlinear analyses. The geometry of the two-way slab is
defined in Fig. 3.6. The square slab is reinforced in two directions at 75% of its depths.
The reinforcement ratio (volume of slab/volume of concrete) is 8.5x10-3 in each
direction. The summary of the material properties have been taken from Gilbert and
Warner (1978) and used in the present FE modelling is shown in Tables 3.4 and 3.5.
Detail material properties with damage parameters as used in Concrete Damage
Plasticity model are presented in Table B.2 of Appendix-B. Typical concrete mesh (size
2.78% of span) is shown in Fig. 3.7. Symmetry conditions allow to model one-quarter
of the slab. The concrete part of this slab has been modeled by four noded doubly
curved thin shell elements (S4R). This shell element has six degree of freedoms at each
node. Nine integration points have been used through the thickness of the concrete to
ensure that the development of plasticity and failure is modeled adequately.
Reinforcement (rebar) is homogeneously connected with shell elements of concrete, so
the two-way reinforcement has been modeled by using rebar layers option of ABAQUS
(2012). Symmetry boundary conditions have been applied on the two edges of the
mesh, and the corner point has been restrained in the transverse direction.
Table 3.4: Material property of reinforcement
Reinforcement ratio
Density
ρ
(lb-sec2/in4)
Poison‟s Ratio
ν
Young's
Modulus
(ksi)
Yield
Stress
(ksi)
Yield
Strain
x10-3
Ultimate
Strength
(ksi)
Ultimate
Strain
x10-3
8.5x10-3 7.3x10-4 0.29 29000 50 1.72 50 80
Table 3.5: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable elastic stress
(ksi)
Allowable elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
Cracking failure stress
(ksi)
2.25x10-4 4150 0.15 3 7.23x10-4 5.5 0.0015 0.459
61
Figure 3.6: Geometry of RC slab tested by McNeice (1967)
Figure 3.7: FE model of one-quarter of RC slab tested by McNeice (1967)
62
The comparison between load deflection diagrams for McNeice slab (Test) and
presently FE analyzed slab (FE) is presented in Fig. 3.8. It is observed that two the
curves exhibit a linear elastic behaviour at the initial stage. After some time with the
increase of load, it is followed a gradual development of nonlinear response and have
some difference in the load-deflection behaviour between present FE analysis and
experimental results but maximum load for both slabs is about 3.36 kips.
Figure 3.8: Comparison of Load-deflection diagram of numerically analyzed slab with
test result observed by McNeice (1967)
3.5 Validation of RC Beam Modelling with Theoretical Result
The main purpose of the present section is to verify the modelling technique of RC
members through comparing the theoretical result and FE analysis result of under RC
beam.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 0.1 0.2 0.3 0.4 0.5
Load
, kip
Deflection, in
FE TEST
63
The basic characteristic of the under RC beam is ductile behavior, i.e. the reinforcement
of the beam yield before crushing of the concrete. This type of beam will undergo large
deformation before failure. Adequate shear reinforcement of beam is required to
provide for preventing shear failure of beam before flexural failure. In the present
section, the objective is to predict and compare different stage of loading of RC beam
as shown in Fig. 3.9. For this, loading stage can be divided into three different stages,
i.e. a) stress elastic and section uncracked, b) stress elastic and section cracked, and c)
loading which produce nominal moment, Mn (stress become plastic).
Figure 3.9: Dimensional view of RC beam
The beam, shown in Fig. 3.9, is 12 ft 4 in long and cross sectional dimension is 10 in x
15 in. This beam is supported by two steel bars and loaded at centre by another steel
bar. The dimension of supporting and loading bars is 1 in x 2 in x 10 in. The beam is
reinforced by two 20mm diameter reinforcement as bottom flexural reinforcement and
two 10mm diameter reinforcement as top flexural reinforcement which provide support
for shear reinforcement. The beam also has two-legged closed tie as shear
reinforcement and spacing of the tie bars is 6in centre to centre throughout the beam.
The clear cover of the beam from centre of flexure reinforcement is 1.5 in. The
reinforcement ratio of the beam, 0.0065, is much less than the balance reinforcement
ration for same beam, 0.032. So, the beam is under RC beam.
The summary of the material properties used in the present FE modelling is shown in
Tables 3.6 and 3.7. Detail material properties with damage parameters as used in
Concrete Damage Plasticity model are presented in Table B.3 of Appendix-B. The
concrete beam and the reinforcement have been modeled by eight-node continuum
solid elements (C3D8R) and two nodded truss element (T3D2), respectively. The steel
supports and steel loading bar have been modeled with continuum solid element
(C3D8R). Surface to surface contact (Explicit) technique has been used to create proper
64
interaction between concrete beam and steel supports and embedded technique has
been used to constraint the two node truss element (steel reinforcement) into solid
element (concrete beam) in order to create a perfect no slip action.
The concrete beam has been divided into 22200 elements with size of 0.69% span of
beam. Each of three supporting steel bar were also meshed with 20 elements of same
size. All reinforcements have been divided by link elements with length of 0.69% span
of beam. Figures 3.10 and 3.11 show the concrete beam and reinforcement after divided
into Finite elements.
Figure 3.10: Concrete modelling with C3D8R Brick elements
Table 3.6: Material property of reinforcement
Density
ρ
(lb-sec2/in4)
Poison‟s Ratio
ν
Young's
Modulus
(ksi)
Yield
Stress
(ksi)
Yield
Strain
x10-3
Ultimate
Strength
(ksi)
Ultimate
Strain
7.3x10-4 0.29 29000 50 1.72 65 0.11
65
Figure 3.11: Reinforcement modelling with T3D2 truss elements
Table 3.7: Material Property of Concrete
Density
ρ
(lb-
sec2/in4)
Modulus
of
elasticity
E (ksi)
Poison‟s
Ratio
ν
Allowable
elastic
stress
(ksi)
Allowable
elastic
Strain
(in/in)
Ultimate
Stress
(ksi)
Ultimate
Strain
(in/in)
Modulus
of
rupture
(ksi)
2.25x10-4 4150 0.15 1.89 4.6x10-4 5.44 0.0025 0.508
3.5.1 Response of beam
The response of analyzed beam has been discussed in the present section by focusing
three different stage of loading of under RC beam.
a) Stress elastic and section uncracked: At low stage of loading which has
produced tensile stress at bottom fiber of concrete beam is less than modulus of rupture
of concrete, so that, no tension cracks have developed. For this inspection, transformed
section of concrete beam is required and finds the maximum load at which no crack
66
will be appeared at bottom face of beam. Transformed section of this beam is shown in
Fig. 3.12.
Figure 3.12: RC beam (a) cross section (b) transformed section
Modular ratio,
Moment of Inertia of transformed section, I = 3050.1 in4
From ABAQUS (2012) results, it has been found that upto 5.78 kip load, there is no
flexural crack.
The calculated moment at load 5.78 kip is
(3.1)
So, tensile stress at bottom fiber of concrete is
, which is almost
equal to modulus of rupture of concrete (0.508 ksi). So, theoretically no cracks have
developed at bottom fiber of RC beam. The stress block across the beam remained
elastic and tensile stress at fiber of beam is 0.418 ksi. The stress distribution at
midspan of beam at 5.78 kip is shown in Fig. 3.13.
b) Stress elastic and section cracked: When the tensile stress of the beam has
exceeded the modulus of rupture, cracks from. From ABAQUS (2012) results, it has
been found that at 6.84 kip concentrated load at centre of beam, first flexural cracks
have appeared.
The calculated moment at midspan of the beam for 6.84 kip concentrated load is
67
(3.2)
So, tensile stress at bottom fiber of concrete is
, which is greater
than modulus of rupture of concrete (0.508 k ). So, theoretically cracks have
developed at bottom fiber of RC beam. The stress block across the beam remained
linear. The stress distribution and crack pattern at midspan of the analyzed beam at 6.84
kip concentrated load at centre of beam is shown in Figs. 3.14 and 3.15 respectively.
Here it is also be found that, tensile stress in concrete decrease due to development of
tension crack at bottom of concrete.
Now it is required to find maximum load at which stress across the beam remain linear
after crack formed and for this concrete compressive stress is less than approximately
half of [H. Nilson et. al. (2010)]. At this stage it is assumed that tension cracks have
progressed all the way to the neutral axis. The depth of stress block is “kd”
Where, √ (3.3)
So, depth of stress block “kd” is 3.51 in.
The compressive stress equal to half of (5.44 ksi) at top face of beam has been
produced by the moment “ ”. So,
(3.4)
This moment will be produced when concentrated load at midspan of beam is
(3.5)
The stress and crack pattern at 16.90 kip concentrated load at midspan of beam shows
in Figs. 3.16, 3.17 and 3.18. Stress across the beam remains linear and depth of the
stress block is 3.5 in which is similar to the calculated value of depth of stress block.
c) Loading which produce nominal moment (section become plastic): If the load
increases further, stress across the beam will not remain elastic, it becomes plastic.
Then linear procedure is not applicable for stress calculation.
68
The stress block depth,
so, ⁄ (3.6)
And corresponding nominal moment, is
( ⁄ ) (3.7)
This moment will be produced when concentrated load at midspan of beam is
(3.8)
The stress and crack pattern at 20.44 kip concentrated load at midspan of beam shows
in Figs. 3.19, 3.20 and 3.21
Figure 3.13: The stress distribution in psi at midspan of RC beam at 5.78 kip load
69
Figure 3.14: The stress distribution in psi at midspan of RC beam at 6.84 kip load
(elastic, cracked)
Figure 3.15: Crack pattern of RC beam at 6.84 kip load (elastic, cracked)
70
Figure 3.16: The stress distribution in psi at midspan of RC beam at 16.90kip load
(elastic, cracked)
Figure 3.17: The stress distribution in psi at midspan of RC beam at 16.90kip load
(elastic, cracked)
71
Figure 3.18: Flexural crack pattern of under reinforced beam at 16.90kip load (elastic,
cracked)
Figure 3.19: The stress distribution in psi at midspan of under RC beam at 20.44kip
load (plastic, cracked)
72
Figure 3.20: The stress distribution in psi at midspan of under RC beam at 20.44kip
load (plastic, cracked)
Figure 3.21: Flexural crack pattern of under RC beam at 20.44kip load (plastic,
cracked)
Finally, it is required to observe that the load-deflection diagram of aforesaid under RC
beam in Fig. 3.22. As previous discussion the beam is loaded at midspan and supported
at two ends as simply supported. This allows rotation and translation at two end of the
beam. As loading proceeds, cracks will develop in concrete which visible clearly in the
beam. It has been seen in load deflection diagram that the initial behavior under service
load (6.80 kips) is approximately elastic. The beam has performed as ductile if the load
increased beyond the service load. The reinforcement has yielded before final failure of
the beam by crushing of concrete. This ductile behaviour is the main principle of under
RC beam. The simulated beam has been failed by crushing of concrete at 1 in
deflection at midspan of beam. The failure deflected shape of the analyzed beam is
73
shown in Fig. 3.23. Same analysis has been repeated with a coarser mesh for both
Explicit and Implicit method of analysis. The load-deflections carves are shown in on
the same Fig. 3.23 and both the analysis methods give similar results.
Three different RC beams have also been numerically modeled in the present study.
Among of theses beams, one is over reinforced and other two beams are under
reinforced beam. Again among of these two under reinforced beams, one beam has
only longitudinal reinforcement i.e. this beam is designed as shear critical beam and
other beam has both shear as well as longitudinal reinforcement i.e. this beam acts as
simply under reinforced beam. The responses of these analyzed beams are in well
agreement with the theoretical results of RC beam. The details of these numerical
analyses are shown in Appendix-C.
Figure 3.22: Load-deflection diagram of analyzed RC beam
0
10
20
30
40
50
60
70
0
5
10
15
20
25
0 0.25 0.5 0.75 1
Mn (
anal
ytic
al),
kip-
ft
App
lied
load
, kip
Midspan deflection, in
load vs. deflectionImplicit analysis method (Coarser mesh)Explicit analysis method (Coarser mesh) (analytical)Mn
74
Figure 3.23: Failure deflected shape of analyzed RC beam
3.6 Linear FE Analysis of SDOF System under Dynamic Load
The main purpose of the present section is to verify the modelling technique of SDOF
system under dynamic load through comparing the theoretical result with FE analysis
result.
The essential physical properties of any linearly elastic structural or mechanical system
subjected to an external source of excitation or dynamic loading are its mass, elastic
properties (flexibility or stiffness) and energy loss mechanism or damping. In the
simplest model of a SDOF system, each of these properties is assumed to be
concentrated in a single physical element.
The analyzed SDOF system, shown in Fig. 3.24, is a single mass vertical structure with
one end fixed and other end free for sideway, 12 ft high and 4 in diameter pipe with
0.237 in thickness (I=7.23 in4). The FE model has been created by two noded beam
elements (B31). The mesh consisted of 11 nodes and 10 elements with 14.4 inch in
size. A lamped mass of 13.47 lb-Sec2/in has been assigned at the top node of the
structure, where the rest of the structure has no mass. A fixed-base condition has been
simulated by restraining the nodes corresponding to the base of the structure against
movements and rotation in all directions. In order to facilitate the determination of the
analytical response of the structure, it has idealized as a SDOF system.
75
12’
m= 13.47 lb-Sec2/in
Figure 3.24: Single degree of freedom system (SDOF)
The structure has been subjected to lateral static loads, initial velocity including free
vibrations, dynamic step load and North-south component of ground acceleration of El
Centro Earthquake, Chopra (1995) to determine its analytical and numerical response
under liner elastic conditions. The analyses have been performed by ABAQUS/implicit
and ABAQUS/Modal dynamics (2012) and the results have been compared with the
analytical calculations.
3.6.1 Lateral stiffness of the structure
As mentioned earlier, the structure is idealized as a linear elastic SDOF system, so
material properties are modulus of elasticity „E‟ is assumed as 29000 ksi and poison‟s
ratio „ν‟ is assumed as 0.29. For a vertical structure fixed at the base and subjected to
only lateral displacement but no rotation allowed at the top, the lateral stiffness k is
given as:
, where I and h are the second moment of area and height of
the structure respectively.
In order to verify the calculated stiffness of the structure, a series of lateral static loads
is required to apply at top of the structure and find out the displacement created at same
point. Figure 3.25 shows the load verses displacement at top node of the structure.
76
Figure 3.25: Response of SDOF system under static load
The slope of the curve is the average stiffness of 0.833 , which is equal to the
stiffness calculated by analytical equation. The idealization of the structure as a SDOF
system is thus justified.
3.6.2 Response to free vibration
The equation of motion for the simple system of Fig. 3.24 is most easily formulated by
directly expressing the equilibrium of all forces acting on the mass using d'Alembert's
principle. The differential equation of motion for free vibration of SDOF system is
found to be
(3.9)
Free vibration of a SDOF system is divided into three types depending upon damping
property of the system.
a) Undamped system.
b) Undercritically- damped system.
c) Critically-damped system.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014
App
lied
load
, lb
Lateral diaplacement, in
77
a) Undamped system: For Undamped free vibration, the solution [Clough and
Penzien (2003)] of differential equation of motion of SDOF system is
(3.10)
where and are displacement and velocity at time zero. is the fundamental
natural frequency of the structure.
The analytical dynamic displacement response of the structure has been determined by
assigning an initial velocity of 2 in/sec and initial displacement of zero to the lumped
mass at the top node of structure. The natural fundamental circular frequency could
be calculated based on stiffness , and it is found to be 7.865 rad/sec.
A dynamic analysis using ABAQUS/Implicit (2012) for the structure has been carried
out, which is compared with the exact analytical response in Fig. 3.26. As seen in the
Figure, the two solutions are in very good agreement.
Figure 3.26: Comparison of analytical and numerical Undamped free vibration
response
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 1 2 3 4
Dis
plac
emen
t, in
Time, sec
ABAQUSAnalytical
78
b) Undercritically-damped system: For undercritically-damped free vibration, the
solution [Clough and Penzien (2003)] of differential equation of motion of SDOF
system is
* (
) + (3.11)
where and are displacement and velocity at time zero. is the fundamental
natural frequency of the structure and √
The damping property of the structure is assumed as 2% of critical damping. The
analytical dynamic displacement response of the structure has been determined by
assigning an initial velocity of 2 in/sec and initial displacement of zero to the lumped
mass at the top node of structure. The natural fundamental circular frequency could
be calculated based on stiffness , and it is found to be 7.865 rad/sec.
The value of is found by using and its value is 7.863 rad/sec. Rayleigh
damping factors [Wilson E. L. (2004)] have been used in this analysis to introduce the
damping property ( of critical damping) of the analyzed structure. To find the
Rayleigh damping factors, it is required to find the fundamental natural frequency of
the structure for first two modes. The fundament frequency of the analyzed column is
shown in Table. 3.8.
Table 3.8: Fundamental natural frequency of the column
Mode no. Circular frequency, (rad/Sec.)
1 7.865
2 317.31
So, Rayleigh damping factors for 2% critical damping are:
(3.12)
α = (3.13)
79
A dynamic analysis using ABAQUS/Implicit (2012) for the structure has been carried
out, which is compared with the exact analytical response in Fig. 3.27. As seen in the
Figure, the two solutions are in very good agreement.
Figure 3.27: Comparison of analytical and numerical 2% critically-damped free
vibration response
c) Critically-damped system: For critically-damped free vibration, the solution
[Clough and Penzien (2003)] of differential equation of motion of SDOF system is
[ ] (3.14)
where and are displacement and velocity at time zero. is the fundamental
natural frequency of the structure.
The damping property of the structure is assumed as 100% of critical damping. The
analytical dynamic displacement response of the structure has been determined by
assigning an initial velocity of 2 in/sec and initial displacement of zero to the lumped
mass at the top node of structure. The natural fundamental circular frequency could
be calculated based on stiffness , and it is found to be 7.865 rad/sec.
A dynamic analysis using ABAQUS/Implicit (2012) for the structure has been carried
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0 2 4 6 8 10 12 14
Dis
plac
emen
t, in
Time, Sec
ABAQUSAnalytical
ξ = 2%
80
out, which is compared with the exact analytical response in Fig. 3.28. As seen in the
Figure, the two solutions are in very good agreement.
Figure 3.28: Comparison of analytical and numerical 100% critically-damped free
vibration response
3.6.3 Response to step load
A step load jumps suddenly from zero to certain value (say 1kips) and stays constant at
that value as shown in Fig. 3.29. The equation of motion (Eqn. 3.9) has been solved by
using Duhamel‟s integral [Chopra (1995)] and response of structure to step load has
been found as bellow:
[ (
√ )] (3.15)
where is static displacement. is the fundamental natural frequency of the
structure and √
The damping property of the simple structure as shown in Fig. 3.24 is assumed as 3%
of critical damping. The analytical dynamic displacement response of the structure has
been determined by assigning a step loading of 1kips to the lumped mass at the top
node of structure. The first two natural fundamental circular frequency and of
-0.12
-0.10
-0.08
-0.06
-0.04
-0.02
0.00
0.02
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50
Dis
plac
emen
t, in
Time, Sec.
ABAQUSAnalytical
ξ = 100%
81
the analyzed SDOF system are 7.865 rad/sec and 10034 rad/sec respectively. On the
basis of these fundamental natural frequencies the Rayleigh mass damping factor for
3% critical damping is 0.472. Dynamic analysis of the SDOF system has been carried
out by using ABAQUS/Modal dynamics (2012) and compared with the exact analytical
response in Fig. 3.30. As seen in the Figure, the two solutions are in very good
agreement.
Figure 3.29: Step loading acting on SDOF System
Figure 3.30: Comparison of analytical and numerical response to step loading response
for 3% critically-damped system
-1.2
-1
-0.8
-0.6
-0.4
-0.2
00 0.5 1 1.5 2 2.5 3 3.5
P(t),
kip
Time, Sec
-2.5
-2
-1.5
-1
-0.5
00 0.5 1 1.5 2 2.5 3 3.5
Dis
plac
emen
t, in
Time, Sec
ABAQUS Analytical
82
3.6.4 Validation of dynamic equation of equilibrium
The differential equation of motion for SDOF system subjected to step loading is found
to be:
(3.16)
where p(t) is a step loading.
The analyzed SDOF system, shown in Fig. 3.24, is a single mass vertical structure. This
SDOF system has been subjected to step loading shown in Fig. 3.29. The damping
property of this simple structure is assumed as 3% of critical damping. So the Rayleigh
mass damping factor for 3% critical damping is 0.472. Dynamic analysis of the SDOF
system has been carried out by using ABAQUS/Modal dynamics (2012). The mass,
damping and stiffness parts of equation of motion have been calculated separately
based on FE analysis as shown in Fig. 3.31. The summation of these three parts of force
has been compared with the applied step loading and the two curves are in very good
agreement.
Figure 3.31: Dynamic equilibrium of SDOF system to step load
The analytical dynamic displacement response of SDOF system has also been
determined by assigning the North- South ground acceleration of El Centro earthquake.
The responses of this analyzed SDOF system for different damping property are shown
in Appendix-D
-2500
-2000
-1500
-1000
-500
0
500
1000
1500
0 0.5 1 1.5 2 2.5 3 3.5
Forc
e, k
ips
Time, Sec
"mv (t)" "cv (t)" "kv(t)" mv (t)+cv (t)+kv(t) p(t)
83
3.7 Response of Nonlinear SDOF System to Free Vibration
The main purpose of the present section is to verify the modelling technique of RC
column under dynamic load through comparing the theoretical result with FE analysis
result.
The analyzed system, shown in Fig. 3.32, is a lump mass vertical structure with fixed
base, 12 ft high and 8 in x 8 in RC column. The Column is reinforced by four 16 mm
diameter reinforcement as main reinforcement. The beam also has 10 mm diameter
closed shear reinforcement and spacing is 6in centre to centre throughout the column.
The clear cover from centre of main reinforcement is 1.0 in. The top node of the
structure has assigned a lumped mass of 12.953 lb-Sec2/in.
The concrete and reinforcement of the analyzed column has been modeled by CDP
model and elastic-plastic model respectively. The material properties of concrete and
reinforcement are shown in Tables 3.9 and 3.10 respectively. Detail material properties
with damage parameters as used in Concrete Damage Plasticity model are presented in
Table B.3 of Appendix-B.
Table 3.9: Material property of reinforcement
Density
ρ
(lb-sec2/in4)
Poison‟s Ratio
ν
Young's
Modulus
(ksi)
Yield
Stress
(ksi)
Yield
Strain
x10-3
Ultimate
Strength
(ksi)
Ultimate
Strain
7.3x10-4 0.29 29000 50 1.72 65 0.11
Table 3.10: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable elastic stress
(ksi)
Allowable elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
Modulus
of rupture
(ksi)
2.25x10-4 4150 0.15 1.89 4.6x10-4 5.44 0.0025 0.508
84
Figure 3.32: Dimensional view of RC column
The concrete column has been modeled by eight-node continuum solid elements
(C3D8R) and the reinforcement has been modeled by two nodded truss elements
(T3D2). The embedded technique has been used to constraint the two noded truss
elements (steel reinforcement) into solid elements (concrete column) in order to create
a perfect bonding with no slip action. The concrete column has been divided into 9216
elements with size as 0.69% of height of column. All reinforcement has been divided
by link elements. Figures 3.33(a) and (b) show the concrete column and reinforcement
after divided into Finite elements.
85
(a) (b)
Figure 3.33: FE Model of (a) concrete and (b) reinforcement of column
The analyzed column has shown elastic behavior at low initial velocity (0.5 in/Sec2)
due to material of concrete remain elastic. If the initial velocity increases the
nonlinearity of column will show.
For undercritically-damped free vibration, the solution of differential equation of
motion of SDOF system is as Eqn. 3.11.
The damping property of the structure is assumed as 5% of critical damping and the
fundament frequency of the analyzed column is shown in Table. 3.11.
Table 3.11: Fundamental natural frequency of the column
Mode no. Circular frequency, (rad/Sec.)
1 22.179
2 341.79
86
So, Rayleigh damping factors for 5% critical damping have been calculated by using
Eqns. 3.12 and 3.13 and values are:
α =
The analytical dynamic displacement response of the structure has been determined by
assigning an initial velocity of 0.5 in/sec and initial displacement of zero to the lumped
mass at the top node of structure. The natural fundamental circular frequency is
found to be 22.179 rad/sec. The value of is found by using and its value is
22.151 rad/sec. A dynamic analysis using ABAQUS/Implicit (2012) for the structure
has been carried out, which is compared with the exact analytical response in Fig. 3.34.
As seen in the Figure, the two solutions are in very good agreement because the column
is remaining elastic and no tension cracks have developed in the column.
Figure 3.34: Comparison of analytical and numerical undercritically-damped free
vibration response (Nonlinearity not triggered due to low stress level)
Again the analytical dynamic displacement response of the structure has been
determined by assigning an initial velocity of 25 in/sec and initial displacement of zero
to the lumped mass at the top node of structure. A dynamic analysis using
ABAQUS/Implicit (2012) for the structure has been carried out, which has been
compared with the exact analytical response in Fig. 3.35. As seen in the Figure, the two
solutions are not agreement because the columns material properties become nonlinear
and tension cracks have developed in the column as shown in Fig. 3.36.
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
0 0.5 1 1.5 2
Dis
plac
emen
t, in
Time, Sec.
Initial Velocity 0.5in/sec Linear_Analytical
87
Figure 3.35: Comparison of analytical and numerical undercritically-damped free
vibration response (involving nonlinearity)
Figure 3.36: Tensile crack develop at RC column
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 0.5 1 1.5 2
Dis
plac
emen
t, in
Time, Sec
Initial Velocity 25 in/sec Linear_Analytical
88
3.8 FE Analysis of SDOF Beam under Impact Load
A SDOF system under impact load has been modeled in the present section to verify
the modelling technique of any structure under impact load with respect to theory of
dynamics.
The steel beam has been loaded by impact load using drop-weight system. The beam is
144 in long with 8 in x 8 in cross-section and pin-end supported at two ends. A lumped
mass of 10 lb-Sec2/in has been assigned at centre node of midspan of the beam. The
analyzed beam has been subjected to an impact load, shown in Fig. 3.37, generated by a
large mass low velocity impactor. The impactor has a mass of 3 lb-Sec2/in and
impacted the beams at a velocity of 300 in/sec. The details of the tested beam are given
in Fig. 3.37. Material property of this beam has been defined by Modulus of Elasticity
of 29000 ksi and poison‟s ratio of 0.29. Stiffness of the beam is 335.82 kip . The
dynamic equation of motion for undamped system is
(3.17)
Figure 3.37: Schematic diagram of the beam
Dynamic analysis of the SDOF system has been carried out by using
ABAQUS/Explicit (2012).
Beam
Impactor
Support
89
The impact loading with respect to time of FE analysis at contact surface of impactor
and target beam is shown in Fig. 3.38. This impact force has a pick load of 325.88 kip.
The mass and stiffness parts of equation of motion have been calculated separately
based on FE analysis as shown in Fig. 3.39. The summation of these two parts of force
has been compared with the impact loading and the curves are in very good agreement.
Figure 3.38: Impact load generated between two surfaces of beam and impactor
Figure 3.39: Equilibrium of motion for SDOF beam to impact load
-350
-300
-250
-200
-150
-100
-50
0
50
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Impa
ct lo
ad, k
ips
Time, Sec
-350
-300
-250
-200
-150
-100
-50
0
50
100
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07
Forc
e, k
ips
Time, Sec
"mv (t)" kv(t) mv (t)+kv(t) p(t)
90
3.9 Summary
The present study has been performed to verify the finite element modelling technique
of different RC structural elements i.e. beam, column and slab by most convenient
software ABAQUS (2012). After successful investigation of these FE models, the
results are concluded as follows:
The structural behaviour of RC beam i.e. mode of failure, maximum load
carrying capacity and load-deflection diagram at centre of beam due to applied
static load as observed by FE model shows a good agreement with the test
results found by Saatci (2007). Flexibility of the beam after damage depends on
the tension softening parameters of concrete. Theses parameters have been
selected according to ABAQUS (2012) and post damage behaviour of beam
compares well with test result.
The load-deflection diagram due to a static concentrated load at centre of RC
slab as found by nonlinear FE analysis is in good agreement with the test result
obtained by McNeice (1967).
The nonlinear FE modelling of an under-reinforced beam under different stages
of loading i.e. a) stress elastic and section uncracked, b) stress elastic and
section cracked, and c) loading which produce nominal moment (stress become
plastic) show structural responses which match very closely with the analytic
results.
The elastic responses of linear SDOF system with different damping properties
subjected to an initial velocity as well as a step load at lumped mass point are in
good agreement with analytic result.
For north-south ground acceleration of El Centro earthquake, the elastic
deflection response of a SDOF system is in good agreement with result found
by Chopra (1995) and it is observed that the response depends only on the
natural vibration period of the system and its damping ratio.
The nonlinear FE modelling of an RC column under dynamic loading shows
linear response upto elastic range of column and this response is in good
agreement with linear analytic result.
The impact load generated due to direct collision between an impactor and RC
target structure had been found to be equal to the summation of stiffness, inertia
and damping forces of structure.
91
Finally, from all these analysis it would seem that these FE model techniques of RC
structure by ABAQUS (2012) can be used with confidence in this research work
regarding behaviour of RC structure under impact load.
92
Chapter 4: FE modelling of RC Beam and Slab under Impact
FE modelling of RC Beam and Slab under Impact
4.1 Introduction
In Chapter 3, nonlinear finite element (FE) analysis of different RC member using
software ABAQUS (2012) has been extended to model impact behaviour of reinforced
concrete (RC) beams and slabs. The present chapter describes the effect of large mass,
low velocity impact on RC structural elements i.e. beam and slab. FE analysis has been
performed on RC beam and slab subjected to low velocity large mass impact load to
validate numerical modelling. The results of the analysis will be compared with the
tests results to demonstrate the validation and suitability of the analysis procedure
employed.
4.2 FE Analysis of RC Beam under Impact Load
In the complete setup, the RC beam has been subjected to impact loading using drop-
weight system. The beam is 118 in long with a 4 in x 8 in cross-section and pin-end
supported at 6 in apart from the ends. The reinforced consisted of two 12 mm diameter
high yield steel bars at the bottom and two 6 mm diameter mild bars at the top, 6 mm
diameter mild steel shear bars at 8 in centre with 1in concrete cover to the main steel,
shown in Fig. 4.1. The analyzed beams have been divided into two types. First type of
beam has a 12 mm thick plywood pad placed at interface of beam and impactor. The
second type of beam has been impacted by impactor directly. The mass of impactor is
98 kg and impacted to the beams at an impact velocity of 287.33 in/sec.
Figure 4.1: Detail of beam tested by Chen and May (2009)
93
The beam has pin-end support at two ends, as shown in Fig. 4.2, which will allow
rotation of the beam and provide axial and vertical restraints. The details of the tested
beam are shown in Fig. 4.1.
Figure 4.2: Pin-ended support used by Chen and May, (2009)
4.3 Simulations of Beam with Plywood Pad at Interface of Beam and Impactor
The simulated RC beam has been divided into finite elements by using three
dimensional solid elements. The complete process of modelling for FE analysis of
beam has been divided into steps such as element‟s modelling, parts interaction,
modelling of material properties, application of load and boundary conditions, analysis
method, output requests etc.
4.3.1 Element’s modelling
The concrete portion of the beam has been modeled by eight-noded continuum solid
elements (C3D8R) and the steel reinforcements have been modeled by two-noded truss
elements connected to the nodes of adjacent solid elements of concrete. The plywood
pad between beam and impactor has also been modeled by eight-noded continuum solid
elements (C3D8R). The support has been modeled by steel box of 1 in thick plates with
rotating arrangement at the two ends of the box as shown in Fig. 4.3. Vertical and
horizontal restraints have been applied to the nodes of the beam at the support position
by the support arrangement to stop translation of the beam at any direction. The model
has been created for the entire beam so that effects of non-symmetry because of support
94
conditions has been neglected. The impact load (drop-weight) has been developed by
continuum solid and assigning an initial velocity of 287.33 in/sec.
Figure 4.3: FE Modelling of pin-ended support used by Chan and May (2009)
4.3.2 Parts interaction
For the numerical simulations involving contact, the selection of appropriate contact
algorithm is vital to the stability of the solution, ABAQUS (2012) offers various
contact interface algorithms, as discussed in Section 2.9.4, which have been
incorporated for use in the simulations. For the analysis presented here, surface-to-
surface contact (Explicit) algorithm based on a penalty formulation [Section 2.9.4] has
been used for the interface between the concrete beam and the impactor. Friction
between the two surfaces has been neglected. Acceleration due to gravity is also
included in the analysis. Figure 4.4 shows the schematic diagram of the beam.
4.3.3 Material property
The constitutive property of Concrete Damage Plasticity (CDP) model has already been
discussed in the Section 2.10.1. The material properties of concrete used to model the
beam is summarized in Table 4.1. Detail material properties with damage parameters as
used in Concrete Damage Plasticity model are presented in Table B.1 of Appendix-B.
The constitutive property of high yield and mild reinforcements used to model the
reinforcement as summarized in Tables 4.2 and 4.3.
Rotating arrangement
ux, uy, uz
Fixed
ux, Fixed
95
Table 4.1: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable Elastic stress
(ksi)
Allowable Elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
2.25x10-4 4857 0.19 2.18 4.49x10-4 7.25 0.0012
Figure 4.4: Schematic diagram of tested beam by Chan and May (2009)
Table 4.2: Material Property of high yield reinforcement
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 27.56x103 0.29 44.18 1.73x10-3 84.12 4.46x10-3
Beam
Impactor
Plywood support
Supporting Arrangement
96
Table 4.3: Material Property of mild reinforcement
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 29x103 0.29 50.76 1.75x10-3 65 0.084
The constitutive property of plywood material used to model the reinforcement as
summarized in Table 4.4.
Table 4.4: Basic property of Plywood
Density
ρ
(lb-sec2/in4)
Modulus of elasticity
E (ksi)
Poison‟s Ratio
ν
1.12x10-4 1450 0.15
4.4 Mesh Sensitivity of Beam with Plywood Pad at Interface of Beam and
Impactor
A number of models have been created by ABAQUS (2012) for each series using
different mesh sizes to investigate the sensitivity of mesh discretization. Three different
size of elements denoted as mesh-1, mesh-2 and mesh-3 have been used to discretize
the analyzed RC beam. The number of elements along length, width and depth of beam
for different mesh size are presented in Tables 4.5 and 4.6 for linear and nonlinear
analysis respectively. Figure 4.5 shows the FE model of beam, showing meshing for
RC Beam and reinforcement arrangement.
97
(a)
(b)
Figure 4.5: FE model of beam (a) complete beam mesh (b) reinforcement mesh
To find the effect of nonlinearity on mesh sensitivity of beam, it is required to model
two series of beam for different element size. First series of beam will present the linear
analysis and second series will present the nonlinear analysis of beam.
4.4.1 Sensitivity analysis for linear material properties of beam
Figure 4.6 shows the transient displacement histories obtained at the centre of the beam
from the linear analysis using three different mesh sizes. The model appears to be
insensitive to mesh size, as shown by the no difference in the peak displacement and
periods. The peak displacement is 0.85 in at 8 ms for three different mesh sizes.
98
Figure 4.6: Comparison of transient displacement histories of linear RC beam for
different mesh size
Figure 4.7 shows the impact force histories obtained at the centre of the beam from the
linear analysis using three different mesh sizes. The model appears to be also
insensitive to mesh size at peak impact force, as shown by the no difference in the peak
impact force and periods. The peak impact force is 228.75 kip at 0.25 ms for three
different mesh sizes.
Based on the findings of this investigation, a mesh size denoted as mesh-1 is considered
to be relatively insensitive to mesh discretization for linear analysis of the present RC
beam.
4.4.2 Sensitivity analysis for nonlinear material properties of beam
Figure 4.8 shows the transient displacement histories obtained at the centre of the beam
from the nonlinear analysis using three different mesh sizes. The model appears to be
moderately sensitive to mesh size, as shown by the difference in the peak displacement
and periods. The coarse mesh-1 predicted a peak displacement of 3.48 in at 31.75 ms,
against 4.02 in at 35.25 ms for mesh-3 and 4.16 in at 36.5 ms for mesh-2. The
displacements of analyzed beam for all mesh sizes upto 10ms period of time are almost
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0 10 20 30 40 50 60
Dis
plac
emen
t, in
Time, ms
Mesh-1 Mesh-2 Mesh-3
99
same to each other. After that period of time, the displacement for mesh-2 and mesh-3
are slightly higher than displacement for mesh-1.
Figure 4.7: Comparison of transient impact force histories of linear RC beam for
different mesh size
Table 4.5: Mesh data for linear analysis of beam
Mesh-1 Mesh-2 Mesh-3
Solid elements 3392 6625 30208
Truss elements 696 870 1392
Element
106 nos. elements along length x 8 nos.
elements along depth x 4 nos. elements along width of beam
133 nos. elements along length x 10
nos. elements along depth x 5 nos. elements along width of beam
212 nos. elements along length x 16
nos. elements along depth x 8 nos. elements along width of beam
0
50
100
150
200
250
0 10 20 30 40 50
Impa
ct lo
ad, k
ip
Time, ms
Mesh-1 Mesh-2 Mesh-3
100
Table 4.6: Mesh data for nonlinear analysis of beam
Mesh-1 Mesh-2 Mesh-3
Solid elements 3392 30208 59000
Truss elements 696 1392 1740
Element
106 nos. elements along length x 8 nos.
elements along depth x 4 nos. elements along width of beam
212 nos. elements along length x 16
nos. elements along depth x 8 nos. elements along width of beam
265 nos. elements along length x 20
nos. elements along depth x 10 nos. elements along width of beam
Figure 4.8: Comparison of transient displacement histories of nonlinear RC beam for
different mesh size
Figure 4.9 shows the impact force histories obtained at the centre of the beam from the
nonlinear analysis using three different mesh sizes. The model appears to be highly
sensitive to mesh size at peak impact force, as shown by the difference in the peak
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
00 10 20 30 40 50
Dis
plac
emen
t, in
Time, ms
Mesh-1 Mesh-2 Mesh-3
101
impact force and periods. The coarse mesh-1 predicted a peak Impact force of 96 kip at
0.25 ms, against 76 kip at 0.25 ms for mesh-2 and 52 kip at 0.25 ms for mesh-3.
The variations in the peak displacements and time periods are smaller between mesh-2
and mesh-3 as compared to those with mesh-1. Based on the findings of this
investigation, mesh-3 is considered to be relatively insensitive to mesh discretization.
Figure 4.9: Comparison of transient impact force histories of nonlinear RC beam for
different mesh size
4.5 Validation of FE Analysis Results
The beam has been impacted by a single impact of 98 kg in mass at the centre of the
beam with an initial velocity of 287.33 in/sec. The beam has been modeled by using a
mesh-3 as per mesh sensitivity studies. The FE analysis has been performed for a
period of 50 ms. For comparison with the test, transient impact force histories, and
crack patterns are used.
4.5.1 Transient impact force
The transient impact force history in the test, conducted by Chan and May (2009), was
recorded using a load cell placed between the impactor mass and the impactor head. In
the FE analysis, the impactor force history has been obtained by using the interface
0
20
40
60
80
100
120
0 10 20 30 40 50
Impa
ct lo
ad, k
ips
Time, ms
Mesh-1 Mesh-2 Mesh-3
102
forces generated when the impactor contacted with the beam. These forces are
dependent on the type of contact and the stiffness of the interface. As described in
Section 2.9.4, surface-to-surface contact (Explicit) algorithm based on a penalty
formulation has been used for present model. In this type of contact, master and slave
surfaces are defined using surface segments. The impactor surface is treated as master
surface and the plywood [Table 4.4] placed between beam and impactor is considered
as slave surface.
Figure 4.10 shows a comparison of transient impact forces history. The transient force
history for this beam is shows that peak impact force of 51.55 kip occurs at 0.25 ms and
the test conducted by Chen and May, (2009) shows a peak impact force of 52.60 kip at
1.54 ms, which is reasonable difference for the peak forces. The shape of the impact
force history for FE analysis is in reasonable agreement with the test. However, the
impact force occurs earlier in the analysis than the experiment. This time lag is very
small i.e. 1.3 ms. This difference in time may be due to measuring arrangement of the
test.
Figure 4.10: Comparison of impact force history of numerically simulated beam with
tested response by Chan and May (2009)
-20
-10
0
10
20
30
40
50
60
0 10 20 30 40 50
Impa
ct lo
ad, k
ip
Time, ms
TestFE AnalysisReaction force
103
Figure 4.10 also shows time history for the reaction force obtained from the FE analysis
of beam. The reaction force has been determined on the two supports and found a peak
value 11.92 kip at 7.5 ms. It can be seen in Fig. 4.10 that the reaction force reached its
peak value when the impact force has already passed its peak at 0.25 ms resulting in a
time lag of 7.25 ms.
4.5.2 Crack patterns and damage
The numerical simulation presented here has been performed by employing the CDP
model for concrete, Section 2.10.1. It provides a useful assessment to the extent of
cracking and their locations. The crack patterns and damage obtained from the
experimental test has been compared with the patterns from the simulation. Figure 4.11
presents the crack patterns for the beam from the test and the analysis. As mentioned
earlier, this beam has been impacted by an impactor with plywood placed between the
beam and the striker.
(a)
(b)
(c)
Figure 4.11: Comparison of crack and damage patterns of numerically simulated beam
with test result observed by Chan and May (2009): (a) tested beam, (b)
tension damage and (c) compression damage of analyzed beam.
104
The crack patterns predicted by the analysis shows the development of diagonal cracks,
originating at the impact point and propagating downwards with an angle of
approximately 45 degrees forming a shear-plug. This was also observed in the test
result. Vertical cracks starting from the top of the beam away from the impact zone
have also formed in the analysis.
4.5.3 Correlation between transient load and crack development
In Fig. 4.12, the impact force time history together with a series of images at different
time intervals are presented from the test and analysis on beam, to show the correlation
between the impact force and crack development in the impact zone. Figure 4.12(a)
shows initial diagonal shear cracks forming on beam as the load reached its maximum
at 2 ms. The analysis shown in Fig. 4.12(b) almost exactly reproduces the initial
diagonal cracks at the point of maximum impact load at 0.25 ms. Vertical flexural
cracks start to develop at about 5 ms followed by a short period of separation between
the impactor and the beam at about 10ms as the beam deformed at a faster rate. The
simulation also picks up this phenomenon, the striker almost separate indicated by low
impact load is predicted at about 6ms. Both the beam and the striker continue to move
downwards. From 11.75 ms to 35.65 ms and 12 ms to 40 ms, the beam and the
impactor regained the contact resulting in more cracking in the test and simulation,
respectively. At about 50 ms, the impactor started to move upwards and the beam and
impactor separated againch in the FE analysis. The correlation between the impact load
and crack development in the test and simulation are remarkably similar. Formation of
initial diagonal shear cracks, followed by vertical flexural cracks are picked up
accurately by the analysis.
4.6 Computational Nonlinear Simulations of Beam without Plywood Pad
The simulation technique for RC beam without any pad between impactor and beam
surface is almost similar to the simulation technique for RC beam with plywood pad
[Section 4.3]. The initial i.e. hitting velocity of impactor is also equal to 287.33 in/sec.
Surface-to-surface contact (Explicit) algorithm based on a penalty formulation [Section
2.9.4] has also been used for the interface between the concrete beam and the impactor
in the present FE analysis. Acceleration due to gravity also included in the analysis.
Figure 4.13 shows the schematic diagram of the beam.
105
(a) From test by Chen and May (2009)
(b) From FE analysis
Figure 4.12: Correlation between impact load and crack propagation for beam from
tested by Chan and May (2009) and numerical analysis
0
10
20
30
40
50
60
0 10 20 30 40 50
Impa
ct lo
ad, k
ip
Time, ms
-10
0
10
20
30
40
50
60
0 10 20 30 40 50 60
Impa
ct lo
ad, k
ip
Time, ms
106
Figure 4.13: Schematic diagram of tested beam by Chan and May (2009)
The constitutive properties of beam, reinforcement, impactor, and supports are also
similar to the materials used for modelling of the beam with plywood pad between
beam and impactor, which are presented in the Tables 4.1, 4.2 and 4.3. The mesh
sensitivity analysis for the present FE analysis has also been successfully completed
and the element size denoted as mesh-3 in Table 4.6 has been decided to use in FE
modelling for the present analyzed beam.
4.7 Validation of FE Analysis Results
The beam subjected to a single impact of 98 kg mass impacting at the centre of the
beam with an initial velocity of 287.33 in/sec. For comparison with the test, transient
impact force histories, and crack patterns are used.
4.7.1 Transient impact force
In the present analysis, the impactor force history has been plotted by using the
interface forces generated, with respect to times, between the impactor and target beam.
Figure 4.14 shows a comparison of transient impact forces history between FE analysis
and test result. The transient force history for the analyzed beam is shows that a peak
impact force of 28 kip at 0.25 ms and the test conducted by Chen and May, (2009)
shows a peak impact force of 40.75 kip at 1.34 ms. The difference of peak impact force
between FE analysis and test is 12.75 kip.
Beam
Impactor
Supporting Arrangement
107
Figure 4.14: Comparison of impact force history of numerically simulated beam with
tested response by Chan and May (2009)
Figure 4.14 also shows the time history of reaction force obtained from the FE analysis
of beam. Peak value of the reaction force is 11.95 kip at 7.5 ms. It can be seen in Fig.
4.14 that the reaction force reached its peak value when the impact force has already
passed its peak at 0.25 ms resulting in a time lag of 7.25 ms.
4.7.2 Crack patterns and damage
Figure 4.15 presents the crack patterns for the beam from the test and the analysis. The
crack patterns predicted by the analysis shows the development of diagonal cracks,
originating at the impact point and propagating downwards with an angle of
approximately 45 degrees forming a shear-plug. This was also observed in the test
result. Vertical cracks starting from the top of the beam away from the impact zone
have also formed in the analysis.
-20
-10
0
10
20
30
40
50
0 10 20 30 40 50 60
Impa
ct lo
ad, k
ip
Time, ms
Test FE Analysis Reaction
108
(a)
(b)
(c)
Figure 4.15: Comparison of crack and damage patterns of numerically simulated beam
with test result observed by Chan and May (2009): (a) tested beam, (b)
tension damage and (c) compression damage of analyzed beam.
4.8 FE Analysis of RC Slab under Impact Load
The methodology developed for the three dimensional analysis of RC beam, as
described in the previous sections of this chapter, has been applied to RC slab. The
results obtained from the numerical simulations have been compared with the tests that
were performed by Chen and May (2009).
4.8.1 Description of slabs tested by Chen and May (2009)
A series of experiment studies to investigate the large mass low velocity impact
performance of RC slabs was performed by Chan and May (2009). Here only test result
for slab-2 from that series of test has been analyzed by FE software ABAQUS(2012).
109
In these slab tests, slab denoted as slab-2 was 30 in square in size and 3 inch in
thickness which was tested under drop-weight loads of 98.7 kg with velocity of 256
in/sec. The detail of slab is given in Fig. 4.16. Ultimate crash strength and modulus of
rupture for concrete were 8.7 ksi and 0.59 ksi respectively. The slab was reinforced
with 6 mm diameter high yield steel bar as top and bottom reinforcement. The concrete
cover between the main bars and top or bottom edges of the slab was 0.5 in. The main
reinforcement bars are separated at 2.4 inch intervals. The impact mass was cylindrical
in size with 8 inch in height and 4 inch in diameter. This cylinder contains a mass cell
of 98.7 kg. The impact velocity of mass cell was 255 in/sec i.e. the height of that
falling mass was exactly 7.05 ft and it falls by gravity load to gain the hitting velocity
of 255 in/sec. This slab was supported by steel plates at four corners.
Figure 4.16: Dimension of slab tested by Chen and May (2009)
4.8.2 Experimental result of tested slab
Two same types of slab were tested by Chen and May (2009). Only difference was the
impactor head size. They use two types of impactor with two head shape i.e. flat headed
and hemispherical headed impactor. Table 4.7 is presenting the slab details.
The output from the tests included time histories of impact force. Figure 4.17 shows the
impact force time histories for two slabs. For the slab test, a peak load of 39.12 kip was
recorded in slab-2 at 1.1 ms. The shapes of the transient impact histories for two slabs
are similar.
110
Table 4.7: Details of slab tests
Slab No.
Striker Mass (kg)
Impactor head Impact velocity(in/sec)
Steel ratio
f'c (ksi)
fr (ksi)
Slab-1 98.7 Hemispherical 256 0.6 8.7 0.59
Slab-2 98.7 Flat 256 0.6 8.7 0.59
Figure 4.17: Impact force histories of slabs tested by Chen and May (2009)
Figure 4.18 shows the damage, after the impact, to the top and bottom faces of slabs.
There was a significant amount of penetration of the impactor into the slabs.
4.9 Computational Nonlinear FE Analysis of Slab-2
The simulation technique of FE models of RC slab is similar to the modelling
technique for RC beam presented at Section 4.3.
0
5
10
15
20
25
30
35
40
45
0 2 4 6 8 10 12 14 16 18
Impa
ct lo
ad, k
ip
Time, ms
Slab-1 Slab-2
111
(a) Slab-1
(b) Slab-2
Figure 4.18: Damage at top and bottom faces of slabs tested by Chen and May (2009)
4.9.1 Element’s modelling of RC slab
The concrete portion of the slab and impactor have been modeled by eight-noded
continuum solid elements (C3D8R) and the steel reinforcements have been modeled by
two-noded truss elements connected to the nodes of adjacent solid elements of concrete
by embedded technique of FE software ABAQUS (2012). The slab has been supported
at four corners by eight 3 in x 3 in x 0.5 in steel plates at top and bottom face of slab.
The supporting steel plates have also been modeled by eight-noded continuum solid
elements (C3D8R). The impact load (drop-weight) has been developed by continuum
solid impactor with an initial velocity of 255 in/sec.
Top face Bottom face
Top face Bottom face
112
4.9.2 Parts interaction
For the present analysis of slab, surface-to-surface contact (Explicit) algorithm based
on a penalty formulation is used for the interface between the concrete slab and the
impactor. The interaction between support plate and slab has also been modeled by
surface-to-surface contact (Explicit) contact method. Friction between the two surfaces
is neglected. Acceleration due to gravity also included in the analysis. Figure 4.19
shows the schematic diagram of the slab.
Figure 4.19: Schematic diagram of tested Slab-2 by Chan and May (2009)
4.9.3 Material property
Tables 4.8 and 4.9 shows the material properties for concrete and reinforcement used to
model the RC slab. Detail material properties with damage parameters as used in
Concrete Damage Plasticity model are presented in Table B.4 of Appendix-B.
Table 4.8: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable Elastic stress
(ksi)
Allowable Elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
2.25x10-4 4857 0.15 4.12 8.48x10-4 8.63 0.002
Slab
Impactor
Top Support Plate
Bottom Support Plate
113
Table 4.9: Material Property of high yield reinforcement
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 27.56x103 0.29 44.18 1.73x10-3 84.12 4.46x10-3
4.10 Mesh Sensitivity Analysis of Slab
In order to study the effect of element mesh sizes on the analysis, three mesh sizes have
been used to model the slabs. Table 4.10, shows the details of the mesh data for slab. In
a typical model of 30 in x 30 in x 3 in square slab, there are 5274 solid elements of
concrete and 1872 truss elements of reinforcement for mesh size denoted as mesh-1.
For mesh-2, the number of solid elements and truss elements are 12500 and 2496
respectively. Similarly for mesh-3, the number of solid elements and truss elements are
42187 and 3744 respectively. Figure 4.20 shows the FE model of slab showing
meshing for complete arrangement of slab and reinforcement arrangement.
Table 4.10: Mesh data for slab
Mesh-1 Mesh-2 Mesh-3
Solid elements 5274 12500 42187
Truss elements
1872 2496 3744
Element
38 nos. elements along length x 4 nos.
elements along depth of slab
50 nos. elements along length x 5 nos.
elements along depth of slab
75 nos. elements along length x 8 nos.
elements along depth of slab
114
(a)
(b)
Figure 4.20: FE model of slab-2 (a) complete slab with impactor and support mesh (b)
reinforcement mesh
115
To find the effect of nonlinearity on mesh sensitivity of beam, it is required to model
two series of slabs for different element size. First series of slabs will present the linear
analysis and second series will present the nonlinear analysis of beam. The analyses
have been carried out by using the refined mesh sizes and the results have compared for
any variation in the impact force and displacement histories.
4.10.1 Sensitivity analysis for linear material properties of slab
Figure 4.21 presents the transient displacement histories obtained at the centre of the
slab from the linear analysis using three different mesh sizes. The model appears to be
insensitive to mesh size, as shown by the very small difference in the peak
displacement and periods. The peak displacement is 0.2 in at 1.5 ms for three different
mesh sizes.
Figure 4.21: Comparison of transient displacement histories of linear RC slab for
different mesh size
Figure 4.22 shows the Impact force histories obtained at the centre of the slab from the
linear analysis using three different mesh sizes. The model appears to be also
insensitive to mesh size at peak impact force, as shown by the no difference in the peak
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0 2 4 6 8 10 12 14 16
Dis
plac
emen
t, in
Time, ms
Mesh-1 Mesh-2 Mesh-3
116
impact force and periods. The peak impact force is 235.63 kip at 1.95 ms for three
different mesh sizes.
Based on the findings of this investigation, mesh-3 is considered to be relatively
insensitive to mesh discretization for linear analysis of this RC slab.
Figure 4.22: Comparison of transient impact force histories of linear RC slab for
different mesh size
4.10.2 Sensitivity analysis for nonlinear material properties of slab
Figure 4.23 shows the transient displacement histories obtained at the centre of the slab
from the nonlinear analysis using three different mesh sizes. The model appears to be
moderately sensitive to mesh size, as shown by the difference in the peak displacement
and periods. The peak displacements are 1.66 in, 1.81 in and 1.94 in for mesh-1,
mesh-2 and mesh-3 respectively. The peak displacements and peak times for the three
mesh sizes are not match to each other. Figure 4.24 shows the comparison of transient
impact force histories for the three mesh size of slab. The three curves are very similar
0
50
100
150
200
250
0 2 4 6 8 10 12 14 16
Impa
ct lo
ad, k
ips
Time, ms
Mesh-1 Mesh-2 Mesh-3
117
in shape with small variation in the peak force. From this sensitivity analysis fine
mesh-3 has been incorporated in the present FE analysis of the slab.
Figure 4.23: Comparison of transient displacement histories of nonlinear RC slab for
different mesh size
Figure 4.24: Comparison of transient impact force histories of nonlinear RC slab for
different mesh size
-2.5
-2
-1.5
-1
-0.5
00 5 10 15 20 25 30
Dis
plac
emen
t, in
Time, ms
Mesh-1 Mesh-2 Mesh-3
0
5
10
15
20
25
30
35
40
45
50
0 5 10 15 20 25 30 35
Impa
ct lo
ad, k
ips
Time, ms
Mesh-1 Mesh-2 Mesh-3
118
4.11 Comparison of FE Analysis Results of Slab with Test Results
The slab has been impacted by a single impact of 98 kg mass at the centre of the slab
with an initial velocity of 256 in/sec. The FE analysis has been performed for 30 ms
period of time and the results have been compared with the test results of Slab-1 and
Slab-2. For comparison with the test, transient impact force histories, and crack patterns
are used.
4.11.1 Transient impact force
The transient impact force history in the test was recorded using a load cell placed
between the impactor mass and the impactor head. In the present analysis, the impactor
force history has been obtained by using the interface forces generated when the
impactor contacted with the beam. These forces are dependent on the type of contact
and the stiffness of the interface. Figure 4.25 shows a comparison of transient impact
forces history. The transient force history for the analyzed slab is shows that a peak
impact force of 47.33 kip at 0.3 ms and the test conducted by Chen and May, (2009)
shows a peak impact force of 38.30 kip at 1.15 ms.
Figure 4.25: Comparison of impact force history of numerically simulated slab with
tested response by Chan and May (2009)
-10
0
10
20
30
40
50
0 5 10 15 20 25 30 35
Impa
ct lo
ad, k
ip
Time, ms
Test (Slab-1)
Test(Slab-2)
FE Analysis (Slab-2)
119
4.11.2 Crack patterns and damage
The crack patterns and damage obtained from the experimental test has been compared
with the patterns from the simulation. Figure 4.26 presents the damage and crack
patterns for the slab from the test and the analysis. On the top face of slab, a similar
zone of damage at the point of impact can be observed as in the test. A small
penetration and damaged at bottom face of slab has been created by impactor as
observed in the test.
Figure 4.27 presents the plot of axial stresses in the reinforcement embedded into
concrete of the RC slab.
Figure 4.26: Comparison of crack and damage patterns of numerically simulated slab
with test result observed by Chan and May (2009)
Top face Bottom face
Top face Bottom face
120
Figure 4.27: Stresses distribution of reinforcement of slab in psi
4.12 Summary
In the current Chapter, impact behaviour of RC beam and slab tested under low
velocity large mass impact has been numerically simulated using damage plasticity
model of FE software ABAQUS (2012). After successful investigation of these impact
behaviour of RC beams and slabs, the results are concluded as follows:
The transient impact force histories obtained from the nonlinear FE analyses are
in reasonable agreement with the impact force histories obtained for tests of RC
beam and slab under impact load [Chan and May (2009)].
The peak reaction force obtained from the nonlinear FE analysis shows a time
lag to the peak transient impact force of actual test.
The crack patterns obtained from the analysis using the CDP model of
ABAQUS (2012) matched reasonably well with the cracks and damage patterns
observed in the tests.
Based on findings of this chapter, it can be concluded that the CDP model provide
consistent results for the impact analysis of RC elements such as beam, slab etc.
subjected to low velocity large mass impact.
121
Chapter 5: Behaviour of RC Structure under Impact Load
Behaviour of RC Structure under Impact Load
5.1 Introduction
The purpose of this chapter is to conduct a thorough parametric study to identify the
effects of impactor‟s mass and velocity on wide range of impactor structure i.e.
reinforced concrete (RC) beam and column. At the beginning of the chapter, a series of
low speed large mass impact on RC beams has been modeled and analyzed by finite
element (FE) package ABAQUS (2012). This analysis will help to establish a basic
relation between static and dynamic capacities of RC beam under impact load at failure
condition. After that another series of RC beam under impact load has been modeled
and analyzed to establish some numerical equations of beam which will help to find the
structural condition of beam after application of impact load i.e. failed or not. After
successful analysis of RC beam, same type of analysis has been conducted for RC
column which will help to modify the interaction diagram of RC column under impact
load.
At the latter part of this chapter, some practical structural component as case study such
as RC column with elastic footing foundation under axial impact load and impact load
created by direct hit of bus or any other vehicle on pier of flyover have been studied.
5.2 RC Beam under Impact Load
Generally, the damage forms of RC beam by impact loads are penetration, spalling,
scabbing, perforation, flexural failure, shear failure etc. Among of these failures some
are local damage and some are global damage. Details of failure mechanism of RC
structure have been presented in Sections 2.2 and 2.3. In the present study global failure
mechanism i.e. flexural failure of beam has been considered. So selection of beam and
impactor property (mass and velocity) have been done in such a way that only global
damage will be occurred, but in practical field all type of failure, local and/or global
damage, may occur at beam under impact load. Basically high velocity small mass
impactor may create local damage at RC structure during impact which has not been
considered in the current work.
Two series of RC beams have been modeled and analyzed in the present study. The
property of beam and impactor of first series of beam are similar to the experiments on
122
RC beams were performed by Tachibana et al. (2010). The dimension, material
property of beams and mass, velocity, material property of impactors were selected by
Tachibana et al. (2010) in such a way that all beam are failed under selected impact
load. This first series of beam will help to define the relationship between static and
dynamic ultimate capacity of RC beam under direct low velocity large mass impact
load. Here it is very important that all beam are failed by flexural damage, no shear
failure will be occurred because shear capacity of these beams are much higher than the
flexural capacity.
Second series of beams have been selected randomly in this study. This beam series has
been experienced different magnitude of impact load by impactor but all beam are not
failed. However, all beams of second series have some permanent deflection but
magnitude of permanent deflection is very low than the deflection at failure stage of the
beam. This second series of beam will be helped to develop the numerical equations to
calculate the developed impact load for different impactor impacted on individual RC
beam.
Finally, combination of results from numerical model and analysis of these two series
of beam will be helped to calculate whether any RC beam is failed or not by direct
impact of defined impactor.
5.3 Description of Beam Used in Analysis
As discuss above, the relationship between static and dynamic capacities of RC beam
have been developed by numerical Modelling of the series of beams which are similar
to the beams tested by Tachibana et al. (2010). The main reason to use same series of
beam in the present study is it helps to verify the numerical Modelling of RC beam as
well as fulfill the objective of Modelling. This series of beam has been failed by
corresponding impactor with selected mass and velocity which help to find the relation
between static and dynamic capacity of beam.
5.3.1 Dimension of beams
Several series of impact tests were carried out by Tachibana et al. (2010) using various
RC beams with shear reinforcement which have been used for present study. Details of
the beams and the reinforcement arrangement are shown in Fig. 5.1 and Table 2.1
shows the design values of the beams.
123
Figure 5.1: Details of RC beams tested by Tachibana et al. (2010)
124
Table 5.1: Design values of RC beams
Beam Type
width x Height x Span
(in)
Bottom Bar (mm)
Shear bar (mm)
Ultimate load capacity at midspan (kip)
Corresponding to banding, Pu
Corresponding to Shear, Vu
A1 6x10x40 2#13 6@2in C/C 16.10 92.05
A2 6x10x80 2#13 6@2in C/C 8.05 92.05
A4 6x10x160 2#13 6@2in C/C 4.03 92.05
B 12x6X80 4#13 6@2in C/C 7.70 50.51
C 6x12x80 2#16 6@2in C/C 11.66 92.05
D 6x12x80 2#10 6@2in C/C 4.45 92.05
E 6x16x80 2#13 6@2in C/C 14.00 155.27
F 6x16x80 2#10 6@2in C/C 7.62 155.27
All beams have rectangular sections with the main reinforcement arranged at the top
and bottom sides and a shear reinforcement of 6mm diameter. The beam type A, C and
D have the same sections with width of 6 in and a height of 10 in. For type A1, A2 and
A4, the span length is 40 in, 80 in and 160 in, respectively. For the beam type A, B and
E, the diameter of main reinforcement is 13 mm. Diameter of 16 mm are used for type
C and diameter 10 mm for types D and F.
5.3.2 Material of beams
The design strength of concrete is 3.48 ksi. The yield stress of the reinforcement is
50 ksi for the bending bars and 40 ksi for the stirrups, respectively. The static ultimate
bending capacities Pu of the beam types B, F and A2 are comparable, while bending
capacities of beam types C and E are larger and the one of type D is smaller. The
ultimate shear capacity Vu in all beams is larger than the ratio of capacity
. Namely, the bending failure is preceding the shear failure for static load
in all cases. Material properties of concrete and reinforcement are presented in Tables
5.2, 5.3 and 5.4. Detail material properties with damage parameters as used in Concrete
Damage Plasticity model are presented in Table B.5 of Appendix-B.
125
Table 5.2: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable elastic stress
(ksi)
Allowable elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
2.25x10-4 3360 0.15 1.4 4.2x10-4 3.48 0.002
Table 5.3: Material Property of bending reinforcement
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 29x103 0.29 50 1.72x10-3 50 0.156
Table 5.4: Material Property of shear reinforcement
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 29x103 0.29 40 1.38x10-3 40 0.274
5.3.3 Overview of impactor and respective beam
The RC beams are impacted by steel mass, which is dropped from specific height i.e.
every impactor have a fixed hitting velocity which is directly proportional to drop
height. The impactors used in the numerical models have a flat contact surface with a
length of 6 in, a radius of 3 in and masses of 150 kg, 300 kg or 450 kg. The mass of
126
cylindrical impactor with 6 inch in height and 3 inch in radius is 10.4 kg, so the
additional mass of impactor has been applied to the impactor by mass distribution
application of ABAQUS (2012).
The results of all analyses in the present study have been summarized at Table 5.5. The
velocity and mass of impactor have been varied with beams which ensure that all
beams will be failed under that amount of momentum of impactor.
5.4 Numerical Modelling of Beam
Numerical analysis by FE method has been carried out for the beams presented in Table
5.5. Modelling techniques of all the beams are similar, so detail Modelling technique
for beam no A-2(13) has only been presented in the present section.
The dimension of selected beam A-2(13) has already been presented in Table 5.1. The
impactor with defined mass (300 kg) and velocity (5 m/sec) has impacted at midspan of
the beam. The schematic view of model is shown in Fig. 5.2. The concrete part of this
beam has been modeled by eight noded brick elements (C3D8R). The brick element has
three degree of freedoms at each node. The reinforcement of this beam has been
modeled by two noded truss elements (T3D2). This truss element, T3D2, is capable to
take only tension and compression forces.
The individual parts of the model such as concrete, reinforcement, impactor should be
connected properly to each other. The bending and shear reinforcements are connected
with surrounding concrete at each intersection points of the concrete and the
reinforcement elements by embedded model technique of ABAQUS (2012). Full bond
is always assumed between the reinforcement and the concrete. Surface to surface
(Explicit) contact method has been used to model the contact behaviour between RC
beam and steel impactor surface. The penalty method, as discuses in Section 2.9.4, has
been used for contact algorithm. The impactor could be separated from RC beam after
contact which is helpful to create the actual behaviour of impact.
The constitutive material property of concrete has been modeled by Concrete Damage
Plasticity CDP model of ABAQUS (2012). The CDP model helps to visualize the
damage level of concrete elements as tension and compression damage. Details of CDP
model have already been presented at Section 2.10.1. Other material of RC beam i.e.
reinforcement has been modeled by simple elastic-plastic material of ABAQUS (2012).
127
Table 5.5: Overview of impactors and beams
Model No.
Beam Type
Impactor Mass
m
Impactor Velocity
Vcol
(in/sec)
Kinetic Energy of Impactor
Ecol
(lb-in)
Momentum
Mcol
(lb-sec)
(kg) lb-sec2/in
1 A2 150 0.857 137.76 8131.99 118.06
2 A2 300 1.713 94.46 7642.29 161.81
3 A2 450 2.570 78.72 7962.94 202.31
4 A2 150 0.857 192.86 15938.05 165.28
5 A2 300 1.713 137.76 16254.50 235.98
6 A2 450 2.570 110.21 15607.92 283.24
7 A2 150 0.857 236.16 23898.11 202.39
8 A2 300 1.713 165.31 23405.91 283.18
9 A2 450 2.570 137.76 24386.50 354.04
10 A2 300 1.713 78.72 5307.59 134.85
11 A2 300 1.713 118.08 11942.08 202.27
12 A2 300 1.713 157.44 21230.37 269.69
13 A2 300 1.713 196.8 33172.45 337.12
14 A1 300 1.713 196.8 33172.45 337.12
15 A4 300 1.713 196.8 33172.45 337.12
16 B 300 1.713 196.8 33172.45 337.12
17 C 300 1.713 196.8 33172.45 337.12
18 D 300 1.713 196.8 33172.45 337.12
19 E 300 1.713 196.8 33172.45 337.12
20 F 300 1.713 196.8 33172.45 337.12
The beam has been modeled with full scale to ensure the actual behaviour of impact
test. Support condition of the beam is pin support at two ends of beam span. The initial
128
velocity of the impactor was 5 m/sec (196.8 in/sec) and total mass of the impactor was
300 kg (1.713 lb-sec2/in).
The element size of 1 in (25 mm) for concrete beam, reinforcement and impactor has
been proved most reasonable element size from mesh sensibility analysis. Figures 5.3
and 5.4 are showing concrete beam element mesh and reinforcement elements,
respectively.
Figure 5.2: Schematic view of RC beams with impactor
Impactor
Beam
129
Figure 5.3: Concrete beams with C3D8R brick element mesh
Figure 5.4: Reinforcement represented by T3D2 truss elements
5.5 Result of Beam Analysis
The impact force which has been generated at contact surface between impactor and
beam is divided into three different form of forces i.e. inertia force (ma), damping force
(cv) and stiffness force (kd). As impact force has been generated within very short
duration of time, so damping force is negligible amount in total contact force. The
stiffness force is only responsible part of force which creates damage at RC beam.
Another thing to be noted that total reaction force of the beam at any instant of time is
equal to the stiffness force if any other external force is not applied to the beam.
Figure 5.5 shows the time response of stiffness part of impact force (reaction force).
Some characteristic values resulted from this time-force curve:
Impulse : Total area under time-force curve is defined as impulse i.e. impulse of
impact has been calculated by integration of time-force curve.
∫
Mean Impact force Pm: Impulse, Ip divided by duration of impact force, Td.
130
Mean impact force Pm at failure condition of RC beam is considered as ultimate
dynamic flexural capacity of beam under impact load.
Figure 5.5: Time response of stiffness force (reaction)
From the Fig. 5.5 it has been seen that reaction force (stiffness force) has contained
negative value at just beginning of impact. This phenomenon has been observed during
impact analysis because beam has tried to bounce off to the opposite direction of
impactor.
-5
0
5
10
15
20
25
30
35
0 5 10 15 20 25 30 35
Impa
ct F
orce
, P ,
kip
Time, T, ms
Time Vs Stiffness force
Mean Impact force, Pm
131
Figure 5.6: Comparison of the displacement history at midspan for beam A-2(13) with
test result conducted by Tachibana (2012).
Figure 5.6 shows the time response of the displacement at the midspan for the beam
A-2(13) impacted by a mass of impactor 300 kg (1.71 lb-sec2/in) and a velocity of
5 m/sec (196.8 in/sec). Large plastic strains are observed in the main reinforcement that
resulted in large permanent displacement of 2.28 in and considered the beam is failed.
The same RC beam was tested by Tachibana et al. (2010) and observed displacement
was 2.30 in which matched reasonably well with ultimate deflection of numerically
analyzed beam.
The numerical modellings and analyses have been conducted for all other beams
presented in Table 5.5. Table 5.6 summaries the maximum impact force, , impulse,
, duration of impact force, , mean impact, , maximum displacement, . The
comparison of deflection of numerically analyzed beams with deflection observed
during test carried by Tachibana et al. (2010) is also shown in Table 5.6.
0
0.5
1
1.5
2
2.5
0 10 20 30 40 50
Dis
plac
emen
t, in
Time, T, ms
FE Analysis
Max. Deflection observed byTachibana (2010)
132
Table 5.6: Numerical result of all analyzed beams
Model No.
Beam Type
Maximum impact force
(kip)
Impulse
(lb-sec)
Duration of
impact force
(ms)
Mean impact force
(kip)
Maximum displacement
(in)
Maximum displacement observed by Tachibana
(2010)
(in)
1 A2 45.44 169.27 15.00 11.28 0.50 0.54
2 A2 41.46 214.21 20.00 10.71 0.53 1.00
3 A2 24.98 252.53 23.00 10.98 0.58 1.46
4 A2 29.89 211.38 19.50 10.84 1.00 0.64
5 A2 38.51 282.74 25.50 11.09 1.12 1.24
6 A2 42.13 318.17 30.00 10.61 1.07 1.72
7 A2 28.02 240.05 21.50 11.17 1.56 0.70
8 A2 30.72 320.93 29.00 11.07 1.64 1.31
9 A2 31.61 383.82 34.50 11.13 1.66 1.91
10 A2 39.54 190.09 17.50 10.86 0.38 0.50
11 A2 44.75 253.97 23.00 11.04 0.82 1.06
12 A2 30.91 310.15 27.50 11.28 1.47 1.63
13 A2 39.51 352.93 32.00 11.03 2.28 2.30
14 A1 96.88 352.92 14.00 25.21 0.83 0.95
15 A4 18.72 318.82 66.00 4.83 4.23 4.52
16 B 38.47 368.60 35.00 10.53 2.17 3.03
17 C 36.51 370.92 26.00 14.27 1.69 1.67
18 D 23.77 309.35 47.00 6.58 2.90 3.70
19 E 59.09 370.53 21.00 17.64 1.10 1.15
20 F 40.33 527.51 50.00 10.55 1.52 1.73
133
Figure 5.7 shows the impulse, in relation to the ultimate bending strength, . The
impulse are about 370 lb-sec for beam types A-2(13), A-1(14), A-4(15), B(16), C(17),
D(18), E(19) and F(20) with constant momentum of impactors 337.12 lb-sec and do not
vary with type of beams. Therefore, it can be expected that the impulse depends on the
momentum of impactor only at the time of collision. Figure 5.8 shows relation between
impulse, and momentum of impactor and it is found that, impulse is directly
proportional to the momentum of impactor.
The relationship between impact force duration and ultimate bending capacity of
beam types A-2(13), A-1(14), A-4(15), B(16), C(17), D(18), E(19) and F(20) beams is
shown in Figure 5.9. It is observed from this relation that the impact force duration
is decreasing with the static ultimate bending capacity is increasing. It has also revealed
that the impact force duration is proportional to the momentum of impactor at the time
of collision. The impact force duration is proportional to the ratio of the momentum of
the impactor to the ultimate bending capacity of beam i.e. ⁄ as shown in Fig.
5.10.
Figure 5.7: Relationship between static bending capacity and impulse for numerically
analyzed beam number A-2(13), A-1(14), A-4(15), B(16), C(17), D(18),
E(19) and F(20)
0
100
200
300
400
500
600
0 2 4 6 8 10 12 14 16 18
Impu
lse,
Ip,
lb-s
ec
Static bending capacity, Pu, kip
A1 A4 B C D E F
134
Figure 5.8: Relationship between momentum of impactor and impulse observed from
FE analysis of all beams
Figure 5.9: Relationship between static bending capacity and duration for numerically
analyzed beam number A-2(13), A-1(14), A-4(15), B(16), C(17), D(18),
E(19) and F(20)
0
100
200
300
400
500
600
0 50 100 150 200 250 300 350 400
Impu
lse,
Ip,
lb-s
ec
Momentum of impactor, Mcol, lb-sec
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16 18
Impa
ct fo
rce
dura
tion,
Td,
ms
Static bending capacity, Pu, kip
A1 A4 B C D E F
135
Figure 5.10: Relationship between Mcol/Pu and duration of impact force observed
from FE analysis of all beams
Figure 5.11: Relationship between static bending capacity and mean impact force
0
10
20
30
40
50
60
70
0 20 40 60 80 100
Impa
ct fo
rce
dura
tion,
Td,
ms
Mcol/Pu
Pm = 1.37Pu
0
5
10
15
20
25
30
0 5 10 15 20
Mea
n Im
pact
For
ce, P
m, k
ip
Static bending Capacity, Pu, kip
136
Figure 5.11 shows the relation between the mean impact force, Pm and the static
ultimate bending capacity Pu for all analyzed RC beams. This relationship can be
expressed as follows:
From this expression it can be concluded that ultimate capacity of any RC beam under
impact load, stiffness load generated between impactor and beam due to collision, will
be increased by 1.37 times of ultimate static bending capacity of that beam. So any RC
beam is completely failed when mean impact force, Pm, will become more than 1.37
times of ultimate static bending capacity of the beam.
5.6 Evaluation of Damage Level for RC Beam
The selection of RC beams and corresponding impactors (mass and velocity) to find
ultimate capacity under impact load has been done in the Section 5.3 in such a way that
all beams are failed with respect to selected impactors. Now, impactors and beams have
been selected randomly to evaluate the damage level of RC beam. So new series of
beams will be damaged or partially damaged under selected impactor load. The results
for beams which have already been modeled to find the relation between ultimate static
and dynamic capacity of beam will also be used to find damage level for RC beam.
The dimension of new series of beam are similar to the dimension of beam used at
Section 5.3 but only impactors properties have been changed for present series of
models.
Table 5.7 summarizes the conducted impact models and analysis in the present section.
The velocity and mass of impactor have been varied with beams. The newly selected
beams could be failed or partially damaged by impact load.
5.6.1 Result of beam analysis
The maximum impact force , impulse , duration of impact force , mean
impact , maximum displacement for newly modeled twenty one numbers
beams are summarized in Table 5.8. The results in Table 5.6 for previously modeled
beams for find out the ultimate static and dynamic capacity of RC beam have also been
used to find out the numerical equation for impulse as well as duration of impact for
any combination of beams and impactors (mass and velocity). This numerical equation
will be helped to predict the damage level of RC beam for any impact case.
137
Table 5.7: Overview of impactors and beams
Model No.
Beam Type
Impactor Mass
m
Impactor Velocity
Vcol
(in/sec)
Kinetic Energy of Impactor
Ecol
(lb-in)
Momentum
Mcol
(lb-sec)
(kg) lb-sec2/in
1 A2 150 0.857 98.4 4147.61 84.30
2 A2 200 1.142 78.72 3539.30 89.92
3 A2 250 1.428 78.72 4424.12 112.40
4 A2 150 0.857 157.44 10617.89 134.88
5 A2 250 1.428 137.76 13548.87 196.70
6 A2 400 2.285 78.72 7078.60 179.84
7 A2 150 0.857 118.08 5972.56 101.16
8 A2 300 1.713 78.72 5308.95 134.88
9 A2 400 2.285 118.08 15926.84 269.76
10 A2 300 1.713 39.36 1327.24 67.44
11 A2 150 0.857 78.72 2654.47 67.44
12 A2 200 1.142 118.08 7963.42 134.88
13 A2 200 1.142 157.44 14157.19 179.84
14 A2 150 0.857 196.8 16590.46 168.60
15 A1 200 1.142 118.08 7963.42 134.88
16 A4 200 1.142 118.08 7963.42 134.88
17 B 200 1.142 118.08 7963.42 134.88
18 C 200 1.142 118.08 7963.42 134.88
19 D 200 1.142 118.08 7963.42 134.88
20 E 200 1.142 118.08 7963.42 134.88
21 F 200 1.142 118.08 7963.42 134.88
138
Table 5.8: Numerical result of all analyzed beams
Model No.
Beam Type
Maximum impact force
(kip)
Impulse
(lb-sec)
Duration of impact force
(ms)
Mean impact force
(kip)
Maximum displacement
(in)
1 A2 36.82 113.36 14 8.10 0.20
2 A2 36.81 122.20 17 7.19 0.18
3 A2 23.12 134.67 19 7.09 0.22
4 A2 40.69 180.86 18 10.05 0.49
5 A2 32.89 247.14 23 10.75 0.63
6 A2 39.60 235.54 22 10.71 0.35
7 A2 42.45 126.28 17.7 7.13 0.28
8 A2 39.54 305.74 30 10.19 0.27
9 A2 35.21 383.82 34.5 11.13 0.75
10 A2 28.40 106.17 13 8.17 0.10
11 A2 36.61 89.26 15 5.95 0.14
12 A2 46.01 178.81 18.5 9.67 0.37
13 A2 33.81 219.37 23 9.54 0.64
14 A2 26.04 186.98 23.5 7.96 0.74
15 A1 64.02 158.09 9 17.57 0.13
16 A4 15.88 190.86 45 4.24 0.68
17 B 41.70 166.41 24 6.93 0.39
18 C 51.05 181.49 17.5 10.37 0.33
19 D 17.91 152.14 22 6.92 0.48
20 E 78.91 188.03 17 11.06 0.17
21 F 42.73 166.44 16 10.40 0.19
139
Figure 5.12 shows the relation between Impulse, and momentum of impactor, .
This relationship has been developed for results from both series of beam and can be
expressed with the following equation:
Figure 5.12: Relationship between momentum of impactor and impulse
The duration of impact is directly proportional to the momentum of impactor, and
inversely proportional to the ultimate static bending capacity of beam as discussed in
Section 5.5. The relation between the duration of impact and ratio of momentum of
impactor to ultimate static bending capacity of beam is shown in Fig. 5.13.
This relationship has also been developed for results from both series of beam as
similar for developing impulse and momentum relationship. This relationship can be
expressed with the following equation:
Ip = 0.9759Mcol + 47.282
0
50
100
150
200
250
300
350
400
450
0 50 100 150 200 250 300 350 400
Impu
lse,
Ip, l
b-s
Momentum of weight, Mcol, lb-s
140
Figure 5.13: Relationship between Mcol/Pu and duration of impact force
The mean impact force, for any collision between impactor and beam can be
calculated by divided impulse, with duration of Impact force, as per equation 5.2.
Ultimate dynamic capacity of beam i.e. ultimate mean impact force for any beam is
1.37 times more than ultimate static bending capacity of that beam as per equation 5.3.
If the calculated mean impact force is exceed 1.37 times of ultimate static bending
capacity of the beam then this beam will be failed completely. On the other hand, if the
calculated mean impact force is not exceed 1.37 times of ultimate static bending
capacity of the beam then it is consider that the beam is damaged partially. The
rectification or other type of precaution could be taken on damaged beam which depend
upon the level of damage.
5.7 RC Column under Impact Load
The failure patterns of RC column under impact load are similar to the failure pattern of
RC beam i.e. penetration, spalling, scabbing, perforation, flexural failure, shear failure,
bearing failure etc. As similar to the RC beam, only global failure pattern has been
Td = 0.6278Mcol/Pu + 8.5772
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40 50
Impa
ct fo
rce
dura
tion,
Td,
ms
Mcol/Pu
141
considered for RC column in the present study. So selections of column and impactor
properties (mass and velocity) have been done in such a way that only global damages
will occur.
The strength interaction diagram for RC column is defined the failure load and failure
moment for the full range of eccentricities from zero to infinity. The interaction
diagram for RC column is divided into two range of failure, one is compression failure
range and another is tension failure range. The column with zero axial load acts as
simple beam and failed by elongation of flexural reinforcement.
In the present study, a RC column has been loaded laterally by an impactor. The mass
and velocity of the impactor has been selected in such a way that the column will be
failed by global damage. The column has also been loaded at axial direction with
different level of constant load during impact analysis of that column.
Finally the column has been modeled with considering no load along transvers
direction i.e. no moment and the column has been loaded along axial direction only by
impactor.
5.7.1 Dimension and material properties of column
The width and depth of selected RC column are 10 in and 12 in respectively. This
analyzed column is 10 ft in height and reinforced with four numbers 20 mm bars at four
corner of tie bar. The longitudinal reinforcements of this column are tied up with
10 mm tie bars with 10 in spacing. The details of the column and reinforcement
arrangement are shown in Fig. 5.14. One end of the column is fixed supported and
other end is free.
The ultimate strength of concrete and yield strength of main reinforcement bar are
3.48 ksi and 50 ksi respectively. The yield strength of tie bar is 40 ksi. Material
properties of concrete and reinforcement are presented in Tables 5.9, 5.10 and 5.11.
Detail material properties with damage parameters as used in Concrete Damage
Plasticity model are presented in Table B.5 of Appendix-B.
142
Figure 5.14: Details of RC column
Table 5.9: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable elastic stress
(ksi)
Allowable elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
2.25x10-4 3360 0.15 1.4 4.2x10-4 3.48 0.002
143
Table 5.10: Material Property of longitudinal reinforcement
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 29x103 0.29 50 1.72x10-3 50 0.156
Table 5.11: Material Property of tie bar
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 29x103 0.29 40 1.38x10-3 40 0.274
5.7.2 Overview of impactor
The RC column has been impacted by steel mass with fixed collision velocity. In the
present study six numbers of impactors with different masses and velocities have been
used to find out interaction diagram of selected RC column under impact load. Table
5.12 summaries the conducted impact models and analysis of RC column by using six
different types of impactor. The material properties of impactors are similar to the
material properties of longitudinal reinforcement used to model RC column. A
rectangular steel impactor with 12 inch in height and length and 10 inch in width has
been used for all impact models of RC beam. In model no. 6, an impactor with 2500 kg
(14.279 lb-sec2/in) mass and 118.08 in/sec velocity has impacted along axial direction
of column at free end.
144
Table 5.12: Overview of impactors and column
Model No.
Impactor Mass
m
Impactor Velocity
Vcol
(in/sec)
Kinetic Energy of Impactor
Ecol
(lb-in)
Momentum
Mcol
(lb-sec)
Axial load of column
Pa
(kip) (kg) lb-sec2/in
1 400 2.285 393.60 176964.89 899.21 0
2 400 2.285 236.16 63707.36 539.53 44.3
3 600 3.427 157.44 42471.57 539.53 132.9
4 600 3.427 137.76 32517.30 472.09 208.21
5 400 2.285 196.80 44241.22 449.61 310.1
6 2500 14.279 118.08 99542.75 1686.02 Impactor
5.8 Numerical Modelling of Column
Numerical analysis by FE method has been carried out for the column with different
impactors presented in Table 5.12. The RC column has been impacted at mid height by
selected impactor. The schematic view of models is shown in Figs. 5.15 and 5.16. The
concrete part of column and impactor have been modeled by eight noded brick
elements (C3D8R) whereas reinforcement of column has been modeled by two noded
truss elements (T3D2). A 12 in steel plate with same cross sectional dimension of
column has been used at both ends of column for model no. 6. These end plates are
helped to protect the concrete of column from bearing failure. Embedded model
technique of ABAQUS (2012) has been used to create perfect bonding with no slip
between reinforcement and concrete of column. Surface to surface (Explicit) contact
method has been used to model the contact behaviour between RC column and steel
impactor surface.
145
Figure 5.15: Schematic view of RC column with impactor for model no. 1 to 5
Figure 5.16: Schematic view of RC column with impactor for model no. 6
Impactor
Column
Fixed End
Free End
Impactor
Column
Fixed End
End Plate
146
5.9 Result of Column
The maximum impact force , impulse , duration of impact force , mean
impact and support moment M for RC columns for six different models are
summaries in Table 5.13.
Table 5.13: Numerical result of all analyzed models
Model No.
Maximum impact force
(kip)
Impulse
(lb-sec)
Duration of impact force
(ms)
Mean impact force
(kip)
Moment
at column support
(kip-in)
1 24.56 585.89 75 7.81 468.60
2 35.08 314.24 30 10.47 628.20
3 41.06 258.73 17 15.22 913.20
4 50.92 163.03 11.5 14.18 850.80
5 49.50 105.89 10.5 10.08 605.09
6 517.81 1698.89 4.25 399.74 Axial Impact
* “H” is the height of column.
Analyses of model no. 1 to 5 have been conducted to find the failure moment at support
of column by impact of selected impactor at mid height of column. In these models
column have been experienced constant axial load during the analysis but in model no.
6 column has not been experienced any axial load and impactor hit the column at top
free end. It is possible to draw an interaction diagram by failure pairs of moment and
axial load of column which has already been presented in Tables 5.12 and 5.13.
Figure 5.17 shows the compression of strength interaction diagram of analyzed RC
column under static load and strength interaction diagram of same RC column under
impact load by low velocity large mass impactors. In the present study, analysis of
model no. 1 to 5 have been conducted to find out the maximum lateral mean impact
load carrying capacity of column whereas model no. 6 has been used to find out the
maximum axial mean impact load carrying capacity. Table 5.14 shows the comparison
147
of load carrying capacity of analyzed RC column during application of static and
impact load.
From the comparison of interaction diagram of RC column for both static and impact
load, it is observed that during application of impact load, any RC column is capable to
carry about 1.37 times more load than the static lateral load but the column is not
capable to carry full range of axial load for which it is designed. When failure is
governed by tension failure, capacity increases. However, capacity is reduced under
impact when failure is by crushing of concrete before tension yielding.
Table 5.14: Comparison of load carrying capacity of analyzed RC column
Model No.
Static axial load (kip)
Static failure
moment
(kip-in)
Applied axial load
(ms)
Mean impact force
(kip)
Moment
at column support
(kip-in)
Load increasing
factor
1 0 337 0 7.81 468.60 1.39
2 44.59 468.25 44.3 10.47 628.20 1.34
3 132.14 662.00 132.9 15.22 913.20 1.38
4 208.81 597.33 208.21 14.18 850.80 1.42
5 310.04 448.31 310.1 10.08 605.09 1.35
6 437.75 0 - 399.74 Axial Impact 0.91
148
Figure 5.17: Compression of strength interaction diagram of column under static and
impact load.
A flexible foundation, supporting single column, has also been modeled and analyzed
by FE method in the present study. This model helps to visualize the realistic picture of
RC column with foundation. The details of this FE analyzed flexible foundation is
shown in Appendix-E
5.10 Impact Load on Flyover Pier
The pier of flyover is such a structural element that it is always under threat of lateral
impact load produced by vehicle colliding across the flyover. The lateral capacity of
such type of flyover pier is much higher than impact load by vehicle accident but
sometime such type of lateral load causes major damage of flyover piers. So during
design of pier of flyover such accidental load is always needed to be considered.
A series of vehicle (school bus, single unit truck, pickup etc.) crash tests into roadside
barriers were carried out by Texas Transportation Institute from 1980 to 1988 [Neol et
0
50
100
150
200
250
300
350
400
450
500
0 200 400 600 800 1000
Axi
al lo
ad, P
, kip
Moment, M, kip-in
Static load Impact load
149
al. (1981)] to find out the possible maximum impact load. In the present section, the
results of tests carried out by Texas Transportation Institute have been used to check
the performance of a typical flyover pier, located at Dhaka city of Bangladesh. The
mass, velocity, impact angle and impact load produced by a school bus and single unit
truck during collision with concrete barrier are summarized in Table 5.15.
Table 5.15: Impact test conducted by Texas Transportation Institute (1980 to 1988)
Type of Vehicle
Mass
m (kg)
Impact Velocity
V (km/h)
Impact angle
(degrees)
Maximum impact load
(kN)
Reference
School bus 9094 93
(84.73ft/sec.)
15 328.4(73.83kip) Neol et al.
(1981)
Single unit Truck
8172 80.5
(73.34ft/sec)
14 368.77(82.91kip) (parallel to
barrier)
136.28(30.64kip) (perpendicular to
barrier)
Buth et al.
(1990)
The selected typical pier of flyover is a 95 in x 95 in rectangular RC column, reinforced
by 58 numbers 40 mm bars. This pier is fixed by pile cap and axially loaded by
1000 ton load. Schematic view of this pier is shown in Fig. 5.18. The numerical model
of this pier has been developed by ABAQUS (2012) software. Concrete portion and
reinforcement of this RC column have been modeled by continuum solid elements,
C3D8R and truss elements, T3D2 respectively. Embedded technique has been used to
create perfect bonding between concrete and reinforcement. From mesh sensitivity
analysis, element size of 100mm has proved as a best element size to discretize this RC
column model. Material properties of concrete and reinforcement have been used for
the modeled flyover pier is shown in Tables 5.16 and 5.17 respectively. Detail material
properties with damage parameters as used in Concrete Damage Plasticity model are
presented in Table B.6 of Appendix-B.
150
Figure 5.18: Schematic view of analyzed flyover pier
Table 5.16: Material Property of Concrete
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Allowable Elastic stress
(ksi)
Allowable Elastic Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
2.25x10-4 3580 0.15 1.6 4.47x10-4 4 0.0022
Table 5.17: Material Property of reinforcement
Density
ρ
(lb-sec2/in4)
Modulus of
elasticity
E (ksi)
Poison‟s Ratio
ν
Yield stress
(ksi)
Yield Strain
(in/in)
Ultimate Stress
(ksi)
Ultimate
Strain
(in/in)
7.45x10-4 29x103 0.29 60 2.07x10-3 60 0.274
151
The present impact analysis of flyover pier has been conducted for two types of vehicle
crash, one for a standard school bus and another for a single unit truck. The school bus
is 9094 kg in mass and velocity is 84.73 ft/sec. Impact angle of this bus is 15 degrees.
The maximum impact load was produced by this bus during crash test at Texas
Transportation Institute is 328.4 kN (73.83 kips) and for present analysis considering
that maximum impact load will be developed between 25 ms.
The second vehicle has been selected for other test is single unit truck of 8172 kg in
mass and impact velocity of the truck is 73.74 ft/sec. The impact angle at time of
collision is 14 degree. This impact test was also conducted by Texas Transportation
Institute and maximum impact load was founded as 368.77 kN (82.91 kip) along
direction of flyover and 136.28 kN (30.64 kip) along perpendicular to the flyover. This
maximum impact load has been used at present analysis.
The main focus of these two vehicles collision models is to examine the capacity of
pier to withstand these vehicles impact load. Figures 5.19 and 5.20 shows the stress and
deflection contours of flyover pier under bus and truck created impact load,
respectively.
(a) (b)
Figure 5.19: Flyover pier collided by bus (a) Stress contours in psi unit and (b)
deflection contours in inch unit
152
(a) (b)
Figure 5.20: Flyover pier collided by truck (a) Stress contours in psi unit and (b)
deflection contours in inch unit
From these analyses it is observed that the designed pier of flyover is sufficiently
strong to withstand the impact load, produced by collision of selected bus and truck,
without any significant damage.
At some location of Dhaka city, flyover runs across the railway track. So it is also
possible to generate a collision between train and pier of flyover. Now, the main focus
of present analysis is to observe the resisting capacity of flyover pier against impact
load generated by collision with derailed train locomotive.
A typical railway track map and possible direction of derailment of railway locomotive
are shown in Fig. 5.21. Here, the locomotive is 160 Ton in mass and velocity is
40 km/hr along direction of track. The angle of collision with track or face of flyover
pier is 15°.
Figures 5.22(a) and 5.22(b) are showing RC flyover pier and railway locomotive mesh
at just before of collision from top and side view, respectively.
153
Figure 5.21: Typical railway track map and possible direction of derailment
Figures 5.23(a), 5.23(b) and 5.24 shows the stress, deflection and damage contours of
flyover pier under impact load is by collision with derailed locomotive. From the FE
analysis it is observed that the flyover pier is almost damaged during collision.
Finally, it is noted that, the designed flyover pier is capable to resist the impact load
generated by bus and truck collision but same pier is very weak against train generated
impact load.
(a)
(b)
Figure 5.22: RC flyover pier and railway locomotive mesh at just before of collision
(a) top view and (b) side view
154
(a) (b) Figure 5.23: Flyover pier collided by train (a) Stress contours in psi unit and (b)
deflection contours in inch unit
(a) (b)
Figure 5.24: Flyover pier collision with train (a) tension damage contours and
(b) compression damage contours
155
5.12 Summary
In this chapter the impact behaviour of RC beam, column subjected to low velocity
large mass impact loads have been studied. After successful investigation of these
impact behaviour of RC elements, the results are concluded as follows:
The impulse i.e. total area under time force curve of large mass low velocity
impact, only depends on the momentum of impactor at time of collision.
The impact force duration is proportional to the ratio of the momentum of the
impactor to ultimate bending capacity of the target beam.
The mean impact force at failure, ratio of impulse and duration of impact, is
1.37 times higher than the ultimate bending capacity of RC beam.
The relation between impulse, and momentum of the impactor, for any
combination of mass and velocity of impactor collided on RC beam can be
expressed as following numerical equation:
The relation between duration of impact, and ratio of impactor momentum,
and ultimate static bending capacity of beam, for any combination of
mass and velocity of impactor impacted on RC beam can be expressed as
following numerical equation:
If calculated mean impact force from above two numerical equations exceeds
1.37 times of ultimate static bending capacity of beam, then this beam will be
completely failed.
When failure is governed by tension failure, the capacity of RC column is
increased by 1.37 times of static lateral load. However, axial capacity is reduced
under impact by 0.91 times of axial actual capacity, when failure is by crushing
of concrete before tension yielding.
RC column supported by flexible foundation is capable to absorb some impact
load before failure of loaded column.
156
The selected typical flyover pier in the present study is capable of withstanding
the impact load created by bus and truck but same pier is very weak against
train generated impact load.
All results and developed numerical equations in the present study will not be
applicable for any local damage of RC structural elements.
157
Chapter 6: Conclusion and Recommendation
Conclusion and Recommendation
6.1 Introduction
An investigation into the impact behaviour of reinforced concrete (RC) members has
been described in this research work. The study mainly focused on the large mass low
velocity impacts on RC beams and column to study their global response. The
investigations involved numerical modelling using a commercial finite element (FE)
software ABAQUS (2012). The numerical model is capable of predicting the response
of RC member under large mass low velocity impact load for linear as well as
nonlinear stage through development of stresses, damage of concrete and degradation
of strength. A limited parametric study has been carried out to identify the effects of
impactors mass and velocity on RC beam and column. This study identifies the
parameters affecting the failure behaviour of RC beams and columns. It also observes
the level of damage due to different impact load. The findings of this research work
will be helpful to the designer to establish some rules for designing the RC member
under large mass low velocity impact load.
6.2 Findings of Work
The following findings are observed in this research work:
The structural behaviour of RC beam i.e. mode of failure, maximum load
carrying capacity and load-deflection diagram at centre of beam due to applied
static load as observed by FE model shows a good agreement with the test
results found by Saatci (2007). Flexibility of the beam after damage depends on
the tension softening parameters of concrete. Theses parameters have been
selected according to ABAQUS (2012) and post damage behaviour of beam
compares well with test result.
The load-deflection diagram due to a static concentrated load at centre of RC
slab as found by nonlinear FE analysis is in good agreement with the test result
obtained by McNeice (1967).
The nonlinear FE modelling of an under-reinforced beam under different stages
of loading i.e. a) stress elastic and section uncracked, b) stress elastic and
section cracked, and c) loading which produce nominal moment (stress become
158
plastic) show structural responses which match very closely with the analytic
results.
The elastic responses of linear SDOF system with different damping properties
subjected to an initial velocity as well as a step load at lumped mass point are in
good agreement with analytic result.
For north-south ground acceleration of El Centro earthquake, the elastic
deflection response of a SDOF system is in good agreement with result found
by Chopra (1995) and it is observed that the response depends only on the
natural vibration period of the system and its damping ratio.
The nonlinear FE modelling of an RC column under dynamic loading shows
linear response upto elastic range of column and this response is in good
agreement with linear analytic result.
The impact load generated due to direct collision between an impactor and RC
target structure had been found to be equal to the summation of stiffness, inertia
and damping forces of structure.
The transient impact force histories obtained from the nonlinear FE analyses are
in reasonable agreement with the impact force histories obtained for tests of RC
beam and slab under impact load [Chen and May (2009)]. The peak reaction
force obtained from the nonlinear FE analysis shows a time lag to the peak
transient impact force of actual test. This difference in time may be due to
measuring arrangements of the test.
The crack patterns obtained from the analysis using the concrete damage
plasticity model of ABAQUS (2012) matched reasonably well with the cracks
and damage patterns observed in the tests.
The impulse i.e. total area under time force curve of large mass low velocity
impact, only depends on the momentum of impactor at time of collision.
The impact force duration is proportional to the ratio of the momentum of the
impactor to ultimate bending capacity of the target beam. The mean impact
force at failure, ratio of impulse and duration of impact, is 1.37 times higher
than the ultimate bending capacity of RC beam. A similar value was obtained
by Tachibana et al. (2010).
159
The relation between impulse, and momentum of the impactor, for any
combination of mass and velocity of impactor collided on RC beam can be
expressed as following numerical equation:
The relation between duration of impact, and ratio of impactor momentum,
and ultimate static bending capacity of beam, for any combination of
mass and velocity of impactor impacted on RC beam can be expressed as
following numerical equation:
If calculated mean impact force from above two numerical equations exceeds
1.37 times of ultimate static bending capacity of beam, then this beam will be
completely failed.
When failure is governed by tension failure, the capacity of RC column is
increased by 1.37 times of static lateral load. However, axial capacity is reduced
under impact by 0.91 times of axial actual capacity, when failure is by crushing
of concrete before tension yielding.
RC column supported by flexible foundation is capable to absorb some impact
load before failure of loaded column.
The selected typical flyover pier in the present study is capable of withstanding
the impact load created by bus and truck without significant damage but same
pier is severely damaged due to train generated impact load.
6.3 Summary
The following conclusions may be derived from this research work:
Numerical modelling of large mass low velocity impact load on RC member by
using FE software ABAQUS (2012) based on nonlinear FE method has been
done successfully and the numerical results have shown a good correlation with
available experiment.
Transient impact force histories and crack patterns obtained from FE analyses
of RC structural elements i.e. beam and slab under impact load match
reasonably well with the test results but a time lag has been observed between
peak impact forces for FE analysis and test result.
160
If only global damage is under consideration and analyzed RC beam is failed
completely due to large mass low velocity impactor‟s load then impulse of
impact load history will only depend upon the momentum of impactor and the
duration of impact load varies proportionally with the ratio of momentum of
impactor to ultimate bending capacity of beam. The beam will fail completely,
if the mean impact force exceeds 1.37 times of its ultimate bending capacity.
The bending capacity of RC column will be increased by 1.37 times of its actual
capacity, if the failure is governed by tension. But axial capacity will be reduced
by 0.91 times when failure is by crushing of concrete before tension yielding.
6.4 Recommendation for Future Studies
Based on the findings of the present research work, further areas of studies can be
identified to help understand the impact response of RC members. The following
recommendations are made for future studies:
An experimental programme to investigate the actual residual capacity of the
RC member following an impact event can be undertaken. This could be very
helpful, especially for restrengthening and retrofitting of structures subjected to
highly dynamic accidental loading.
The present study is only done to focus the global damage of RC members and
all developed numerical equations are valid for globally damaged RC members.
So a research work considering local as well as global damage of RC members
could be very useful to understand the complete damage/failure pattern of RC
members under impact load.
Small mass high velocity impact load on RC members can also be modeled
numerically for the future research work.
The effect of prestressing on the impact behaviour of beams may become an
important research field because prestressing is very common for long-span
structures like bridges.
A perfect bond has been considered between reinforcement and concrete during
modelling of all RC members in the present study. The perfect bond is not
capable to show the actual failure response of RC member. In future analysis,
the bond-slip models should be incorporated to improve the prediction of failure
response.
161
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Vonk, R. A. (1993), “A micromechanical investigation of softening of concrete loaded
in compression”, Heron Vol. 38 (3), pp. 3-94.
William, M.S. (1994), “Modelling of local impact effects on plain and reinforced
concrete”, ACI Structural Journal, 91, pp. 178-187.
Wilson, E. L. (2004), “Static and Dynamic Analysis of Structures”, Fourth Edition.
Yankelevsky, D.Z. (1997), “Local response of concrete slabs to low velocity missile
impact”, International Journal of Impact Engineering, 19, pp. 331-343.
167
Zineddin, M. and Krauthammer, T. (2007), “Dynamic response and behaviour of
reinforced concrete slabs under impact loading”, International Journal of Impact
Engineering, 34, pp. 1517-1534.
168
Appendix-A
A.1 Concrete Damage Plasticity Model
Under uniaxial tension the stress-strain response follows a linear elastic relationship
until the value of the failure stress, is reached. The failure stress corresponds to the
start of micro-cracking in the concrete material. Beyond the failure stress the formation
of micro-cracks is represented macroscopically with a softening stress-strain response,
which induces strain localization in the concrete structure. Under uniaxial compression
the response is linear until the value of initial yield, In the plastic regime the
response is typically characterized by stress hardening followed by strain softening
beyond the ultimate stress, This representation, although somewhat simplified,
captures the main features of the response of concrete.
It is assumed that the uniaxial stress-strain curves can be converted into stress versus
plastic-strain curves. (This conversion is performed automatically by ABAQUS (2012)
from the user-provided stress versus “inelastic” strain data, as explained below.) Thus,
Where the subscripts t and c refer to tension and compression, respectively; and
, are the equivalent plastic strains,
and are the equivalent plastic strain
rates, is the temperature, and are other predefined field variables.
As Shown in Fig. 2.34, when the concrete specimen is unloaded from any point on the
strain softening branch of the stress-strain curves, the unloading response is weakened:
the elastic stiffness of the material appears to be damaged (or degraded). The
degradation of the elastic stiffness is characterized by two damage variables, and ,
which are assumed to be functions of the plastic strains, temperature, and field
variables:
169
The damage variables can take values from zero, representing the undamaged material,
to one, which represents total loss of strength.
If is the initial (undamaged) elastic stiffness of the material, the stress-strain relations
under uniaxial tension and compression loading are, respectively:
We define the “effective” tensile and compressive cohesion stresses as
The effective cohesion stresses determine the size of the yield (or failure) surface.
Although there are different possibilities to describe the complex, nonlinear material
behaviour of concrete in ABAQUS (2012), suitable and admissible results for the
simulation of the three-dimensional state of stress corresponding to the failure can only
be derived from the elasto-plastic damage model “concrete damaged plasticity”
ABAQUS (2012).
Developed by (Lubliner, 1989) and elaborated by (Lee, 1998), the material model
assumes a non-associated flow rate as well as isotropic damage. For the implementation
of the concrete model, two types of material functions have to be defined. In this
regard, stress-strain relations represent the uniaxial material behaviour under
compressive and tensile loadings, which also includes cyclic un- and reloading. Among
other things, suitable formulations for the stress-strain relations of concrete are given in
(CEB-FIP, 1993; Pölling, 2000; Mark, 2006). Functions, which represent the evolution
of the damage variables under compressive loadings dc and tensile loadings dt are
shown in Eqn. A.9 and Eqn. A.10, respectively.
⁄
170
⁄
According to (CEB-FIP, 1993), the stress-strain relation behaviour of concrete under
uniaxial compressive loading can be divided into three domains. As shown in Fig. A.1,
the first section represents the linear-elastic branch, which can be formulated as a
linear-elastic function of the secant modulus of elasticity Ec :
Figure A.1: Stress-strain relation for uniaxial compressive loading
While according to (CEB-FIP, 1993) the linear branch ends at , the
stress-strain relations are negligible modified and the first section is expanded up to
. This allows the advantage of using the secant modulus as a material
parameter that complies with the standards. Anymore, for verification of experiments
with partial given material parameters, the modulus of elasticity can be taken from code
approximations.
Eqn. A.12 describes the ascending branch of the uniaxial stress-strain relation for a
compression loading up to the peak load at the corresponding strain level .
(
)
(
)
According to this, the modified parameter Eci corresponds to the modulus of elasticity
in Eqn. A.13 (CEB-FIP, 1993; Mark, 2006) and can be calculated from:
171
(
)
Section three in Figure 2.36 represents the post-peak branch and is described by
Eqn. A.14:
(
)
The post-peak behaviour depends on the descent function :
[ (
)]
Basing on the assumption that the constant crushing energy (Pölling, 2000) is a
material property, Eqn. A.16 considers its dependency on the geometry of the tested or
simulated specimen (Vonk, 1993; Van Mier, 1984) to almost eliminate mesh
dependencies of the simulation results:
Herein represents the characteristic length of the simulated or tested specimen. In the
literature a wide differing range of values for the crushing energy can be found. In
this regard, simulations of path controlled uniaxial compression tests are used to
validate an admissible parameter for . The best approximation was found using a
crushing energy of
The description of the stress-strain relation for tensile loading is divided into two
sections. Up to the maximum concrete tension strength, the linear part is calculated
from:
172
Figure A.2: Stress-strain relation for uniaxial tension loading
The descent branch of the stress-strain relation of concrete loaded in uniaxial tension
can be derived from a stress-crack opening relation (Eqn. A.18) according to (Hordijk,
1992), basing on the fictitious crack model of (Hillerborg, 1983)
( (
)
)
The free parameters could be experimentally determined to and
(Hordijk, 1992). The function (Eqn. A.10) regulates the tension damage and
includes the experimentally determined parameter .
A.2 Raleingh Damping Matrix Formation
During formation, the damping matrix is assumed to be proportional to the mass and
stiffness matrices, Wilson (2004), as follows:
Where:
is the mass-propotional damping coefficient; and
is the stiffness-propotional damping coefficient.
Relationship between the modal equations and orthogonality conditions allow this
equation to be rewritten as:
173
Where:
is the critical-damping ratio; and
is the natural frequency .
Here, it can be seen that the critical-damping ratio varies with natural frequency. The
values of and are usually selected, according to engineering judgment, such that the
critical-damping ratio is given at two known frequencies.
If the damping ratios (ξi and ξj ) associated with two specific frequency (ωi and ωj), or
modes, are known, the two Rayleigh damping factors ( and ) can be evaluated by the
solution of pair of simultaneous equations, given mathematically by:
[ ]
⌈⌈⌈⌈
⌉⌉⌉⌉
* +
When damping for both frequencies is set to an equal value, the conditions associated
with the proportionality factors simplify as follows:
174
Appendix-B
Table B.1: Material Property with damage parameter of Concrete
Density, ρ (lb-sec2/in4) 2.25x10-4 Modulus of
Elasticity, E (ksi) 4974
Poison’s Ratio, ν 0.15
Dilation angle, Ψ 38° Eccentricity 0.1
⁄ 1.16 K 0.67
Compression Behavior Compression Damage
Yield stress (ksi) Inelastic strain Damage parameter Inelastic strain
2.17 0.00x100 0 0.00x100
2.93 7.47x10-5 0 7.47x10-5
4.35 9.88x10-5 0 9.88x10-5
5.85 1.54x10-4 0 1.54x10-4
7.25 7.62x10-4 0 7.62x10-4
5.84 2.56x10-3 0.20 2.56x10-3
2.93 5.68x10-3 0.60 5.68x10-3
0.76 1.17x10-2 0.89 1.17x10-2
Tension Behavior Tension Damage
Yield stress (ksi) Cracking strain Damage parameter Cracking strain
0.29 0.00x100 0 0.00x100
0.41 3.33x10-5 0 3.33x10-5
0.27 1.60x10-4 0.41 1.60x10-4
0.13 2.80x10-4 0.70 2.80x10-4
0.03 6.85x10-4 0.92 6.85x10-4
0.01 1.09x10-3 0.98 1.09x10-3
175
Table B.2: Material Property with damage parameter of Concrete
Density, ρ (lb-sec2/in4) 2.25x10-4 Modulus of
Elasticity, E (ksi) 4150
Poison’s Ratio, ν 0.15
Dilation angle, Ψ 38° Eccentricity 0.1
⁄ 1.16 K 0.67
Compression Behavior Compression Damage
Yield stress (ksi) Inelastic strain Damage parameter Inelastic strain
3.00 0.00x10+0 0 0.00x10+0
3.99 3.76x10-4 0 3.76x10-4
4.81 7.52x10-4 0 7.52x10-4
5.32 1.13x10-3 0 1.13x10-3
5.50 1.51x10-3 0 1.51x10-3
5.40 2.01x10-3 0.25 2.01x10-3
5.13 2.51x10-3 0.33 2.51x10-3
4.74 3.01x10-3 0.41 3.01x10-3
3.81 4.01x10-3 0.56 4.01x10-3
2.94 5.01x10-3 0.68 5.01x10-3
1.98 6.51x10-3 0.81 6.51x10-3
1.23 8.51x10-3 0.90 8.51x10-3
Tension Behavior Tension Damage
Yield stress (ksi) Cracking strain Damage parameter Cracking strain
0.46 0.00x10+0 0 0.00x10+0
0.00 1.00x10-3 0.85 1.00x10-3
176
Table B.3: Material Property with damage parameter of Concrete
Density, ρ (lb-sec2/in4) 2.25x10-4 Modulus of
Elasticity, E (ksi) 4150
Poison’s Ratio, ν 0.15
Dilation angle, Ψ 38° Eccentricity 0.1
⁄ 1.16 K 0.67
Compression Behavior Compression Damage
Yield stress (ksi) Inelastic strain Damage parameter Inelastic strain
1.89 0.00x10+0 0 0.00x10+0
2.90 7.00x10-4 0 7.47x10-5
3.48 1.00x10-3 0 9.89x10-5
5.44 2.00x10-3 0 1.54x10-4
3.26 3.40x10-3 0 7.62x10-4
2.32 5.00x10-3 0.20 2.56x10-3
0.60 5.68x10-3
0.89 1.17x10-2
Tension Behavior Tension Damage
Yield stress (ksi) Cracking strain Damage parameter Cracking strain
0.51 0.00x10+0 0 0.00x10+0
0.25 1.50x10-4 0 3.33x10-5
0.12 3.50x10-4 0.41 1.60x10-4
0.04 6.00x10-4 0.70 2.80x10-4
177
Table B.4: Material Property with damage parameter of Concrete
Density, ρ (lb-sec2/in4) 2.25x10-4 Modulus of
Elasticity, E (ksi) 4857
Poison’s Ratio, ν 0.15
Dilation angle, Ψ 38° Eccentricity 0.1
⁄ 1.16 K 0.67
Compression Behavior Compression Damage
Yield stress (ksi) Inelastic strain Damage parameter Inelastic strain
4.12 0.00x10+0 0 0.00x10+0
5.71 4.25x10-4 0 4.25x10-4
7.12 8.50x10-4 0 8.50x10-4
8.18 1.27x10-3 0 1.27x10-3
8.63 1.70x10-3 0 1.70x10-3
7.72 2.20x10-3 0.07 2.20x10-3
5.87 2.70x10-3 0.16 2.70x10-3
4.19 3.20x10-3 0.27 3.20x10-3
2.19 4.20x10-3 0.51 4.20x10-3
1.27 5.20x10-3 0.69 5.20x10-3
0.68 6.70x10-3 0.85 6.70x10-3
0.36 8.70x10-3 0.93 8.70x10-3
Tension Behavior Tension Damage
Yield stress (ksi) Cracking strain Damage parameter Cracking strain
0.51 0.00x10+0 0 0.00x10+0
0.25 1.50x10-4 0 3.33x10-5
0.12 3.50x10-4 0.41 1.60x10-4
0.04 6.00x10-4 0.70 2.80x10-4
178
Table B.5: Material Property with damage parameter of Concrete
Density, ρ (lb-sec2/in4) 2.25x10-4 Modulus of
Elasticity, E (ksi) 3360
Poison’s Ratio, ν 0.15
Dilation angle, Ψ 38° Eccentricity 0.1
⁄ 1.16 K 0.67
Compression Behavior Compression Damage
Yield stress (ksi) Inelastic strain Damage parameter Inelastic strain
1.39 0.00x10+0 0 0.00x10+0
2.35 3.96x10-4 0 3.96x10-4
3.00 7.92x10-4 0 7.92x10-4
3.37 1.19x10-3 0 1.19x10-3
3.48 1.58x10-3 0 1.58x10-3
1.85 3.08x10-3 0.62 3.08x10-3
0.77 4.58x10-3 0.86 4.58x10-3
0.39 6.08x10-3 0.94 6.08x10-3
0.23 7.58x10-3 0.97 7.58x10-3
0.15 9.08x10-3 0.98 9.08x10-3
Tension Behavior Tension Damage
Yield stress (ksi) Cracking strain Damage parameter Cracking strain
0.51 0.00x10+0 0 0.00x10+0
0.25 1.50x10-4 0 3.33x10-5
0.12 3.50x10-4 0.41 1.60x10-4
0.04 6.00x10-4 0.70 2.80x10-4
179
Table B.6: Material Property with damage parameter of Concrete
Density, ρ (lb-sec2/in4) 2.25x10-4 Modulus of
Elasticity, E (ksi) 3580
Poison’s Ratio, ν 0.15
Dilation angle, Ψ 38° Eccentricity 0.1
⁄ 1.16 K 0.67
Compression Behavior Compression Damage
Yield stress (ksi) Inelastic strain Damage parameter Inelastic strain
1.60 0.00x10+0 0 0.00x10+0
2.72 4.38x10-4 0 4.38x10-4
3.46 8.76x10-4 0 8.76x10-4
3.87 1.31x10-3 0 1.31x10-3
4.00 1.75x10-3 0 1.75x10-3
3.94 2.25x10-3 0.13 2.25x10-3
3.76 2.75x10-3 0.17 2.75x10-3
3.49 3.25x10-3 0.22 3.25x10-3
2.85 4.25x10-3 0.33 4.25x10-3
2.23 5.25x10-3 0.45 5.25x10-3
1.53 6.75x10-3 0.61 6.75x10-3
0.96 8.75x10-3 0.77 8.75x10-3
Tension Behavior Tension Damage
Yield stress (ksi) Cracking strain Damage parameter Cracking strain
0.51 0.00x10+0 0 0.00x10+0
0.25 1.50x10-4 0 3.33x10-5
0.12 3.50x10-4 0.41 1.60x10-4
0.04 6.00x10-4 0.70 2.80x10-4
180
Appendix-C
C.1 FE Analysis of Over-Reinforced Concrete Beam under Static Load
The reinforced concrete beam is defined as over-reinforced concrete beam when it
shows almost brittle behavior under load larger than service load. Such type of beam is
failed by sudden crushing failure of concrete before yielding of reinforcement. The
constitutive material property has been used to simulate the over-reinforced concrete
beam is same as used to model the under-reinforced concrete beam, Section 3.5.
The over-reinforced beam will undergo comparatively small deformation before failure.
Adequate shear reinforcement has been used to prevent shear failure of beam before
flexural failure.
The beam, shown in Fig. C.1 is 12 ft 4 in long and cross sectional dimension is 10 in x
15 in. This beam is supported by two steel bars and loaded at center by another steel
bar.
Figure C.1: Dimensional view of over-reinforced concrete beam
The supporting and loading bars are 1 in x 2 in x 10 in. The beam is reinforced by
six no. 29 mm diameter reinforcement bar in two layers as bottom flexural
reinforcement and two no. 10 mm diameter reinforcement as top flexural reinforcement
which provide support for shear reinforcement. The distance between two layers of
reinforcement at bottom of the beam is 2 in. The beam also has two-legged closed tie as
shear reinforcement and spacing of the tie bars is 6 in center to center throughout the
beam. The clear cover of the beam from center of flexure reinforcement is 1.5 in. The
material properties and simulation technique of finite element models of this beam is
same as under-reinforced concrete beam [Section 3.5].
181
Figure C.2 shows the load-deflection diagram of the analyzed over reinforced concrete
beam under static beam. It has been seen in load deflection diagram that the initial
behavior of the beam bellow service load (8.25 kip) is approximately elastic. Beyond
the service load the crashing of concrete has been shown some limited ductility.
However the steel reinforcement has been remained elastic and it has not been
contributed the ductile behavior. This beam has been suddenly failed at ultimate failure
load (73.43 kip).
Figure C.2: Load-deflection diagram of FE analyzed over-reinforced concrete beam
C.2 FE Analysis of Reinforced Concrete Beam for Shear Failure under Static
Load
The failure patterns of reinforced concrete beam are basically divided into types, one is
shear failure and another is flexural failure. If the flexural capacity of a beam is larger
than the shear capacity, the beam is failed by shear failure. Shear capacity of a beam is
provided by concrete of beam itself and additional shear reinforcement. Beams with
and without shear reinforcement have been modeled by FE software ABAQUS (2012),
which will help to observe the actual failure patterns of these beam under static load.
0
10
20
30
40
50
60
70
80
0 0.2 0.4 0.6 0.8 1 1.2 1.4
App
lied
load
, kip
s
Midspan deflection, in
182
C.2.1 Beam without shear reinforcement
The beam has been designed in such a way that flexural capacity of the beam is higher
than the shear capacity.
Figure C.3 shows a 12 ft 4 in long beam with cross sectional dimension is 10 in x 15 in.
The beam is supported by two steel bars and loaded at two points by another two steel
bars.
Figure C.3: Dimensional view of reinforced concrete beam without shear bars
The supporting and loading bars are 1 in x 2 in x 10 in. The beam is reinforced by
two no. 25 mm diameter reinforcement as bottom flexural reinforcement. No shear
reinforcement has been used in the beam. The clear cover from center of flexure
reinforcement is 1.5 in. The constitutive material property has been used to simulate
this RC beam is same as used to model the under-reinforced concrete beam,
Section 3.5.
The nominal moment capacity of the beam due to flexural reinforcement is
105.86 kip-ft. This moment will be produced at middle one-third portion of beam, if
concentrated load at two loading cell become Pn = 35.29 kip.
Hear the shear capacity of the beam is Vc = 20.41 kip and it is provided by the concrete
only. Since, , the beam is failed by shear failure of concrete.
From the FE analysis of this beam, it is observed that shear cracks have been initiated
at applied load of 20.41 kip. If the load increased farther the shear crack will try to
reach the top face of the beam and sudden failure will occur. Figure C.4 shows the load
deflection diagram of this beam
183
Figure C.4: Load-deflection diagram of simulated reinforced concrete beam without
shear bar
C.2.2 Beam with shear reinforcement
The beam has been modeled in such a manner that flexural capacity of the beam is less
than the shear capacity of the beam. So the beam behaves as like as under-reinforced
concrete beam.
The beam, shown in Fig. C.5, is 12 ft 4 in long and cross sectional dimension is 10 in x
15 in. Support condition of this beam is same as beam without shear reinforcement.
The constitutive material property has been used to simulate this RC beam is same as
used to model the under-reinforced concrete beam, Section 3.5. The beam is reinforced
by two no. 25 mm diameter reinforcement as bottom flexural reinforcement and two
no. 10 mm diameter reinforcement as top reinforcement which support the shear
reinforcement. The beam also has two legged closed tie as shear reinforcement and
spacing of the tie bars is 6 in center to center throughout the beam. The clear cover
from center of flexure reinforcement is 1.5 in.
0
5
10
15
20
25
30
35
0 0.2 0.4 0.6 0.8 1 1.2
App
lied
Load
, kip
s
Midspan deflection, in
184
Figure C.5: Dimensional view of reinforced concrete beam with shear bars
The nominal moment capacity of the beam due to flexural reinforcement is
105.86 kip-ft. and corresponding applied load is Pn = 35.29 kip.
Hear the shear capacity of the beam is provided by both shear reinforcement and
concrete itself. Total shear capacity of the beam is Vc +Vs = 52.55 kip. Since the
beam will show as ductile behavior as under-reinforced concrete beam. The complete
picture of failure has been presented by the load deflection diagram of this beam as
shown on Fig. C.6 and it is just like as under-reinforced concrete beam [Section 3.5.1].
Figure C.6: Load-deflection diagram of simulated reinforced concrete beam with shear
bar
0
5
10
15
20
25
30
35
40
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
App
lied
load
, kip
s
Midspan deflection, in
185
Tables C.1 and C.2 show the type of different RC beams with reinforcement
arrangement and comparison between calculated and observed responses of these
beams.
Table C.1: Summary of reinforced concrete beam under static load
Beam No.
Bottom Reinf.
Top Reinf.
Shear Reinf. Remarks
Beam-1 2#20mm 2#10mm 10mm@6 inC/C Under-reinforced
concrete beam
Beam-2 6#29mm 2#10mm 12mm@6 inC/C Over-reinforced concrete beam
Beam-3 2#25mm - - Under-reinforced
concrete beam
Beam-4 2#25mm 2#10mm 10mm@6 inC/C Under-reinforced
concrete beam
Table C.2: Summary of responses for different type of reinforced concrete beam under
static load
Beam No.
Failure type Designed Mn (kip-ft)
Designed Vn (kip)
Observed Mn (kip-ft)
Observed Vn (kip)
Beam-1 Through yielding of flexural
reinforcement i.e. ductile failure
61.33 - 64.28 -
Beam-2 Through crushing of concrete i.e. brittle failure
226.44 - 220.29 -
Beam-3 Shear failure of beam
105.86 20.41 88.58 29.53
Beam-4 Flexural failure of beam
105.86 52.55 104.97 34.99
186
Appendix-D
D.1 Response to North-South Ground Acceleration of El Centro Earthquake
Fundamental natural frequency of any structure depends upon the mass and stiffness of
the structure. If the stiffness of the structure is remains constant then the fundamental
natural frequency of the structure is only changed by the mass of the structure. For a
given ground motion, the deflection response of a SDOF system depends only on the
natural vibration period of the system and its damping ratio [Chopra (1995)].
The analytical dynamic displacement response of the structure has been determined by
assigning the North- South ground acceleration, shown in Fig. D.1, of El Centro
Earthquake [Chopra (1995)] at the bottom node i.e. base of structure. The natural
vibration period is fixed as 2 Sec. So the mass of the structure could be calculated
based on stiffness , and it is found to be 84.29 lb-Sec2/in. The
damping property of the system is varied as . The dynamic
analyses using ABAQUS/Modal dynamics (2012) for the structure of different
damping property have been carried out, which has been compared with the response
calculated by Chopra (1995) in Figs. D.2, D.3 and D.4. As seen in the Figures, the
maximum displacement at top node i.e. mass point of structure are 9.91 in, -7.46 in and
5.37 in for damping property of 0%, 2% and 5% respectively. These displacement
values are in very good agreement with results found by Chopra (1998).
Figure D.1: North-South component of horizontal ground acceleration of El Centro Earthquake of May 18, 1940
-0.40
-0.20
0.00
0.20
0.40
0 5 10 15 20 25 30 35
Gro
und
Acc
eler
atio
n, g
Time, Sec
187
Figure D.2: Displacement response of SDOF Systems to El Centro Earthquake (T = 2
Sec. and ξ = 0%)
Figure D.3: Displacement response of SDOF Systems to El Centro Earthquake (T = 2
Sec. and ξ = 2%)
Figure D.4: Displacement response of SDOF Systems to El Centro Earthquake (T = 2
Sec. and ξ = 5%)
9.91
-10
-5
0
5
10
0 5 10 15 20 25 30 35
Dis
plac
emen
t, in
Time, Sec.
-7.46 -10
-5
0
5
10
0 5 10 15 20 25 30 35
Dis
plac
emen
t, in
Time, Sec.
5.37
-6
-4
-2
0
2
4
6
0 5 10 15 20 25 30 35
Dis
plac
emen
t, in
Time, Sec.
188
Appendix-E
E.1 RC Column with Elastic Foundation under axial impact load
The design procedure of RC foundation is basically divided into two types, rigid
foundation and flexible foundation, depending upon the distribution of bearing pressure
that act as upward loads on the foundation.
For compressible soil, it is assumed that, the deformation or settlement of soil at a
given location and bearing pressure at that location are proportional to each other.
If the foundation is quite rigid the settlements in all portions of the foundation will be
substantially the same and upward subgrade reaction on foundation will be same i.e.
uniformly distributed.
On the other hand, if the foundation is relatively flexible, settlement of foundation will
no longer uniform i.e. subgrade reaction will not be uniform. In this situation, normally,
subgrade reaction will be higher at beneath of column position and it will be decrease
with increasing distance from column position.
In the present section a flexible foundation, supporting single column, has been
modeled and analyzed by FE method. This model helps to visualize the realistic picture
of RC column with foundation. The selected RC column for the present analysis is
similar to the RC column used in model no. 6 of Section 5.6.
The foundation of selected RC column has been designed for maximum axial load
carrying capacity of that RC column. The cross sectional dimension of selected RC
column is 10 in x 12 in with four numbers 20 mm main reinforcement as shown in Fig.
5.14 and material properties of concrete and reinforcements are presented in Tables 5.9,
5.10 and 5.11. So the axial capacity of this RC column becomes 437.75 kip. On the
basis of this ultimate axial capacity of RC column and 2 ksf bearing capacity of soil,
the size of squire foundation becomes 15ft. The calculated depth for this foundation is
25 in with considering 2.5 in clear cover at bottom of foundation. The foundation has
been reinforced by 20mm reinforcement as flexural bar with 5 in spacing at both
direction of foundation. The detail schematic view of foundation is shown in Fig. E.1.
189
The impact load has been applied at top face of RC column by an impactor of mass and
velocity are 2500 kg and 118.08 in/sec respectively. The top face of RC column has
been protected by a steel end plate of 1in thickness.
The compressible soil property, subgrade modulus of 100 kcf, has been modeled by
springs, having stiffness of 900 kip/ft, at bottom face of foundation. The contributed
area of foundation for each spring is 9 sft. So stiffness of springs has positioned at edge
and corner of foundation will be decrease with decreasing contribution area of
foundation.
Figure E.1: Details of RC foundation
190
The Modelling technique of RC foundation is similar to that of for RC column or beam
as discussed. The concrete part and impactor have been modeled by continuum solid
element C3D8R and for reinforcement, truss element T3D2 has been used. Embedded
technique and surface to surface (Explicit) contact method have been used to model
reinforcement to concrete interaction and impactor to RC column respectively. The
complete model after discretized into Finite elements is shown in Fig. E.2.
Figure E.2: Complete FE model of Foundation
Figure E.3 shows the time response of impact reaction force at bottom of flexible
foundation and also shows the comparison between time responses of impact forces
which have been created by impactors on top free end of flexible foundation supported
Column
Impactor
Foundation
Spring
Support
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RC column and simply fixed ended column. RC column with flexible foundation has
presented the most realistic picture.
The mean impact force for flexible foundation supported RC column and simply fixed
ended RC column are 145 kip and 399.75 kip respectively. The mean impact force for
flexible foundation supported column is less than that of for fixed ended column
because the soil, represented by spring, has been absorbed some impact force by elastic
settlement of foundation. So depending upon the compressibility of soil impact load
and damage level of RC column, impacted by any impactor, will be varied. For hard
soil, impact load will be higher than that of for soft compressible soil.
Figure E.3: Comparison of reaction force histories of fixed ended column with flexible foundation supported column.
Damage patterns of foundation and column are shown in Fig. E.4. Flexural crack has
been initiated at bottom face of foundation by impact load. After that these flexural
cracks have been propagated to the top face of foundation. This cracking pattern of
foundation has agreed the design concept of foundation. The RC column has been
damaged by creating some tension crack throughout the height of column but
compression damage has not been developed by selected impactor‟s load, whereas
-300
-200
-100
0
100
200
300
400
500
600
0 10 20 30 40 50
Rea
ctio
n lo
ad, k
ip
Time, ms
Fixed ended columnFlexible Foundation supported column
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same type of impactor is enough to damage the fixed ended column by compression
damage. So it is proved that, RC column with flexible foundation is capable to
withstand heavier impactor‟s load.
(a)
(b)
Figure E.4: Tension damage pattern at (a) perspective view and (b) bottom face of analyzed foundation