Analysis of the Deformation and Failure of Blast Loaded Unstiffened and Stiffened Plates By N A G A R A J A R U D R A P A T N A B.E., Bangalore University, Bangalore, India, 1987 M.Tech., Indian Institute of Technology, Madras, India, 1990 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE F A C U L T Y OF G R A D U A T E STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1997 © N A G A R A J A R U D R A P A T N A , 1997 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract A numerical investigation into the deformation and failure of clamped unstiffened and stiffened mild steel plates is carried out. Studies indicate that with increasing load intensity, simple plate structures under blast loading exhibit the three general modes of failure: mode I (large inelastic deformation), mode II (tensile tearing and deformation) and mode III (shear rupture). The failure analysis of plate structures subjected to blast loading is still considered a difficult task. These problems are highly nonlinear due to the combined effect of large deformation, material plasticity and high strain-rates. The present work develops a semi-numerical model for failure prediction which is aimed at providing a simple tool for preliminary design/analysis of such structures. Analytical failure models have been incorporated in an existing finite element code which handles the large deformation, elastic-plastic transient behaviour of unstiffened and stiffened plate structures. The finite element formulation employs existing super finite plate and beam elements. An interactive failure model is proposed to predict the tearing and rupture of thin steel plates and stiffened plate structures under blast loading. The model accounts for the membrane and bending strains as well as the transverse shear stress experienced by the structure under the applied load. The interaction between the tensile tearing and shearing mode of failure is considered via an interaction relation between the strain and stress ratios. Two interactive failure criteria are considered, either Linear (LIC) or Quadratic (QIC) based on the way the ratios are added. The bending strain is estimated by assuming ii that a plastic hinge line develops at the boundary, while the membrane strain is calculated using the finite element prediction of the deformed profile. The total strain, composed of membrane and bending components, is then divided by a specified rupture strain for the material, to obtain the strain ratio. Since the finite element formulation is based on Kirchhoff plate bending theory, there is no direct prediction of shear strains (stresses). In order to achieve a continuous estimation of the shear force and stress along the plate boundary, a series of very stiff springs are introduced there. The use of high stiffness values effectively simulates the clamped boundary condition of the problem. The estimated shear stress is then compared to the ultimate shear strength of the material to form a stress ratio. A node release algorithm is developed to simulate the progression of rupture. The analysis is continued in the post-failure phase to account for the deformation which continues during the free flight of the torn plate. The predicted failure modes using the above model for blast loaded plate structures are presented and compared with previously published experimental data. The Quadratic Interaction Criterion predicts consistently better results than the Linear Interaction Criterion as compared to the experimental results. The results clearly indicate the influence of shear on the failure mechanism not only for mode III, but also for mode II. The results confirm the importance of the interaction effects of tensile and bending strain on tearing and shear failure. iii Table of Contents Abstract ii Table of Contents iv List of Tables ix List of Figures xi Nomenclature xix Acknowledgments xxvi 1 Introduction 1 1.1 Background 1 1.2 Purpose and Scope 2 2 Review of Literature 4 2.1 Mode II Failure 6 2.1.1 Classical Approach 6 2.1.1.1 Fracture Mechanics 6 2.1.1.2 Void Coalescence Mechanism 7 2.1.1.3 Damage Mechanics 7 2.1.1.4 Forming-Limit Diagrams 8 2.1.2 Analytical Failure Models 8 2.1.2.1 Strain Based Model 9 IV 2.1.2.2 Stress Based Approach 9 2.1.2.3 Plastic Work Density Approach 10 2.1.3 Computational Failure Models . 10 2.1.3.1 Strain Based Model 11 2.2 Mode III: Transverse Shear Failure 11 2.3 Interactive Failure Criteria for Predicting Failure Modes 12 2.4 Post-Failure Analysis 13 2.5 Experimental Results 14 3 Deformation and Failure Model Formulation 17 3.1 Introduction 17 3.2 Model Philosophy 17 3.3 Deformation Response 19 3.3.1 Equations of Motion 20 3.3.2 Finite Element Formulation 22 3.3.3 Damping Matrix 25 3.3.4 Solution Procedure 26 3.4 Failure Model 28 3.4.1 Introduction .28 3.4.2 Mode II Failure 29 3.4.2.1 Unstiffened Plate 30 3.4.2.2 Stiffened Plate 32 3.4.3 Mode III Failure 33 V 3.4.3.1 Unstiffened Plate 33 3.4.3.2 Stiffened Plate 34 3.4.4 Mode II-III Interaction 35 3.4.5 Post-Failure Analysis 36 3.5 Computer Implementation 37 Square Plate Analysis Results 43 4.1 Introduction . 43 4.2 Experimental Observations 43 4.3 Verification of Spring Model: Response without Failure 45 4.3.1 Static Analysis Results 45 4.3.2 Dynamic Analysis Results 46 4.4 Failure Analysis Results 46 4.4.1 Model 48 4.4.2 Mode II . . . .51 4.4.2.1 Mode II* 52 4.4.2.2 Mode Ha 53 4.4.2.3 Mode lib 53 4.4.3 Mode III 53 4.4.4 Other Failure Parameters 54 4.4.4.1 Strain and Stress Ratios to First Element Failure 54 4.4.4.2 Time to Failure 55 4.4.4.3 Permanent Central Displacement (Deflection-to-Thickness Ratio 55 vi 4.4.4.4 Side Pull-in 57 4.4.4.5 Residual Kinetic Energy 57 4.4.4.6 Centreline Failure Profiles 58 5 Sensitivity Studies 88 5.1 Introduction 88 5.2 Mesh Density 88 5.3 Time Step Size 91 5.4 Plate Thickness 93 5.5 Strain-Rate 95 5.6 Pulse Shape 97 6 Stiffened Plate Analysis Results 115 6.1 Introduction 115 6.2 Experimental Observations 116 6.3 Spring Model Verification: Response without Failure 117 6.3.1 Static Analysis Results 117 6.3.2 Dynamic Analysis Results 118 6.4 Onset of Failure 119 6.4.1 Uniform Load 119 6.4.1.1 Model 120 6.4.1.2 Mode II : 125 6.4.2 Nonuniform Load 128 6.4.2.1 Model 129 vii 6.4.2.2 Mode II 132 6.5 Post-Failure Analysis 134 6.5.1 Mode II* 135 6.5.2 Modella 136 6.5.3 Permanent Central Displacement 137 6.5.4 Side Pull-In and Residual Kinetic Energy 137 6.5.5 Centreline Displacement Profiles 138 7 Summary, Conclusions and Suggestions 176 7.1 Summary 176 7.2 Conclusions 178 7.3 Suggestions for Future Work 180 Bibliography 181 Appendices 190 A Transverse Shear Stress in Plates through Equilibrium Equations 190 B Displacement Functions 193 C Shape Functions 196 D Spring Stiffness Matrix 200 E Global Force Balance Method 202 viii List of Tables 4.1 Displacement and shear force for a clamped plate with different grid sizes 60 4.2 Comparison of results with different spring stiffnesses 60 4.3 Strain distribution of plate along the boundary in mode I failure 61 4.4 Central displacement during post-failure analysis 61 4.5 Failure strain proportions to initial failure a) LIC 62 b) QIC 62 5.1 Comparison of results with grid sizes for a plate in mode II* failure 99 5.2 Comparison of results with grid sizes for a plate in mode Ha failure 99 5.3 Comparison of results with time steps for a plate in mode II* failure 99 5.4 Comparison of results with time steps for a plate in mode Ha failure 100 5.5 Comparison of results with plate thickness in different modes of failure a) Mode II* 100 b) Mode Ila 100 c) Mode lib 101 d) Mode III 101 5.6 Cowper-Symonds material constants for calculating dynamic rupture strain. . . .101 5.7 Comparison of results at the threshold impulse to mode II* failure using dynamic rupture strain 102 6.1 Static analysis results of 2-bay stiffened plate 1 139 6.2 Influence of spring stiffness on the linear elastic response of 2-bay stiffened plate I 139 ix 6.3 Dynamic analysis results of 2-bay stiffened plate II 140 6.4 Central displacement and maximum strain at midpoint on boundary parallel to stiffener of stiffened plate with different stiffener sizes under 5 Ns impulse a) 2 x 2 g r i d 140 b) 3 x 3 grid 140 6.5 Stress ratio, strain ratio, time to first element failure and stiffener centre displacement at the threshold impulse to mode II failure for all stiffener sizes using 2 x 2 grid a) LIC 141 b) QIC 141 6.6 Stress ratio, strain ratio, time to first element failure and stiffener centre displacement at the threshold impulse to mode II failure for all stiffener sizes using 3 x 3 grid 141 6.7 Comparison of maximum strain at midpoint on boundary parallel to stiffener under different load distribution 142 x List of Figures 2.1 Forming-limit diagram 16 3.1 Spring model for plate structures (Quarter plate model) a) Stiffened plate 40 b) Unstiffened plate 40 3.2 The super finite elements a) Plate element 41 b) Beam element in x-direction 41 3.3 Flow chart for the analysis 42 4.1 Experimental arrangement (Ref. Nurick et al., 1996) 63 4.2 Failure modes of an explosively loaded square plate (Ref. Nurick et al., 1996). . 64 4.3 Comparison of shear force along the plate boundary 65 4.4 Transient linear elastic response of plate 65 4.5 Transient nonlinear elastic-plastic response of plate 66 4.6 Finite element model of plate (Quarter plate model) 66 4.7 Comparison of displacement-time history of plate in mode I failure 67 4.8 Displacement-time history of plate in mode I failure using spring model 67 4.9 Comparison of permanent displacement profiles of plate in mode I failure 68 4.10 Transient deformation profiles of plate in mode I failure 68 4.11 Permanent displacement profile of plate in mode I failure a) 3-D Profile 69 b) w-displacement contours 69 xi 4.12 Shear force-time history of plate in mode I failure 70 4.13 Comparison of stress ratio-time history for a plate in mode I failure 70 4.14 Time histories of strain and stress ratios and failure function of a plate in mode I failure a) Linear Interaction Criterion 71 b) Quadratic Interaction Criterion 71 4.15 Time history of central displacement, side pull-in, kinetic energy of plate in mode II* failure 72 4.16 Transient deformation profiles of square plate in mode II* failure a) Displacement profiles upto first element failure 73 b) Complete response of plate 73 4.17 Permanent displacement profile of plate in mode II* failure a) 3-D profile 74 b) w-displacement contours 74 4.18 Time history of central displacement, side pull-in, kinetic energy of plate in mode Ha failure 75 4.19 Post-failure deformation profile in mode Ila failure 75 4.20 Permanent displacement profile of plate in mode Ila failure a) 3-D profile 76 b) w-displacement contours 76 4.21 Time history of central displacement and kinetic energy of plate in mode III failure 77 4.22 Transient motion of plate in mode III failure 78 4.23 Post-failure transient deformation profiles of plate in mode III failure 78 4.24 Permanent displacement profile of plate in mode III failure a) 3-D profile 79 b) w-displacement contours 79 4.25 Stress and strain ratios for first element failure for a plate under explosive load • 80 4.26 Initial and final failure time for a square plate under explosive load a) Linear Interaction Criterion 80 xii b) Quadratic Interaction Criterion 81 4.27a Plot of deflection-to-thickness ratio versus impulse for a square plate under explosive load 81 4.27b Threshold impulse and deflection-to-thickness ratio to different modes of failure for a square plate under explosive load using LIC 82 4.27c Threshold impulse and deflection-to-thickness ratio to different modes of failure for a square plate under explosive load using QIC 82 4.28 Side pull-in of a square plate under explosive load 83 4.29 Plot of side pull-in versus deflection-to-thickness ratio 83 4.30 Residual kinetic energy of a square plate under explosive load 84 4.31 Plot of residual kinetic energy versus deflection-to-thickness ratio 84 4.32 Displacement profile at the time of first element failure of a square plate under explosive load a) Linear Interaction Criterion 85 b) Quadratic Interaction Criterion 85 4.33 Permanent displacement profile of a square plate under explosive load a) Linear Interaction Criterion 86 b) Quadratic Interaction Criterion 86 4.34 Comparison of permanent displacement profile of square plate in mode Ila failure 87 5.1 Temporal variation of central displacement of a plate in mode I failure with grid size 103 5.2 Comparison of time histories of strain ratio, stress ratio and failure function of a plate in mode I failure with grid size a) Strain ratio 103 b) Stress ratio 104 c) Failure function 104 5.3 Comparison of permanent deflection profile of a plate in mode II* failure with grid size 105 5.4 Comparison of permanent deflection profile of a plate in mode Ila failure with xiii grid size 105 5.5 Comparison of central displacement time history of a plate in mode II* failure with time step size 106 5.6 Variation of strain ratio, stress ratio and failure function with time step size a) Strain ratio 106 b) Stress ratio 107 c) Failure function 107 5.7 Comparison of permanent deflection profile of a plate in mode II* failure with time step size 108 5.8 Comparison of time history of central displacement of a plate in mode Ha failure with time step size 108 5.9 Comparison of permanent deflection profile of a plate in mode Ha failure with time step size 109 5.10 Comparison of threshold impulses to failure for a plate with different thicknesses 110 5.11 Comparison of deflection-to-thickness ratio versus impulse of plate with different thicknesses 110 5.12 Comparison of residual kinetic energy of plate with different thicknesses I l l 5.13 Threshold impulse to different modes of failure for plate with different thicknesses using QIC 112 5.14 Variation of strain-rate and yield stress with impulse 113 5.15 Dynamic rupture strain and threshold impulse to failure for a plate under explosive load 113 5.16 Plot of deflection-to-thickness ratio versus impulse for a plate under different load-time characteristics 114 6.1 Failure modes of explosively loaded stiffened square plates 143 6.2 Configuration of 2-bay stiffened plate I 144 6.3 Configuration of 2-bay stiffened plate II 145 xiv 6.4 Comparison of linear elastic response of clamped 2-bay stiffened plate II due to step load 146 6.5 Comparison of nonlinear elastic response of clamped 2-bay stiffened plate II due to step load 146 6.6 Configuration and finite element model of one-way stiffened plate a) Configuration of plate 147 b) Finite element model of quarter plate (2x2 grid) 147 6.7 Comparison of mode I displacement time history of 3 x 2 - mm stiffened plate 148 6.8 Comparison of permanent deflection profile of 3 x 2 - mm stiffened plate in mode I failure 148 6.9 Time history of central displacement and kinetic energy of 3 x 2 - mm stiffened plate in mode I failure 149 6.10 Transient deflection profiles of 3 x 2 - mm stiffened plate in mode I failure. . . . 149 6.11 Comparison of central displacement time history of stiffened plate for different stiffener sizes in mode I failure 150 6.12 Predicted permanent deflection profiles for plate with different stiffeners in mode I failure 150 6.13 3-D mode I deflection profiles of stiffened plate a) 3 x 2 - mm stiffener 151 b) 3 x 4 - mm stiffener 151 c) 3 x 5 - mm stiffener 152 d) 3 x 9 - mm stiffener 152 6.14 Comparison of mode I displacement profile for 3 x 2 - mm stiffened plate with different grid sizes 153 6.15 Comparison of strain distribution along the boundary parallel to stiffener for stiffened plates 153 6.16 Plot of strain ratio, stress ratio and failure function with time at the midpoint on boundary parallel to stiffener 154 6.17 Plot of strain ratio, stress ratio and failure function with time at the stiffener centre 154 6.18 Comparison of failure function and components at the threshold impulse to xv mode II failure under uniformly distributed load - LIC a) Strain ratio 155 b) Stress ratio 155 c) Failure function 156 6.19 Comparison of failure function at the threshold impulse to mode II failure under uniformly distributed load - QIC 156 6.20 Comparison of failure function with impulse for plate with different stiffener sizes a) 3 x 2 - mm stiffener 157 b) 3 x 4 - mm stiffener 157 c) 3 x 5 - mm stiffener 158 d) 3 x 9 - mm stiffener 158 6.21 Plot of central displacement versus impulse for 3 x 2 - mm stiffened plate . . . . 159 6.22 Plot of central displacement versus impulse for 3 x 4 - mm stiffened plate . . . . 159 6.23 Plot of central displacement versus impulse for 3 x 5 - mm stiffened plate . . . . 160 6.24 Plot of central displacement versus impulse for 3 x 9 - mm stiffened plate . . . . 160 6.25 Comparison of central displacement time history of stiffened plates under nonuniform load 161 6.26 Predicted mode I deflection profile of stiffened plates under nonuniformly distributed load 161 6.27 Time history of central displacement of 3 x 2 - mm stiffened plate under different load distribution 162 6.28 Predicted mode I deflection profiles for 3 x 2 - mm stiffened plate under different load distribution 162 6.29 Comparison of stiffener centre displacement for plates in mode I failure for different load distribution 163 6.30 Variation of failure function with stiffener depth under nonuniformly distributed load (Nu-2) 164 6.31 Variation of failure function with stiffener size under nonuniformly distributed load (Nu-3) 164 6.32 Variation of failure function with impulse for stiffened plates under xvi nonuniformly distributed load a) 3 x 2 - mm stiffener 165 b) 3 x 4 - mm stiffener 165 c) 3 x 5 - mm stiffener 166 d) 3 x 9 - mm stiffener 166 6.33 Comparison of stiffener centre displacement versus impulse for 3 x 2 - mm stiffened plate under different load distribution 167 6.34 Comparison of stiffener centre displacement versus impulse for 3 x 4 - mm stiffened plate under different load distribution 167 6.35 Comparison of stiffener centre displacement versus impulse for 3 x 5 - mm stiffened plate under different load distribution 168 6.36 Comparison of stiffener centre displacement versus impulse for 3 x 9 - mm stiffened plate under different load distribution 168 6.37 Time history of central displacement, side pull-in and kinetic energy of a 3 x 2 - mm stiffened plate in mode II* failure 169 6.38 Plot of central displacement, side pull-in and kinetic energy versus time for a 3 x 2 - mm stiffened plate in mode II* failure 169 6.39 3-D profile of 3 x 2 - mm stiffened plate in mode II* failure (two views of deformed plate, Impulse 14.1 Ns) 170 6.40 Temporal variation of a central displacement, side pull-in and kinetic energy of 3 x 2 - mm stiffened plate in mode Ha failure 171 6.41 3-D profile of 3 x 2 - mm stiffened plate in mode Ha failure (two views of deformed plate, Impulse 16Ns) 172 6.42 Plot of central displacement versus impulse for a 3 x 2 - mm stiffened plate using QIC 173 6.43 Plot of side pull-in versus impulse for a 3 x 2 - mm stiffened plate using QIC. .174 6.44 Plot of residual kinetic energy versus impulse for a 3 x 2 - mm stiffened plate using QIC 174 6.45 Deflection profile at the time of first element failure for 3 x 2 - mm stiffened plate under uniform load 175 xvii E . l Plot of deflection-to-thickness ratio versus impulse for both failure models . . . .205 xvin Nomenclature Length of plate Area of plate Linear strain-displacement matrix Plate linear strain-displacement matrix Beam linear strain-displacement matrix Damping matrix Consistent plate element damping matrix Consistent beam element damping matrix Nonlinear strain-displacement matrix Plate nonlinear strain-displacement matrix Beam nonlinear strain-displacement matrix Consistent plate element damping matrix Consistent beam element damping matrix Material constants Displacement vector Virtual displacement vector xix Velocity vector Acceleration vector Spring displacement field vector Plate displacement field vector Virtual plate displacement vector Plate velocity vector Plate acceleration vector Beam displacement field vector Virtual beam displacement vector Beam velocity vector Beam acceleration vector Elastic modulus Input energy Residual kinetic energy Tangent modulus Energy ratio Failure function Plate element nodal force vector X X Beam element nodal force vector Global plate internal force vector Global beam internal force vector Resultant spring force Plate thickness Stiffener depth Impulse Threshold impulse to mode II failure Threshold impulse to mode II* failure Threshold impulse to mode Ila failure Threshold impulse to mode III failure Non-dimensional Impulse Non-dimensional threshold impulse to mode II failure Non-dimensional threshold impulse to mode II* failure Non-dimensional threshold impulse to mode Ila failure Non-dimensional threshold impulse to mode III failure Linear stiffness matrix Spring stiffness/unit length Consistent spring stiffness matrix Global stiffness matrix xxi Tangent stiffness matrix Effective stiffness matrix Global spring stiffness matrix Plastic hinge length Length of tear Length of plate or beam Element length Consistent plate element mass matrix Consistent beam element mass matrix Global mass matrix Global plate mass matrix Global beam mass matrix Plate shape functions Beam shape functions Element load vector Global load vector Effective load vector Applied surface traction Shear force/unit length R Total reaction force S Surface area Time Non-dimensional time Time to failure Time to initial failure Time to complete failure u, v In-plane displacement V Volume Vp Velocity of free-flying plate w Transverse displacement Wext External virtual work Wint Internal virtual work x, y, z Rectangular Cartesian coordinates | Virtual nodal displacement vector j 8pe | Virtual plate element displacement vector ]y8he | Virtual beam element displacement vector {S^ Plate nodal displacement vector {8h j Beam nodal displacement vector ^5pe | Plate element displacement vector X X l l l Beam element displacement vector Central deflection of plate Deflection-to-thickness ratio Central deformation at the time of first element failure Mode I displacement i f failure is ignored Side pull-in Time step size Total strain Membrane strain Bending strain Static rupture strain Dynamic rupture strain Strain ratio Strain rate Strain vector Virtual strain vector Curvature of plastic hinge Viscous damping parameter Poisson's ratio xxiv 0 Hinge rotation 0max Maximum slope of the deflected plate element 0sup Rotation of boundary p Mass density cr0 Static yield stress <Jd Dynamic yield stress ax , ay In-plane bending stress {cr} Stress vector \ap | In-plane stress for plate element |cr*| In-plane stress for beam element T Average shear stress TU1I Static ultimate shear stress T^,T_ULT Dynamic ultimate shear stress T0 Load duration xxy In-plane shear stress Txz , r Transverse shear stress r = \ Tay I Stress ratio a>r Frequency associated with r l h mode of vibration Zr Critical damping ratio X X V Acknowledgments I wish to thank my supervisors, Dr. M.D. Olson and Dr. R. Vaziri, for their invaluable guidance, encouragement and interest with regard to this work. I also thank Dr. D.L. Anderson and Dr. R.O. Foschi for their advice and valuable discussions which encouraged me to solve some difficult issues. I would like thank Dr. G.N. Nurick of the University of Cape Town in South Africa, for providing the experimental data and making some constructive suggestions. I would like express my gratitude to Dr. Anoush Poursartip and all the members of the Composites group for their friendship and support during my stay at UBC. I would like to thank Dr. Jiang, Dr. Fagnan and Ms. Qing Luo for the many useful discussions. I would like to greatly acknowledge the Natural Sciences and Engineering Research Council of Canada and the Defence Research Establishment Suffield, Alberta for providing the financial support in the form of a Research Assistantship from the Department of Civil Engineering at the University of British Columbia. I wish to thank Mrs. Leanne Bernaerdt for proof-reading my thesis. I would like to acknowledge Dr. S. Pradhan, Dr. S. Munshi and all other friends for making my stay in Vancouver a pleasant experience. I am grateful to my wife, Anupama, for her invaluable help and understanding. This thesis would not have been possible without her inspiration. xxvi Chapter 1 Introduction 1.1 Background Failure of engineering structures is not uncommon when they are pushed to the limits of their performance capabilities as is the case during earthquakes, military or accidental explosions. An increased understanding of structural failure is necessary in order to reduce both the damage frequency and severity. Many civil, nuclear and ocean engineering structures are subjected to a variety of dynamic loads: wave and wind loads, pile driving, earthquakes, blast loading due to accidental or military explosions, etc. Blast loading can induce a substantial amount of damage in simple structural elements like beams and plates. Understanding the deformation and failure mechanisms involved in explosive loading situations is important in effectively designing a system to withstand these loads. A designer requires information on the failure characteristics of a structure to accurately assess the safety factors and margins against failure. For example, it is important for many practical applications to have the capability for predicting the damage or permanent deformations of a ductile structural member when subjected to large dynamic loads. 1 Chapter 1. Introduction 2 The description of dynamic material behaviour, especially material failure under high loading rates, is a difficult task. Also, the response and failure of structural elements under dynamic loading are complicated processes that are difficult to analyse. A comprehensive analytical solution for structures subjected to explosive loading is practically impossible. Despite the uncertainties about the short-duration loading response and failure characteristics of materials, there are several situations in which it is necessary to anticipate the extent of damage. Numerical techniques offer a means of obtaining a complete solution to these problems. 1.2 Purpose and Scope It is the objective of this thesis to develop a simplified numerical model which is capable of accounting for the different modes of failure that a structure would experience under explosive loading situations. The main focus of the thesis is to consider structural type failures that originate from failure in the material. The simple failure model which requires much reduced input data makes it possible to carry out design oriented analysis. Currently, a host of commercial finite element software packages are available for nonlinear elastic-plastic transient analysis of plate structures. However, very few of them have the capability to predict material failure. The work presented in this thesis is aimed at providing a numerical tool to predict the tearing and rupture of metallic plate and stiffened plate structures. This will help to take these software programs to the next stage of analysis/design profile. A literature review of various approaches used to model the fracture and rupture of metallic structures are presented in Chapter 2. Also, recently developed analytical and Chapter 1. Introduction 3 numerical failure models are discussed in this chapter. In Chapter 3, the development of the failure model is described and the philosophy behind this simple approach is discussed. The mathematical derivation of the governing equations involved in the failure model are presented in detail. This chapter also highlights in some depth the computer code developed to implement the theoretical procedure, including the post-failure analysis. The numerical analyses carried out to verify the new failure model are presented in Chapters 4, 5 and 6. Chapter 4 deals with the tearing and rupture of thin, square, clamped plates, while Chapter 6 is concerned with failure of stiffened plates. Experimental results are used for the verification of the failure model. Chapter 5 is devoted to sensitivity analysis wherein the influence of different parameters on failure prediction is studied in detail. A summary and conclusion from the current study is presented in Chapter 7 along with some suggestions for further research. Chapter 2 Review of Literature The impact behaviour of structures has been a critical design consideration for many practical problems. Over the last two decades, it has received considerable attention by researchers worldwide. Extensive literature review has been done on the dynamic plastic behaviour of structures (Jones, 1975, 1978, 1981, 1985b, 1989a, 1996, Stronge, 1993 and Bangash, 1993). Comprehensive reviews of the state-of-the-art in response predictions for structures subjected to air-blast loads have been given by Ari-Gur et al (1983) and Olson (1991). Menkes and Opat (1973) were the first to mention the failure modes for a blast loaded clamped beam. They conducted experiments on clamped A l 6061-T6 beams subjected to surface explosive charges. They defined three failure modes with increasing impulses as: Mode I: large inelastic deformation; Mode II: tensile tearing and deformation; Mode III: transverse shear failure. Nurick et al (1996) conducted experiments on mild steel plates and observed the same basic modes of failure in plate structures. They suggested further divisions in mode 4 Chapter 2. Review of Literature 5 II failure regime to account for the influence of geometry of these plates. Thus, for plates with square geometry, they reported the following subgroups in mode II failure: Mode II* - Partial failure; Mode Ha - Complete tearing with increasing midpoint deflection; Mode lib - Complete tearing with decreasing midpoint deflection. In mode I, there is no physical fracture of material, but only a large amount of plastic deformation. Jones (1971, 1989b) used the rigid plastic method of analysis to study the behaviour of beams and plates to impulsive loading. A review of theoretical and experimental studies on the deformation of thin plates subjected to impulsive loading was reported by Nurick and Martin (1989). Most of the structural analysis programs currently available (NASTRAN 1 , A N S Y S 2 , ADINA 3 , A B A Q U S 4 ) have the capability to predict the mode I response. Gupta (1987, Gupta et al., 1987) and Houlston et al (1985, 1987) used ADINA to predict the mode I response of unstiffened and stiffened plates. Khalil et al (1987a, 1987b) used the finite strip method to investigate the response of air-blast loaded plates structures. In this thesis, more emphasis is given to predict and understand the mode II and mode III failure behaviour. The mode II and mode III responses of a structure involve material failure leading to structural failure. The pattern of deformation determines where energy is dissipated at an instant of response. Mode II is tensile tearing of outer fibres at or over the supports and is also associated with large plastic deformation, while mode III is shear dominated 1 N A S A Structural Analysis 2 Swanson Analysis Systems 3 Automatic Dynamic Increment Nonlinear Analysis 4 Hibbitt, Karlsson & Sorenson, Inc. Chapter 2. Review of Literature 6 failure with very little or no significant deformation of the structure. Structural behaviour in these two failure modes is quite different and needs to be evaluated in detail. 2.1 Mode II Failure 2.1.1 Classical Approach Mode II can be characterized as ductile fracture of the material. The basics of ductile fracture phenomenon of metals is explained in the text by Thomason (1990). Several approaches have been used to study the ductile fracture of metal structures. Some of these methods are discussed in the following section. 2.1.1.1 Fracture Mechanics A l l metal structures to some extent have either some pre-existing flaws or stress concentrations. During their life-time of usage upon exposure to hostile environments, cracks will initiate and grow. Classical fracture mechanics principles allow us to understand the fracture behaviour of metals. Griffith (1921) developed the basic equations of fracture mechanics. He considered a plate with a transverse crack. When the crack grows by a small amount, strain energy stored in the plate is reduced or released. According to Griffith, the crack will grow if the energy required for crack growth is equal to the energy available or energy released. A huge volume of work on crack initiation and propagation can be found in fracture mechanics literature (Broek, 1986). Recently, Ewing et al (1995) proposed a new Chapter 2. Review of Literature 7 failure model for the rupture of pipes. It is based on the dynamic crack growth rate. They used M S C / D Y T R A N software and added a failure model to predict crack initiation and propagation. A crack is assumed to grow at a constant rate. 2.1.1.2 Void Coalescence Mechanism This method approaches the fracture from a micro or material level. Here, materials are considered to have voids, the growth and coalescence of which leads to fracture in ductile metals. McClintock (1968) proposed a criterion for ductile fracture by the growth and coalescence of cylindrical holes in ductile metals. In this procedure, the growth of holes depends on the entire history of stress, strain and rotation. This method results in a very complicated analytical procedure. A summary of analytical and experimental studies on the strain localization and subsequent ductile failure due to void coalescence mechanism can be found in the works of Theocaris (1986) and Wilson and Acselrad (1984). 2.1.1.3 Damage Mechanics Lemaitre (1985) proposed a continuous damage mechanics model for ductile fracture using thermodynamics and an effective stress concept. This approach defines a damage variable as an effective surface density of cracks or cavity intersections with a plane. The damage process gives rise to initiation of a macro crack for a critical value, which is characteristic for each material. Chapter 2. Review of Literature 8 2.1.1.4 Forming-Limit Diagrams Sheet-metal forming industry has stimulated the development of mathematical models for understanding and obtaining biaxial plastic tensile instability information. Ferron and Zeghloul (1993) reviewed the topics of strain localization and fracture in metal sheets and thin-walled structures mainly from a metal forming point of view. Marciniak and Kuczynski (1967) analysed the loss of stability (localization of strains) for sheet metal subjected to a range in biaxial tension ratios. Their conclusion was that the fracture of metals depend only on a local discontinuity or concentration of strains. Another important factor in stretching a metal is the total strain it would experience before it fails or tears. Hecker and Ghosh (1976) used the concept of empirical forming-limit diagram. A forming-limit diagram is an experimentally determined curve in principal strain space for the locus of points representing the onset of localized necking. As reproduced in Figure 2.1, this diagram represents the limiting strains in biaxial stretch forming operations. 2.1.2 Analytical Failure Models Considerable effort has been put in the last two decades to develop theoretical methods for the dynamic plastic behaviour of simple structural members like beams, plates, etc (Yu and Chen, 1992, Westine and Baker, 1974 and Gupta, 1985). Simplified rigid-plastic analysis procedure is obtained by neglecting the material elasticity (Jones, 1989b). Rupture of a ductile member initiates when either the strain or the density of dissipated energy at a section exceeds some characteristic value. Chapter 2. Review of Literature 9 2.1.2.1 Strain Based Model Jones extended the rigid-plastic analysis of structures to predict failure (1976, 1989c). He obtained a good comparison with experimental data for tearing of outer fibres at the supports by analytically determining the maximum strain occurring at the supports and equating the strain to the uniaxial failure strain. This is effectively a plastic tensile instability criterion of failure. Subsequently, this criterion has been applied for predicting the onset of mode II failure for circular plates and cylindrical shells also. In this model Jones neglected the elastic deformation and the effect of transverse shear on deformation. Wen (1996) used an effective strain failure criterion to predict the tearing of beams subjected to uniformly distributed impulsive loading. This criterion considered the influence of transverse shear on the axial tensile strain. Good agreement with the experimental data is reported by the authors. 2.1.2.2 Stress Based Approach Duffy (1989) explored the idea of generating the strain based failure curves corresponding to general failure curves in stress space. He considered both the von-Mises and Tresca failure surface and mapped them onto the strain space. The general shape of failure curve of biaxial strain failure data showed excellent qualitative agreement for the Tresca failure curve, while little resemblance was observed in the case of von-Mises curve. In general, the Tresca failure shape showed better qualitative agreement than that of the von-Mises failure curve. Chapter 2. Review of Literature 10 2.1.2.3 Plastic Work Density Approach A more universal energy criterion to predict the inelastic behaviour and failure modes of beams loaded impulsively was suggested by Jennings et al (1991), Jones and Shen (1992, 1993). In this approach, onset of mode II failure is identified at a point in a structure when the specific dissipation (density of plastic work) at that point reaches a critical value. This approach is also used for the prediction of tearing in circular plates (Shen and Jones, 1993). 2.1.3 Computational Failure Models Even though abundant data is available on the dynamic elastic-plastic analysis of structures (Fredriksson et al., 1983, Noor, 1981, Yagawa et al., 1984 and Fong, 1982), prediction of the point of structural failure by material separation is still a challenging task. A large number of studies (Hecker, 1976) have been reported on the biaxial plasticity while ductile failure of a material under multi-axial loading state is still a developing field. The relative lack of multi-axial material failure information, compared to the multi-axial plasticity information, is readily apparent when we look at the available computer programs. The most serious limitation to the extensive use of computational techniques is due to the inadequacy of models describing material failure. Furthermore, in a few computer programs where a failure criterion is included, the criterion is typically based upon equating equivalent plastic strain to the failure strain in simple tension (Hallquist, 1981, Anderson et al., 1988) or when it reaches a critical failure strain that is a function of the stress triaxiality (Holmes et al., 1993). Chapter 2. Review of Literature 11 Bammann et al (1993) developed an internal state variable type of criterion specially for numerical applications. The formulation is based on basic theoretical aspects that introduce internal state variables to track plasticity and damage. 2.1.3.1 Strain Based Model Olson et al (1993) used finite element computations to investigate the different failure modes of blast loaded square plates. They assumed that a plastic hinge line develops along the clamped boundary of a plate subjected to blast loading. Following Jones (1976, 1989c), they used the plastic hinge model to calculate the maximum strain at the support. They defined mode II failure as the instant when the maximum strain reaches the rupture strain in the static uniaxial test. Mode I predictions showed good correlation with the experimental results, while mode II results were not as satisfactory. The predictions were limited to the onset of tearing only. Later, they extended this idea to predict the onset of mode II failure for the stiffened square plates and circular plates also (Nurick et al, 1995 and Olson et al, 1994). Other numerical results can also be found in the literature for the failure analysis of blast loaded beams (Yu and Jones, 1989). 2.2 Mode III: Transverse Shear Failure Jones developed his rigid plastic model to predict the mode III threshold also. To estimate the threshold for mode III failure, the concept of shear slide (analogous to the concept of a bending hinge) is specified as an idealization of rapid changes of slopes across a short length of beam. Mode III failure is said to occur when the amount of shear sliding at the supports reaches the beam thickness. Chapter 2. Review of Literature 12 Jones also predicted mode III threshold using the plastic work density approach (Jones and Shen, 1993). They included the transverse shear in the yield criterion and computed the amount of work done by bending and shear. For aluminum alloys, they calculated the threshold to occur when the energy ratio (bending to shear) is 45%. Olson et al (1993) were able to predict the mode III failure by extending the Jones criterion to plates. This was done by doubling the effective area for beam shear failure to account for plate slides. The threshold obtained for mode III impulse showed good agreement with experimental results. 2.3 Interactive Failure Criteria for Predicting Failure Modes The transverse shear effect is believed to have a more important influence on the response of beams when loaded dynamically than statically (Bleich and Shaw, 1960). The effect of shear flexibility and rotatory inertia on the dynamic behaviour under impulsive loading is shown for a variety of structures by different researchers (Jones and de Oliveria, 1978, 1979, 1980, 1983, Quanlin, 1988, Symonds, 1968, Jones and Song, 1986, Jones 1985a and L i et al., 1989, 1995). Except for the plastic work density approach, all other analytical models proposed two different procedures for identifying mode II and mode III failure. Also, the previous models did not include the interaction effects of tensile and shear action while predicting the failure modes. Olson et al (1994) proposed an interactive failure model for circular plates. They defined a failure function which is a linear or quadratic combination of stress and strain ratios. Failure is said to occur when the function attains a value of one. Shear stress was obtained using an equilibrium equation in the transverse direction, since direct shear Chapter 2. Review of Literature 13 estimation was not available because the formulation was based on the Kirchhoff assumption (Fagnan, 1996). This concept was also used by Qing (1994) for predicting the response of clamped beams. Qing used the higher order beam theory to analyse the blast loaded clamped beams. This analysis provided a direct estimation of shear stress in the structure which was used in interactive failure criteria. The model showed a reasonable agreement with experimental results. Qing (1994) also explored the plastic work density approach by incorporating it into the higher order beam model which did not give satisfactory results. 2.4 Post-Failure Analysis The post-failure response can be described in several ways. The failed material may be entirely removed from the computation or be described by a modified constitutive function that describes a weakened material. The basis of fracture modelling is a node release algorithm. Ewing et al (1995) developed a user defined failure routine to be used with D Y T R A N software. In this procedure, nodes along a probable line of failure are doubly defined—one node for each side of potential failure. The subroutine, E X B R K , is called each cycle during solution and is used to determine when nodes should be released, thereby simulating failure. Connolly et al (1986) incorporated a simple crack propagation routine into the existing nonlinear plane strain/stress dynamic transient finite element program to allow for the through thickness crack propagation along a line of symmetry. The crack is advanced one element at a time by releasing the nodal reactions incrementally during time stepping. This is done by applying equal and opposite forces to the nodes, thus Chapter 2. Review of Literature 14 providing an external work component which equals the energy released by the crack during extension. Their study showed good comparisons with experimental results. Fagnan (1996) and Qing (1994) modified the fixed boundary condition to free conditions when failure was identified and continued the post-failure analysis in the circular plate and beam problems respectively. 2.5 Experimental Results The earliest work on explosive loads was by Taylor (1950) during world war II. He conducted experiments to study the response of structures subjected to underwater explosive charges. Earlier experimental work was limited to mode I failure only, to measure the large inelastic deformation. Mode I failure of rectangular and square plates subjected to blast loading has been reported by Jones et al (1970, 1971, 1972), Nurick et al (1986, 1992, 1996), Y u and Chen (1992), Houlston et al (1986), Slater et al (1990) and Zhu (1996). Anderson et al (1988) conducted experiments on circular aluminum plates subjected to explosive loading, while Wierzbicki et al (1996) carried out studies in the mode I range on partially loaded circular plates. Ross et al (1975, 1977) were the first to mention the rupture of plates. They conducted experiments on aluminum plates and reported that failure would start at the middle of clamped boundaries and continue towards the corner. Olson et al (1993) reported a limited set of experiments in the mode II and mode III range in their paper. Recently, Nurick et al (Teeling-Smith and Nurick, 1991, Nurick and Shave 1996) conducted a complete set of experiments on plates of different geometries to estimate the Chapter 2. Review of Literature 15 behaviour beyond mode I stage, and they were the first authors to give quantitative estimation of various parameters. Experimental data on the stiffened plates subjected to blast loading large enough to cause failure is sparse. Nurick et al (1994, 1995) have reported some of these results, which show essentially the same general trends. Some results in the mode I range were also reported by Houlston and Slater (1986). A comprehensive numerical analysis of the behaviour of air-blast loaded unstiffened and stiffened square plates with material, geometric nonlinearities and strain rate sensitivity, which includes the failure modelling in its entirety for all different modes, is still a formidable task and an undeveloped area. It is the purpose of this thesis to propose an approach and a simple numerical failure model for predicting the rupture of thin unstiffened and stiffened plate structures. This model is described in the next chapter. Chapter 2. Review of Literature 01 . 1, _ J __._JL_ L_ _ L _ Jt -60 - 40 - 20 0 20 40 60 80 MINOR STRAIN (PERCENT) Courtesy: Scientific American Figure 2.1: Forming-limit diagram Chapter 3 Deformation and Failure Model Formulation 3.1 Introduction The description and theoretical formulation of the failure model used in the current analysis is presented in this chapter. Also, some aspects of its computer implementation and simulation of post-failure behaviour is described. This chapter is divided into several sections. First, the model philosophy is described in section 3.2. Next, the governing equations of motion are derived in section 3.3. The following subsections present the expressions for the mass, stiffness and damping matrices for the plate and beam structures. The failure model is defined and the two interaction failure criteria are presented in section 3.4. Also, post-failure behaviour is discussed in this section. Finally, the flow chart for the analysis is presented in section 3.5. Since the transient elastic-plastic analysis procedure is widely available in the literature, this thesis focuses more on the failure model and its constituent parameters. 3.2 Model Philosophy The rupture of ductile plates under blast loading occurs as a combination of tensile tearing and shear failure. Hence, a comprehensive analysis of the problem should have 17 Chapter 3. Deformation and Failure Model Formulation 18 information about the bending, membrane, shear strain and/or stress field at all locations of the plate structure. A finite element procedure based on Kirchhoff theory precludes the possibility of obtaining any information regarding shear stress or shear strain directly. Even though equilibrium equations can be used (Canisius and Foschi, 1993) to obtain the transverse shear stress (Appendix A), this procedure poses difficulties for a plate undergoing geometric and materially nonlinear behaviour. The other course of action would require formulation of the problem using Mindlin theory (Mindlin, 1951) or higher order theories (Reddy, 1984a, 1984b, Heyliger and Reddy, 1988). Higher order theories give the nonuniform shear stress distribution across the depth. Qing (1994) used a higher order theory to determine its suitability for blast loaded clamped beams. The shear stress estimated by finite elements fluctuated in the wavelength of about one element size. The shear stress and stress resultants were more accurate at the middle of the element than elsewhere. Thus the shear stress data on the boundary is approximated to be that of the middle value of the first element next to the boundary. The results were not so encouraging from this analysis. On the other hand, in Mindlin theory, the shear strain is assumed to be uniform through the thickness, which may not yield useful results for problems under consideration. In addition, both the Mindlin theory and higher order theories are computationally more intensive and lead to increased costs and time. This does not suit the primary goal of developing a design oriented analysis tool. In view of all these difficulties, a new approach for the solution of this problem is essential. To overcome many of the challenges, a simple methodology based on the concept of a plate supported by springs along the boundary is developed. This technique will be described in the next section. Chapter 3. Deformation and Failure Model Formulation 19 The idea of using springs to model boundary conditions is not new. Quite a few instances can be found in the literature where a variety of edge conditions were simulated using springs. In the current model, only translational springs are considered. The beauty of this idea lies in the fact that the spring forces give a direct estimation of reaction forces at the boundary. The continuous springs at the boundary enable the variation of reaction forces along the boundary to be captured. This is essential in order to model the failure effectively. By having very stiff springs, zero out-of-plane motion of the plate boundary is simultaneously achieved. Thus combining with Kirchhoff theory based analysis, they give a powerful, yet a simple means to analyse these problems. The following assumptions are made to keep the analytical procedure simple. 1. The springs are massless and have infinitesimal length. 2. They have a large stiffness and have translational DOF only. 3. They always behave in a linear elastic fashion, even though the plate and beam exhibit geometric and material nonlinearities. 3.3 Deformation Response The class of problems which are of interest in this work includes tearing and rupture of square unstiffened and stiffened plates. The experimental observation of these explosively loaded plates identifies the boundary of unstiffened plates and the boundary as well as beam-plate interface in the stiffened plate as possible locations of failure. Hence, the numerical analysis should be capable of providing shear stress and force details at the above location. In both types of structures, very stiff springs are used to simulate the rigid (zero out-of-plane motion) boundary condition. The spring model for Chapter 3. Deformation and Failure Model Formulation 20 the plate structures is shown in Figure 3.1. In a stiffened plate, the double node concept is used to define two sets of nodes at the beam-plate junction, one each for beam and plate. Springs are used to connect these beam and plate nodes which will provide an estimation of force at the interface. 3.3.1 Equations of Motion The governing equations of motion are obtained using the principle of virtual work. The principle states that for a body in equilibrium under a set of arbitrary load and boundary conditions, where wext is the external virtual work and wjnt is the internal virtual work of the body. Ignoring body forces other than inertial loads the equation of motion for a body under dynamic loads, can be written as where {<r} and {s} are the stress and strain vectors, p is the mass density, IQ is a viscous damping parameter, q is the applied surface traction, V and S are, respectively, the volume and surface area and {d} is the nodal displacement vector. The tilde(~) is used to denote a virtual change in the given quantity with respect to a generalized displacement and the superimposed dot denotes differentiation with respect to time. For static loads, the velocity and acceleration terms {d} and {d } in Equation (3.2) vanish. As mentioned earlier, the spring element connects the plate nodes to the beam nodes in the stiffened plate problem. The displacement of the spring is therefore (3-1) J pJ/pp} + p}/^{^} + {2r}r{o-} dV- ${d}' {q}ds = 0 (3.2) Chapter 3. Deformation and Failure Model Formulation 21 Kl "({"'}-{"•}) (3.3) where jdp j and are the plate and beam displacement field vectors respectively. In the case of springs at the boundary, the spring displacement is equal to the plate deflection at that point (i.e., the terms associated with jd* j vanish at all these points). The motion of the stiffened plate structure can be expressed as a combination of plate and beam motion. For the plate, the equation of motion can be written as j \dp}'p\dp} + \dp^Kd[dp] + {sp}r{ap} dV-\\dp}' {q)ds + \{dp}'ks[dp}dL - l{db}ks{db}dL = 0 (3.4) while for the beam, f{3')Tr{?) + {a>}\{*} + i?Y{a>} w + \[d"Ys{db)dL - \\dp}Tks{dp}dL - 0 (3.5) where ks is the spring stiffness/unit length and L is the length of the plate or beam structure. For an unstiffened plate, supported by springs at the boundary, Equation (3.4) reduces to J {dp}' p{d^} + \dp}'Kd{dp} + {ep}T{ap] dV - \\d"}r{q}ds \[dp]T ks{dp}dL = 0 (3.6) Chapter 3. Deformation and Failure Model Formulation 22 The governing equations of motion, Equations (3.4), (3.5) and (3.6) are highly nonlinear due to the presence of both geometric and material nonlinearities. Since an analytical solution of these nonlinear equations is impossible, a numerical solution technique has to be employed. Here, the finite element analysis scheme is employed to solve these equations. 3.3.2 Finite Element Formulation Finite element analysis is performed using the super elements shown in Figure 3.2. These new plate and beam elements are used for the large deflection elastic-plastic analysis of unstiffened and stiffened plate structures (Koko, 1990). The displacement fields of the plate and beam elements are represented by polynomial as well as continuous analytical functions (Appendix B). The elements have been specially designed so that one plate element is sufficient to model the deformational response of the entire plate structure. The displacement function for the plate element can be collectively expressed as (Appendix C), |<F j and |£* j are nodal displacement quantities for the plate and beam elements. Similar expressions for a beam oriented in the ^-direction can be obtained by [d") =\vp\= [N"]{5e} (3.7) while for a beam in the x-direction, the displacement functions are (3.8) Chapter 3. Deformation and Failure Model Formulation 23 replacing the terms associated with u by v. The plate element has 55 DOF, while the beam element has 18 DOF. The von-Karman large deflection theory is used to model the geometric nonlinearities. Large deflection effects are taken into account by including first order nonlinearities in the strain-displacement relations. Therefore, the virtual strain vector can be related to virtual nodal displacement vector 18) as where [B] and [C0] are the linear and nonlinear part of the strain-displacement matrix respectively. Material nonlinearities are modeled by the von-Mises yield criterion and its associated flow rule for a bilinear elastic-plastic stress-strain curve. Strain-rate effects are included by adjusting the initial dynamic yield stress, cfy at each Gauss point according to the Cowper-Symonds relation (Jones, 1989b): L | z ) | J where oo is the static uniaxial yield stress, s is the strain-rate and D and n are other material parameters. Using Equations (3.4) and (3.5) and substituting the finite element approximation (3.9) (3.10) of Equations (3.7) and (3.8) for \d"} and \dh], we get V Chapter 3. Deformation and Failure Model Formulation 24 + 8pe)' 'j[N>]TkM[N>]{S'e}dL - {stf \[N']T k,[N>]{5bt}dL = {S/}' \[N"\qds s (3.11) (3-12) Since j ^ / j , | J e * | are arbitrary, Equations (3.11) and (3.12) can be rewritten as • [<]{*}+[< p:}+/({<*/})+{K]{S: } - [K]{S: J ) = M (3.i3) ["f]{*}+KK*}+/(K})+(I<]K}-[*.'K''})={o} (3-14) where [^] is the consistent spring stiffness matrix (Appendix D); / ({^ / j ) a r*d /^j j j ' l jare the plate and beam element internal nodal force vectors; [wtf j , [cfj and |c*j are consistent element mass and damping matrices for the plate and beam respectively; and {p} is the consistent load vector. These are given by [m?] = l[Npfp[Np]dV v K]= j K f p[Nb\dv Chapter 3. Deformation and Failure Model Formulation 25 K ] = 1 K ] 7 ^ K K V V {P} = \\NpJ qds s Combining Equations (3.13) and (3.14) we get "[<] [o]" [k l 4- , Jo] R L * J°] K l » - f - < -MI [M l [Ml .-[*-] [K)J i(°!J r For an unstiffened plate supported by springs at the edges, the above equation reduces to [<]{^ }+[<]{^ } + /({^ }) + [^ ]{^ } = W (3-16) 3.3.3 Damping Matrix The high frequency of spring motion needs to be suppressed so that a smooth distribution of force is obtained. For simplicity, the damping matrix is assumed to be proportional to the stiffness matrix. Thus, M = a,[*] (3-17) Chapter 3. Deformation and Failure Model Formulation 26 where [k] is the linear stiffness matrix, a\ — \cor) E,r and cor are the critical damping ratio and frequency associated with the r t h mode of vibration. 3.3.4 Solution Procedure The element matrix quantities are assembled in the usual finite element fashion to obtain the global equations of motion. For a stiffened plate: \M'\ [o] s"} y + \c] [o] I»]. [ C1J 5P} AW) + . - M M . (3.18) For an unstiffened plate, the above equation becomes [ M p ]{^} + [C"]{^} + F ( { ^ } ) + [A: J = {P} (3.19) A l l matrix quantities are evaluated by Gaussian quadrature. Five integration points are used in each in-plane direction and four through the thickness. Temporal integration is carried out using the Newmark-/? method with Newton-Raphson iteration for the solution of nonlinear equations within each time step. In this implicit solution scheme (Newmark-/?method) Equations (3.18) and (3.19) reduce to n+1 (3.20) where is an effective stiffness matrix given by Chapter 3. Deformation and Failure Model Formulation 27 K —[—\M] + ^-\C] + \KT] (3.21) and 17*1 is the effective load vector given by 2J3 l " V (3.22) With y= 0.5 and /?= 0.25 and substituting for stiffness proportional damping matrix, we get m = At2 L J At ^2^ ^ [KT] + [KT] At -[M] + [Kr] 1 + A^ (3.23) (3.24) and The above theory is codified and incorporated into NAPSSE (Nonlinear Analysis of Plate Structures using Super Elements), a special purpose program developed at U.B.C. for efficient nonlinear analysis of plate structures (Koko, 1990, Koko and Olson, 1991a, 1991b, 1992 and Jiang et al, 1990). This program now has capabilities to perform static, vibration and transient analysis of unstiffened plates and stiffened plates using the spring model. Chapter 3. Deformation and Failure Model Formulation 28 3.4 Failure Model 3.4.1 Introduction Having established the deformation model, analysis and solution procedure, attention will now be focused on developing a model for the failure analysis of plate structures. It is evident that dynamic inelastic rupture of structures is a complex phenomenon. There is a definite need for the development of a reliable criterion that can be used to predict the material rupture. The failure analysis of plates having square geometry involves a) Identifying initial failure b) Failure progression c) Complete or total failure d) Post-failure analysis Initial failure is identified when the failure occurs first under the applied load. Failure progression is achieved by developing a node release algorithm. Complete failure is the stage when the failure reaches the corner of the plate, upon which all nodes are released and the plate becomes free. Post-failure analysis is the phase wherein the free plate flies away from the boundaries. It was observed experimentally (Ross et al., 1975, 1977, Olson et al., 1993, Nurick and Shave, 1996) that the plate deformed substantially before rupture occurs first at the middle of supported boundary (mode II) in an unstiffened plate, for the loading used in this study. For more severe loading, failure begins as shear of the plate at the edges (mode III) before any significant deformation takes place. Chapter 3. Deformation and Failure Model Formulation 29 Although the mode II and mode III are dominated by tensile strain and transverse shear stress respectively, shear effects are significant even at the threshold of mode II failure. Between the threshold values for the two failure modes, the plate fails in a mixed tensile tearing and transverse shearing mode, wherein the shear stress starts to play a dominant role. This indicates that the interaction of tensile strain and transverse shear stress may possibly lead to the rupture of blast loaded ductile plates. This is the idea behind the interactive failure criteria. In the interactive failure criterion, interaction is exhibited by having a failure function as a mathematical function of both tensile strain and transverse shear stress. In this criterion, the contribution of the tensile tearing part is specified by a tensile strain ratio and the extent of the influence of the transverse shear part is taken care of by a shear stress ratio. Both parts are then included in the failure function to predict the rupture of plate structures. In the following sections, the individual components of the failure functions are developed. 3.4.2 Mode II Failure The current model considers the strain effects, following Olson et al (1993). It is impractical to model the highly concentrated plastic deformation of the plate near the boundary with finite elements. A rigorous three dimensional analysis would be necessary to predict the required information accurately and this is not realistic for preliminary design/analysis work. Therefore, an approximation is used to determine the bending strain at the boundary. The rotation of the plastic hinge is assumed to be equal to the Chapter 3. Deformation and Failure Model Formulation 30 maximum slope of the plate along a line perpendicular to the boundary, which usually occurs a short distance from the boundary. 3.4.2.1 Unstiffened Plate In general, the total strain at a point on the boundary of a plate can be expressed as where sm is the membrane strain and sh is the bending strain. In this model the membrane and bending strains are calculated separately from the associated finite element calculation. Membrane Strain: The membrane strain at the boundary is approximated by averaging the membrane strain over the first element adjacent to the boundary. For a plate boundary oriented in the y-direction, the membrane strain in the x-direction is calculated by where Le is the length of the element in the x-direction and u and w are the in-plane and out-of-plane displacements of the midplane. Equation (3.27) is evaluated using numerical integration. Bending Strain: Bending strain, on the other hand, is estimated by assuming that a plastic hinge line forms at the boundary. The maximum bending strain can be mathematically expressed as (3.26) (3.27) hie (3.28) Chapter 3. Deformation and Failure Model Formulation 31 where h is the plate thickness and K is the curvature of the plastic hinge. Furthermore, the curvature K is related to the hinge rotation 0 by 0 K = j (3.29) where / denotes the plastic hinge length. Since the current finite element analysis is unable to model the plastic hinge, approximation is made for the plastic hinge length /. A slip line field analysis (Nonaka 1967) indicates that the hinge length / for a rigid plastic beam of rectangular cross-section varies between H and 2H for maximum transverse displacement between 0 and H (H is the beam depth). This transverse displacement also corresponds to the pure bending behaviour and onset of membrane response of beam respectively under the applied load. Jones (Jones and Shen, 1992,1993) suggested an empirical relation wherein the plastic hinge length changes inversely with the applied impulse. In the present work, it is held constant as there is no theory to predict its value analytically. Even though, the failure predictions are affected somewhat because of this assumption, its effect will diminish as the plate approaches the mode III failure. In the present study, most of the large deflection arises from membrane stretching, and hence the value of / = 2h was used. Therefore, bending strain reduces to 0 eb=j (3.30) In the above equation, the hinge rotation 0 is assumed to correspond to the maximum slope near the boundary of the deflected plate. For the simply supported plate Chapter 3. Deformation and Failure Model Formulation 32 (hinge support), however, the rotation of the plastic hinge must be measured relative to the boundary rotation, i.e., where 0max is the maximum slope of the deflected plate element near the boundary and © is the rotation of the boundary or support. The maximum slope is determined by interpolating across each finite element using the shape functions (Appendix B) available in NAPSSE. These are easily differentiated in the closed form, the location of the maximum slope and its value are determined numerically. 3.4.2.2 Stiffened Plate In the case of stiffened plates, plastic hinge lines are assumed to form in the plate along the supported boundary and also alongside the stiffeners. The hinge rotation at the boundary is again taken to be the maximum slope in the plate adjacent to the fixed boundary and a hinge length of / = 2h is used. However, for the hinge at the stiffener, it is taken to be the algebraic difference between the analogous slope in the plate and the rotation of the stiffener. Once again, Equation 3.27 is used to calculate the membrane strain at the plate boundary. It is suggested that the failure, in the stiffened plates, starts at the middle of the clamped boundary and terminates at the stiffener end (where the stiffener meets the boundary). This means that the stiffener end node would be the last node to fail. The stiffener end node can then be treated like a clamped beam for the purpose of failure analysis. A clamped beam is considered to be in a membrane state once the central displacement is greater than the beam thickness. Jones (1976, 1989) calculated the sup (3.31) Chapter 3. Deformation and Failure Model Formulation 33 bending strain using a value of / = where 1/2 is the half-length of the beam. In the absence of a theory to calculate the plastic hinge length for the stiffener end, the above value is used to estimate the influence of the bending strain on the failure function. Having defined the procedure for calculation of the membrane and bending strains, the total strain in the x-direction is then determined by Equation (3.26). A similar procedure is used to determine the total strain in the >>-direction. Since plastic hinges mainly occur adjacent to the boundaries and beam stiffeners, only the strain in the direction perpendicular to the boundary or stiffener will have a bending strain component at that location. The maximum strain computed using Equation (3.26) at each time step is divided by the specified rupture strain to form the strain ratio, s = e / srup. This is used in calculating the failure function. This ratio will reflect the effect of strain in the failure function. 3.4.3 Mode III Failure 3.4.3.1 Unstiffened Plate In the present model, the springs at the boundary provide a direct estimation of the transverse shear forces. The resultant spring force for an element next to the plate boundary oriented in the x-direction is (3.32) o Chapter 3. Deformation and Failure Model Formulation 34 where Le is the element length, ks is the spring stiffness/unit length and w is the transverse displacement of springs at the boundary. The shear stress, ravg, is then calculated by dividing this reaction force by the total sectional area of each element at the boundary. The sectional area is the product of element length and plate thickness. A stress for the strain-rate effects. The ratio of (CTJCTQ) calculated for the first Gauss point to yield is stored and the static shear strength (TUII) is multiplied by this ratio to obtain the dynamic ultimate shear strength of the material, Xdyn-uit- The static ultimate shear stress is stress at the rupture strain of the material obtained using the bilinear stress-strain relationship. 3.4.3.2 Stiffened Plate Once again, the spring force is used to calculate the shear stress at the boundary and is used to identify the failure. In the case of stiffened plates, the stiffener end node would be the last node to fail during the post-failure analysis (i.e., the entire structure is supported by the stiffener end node just before the complete failure). The shear contribution towards the failure function needs to be evaluated appropriately, for the successful completion of the failure analysis. The shear stress for the stiffener end is obtained via the reaction force estimated at the boundary through the overall structural equilibrium (global force balance method, Appendix E). This value of shear stress is used in calculating the failure function. ratio, T = (^avg^Tdyn-uit)' *s formed. The dynamic ultimate shear strength, rdyn_ult accounts calculated from the ultimate bending strength, i.e., rull = (7u"/ where a,,,, is the Chapter 3. Deformation and Failure Model Formulation 35 3.4.4 Mode II - III Interaction The previous two sections discussed how to evaluate the influence of tensile tearing and transverse shearing on the rupture of plate structures. These two ratios are also reflections of the extent of damage caused by tearing and shearing. With the measure of the two factors causing failure defined, the failure condition may now be specified. Failure is related to tearing and shearing actions and, therefore, the failure condition must be a function of the two ratios, i.e., r f (3.33) V ^ rup 1'dyn-ult J Since both ratios represent the extent of damage to some degree, the above function can be thought of as a general equivalent degree of structural damage. Failure will occur when the extent of damage is 100%, i.e., 1. It is then reasonable to assume that failure of the plates occurs when f e T ^ = 1 (3.34) V £rup *'dyn-ult J Two different failure models are proposed based on the way the ratios are added: Linear Interaction Criterion (LIC) where the ratios are added directly, while in Quadratic Interaction Criterion (QIC) the ratios are squared before being added together. Failure is said to occur when the failure function (Linear or Quadratic sum) reaches a value of one (Equation (3.35) and (3.36)). These models are incorporated into NAPSSE. Linear model: Chapter 3. Deformation and Failure Model Formulation 36 Quadratic model: / = + 'nip 'org dyn-ult = 1 (3.35) f = + \ £ n , p ) • avg V Tdyn-nll J = 1 (3-36) 3.4.5 Post-Failure Analysis Experimental results indicate that the failure of plates in mode II (tensile tearing) starts at the middle of the plate boundary and proceeds towards the corner. Also, an increasing central portion would fail simultaneously with increasing impulse. This behaviour is quite different from the blast loaded clamped beam and circular plates, wherein the first failure also represents the complete failure, due to symmetry of boundary conditions, loading and geometry. Therefore, a means of accounting for failure progression is necessary in the failure analysis of square and rectangular plates. Tearing is a very complicated process and, hence, some simplified assumptions are made in the present analysis. Fracture is assumed to occur instantaneously through the plate thickness. A node release algorithm is developed to simulate the progression of rupture. In this algorithm, the entire element side is released once failure is identified at the corner and midside nodes of that element (i.e., when failure function/= 1). Failure progression is simulated by successive releasing of elements as failure occurs. Once failure reaches the corner of the plate, the plate separates from the boundary and flies freely. Chapter 3. Deformation and Failure Model Formulation 37 However, it should be noted that the transverse displacement of the plate at the instant of complete rupture is usually different from the permanent transverse displacement obtained from the experiments. This is because the plate has some kinetic energy at complete failure since not all the initial input energy has been absorbed by plastic deformations. The kinetic energy at rupture is often significant. With so much energy available at failure, the plate structure is bound to deform until a stable state is reached. As the plate continues to deform after complete severance, part of the kinetic energy is absorbed in further plastic flow until it reaches the steady state where no more plastic deformation occurs. At the steady state, the plate moves with associated rigid body kinetic energy and vibrates with relatively small amounts of energy exchanged between elastic energy and kinetic energy simultaneously. The mid-span transverse deformation at the steady state, which is the average of the maximum and minimum value during the vibration, can be obtained from the post-failure analysis. The idea of the post-failure analysis involves treating the ruptured plate as a free-free plate with initial motion. At this stage, the fixed boundary conditions are changed to free ones and the analysis is continued to account for any deformation during the free flight of the torn plate, until it reaches a steady state. 3.5 Computer Implementation The failure model described in the previous section is incorporated into NAPSSE. The flow chart for the failure analysis of blast loaded plates is given in Figure 3.3. The specified tasks shown in Figure 3.3 can be summarized as follows: Chapter 3. Deformation and Failure Model Formulation 38 Step 1. Input - Data is input to describe the geometry, materials, boundary conditions, loading, failure and solution control parameters. Step 2. Initialize - Initial values are entered into the structural and material data arrays. Step 3. Calculate nodal forces - Forces at the nodes are calculated as the difference between internal and external forces. External loading is due to applied blast pressure. Internal forces are calculated from element stresses. Step 4. Solve equations of motion for displacements - The set of simultaneous equations are solved to obtain the nodal displacements. Step 5. Update velocities and accelerations - Nodal velocities and accelerations are calculated and updated by using the nodal displacement and the current time step. Step 6. Calculate strain increment - From the nodal displacements, strains are obtained at Gauss points, strain rates are calculated and strain increments are obtained. Step 7. Calculate element stresses - Element stresses are calculated in the material constitutive models from the strain increments. Step 8. Calculate internal forces - Nodal forces are calculated by integrating the element stresses. Step 9. Failure model - The strains and stresses at the possible locations of rupture are calculated. Step 10. The failure function is estimated at all locations where failure is likely to occur. The failure is checked at each time step. Step 11. If failure is identified, failure progression is simulated by a nodal release algorithm. When the plate ruptures completely, the plate separates from the Chapter 3. Deformation and Failure Model Formulation 39 boundary. Further temporal progression of finite element analysis for a free-free plate can give information on the motion. Step 12. Output and check for problem termination - Output results i f specified. If the calculation time is less than the specified termination time, return to step 3. Chapter 3. Deformation and Failure Model Formulation Plate boundary also supported by springs (Not shown in this Figure) Plate Springs at the interface of beam and plate Beam a) Stiffened plate y Springs at boundary b) Unstiffened plate Figure 3.1: Spring model for plate structures (Quarter plate model) Chapter 3. Deformation and Failure Model Formulation 1 w)3 t/]0 x,u Degrees of Freedom: At nodes 1,2,3,4 - u,v,w,wx,wy,wxy At nodes 5,7 - u,v,w,wy; plus (ul0, u„) and (w,„ uu), respectively At nodes 6,8 - u,v,w,wx; plus (v„, v,4) and (v,„, v13), respectively At node 9 - u,v,w; plus un, v12, w15, v15 (a) Plate element 1 "5 ut 3 2 x,u Degrees of Freedom: At nodes 1,2 - u,v,w,wx,0,0x At node 3 - u, v, w, &, plus uit us (b) Beam element in x-direction Figure 3.2: The super finite elements Chapter 3. Deformation and Failure Model Formulation 42 Input Initialize (/ = 0) Calculate Nodal Forces Solve Equations of Motion Update Velocities and Accelerations Calculate Strain Increments Begin New Time Step t = t + At Calculate Stresses t Calculate Internal Forces Figure 3.3: Flow chart for the analysis Chapter 4 Square Plate Analysis Results 4.1 Introduction In this chapter, the failure model discussed in Chapter 3 is used to investigate the rupture of unstiffened square plates subjected to explosive loads. First, the experimental observations are presented in section 4.2. In section 4.3, the results from the proposed spring model is verified using the previously published results before attempting the failure analysis. This section presents the results for plates under static and dynamic loads. Finally, the results pertaining to the prediction of rupture of plates are presented in section 4.4. 4.2 Experimental Observations There have been several experimental studies to measure large deformation of plates subjected to blast and impact loading (Nurick and Martin, 1989). However, very few results are available on the tearing and rupture of plate structures (Olson et al, 1993, Nurick and Shave, 1996). Nurick et al (Nurick and Martin, 1989, Nurick, 1987) used the sheet explosive/ballistic pendulum method to investigate the failure mechanism of square mild steel plates. The experimental set up used for the study is shown in Figure 4.1. The test specimen was cut from cold-rolled mild steel plates of 1.6 mm thickness. The plates 43 Chapter 4. Square Plate Analysis Results 44 are clamped between two 20 mm thick steel plate with eight high strength bolts. The clamped assembly was attached to one end of the ballistic pendulum (Figure 4.1). The average static yield stress, (cr0), and rupture strain, {Smp), are obtained from the uniaxial tensile test of mild steel. The explosive used for the blast was arranged in two concentric square annuli placed on a polystyrene pad which was attached to the specimen. The explosive layout is supposed to provide a uniform distribution of loading, although there is no direct measurement to confirm the assumption. In each experiment, the applied impulse; the deformation and side pull-in of plate; and the velocity of free-flying disc are measured. The experimental study on the failure of explosively loaded square plates exhibits the general modes of failure: I (large permanent deformation), II (tensile tearing) and III (shear rupture). In addition, it exhibits several phases in the mode II failure region. These include mode II* which has been defined where only partial tearing of the plate occurs and where the deflection continues to increase for increasing impulse; mode Ila defines the phase where the plate is totally torn from the boundary and the midpoint deflection continues to increase with increasing impulse; and mode lib defines the phase where the plate is completely torn but the midpoint displacement decreases with increasing impulse. These failure modes are shown in Figure 4.2. Chapter 4. Square Plate Analysis Results 45 4.3 Verification of Spring Model: Response without Failure 4.3.1 Static Analysis Results The governing equations of motion for this case reduce to V where | P | is the global load vector and the equations are solved by Newton-Raphson iteration as discussed in Chapter 3. The spring model is first validated by performing some benchmark tests. As a first test, linear elastic static analysis of a clamped plate is carried out to compute the deflection at the plate centre and maximum transverse shear at the boundary. The dimensions and material properties of the square plate are as given below: dimensions = 100 mm x 100 mm x 1 mm elastic modulus, E = 205,000 N/mm 2 Poisson's ratio, v = 0.3 A uniform pressure load of 0.1 N/mm 2 is applied and the plate is modelled using super plate elements. The results are presented in Table 4.1 and compared with analytical solution (Bares, 1969). Convergence of numerical results can be seen with mesh refinement. The variation of shear force along the boundary is shown in Figure 4.3 and is compared with the shear force obtained using the equations of equilibrium (Appendix A). The direct estimation of shear via the spring force predicts a better distribution and is much closer to analytical results. The effect of changing spring stiffness on the results is Chapter 4. Square Plate Analysis Results 46 shown in Table 4.2. Essentially, the stiffer the springs, the closer are the results to that of a fully clamped boundary. 4.3.2 Dynamic Analysis Results The linear elastic dynamic analysis of the above plate subjected to a step load of 268.8 x 10"3 MPa is carried out. The fundamental frequency of the structure predicted by the spring model is 869.55 Hz, which matches the earlier result. The displacement-time history of the plate is shown in Figure 4.4. The result matches with that of a rigidly clamped plate. Next, the nonlinear elastic-plastic transient analysis of a clamped plate subjected to the same step load is carried out using this model and the results are compared with previously published results (Koko, 1990). The time-history of the plate central deflection is shown in the Figure 4.5. The results indicate the predictive capability of the spring model. 4.4 Failure Analysis Results In this section, the results of a clamped plate under blast loading is presented. The failure model and node release algorithm developed in Chapter 3 is used for the analysis. The geometric and material parameters used in the analysis are given below: dimensions 89 mm x 89 mm x 1.6 mm elastic modulus, E 197 GPa tangent modulus, ET 250 MPa Poisson's ratio, v 0.3 Chapter 4. Square Plate Analysis Results 47 static yield stress, cr0 237 MPa density, p 7830 kg/m3 rupture strain, e, •rup 0.3 static ultimate shear stress, r ull 181.4 MPa D 40.4 5 spring stiffness, ks 1 x 1020 N/m Due to symmetry, different finite element grids were used to represent only one quarter of the square plate. The effect of mesh density on failure prediction is discussed in the next chapter. The results reported here are for a 4x4 grid for quarter plate as shown in Figure 4.6, with a time step of 0.5 //sec. A 20% critical damping is used to suppress the high frequency oscillations of spring motion. The burn rate of explosive used in the experiments was approximately 6500 m/sec, providing complete detonation in about 15 //sec. This is higher than the speed of the sound in the material used (steel). Johnson (1972) reported the speed of sound in the following materials: carbon steel 5150 m/sec, aluminium 5700 m/sec, copper 3700 m/sec, and brass 3350 m/sec. It was thus felt that a fair approximation to instantaneous uniform loading was obtained. Thus, the pressure loading was assumed to be a rectangular pulse of 15 //sec duration and uniformly distributed over the entire plate surface. The applied loads are of impulsive nature considering the size of the plate. Non-dimensional parameters are used to represent the results in a very general manner at appropriate places. The parameters used in the current study are as follows: Chapter 4. Square Plate Analysis Results 48 Non-dimensional Impulse (I *) (4.2) Non-dimensional time (t*) t (4.3) Non-dimensional energy (E*) = Residual kinetic energy / Input energy (4.4) Residual kinetic energy is the energy associated with steady state plate motion (plate flying at constant velocity) as explained in section 3.4.5. The residual kinetic energy is calculated from NAPSSE program, while the input energy is related to applied impulse and can be calculated as follows. In the above equations, / is the applied impulse, A is the area of plate, h is the plate thickness, t is the actual time and r0 is the load duration. 4.4.1 Model A typical mode I time history plot for the displacement of the centre of the plate (point C in Figure 4.6) is shown in Figure 4.7. The time history plots for the rigidly clamped plate and the plate supported with very stiff springs at its edges are the same, while the undamped and damped analysis yields virtually identical results as shown in Figure 4.8. The peak displacement occurs at a time of 135-140 //sec (t* « 9). The displacement exhibits an approximately linear increase to a maximum value followed by small elastic vibrations of the order of less than one plate thickness. The permanent deflection profile of the plate for all three cases is shown in Figure 4.9. E I (4.5) 2Aph Chapter 4. Square Plate Analysis Results 49 The analysis is carried out by enforcing the zero slope condition at the boundary. However, all the figures showing the centreline displacement profiles (displacement along C A or CB in Figure 4.6) are obtained by drawing straight lines through the nodal data points and hence shows a nonzero slope at the boundary (location-1.0). Damping does not seem to influence the permanent response, but only helps to remove the high frequency content of spring oscillations. The transient deformation profiles of the plate subjected to an impulse of I* = 0.58 (10 Ns) are shown in Figure 4.10. The dashed lines indicate the profiles of the plate between 0 and 125 //sec at an interval of every 5 /usee. As indicated in this figure, the deflection starts with motion of the entire plate. The boundary is seen as a moving wave or hinge motion toward the centre of the plate. This means that the central portion of the plate remains relatively flat with decreasing size until the hinge has nearly reached the centre of the plate. This central flat portion retains an almost square shape through about half of the deflection process. Then the central portion of the plate begins to bulge uniformly and takes on an almost spherical shape. The centre of the plate continues to deflect, and the spherical portion enlarges slightly, which is clearly seen in Figure 4.11. Thus, the response time of the plate is much longer than the load duration. At the end of the loading phase, the deflection is only of the order of one or two plate thicknesses, which can be seen in Figure 4.10. The variation of shear force with time used in the calculation of shear stress for failure function is shown in Figure 4.12. The resultant shear force calculated using Equation 3.32, for every element shown in Figure 4.6, is compared with that of shear Chapter 4. Square Plate Analysis Results 50 force obtained by global force balance method (Appendix E). The new model captures the variation of force along the boundary. However, the force during the load phase is very much the same for all elements, once again emphasizing the fact that the shear force (stress) is a very important component in predicting the rupture of structures under dynamic loading. The differences in shear forces are more pronounced for elements at the latter stages of response as one moves from the middle of the plate to the corner (from element I to IV along A D or BD in Figure 4.6). Comparison of time history of stress ratio ( r ) for the midpoint on clamped boundary (point A or B in Figure 4.6) is shown in Figure 4.13. The model exhibits the capability of capturing the high shear stress at this location, which leads to a better prediction of threshold impulse to mode II failure. Table 4.3 presents the distribution of the strain along the boundary of an explosively loaded plate under mode I failure. This is for the quadratic failure model at I* = 0.94 (16.3 Ns). It is interesting to note that these strain distributions are very similar in shape to the deformation profile shown in Figure 4.10. The observation of strain and stress distribution along the boundary indicate that mode II failure will occur first at the centre of each side. A plot of strain ratio and stress ratio (s,r ), as defined in section 3.4, versus time at an impulse of I* = 0.58 (10 Ns) for LIC and I* = 0.94 (16.3 Ns) for QIC is shown in Figure 4.14a and 4.14b, respectively. This plot is typical for impulses which cause mode I deformation. Both impulses are less than the critical one required to cause mode II behaviour. The strain ratio increases monotonically to a peak value, then decreases Chapter 4. Square Plate Analysis Results 51 slightly to a lower value and subsequently oscillates around that value. The peak value occurs around 85 //sec (t* « 5.6), which is earlier than the time to reach peak displacement (t « 130-140 //sec or t* « 9). The strain remains constant, although deformation continues. The time to peak strain is much higher than that of the circular plate of equivalent area, which is around 50 //sec (t* = 3.3). The stress ratio plotted in Figures 4.14a and 4.14b shows a more irregular response. The ratio rises steeply to a maximum at 15 //sec (t* = 1) and then falls suddenly when the applied pressure load drops to zero. Subsequently, it rises to another lower maximum around 65-70 //sec (t* = 4.3-4.7) as the inertial forces become significant. Thereafter, it oscillates irregularly about the zero mean position. The corresponding variations of the two failure functions with time can also be seen in these figures. Both functions show a similar variation. That is, an initial sharp peak occurs at about 15 //sec (t* = 1) due to high initial stress ratio followed by a second peak at about 60-65 //sec (t* = 4.3-4.7). The second peak coincides with the peak inertial force and has significant strain contribution. The plate has undergone a large amount of bending rotation. The time to peak failure function is less than the time to peak strain and much less than the time to peak displacement. The relative sizes of these peaks change as a function of the impulse. 4.4.2 Mode II As the impulse is increased, the plate continues to respond in mode I until the threshold impulse for mode II is reached. Failure usually occurs at this point and begins as cracks at Chapter 4. Square Plate Analysis Results 52 the midpoint of the boundary (point A and B). The cracks then grow in both directions along the edges of the plate (AD and BD), meeting at the corners (point D). Predicted mode II threshold impulse is I*m2. = 0.97 (16.8 Ns) for QIC and I*m2. = 0.70 (12 Ns) for LIC. Both the thresholds predicted by this model are less than that predicted by the global force balance method (Appendix E). The lower threshold for mode II failure is a direct result of better representation of shear forces at boundary due to the introduction of springs. 4.4.2.1 Mode II* The time history plot of central deformation and kinetic energy of a plate in mode II* failure is given in Figure 4.15. It shows a linear increase in central displacement, while the energy increases until the end of the loading phase and then decreases continuously. A l l the input energy is absorbed by the plate in undergoing deformation and rupture. Figures 4.16a and 4.16b show the transient deformation profile of the plate. The solid line in Figure 4.16a indicates the profile at the time of first element failure. At this time, the element side is released and the analysis is continued. The time to reach failure (r,-) and the corresponding central deformation ( 4 ) is shown in Table 4.4. The energy is being continuously dissipated into plastic work during rupture progression. The permanent deflection profile of the plate is shown in Figure 4.17. The 3-D picture clearly shows the partial tearing of the plate, which is 75% of the boundary. The ruptured boundary, in addition to being pulled-in, also undergoes transverse deformation. Chapter 4. Square Plate Analysis Results 53 4.4.2.2 Modella The temporal variation of central displacement, side pull-in and kinetic energy of a plate in mode Ha failure is shown in Figure 4.18. The kinetic energy increases monotonically reaching a maximum value at 15 //sec (t* = 1). After the loading phase, it drops continuously, reaches a stable value and oscillates around that value. The time also corresponds to the peak permanent deformation of the plate. Energy dissipation occurs continuously during loading phase as well as after the load drops to zero until 140-150 //sec (t* « 10). Also, this time compares well with the time to peak side pull-in. The displacement profiles of the plate are shown in Figures 4.19 and 4.20. The deformation is seen to continue as a result of stored strain and kinetic energies. By the time of 140 //sec (t* = 9.3), most of the deformation has ceased and the plate continues to travel at constant velocity with some residual elastic vibration. These plots also show the plate pulling-in significantly from the boundary. 4.4.2.3 Mode lib With the increase in applied impulse, the plate enters mode lib failure phase. In this phase, the plate flies away from the boundary at a higher velocity and less energy is dissipated into plastic work. Therefore, the permanent central deformation of the plate decreases continuously with the increase in applied impulse. 4.4.3 M o d e l l l The plot of central displacement and kinetic energy versus time is shown in Figure 4.21 for a plate in mode III failure. Simultaneous and complete rupture of the plate occurs very Chapter 4. Square Plate Analysis Results 54 early in the loading phase. Once separation occurs, the plate begins to travel freely. Less than 15% of the input energy is absorbed by the plate. Figures 4.22 and 4.23 show the distance travelled by the free flying plate, once failure occurs and the actual deformation at each of these instances as it flies away from the boundary. The plate moves like a rigid body with elastic vibration. Figure 4.24 shows the 3-D profile of the plate indicating insignificant plate deformation, a characteristic feature of shear dominated failure. 4.4.4 Other Failure Parameters 4.4.4.1 Strain and Stress Ratios to First Element Failure The variation of strain and stress ratios (s,f) to first element failure with impulse is shown in Figure 4.25. The stress ratio dominance with increasing impulse is evident from this figure. The plots are characterized by a sharp rise or drop at a transition impulse and a level plateau at higher impulses. Both linear and quadratic failure models show the same features. At the threshold impulse to mode II failure, the contributions are roughly 50% for both models. The transition impulse marks the impulse above which the shear stress dominates the failure. Mode III threshold is defined as that impulse for which the shear stress ratio is around one. A strain based model (for example, Olson et al., 1993) fails to capture this significant influence. At mode II threshold, it was observed that both the stress and strain ratio contributions were significant. This indicates that even at the threshold of mode II failure, which has been characterized by several experimentalists as a tensile failure, shear effects Chapter 4. Square Plate Analysis Results 55 are significant. This is consistent with the recent theoretical predictions of Shen and Jones (1992). 4.4.4.2 Time to Failure The time to initial and complete failure (/*„, tc) predicted by both models is shown in Figures 4.26a and 4.26b. Both models show a monotonic decrease in the time to failure with increasing impulse. A sharp transition in the slope of these curves can be seen in these figures, which corresponds to the failure within the loading phase. This is due to the high initial shear stress ratio which can be seen in the previous figure (Figure 4.25). The difference in time between the two curves in each figure presents the amount of time elapsed before the complete rupture of the plate during which considerable energy is dissipated into plastic work. The time difference is largest at the threshold impulse for mode II failure and then decreases monotonically. After the transition, virtually identical lines are also an indication of complete rupture of the plate boundary instantaneously. This is another feature of shear dominated failure. 4.4.4.3 Permanent Central Displacement (Deflection-to-Thickness Ratio) The failure modes of an explosively loaded square plate is presented in Figure 4.27a along with the experimental data. The figure shows a plot of deflection-to-thickness ratio i, Vh) v e r s u s impulse for the experimental results as well as predicted results using the two failure models. The experimental results show that the ratio increases with increasing impulse for mode I, mode II and mode Ha, while a decreasing trend can be observed for Chapter 4. Square Plate Analysis Results 56 mode lib and mode III. Also, an overlap of impulse values and deflection-to-thickness ratios can be seen in the experimental results for various modes of failure. The numerical model depicts all modes of failure including partial failure of the plate. The trend predicted by the two failure models is in agreement with experimental results. The quadratic model consistently gives better results compared to the linear model. The figure shows a very small deformation of the plate at higher impulses which is another characteristic feature of mode III failure. The threshold impulses for mode Ila and mode lib are respectively I*m2a = 0.75 (13 Ns), I*m2b = 1.04 (18 Ns) for LIC and f m 2 a = 1.02 (17.6 Ns), r m 2 b = 1.27 (22 Ns) for QIC (Figures 4.27b and 4.27c). Nurick (1996) performed a regression analysis of experimental results and reported that complete tearing of the plate would occur at I^ 2 a = 0.99 (17 Ns impulse). Earlier numerical results put this value at 1^= 1-16 (20 Ns) for the strain based model (Olson et al., 1993, Jiang et a l , 1993), and at I*m2a= 0.85 and 1.10 (14.6 Ns and 18.9 Ns) for interactive models LIC and QIC respectively (global force balance method, Appendix E). The spring model predicts a value of I ^ 2 a = 0-75 (13 Ns) for LIC. The result of complete failure at I*m2a = 1.02 (17.6 Ns), predicted by the Quadratic Interaction Criterion, is in fact very close to experimental observation. The experimental mode III threshold is at I^ 3 = 2.69 (46.4 Ns). The determination of mode III failure based on permanent central deformation alone is highly subjective. Based on the evaluation of all the parameters presented so far, it can be concluded that mode III failure which is dominated by shear is consistent with early Chapter 4. Square Plate Analysis Results 57 failure time, instantaneous complete failure, very little deformation of plate and no side pull-in. This mode III threshold can be estimated to be at I*m3 = 2.32 (40 Ns). 4.4.4.4 Side Pull-in During the mode II failure, the numerical results indicate the pulling-in of the mid-side of the plate. This pulling-in is due to the deformation of the plate which continues during the time between first tearing of the sides and complete failure of the plate (mode Ha). Figures 4.28 and 4.29 show the plot of side pull-in versus impulse and deflection-to-thickness ratio respectively. Side pull-in increases with increasing impulse and increasing central deformation for mode Ha, as seen in these figures. The numerical results predict a maximum side pull-in of 7.2 mm for I* = 1.16 (20 Ns) impulse with the quadratic model. This compares favourably with the experimental results of 9-9.5 mm for I* = 1.39 (24 Ns impulse). Thereafter, the side pull-in decreases with increasing impulse. For mode III, the experimental results give a value of 0.78 mm corresponding to I* = 2.55 (44 Ns impulse) while the numerical results show no side pull-in. 4.4.4.5 Residual Kinetic Energy The predicted variation of the long-time residual kinetic energy with impulse and deflection-to-thickness ratio is plotted in Figures 4.30 and 4.31, along with the experimental data. The prediction shows an approximately linear increase in residual kinetic energy with impulse, and this is clearly confirmed by the experimental results. However, the predictions are higher than experiments especially at higher impulses. This may be due partly to the neglect herein of any energy loss in the tearing process. Also, Chapter 4. Square Plate Analysis Results 58 some of the experimental kinetic energy may have been in a rotation mode. There is some limited experimental data available indicating this possibility, but no comprehensive results are available to get an estimation of this rotational energy. 4.4.4.6 Centreline Failure Profiles Figures 4.19 and 4.23 show the permanent displacement profiles of the plate at I* = 1.16 and 2.90 (20 and 50 Ns). Comparison of these two figures indicate a flatter permanent profile at higher impulse. The edge of the plate subjected to an impulse I* = 1.16 (20 Ns) shows a larger permanent rotation than that of plate subjected I* = 2.90 (50 Ns). The higher rotation suggests a failure due to tensile tearing or mode II failure at an impulse of I* = 1.16 (20 Ns), whereas the lower rotation for the I* = 2.90 (50 Ns) suggests a failure where shear is more prominent as in a mode III failure. The profiles plotted in Figures 4.32a and 4.32b represent the instantaneous deformation shapes at the times of first element failure in each case for linear and quadratic models respectively. Both models predict the same behaviour, although time to first element failure changes and threshold impulse to failure is different. As mode II failure begins to occur, the plate does not have time to reach the mode I characteristic shape. The central portion of the plate remains relatively flat, and the deformation becomes more concentrated near the boundary. The corresponding numerical results for the failure strain proportions are presented in Table 4.5a and 4.5b for a range of impulses for both the models. It is seen that as the impulse increases, the proportion of bending and membrane to total strain Chapter 4. Square Plate Analysis Results 59 decreases monotonically. This also coincides with simultaneous decrease in the time to failure. Permanent deflection profiles of plates subjected to a range of impulses are shown in Figures 4.33a and 4.33b. The profiles can be divided into 3 groups. At lower impulses, the edge of the plate undergoes a large permanent rotation, suggesting a tensile tearing (mode II) associated with a significant amount of side pull-in. As the impulse increases, the plate enters the mode lib phase wherein the deformation and side pull-in start to decrease. There is also less rotation at the edge. The higher impulses yield a flatter profile, an indication of shear dominance which is characteristic of mode III. Figure 4.34 shows the profile of the plate for I* = 1.16 (20 Ns). The quadratic model predicts a profile closer to the experimental results. Chapter 4. Square Plate Analysis Results 60 Table 4.1: Displacement and shear force for a clamped plate with different grid sizes Grid size Central displacement (mm) Transverse shear force (KN/m) 1 x 1 0.736 4.29 2 x 2 0.741 4.33 4 x 4 0.741 4.39 Analytical Solution 0.746 4.46 Table 4.2: Comparison of results with different spring stiffnesses Spring stiffness Central Transverse shear (N/m) displacement (mm) force (KN/m) 1 x 106 3.329 2.58 1 x 108 0.784 4.18 1 x 1012 0.741 4.39 1 x 1015 0.741 4.39 1 x 102 0 0.741 4.39 Clamped BC 0.741 -Analytical solution 0.746 4.46 Chapter 4. Square Plate Analysis Results 61 Table 4.3: Strain distribution of plate along the boundary in mode I failure Distance from corner £ £ 1.0 0.154 0.064 0.219 0.706 0.875 0.153 0.064 0.218 0.703 0.75 0.152 0.063 0.216 0.697 0.675 0.150 0.063 0.214 0.690 0.5 0.146 0.062 0.208 0.671 0.375 0.136 0.054 0.190 0.613 0.25 0.112 0.034 0.146 0.471 0.125 0.058 0.008 0.065 0.210 0.0 0 0 0 0 Table 4.4: Central displacement during post-failure analysis hear (% of total t, (//sec) A c(mm) length) • 25 58.5 8.52 50 59.0 8.60 75 99.0 . 15.89 Chapter 4. Square Plate Analysis Results Table 4.5: Failure strain proportions to initial failure a) LIC /(Ns) 8 12 0.111 0.038 0.150 13 0.110 0.034 0.144 14 0.108 0.033 0.141 15 0.106 0.032 0.139 16 0.103 0.031 0.134 18 0.048 0.006 0.054 20 0.044 0.005 0.049 25 0.033 0.003 0.036 30 0.012 0 0.012 40 0.003 0 0.003 50 0.003 0 0.003 b) QIC /(Ns) £b £ 16.8 0.152 0.067 0.219 17 0.152 0.067 0.219 18 0.145 0.065 0.210 19 0.143 0.063 0.205 20 0.141 0.062 0.203 21 0.061 0.010 0.071 22 0.060 0.10 0.070 25 0.056 0.008 0.064 28 0.042 0.001 0.018 30 0.017 0.005 0.047 35 0.005 0 0.005 40 0.003 0 0.003 50 0.003 0 0.003 Chapter 4. Square Plate Analysis Results 63 / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / Ceiling 7 Test plate and clamping rig 5Amp fuse wire for velocity determination Wooden batching box 0 Spring steel wire 2624 mm Adjustable screws . Counter balancing £ZL masses I - Beam 7777777777777777777777777777777777777^ Recording ^ ^ p e n Tracing paper s///////////j////)////;/////;////;///////////)/7// Courtesy: International J. of Impact Engg. Figure 4.1: Experimental arrangement (Ref. Nurick et al., 1996) Chapter 4. Square Plate Analysis Results 64 Mode Ila Mode lib Mode III Courtesy: International J Impact Engg. Figure 4.2: Failure modes of an explosively loaded square plate (Ref. Nurick et al., 1996) Chapter 4. Square Plate Analysis Results 65 a> o (0 5000 4000 3000 2000 —\ 1000 -1000 0.0 Analytical result Spring force Equilibrium equations Square plate Linear elastic static analysis Load - 0.1 MPa 0.2 0.4 0.6 0.8 Distance along the boundary 1.0 Figure 4.3: Comparison of shear force along the plate boundary E 4 E c E O ra o. w c O 2 —\ « 1 0 • -1 Square plate Linear elastic analysis Step load - 0.2688 MPa Clamped support Spring support 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Time (msec) 1.4 1.6 Figure 4.4: Transient linear elastic response of plate Chapter 4. Square Plate Analysis Results 66 2.0 0 4 8 12 16 Time (msec) Figure 4.5: Transient nonlinear elastic-plastic response of plate t y A 44.5 mm I II III IV III II I B 44.5 mm Figure 4.6: Finite element model of plate (Quarter plate model) Chapter 4. Square Plate Analysis Results 67 E E, C a> E a> u _ro a. .52 15 c a> O 15 12 H 6 H Square plate I* = 0.58 • • Clamped support Spring support (Undamped) 8 12 Time (t*) 16 20 Figure 4.7: Comparison of displacement-time history of plate in mode I failure E E, c E a> u Q. tn T3 15 O O 15 12 Square plate H I* = 0.58 6 — \ Undamped 20 % damping 8 12 Time (t*) 16 20 Figure 4.8: Displacement-time history of plate in mode I failure using spring model Chapter 4. Square Plate Analysis Results 68 15 C. Square plate 12 — E E r 9 a> E a> o _ « 6 Q. w I* = 0.58 Clamped support Spring support (Undamped) Spring support (20 % damping) 0.0 ~ l ' I ' I ' I F 0.2 0.4 0.6 0.8 Distance along the centreline 1.0 Figure 4.9: Comparison of permanent displacement profiles of plate in mode I failure 0.0 0.2 0.4 0.6 0.8 Distance along the centreline 1.0 Figure 4.10: Transient deformation profiles of plate in mode I failure Chapter 4. Square Plate Analysis Results 69 a) 3-D Profile b) w-displacement contours Figure 4.11: Permanent displacement profile of plate in mode I failure Chapter 4. Square Plate Analysis Results 70 6000 4500 3000 —\ CD O O 1500 re -1500 -3000 Time (t*) Figure 4.12: Shear force-time history of plate in mode I failure o re (A (A CO 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 Square plate I* = 0.58 Clamped support (Global force balance) Spring support " 1 1 1 1 1 ' 8 12 16 20 Time (t*) Figure 4.13: Comparison of stress ratio-time history for a plate in mode I failure Chapter 4. Square Plate Analysis Results 71 -0.2 —\ 0 4 8 12 16 20 Time (t*) a) Linear Interaction Criterion CO -0.2 —\ 0 4 8 12 16 20 Time (t*) b) Quadratic Interaction Criterion Figure 4.14: Time histories of strain and stress ratios and failure function of a plate in mode I failure Chapter 4. Square Plate Analysis Results 72 0 10 20 30 40 Time (t*) Figure 4.15: Time history of central displacement, side pull-in, kinetic energy of plate in mode II* failure Chapter 4. Square Plate Analysis Results 73 "i 1 1 l r 0.0 0.2 0.4 0.6 0.8 Distance along the centreline 1.0 a) Displacement profiles upto first element failure 25 20 ? E, ^ 15 E a> re 10 a. w Q 5 —\ 0.0 Square plate I* = 0.98 -.^%> F i n a l P r o f i l e 0.2 0.4 0.6 0.8 Distance along the centreline 1.0 b) Complete response of plate Figure 4.16: Transient deformation profiles of square plate in mode II* failure Chapter 4. Square Plate Analysis Results b) w-displacement contours Figure 4.17: Permanent displacement profile of plate in mode II* failure Chapter 4. Square Plate Analysis Results 75 Square plate I* = 1.16 0 10 20 30 40 Time (t*) Figure 4.18: Time history of central displacement, side pull-in, kinetic energy of plate in mode Ha failure 30 25 — f 20 — 15 10 5 —{ Permanent deformation Profile at complete failure Profile at the time of first element failure Square plate I* = 1.16 QIC 0.0 0.2 0.4 0.6 0.8 Distance along the centreline 1.0 Figure 4.19: Post-failure deformation profile in mode Ha failure Chapter 4. Square Plate Analysis Results a) 3-D profile J I L 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0:90 b) w-displacement contours Figure 4.20: Permanent displacement profile of plate in mode Ila failure Chapter 4. Square Plate Analysis Results Figure 4.21: Time history of central displacement and kinetic energy of plate in mode III failure Chapter 4. Square Plate Analysis Results 78 70 60 — f 140 usee Square plate I* = 2.90 QIC 50 T3 a> 40 o E a> 30 u c CO w 20 10 0.0 80 usee 60 usee 40|Jsec 20 usee 1.5 |isec n 1 i 1 i 1 r 0.2 0.4 0.6 0.8 Distance along the centreline 1.0 Figure 4.22: Transient motion of plate in mode III failure Permanent deformation Square plate I* = 2.90 QIC 4 — \ ~ ~~ - ^ 140 p-sec 80 usee 60 usee 40 usee 25 ^ ^^5 — r — - r - - - - r — - r — - r - — | — - r - - - - r - — r -0.0 0.2 0.4 0.6 0.8 1.0 Distance along the centreline Figure 4.23: Post-failure transient deformation profiles of plate in mode III failure Chapter 4. Square Plate Analysis Results 79 a) 3-D Profile 0.0 0.1 0.2 0.3 0.4 0.6 0.6 0.7 0.8 0.9 1.0 b) w-displacement contours Figure 4.24: Permanent displacement profile of plate in mode III failure Chapter 4. Square Plate Analysis Results 80 J2 c o C O Q. E o o c o 1.0 0.8 — 0.6 0.4 - 0.2 0.0 Strain ratio (QIC) Stress ratio (QIC) Strain ratio (LIC) Stress ratio (LIC) ~l 1 1 1 1 1 1 [~ 1 2 Impulse (I*) "i 1 r — r Figure 4.25: Stress and strain ratios for first element failure for a plate under explosive load 12 9 —\ E += 6 I— "<5 "i r Initial failure Complete failure i — ~ i r 1 2 Impulse (I*) Figure 4.26a: Linear Interaction Criterion Chapter 4. Square Plate Analysis Results 81 12 E '+= 6 o Initial failure Complete failure b) Quadratic Interaction Criterion Figure 4.26: Initial and final failure time for a square plate under explosive load « 16 —\ w w a> c o c o u 0) Q Figure 4.27a: Plot of deflection-to-thickness ratio versus impulse for a square plate under explosive load Chapter 4. Square Plate Analysis Results 82 16 12 W (0 CD C o ±3 8 c o +3 O O >*-<B Q 4 H Square plate LIC ' Mode I Mode II Threshold to failure modes 0 Mode II* ^ Mode Ma A Mode Mb • Modelll Mode III i 1 1 1 1 1 1 1 1 1 1 1 i r 1 2 Impulse (I*) Figure 4.27b: Threshold impulse and deflection-to-thickness ratio to different modes of failure for a square plate under explosive load using LIC w (A c o 20 « 16 12 —\ c o o Q Square plate QIC Threshold to failure modes 0 Mode II* ^ Mode Ha A Mode lib • Mode III Mode I i 1 r Impulse (I*) Figure 4.27c: Threshold impulse and deflection-to-thickness ratio to different modes of failure for a square plate under explosive load using QIC Chapter 4. Square Plate Analysis Results 83 10 1 6 3 CL CO T3 CO 4 —\ 1 r LIC QIC Exptl data Figure 4.28: Side pull-in of a square plate under explosive load i — r — i — | — i — i — i — | — i — i — i — | — i — i — i — | — i — i — r 0 4 8 12 16 20 Deflection-to-thickness ratio Figure 4.29: Plot of side pull-in versus deflection-to-thickness ratio Chapter 4. Square Plate Analysis Results Figure 4.31: Plot of residual kinetic energy versus deflection-to-thickness ratio Chapter 4. Square Plate Analysis Results 85 6 0.0 0.2 0.4 0.6 0.8 1.0 Distance along the centreline a) Linear Interaction Criterion Figure 4.32: Displacement profile at the time of first element failure of a square plate under explosive load Chapter 4. Square Plate Analysis Results £ E + J c a> E CD o _re Q. w 7 3 + J C CO c re E I— a> Q. 0.0 0.2 0.4 0.6 0.8 Distance along the centreline a) Linear Interaction Criterion 1.0 E E c <D E o u re « c CD c re E V-<D Q. 0.0 0.2 0.4 0.6 0.8 Distance along the centreline b) Quadratic Interaction Criterion 1.0 Figure 4.33: Permanent displacement profile of a square plate under explosive load Chapter 4. Square Plate Analysis Results 87 Figure 4.34: Comparison of permanent displacement profile of square plate in mode I la failure Chapter 5 Sensitivity Studies 5.1 Introduction Sensitivity of the plate response to various geometric, finite element, load and material parameters under explosive load are discussed in this chapter. In section 5.2, the effect of mesh density on the failure prediction and post-failure behaviour is investigated. Section 5.3 discusses the effect of using different time step sizes for the analysis. The influence of plate thickness on the tearing and rupture of plates is presented in the following section (section 5.4). Next, the effect of strain-rate on rupture strain of material and its significance on the threshold impulse to failure is discussed in section 5.5. Finally, the effect of pulse shape on the failure modes of these plates are analysed. 5.2 Mesh Density To check the effect of mesh density on rupture prediction as well as post-failure behaviour, different grids are used to obtain the responses of an 89x89x1.6 mm plate. The material properties are as given in section 4.4. The temporal variation of central displacement of the plate (zl c) is shown in Figure 5.1 for an impulse of I* = 0.58 (10 Ns). This is typical of mode I response, wherein the displacement increases monotonically until t = 135 //sec after which the plate undergoes small elastic oscillations around the 88 Chapter 5. Sensitivity Studies 89 permanent deformation position. The results from all three grid sizes are in excellent agreement. The permanent central deformation of the plate (Ac = 12.7 mm) predicted numerically is in fact close to the experimentally obtained result of 11.1 mm. Figures 5.2a, 5.2b and 5.2c present the time histories of the strain ratio, stress ratio and failure function {s,T,f respectively) for the midpoint of the plate boundary (points A, B in Figure 4.6; first point to fail according to predictions and experiments). The analysis is carried out for an impulse of I* = 0.58 (10 Ns) using the Linear Interaction Criterion. The plate typically responds in mode I failure. The time histories of strain ratio in Figure 5.2a show essentially similar behaviour, and the difference between curves decreases as the mesh becomes denser. That is, the maximum strain at the support is less sensitive to the grid size and converges as finer meshes are used. Therefore, the tensile membrane action coupled with plate bending, which is described by the strain ratio s = s / £" r a p), will converge with the refinement of the grid. Comparisons of stress ratios calculated from different finite element meshes are presented in Figure 5.2b. The same three meshes used above are employed. During most part of the time history, the three curves are almost the same. At the very early stages of the response (0-100 //sec), there is a small difference, but as the mesh is refined the results converge. The results from the 3x3 element grid are very close to the results obtained from the 4x4 element grid. From the point of view of the failure criterion, it is the shear force in the early stages of the response which plays an important role. As discussed, shear stress ratio is almost insensitive to element size. Therefore, the shear contribution to the failure criterion (r = r / Td ull) is independent of mesh size. Chapter 5. Sensitivity Studies 90 Both the strain and shear parts of the failure criterion are almost independent of the grid size. Therefore, the rupture prediction obtained from the interaction criteria are also almost independent of mesh density. For different grid sizes, Tables 5.1 and 5.2 present the results for a plate in mode II* and mode Ha failure under an explosive load. In Table 5.1, the predicted permanent displacement (zl c) converges with finer mesh. The failure ratios required to initiate failure are more or less the same with almost equal contributions from stress and strain ratios. This is consistent with the results discussed in the previous chapter. The results from the three grid sizes in Table 5.1 show partial failure of the plate. This is an indication of the predictive capability of the numerical model. However, the amount of the tear length (/ t e a r) in each case is different. This is due to the fact that at the onset of rupture, failure progression is simulated by releasing the element sides via a node release algorithm. Different element lengths are involved at each release which influences the post-failure behaviour. The other reason for grid dependence on post-failure behaviour is the lack of any energy dissipation mechanism when element sides are released. However, its significance is less when the permanent deflection profiles are compared, as shown in Figure 5.3 for mode II* and Figure 5.4 for mode Ha. It is clear from the results that although the global response of the plate is the same for all three different meshes, better failure progression is simulated with finer mesh. Table 5.2 presents the results of a plate in mode Ha failure. The results indicate that during this failure mode, the strain ratio and shear stress ratio are almost the same regardless of the mesh size used for analysis. Chapter 5. Sensitivity Studies 91 5.3 Time Step Size The analysis using two different time steps is performed and its effect on prediction of different quantities is discussed in this section. To see the effect of time step size on partial failure mode, the analysis of the plate with the same geometric and material properties as in the previous, section, under an impulsive load of I* = 0.70 (12 Ns) is carried out using the Linear Interaction Criterion. The results are shown in Figures 5.5 through 5.7 and in Table 5.3. The analysis predicted partial failure and length of the tear to be 75% in both cases. There is a difference between the predicted values of the side pull-in in Table 5.3. This difference is not observed in other failure modes and at present, there is no explanation for this discrepancy. One possibility is that the time to failure is different and perhaps results in different values of predicted side pull-in. The strain ratio, stress ratio and failure function (e,r,f respectively) leading to first failure, which starts at the middle of the clamped boundary, is shown in Figures 5.6a, 5.6b and 5.6c for the two different time step sizes. The results are identical for the two different time step sizes. The above analyses are carried out using 4x4 grid for the quarter plate. The time history of central displacement and profile of ruptured plate in mode II*, mode Ha failure are shown in Figures 5.5, 5.7 and 5.8, 5.9, respectively. Convergence of results can be observed in these figures. Next the results from mode Ha failure is presented in Table 5.4. The results are almost similar. The results show that the different parameters associated with rupture of plates are within 2-3% for the two time step sizes. Chapter 5. Sensitivity Studies 92 This is very good when we realise that the proposed failure model is very simple and no rigorous fracture mechanics principles are applied to carry out the progression of rupture. The temporal solution procedure used in this analysis is an implicit scheme based on the Newmark-/? method with parameters y = 0.5 and /? = 0.25. The advantage of this method is that it is second order accurate and is unconditionally stable for linear problems; in that, the solution does not grow without bound even when a large time step is used. Unconditional stability of the method also applies for nonlinear problems although there is no rigorous proof in this case (Heppler, 1986). As a consequence of this condition, the method allows for the use of large time steps. Hence, time steps could simply be based on the lowest fundamental frequency (highest period) rather than on the highest frequency (which requires more effort to evaluate accurately) as would be the case i f a conditionally stable explicit scheme is used. This is particularly useful in the present work because with the usage of few elements to model the structure, a larger time step would mean a less intensive computational procedure for the analysis of explosively loaded structures. Thus, a 4x4 grid to model the quarter plate with a time step size of 0.5 //sec is found to be more than sufficient for the current analysis. This is used for the further sensitivity studies presented in the next three sections. A l l these analyses are carried out using the Quadratic Interaction Criterion. Chapter 5. Sensitivity Studies 93 5.4 Plate Thickness Menkes and Opat showed that the threshold impulse to dynamic rupture of beams in mode II and mode III depends only on the thickness of the beam and not on its length. To estimate the influence of geometric parameters on the failure of plates, a series of numerical investigations are carried out on plates having the same planar dimensions, but with different plate thicknesses. The results of this analysis are presented in Table 5.5 (a through d), which provides the various parameters for different modes of failure. Table 5.5a presents the results of a plate in partial failure mode, in which the actual impulse required to cause rupture at the boundary increases with increasing 2 5 < % < 5 0 this trend disappears and thickness. Once the plate is sufficiently thick the threshold impulse to mode II failure decreases. This is because for these plates the shear force plays a dominant role. This trend can also be seen in mode Ha failure given in Table 5.5b. As the plate enters the higher modes of failure, the trend vanishes. In mode III failure, a monotonic decrease in impulse load for increasing thickness can be seen (Table 5.5d). Figure 5.10 shows the threshold impulse (non-dimensional) for various modes of % > 5 0 require the same failure for a plate with different thicknesses. The thin plates critical impulse for the rupture of square plates. This is reflected with the horizontal line for a/h ratios > 50. However, the thick plates tend to have a lower threshold. This is because failure of these plates is still dominated by shear. This trend is also seen for mode Ha. It should be noted that the results were obtained using the Kirchhoff theory which Chapter 5. Sensitivity Studies 94 does not include the shear effects for predicting the response of thick plates. One should be cautious in interpreting the plot shown for alh < 50. The mode lib and mode III thresholds show a monotonic increase with decreasing plate thickness. In these modes, the critical non-dimensional impulse required to cause failure changes inversely with plate thickness. The failure modes of these plates are shown in Figure 5.11. In this figure, the deflection-to-thickness ratio is plotted over a range of impulses. The plot clearly indicates that, with increasing thickness, the plate undergoes less permanent deformation. The thick plates have lower threshold to failure. Also, thick plates do not exhibit the increasing and decreasing trends of permanent deformation after complete failure. The thinner the plate, the greater is the contribution from strain ratio to the failure function in mode II failure. This means that a very thin plate undergoes a large amount of bending rotation in addition to significant stretching. This is because the thin plate develops a sufficiently large inertial force very early in the response resulting in a lesser reaction force at the boundary. Thus, a higher load is required to cause mode III failure. There is a monotonic increase and decrease in the proportions of stress and strain ratio (?, s) contributions respectively with increasing thickness in all failure modes as seen in Tables 5.5a through 5.5d. On the contrary, the thicker plate due to its heavy mass develops a large reaction (shear force) at the boundary very early in the response. Also, thick plates don't show the different sub groups of mode II. Finally, the residual kinetic energy (E*) of the system in different modes of failure is shown in Figure 5.12. The velocity of a free flying plate (Vp) after complete separation Chapter 5. Sensitivity Studies 95 from the boundary is same at the threshold impulse for all thicknesses in mode lib and mode III failure. The 2 mm thick plate in mode Ha failure has a higher residual kinetic energy due to initial failure during the loading period. Figure 5.13 shows a plot of the deflection-to-thickness ratio versus impulse. The solid lines indicate the threshold impulse to the mode II and mode III failure obtained using the Quadratic Interactive Criterion for a series of square plates with different plate thicknesses. This figure can be used as a design chart and is the first of its kind. Similar charts can be obtained for plates with other material properties and can be quite useful in the preliminary design of simple plates to resist blast loads. 5.5 Strain-Rate The tearing and rupture of the square plate is predicted in the previous chapter using the static rupture strain (f„ ( / )) of material. In rate dependent materials, large strain-rates increase the flow stress for plastic deformation. This effect is well documented with sufficient experimental data supporting the effect (Jones, 1989b). However, the corresponding influence of strain-rate on ultimate or rupture strain of material is not well understood. This is partly due to the difficulties associated in conducting experiments at a constant strain-rate over large ranges of strain up to rupture. Some limited observations of experimental results indicate that the rupture strain decreases with increasing strain-rate. Following Jones (1989c, 1989d), a simple method is proposed to estimate this influence. To simplify the analysis, the following assumptions are made: Chapter 5. Sensitivity Studies 96 1. The energy to rupture is invariant. 2. The variation of strain-rate sensitive effect with strain is disregarded. The dynamic flow stress, <rd, is related to static yield strength, <r0, of material, via Cowper-Symonds relation: ad =<Jo Mil 1 + 1^" D-(5.1) where e is the strain-rate, and D and n are material constants. The dynamic rupture strain can be related to the static rupture strain e by an expression similar to the Cowper-Symonds equation: dyn-rup £rup l/n 1 + 1^" D, (5.2) This expression gives a fracture strain that is inversely proportional to dynamic stress to keep the energy absorbed invariant. Three different sets of values for D and n are used in the current study which are shown in Table 5.6. These values were first proposed by Jones (1989d) for steel materials. The square plate which was analysed in Chapter 4 is once again used for the comparative study. The effect of strain-rate on yield stress is indirectly shown in Figure 5.14, while Figure 5.15 shows a plot of rupture strain with impulse. With the increase in applied impulse, the strain-rate goes up leading to an increase in the dynamic flow stress (Figure 5.14) and a corresponding decrease in rupture strain (Figure 5.15). The rupture strain decreases quite sharply at rates of order 102, and it remains almost a constant at strain-rates of 103 and 104 as seen in Figure 5.15. Also in this figure, the threshold Chapter 5. Sensitivity Studies 97 impulse to mode II and mode III failure is shown. The decreasing rupture strain leads to a drop in the threshold impulses to failure in mode II*, while the threshold impulse to mode III remains unchanged. This is mainly because the drop in rupture strain of material due to high strain-rates can have influence only on strain dominated failure modes, which is mode II. The shear stress developed by the plate remains unchanged, as is the threshold impulse to transverse shear failure. A l l analyses are carried out using a Quadratic Interaction Criterion. A failure envelope can be seen which encompasses the entire range over which the material would fail leading to rupture at the boundary. Table 5.7 presents the various parameters associated with the threshold impulse to failure for a plate with different material constants. At the onset of rupture, the plate is ruptured only partially with the tear length ranging from 50-75% of the clamped boundary. The failure is dominated by strain which is reflected by a very high strain ratio to failure. 5.6 Pulse Shape The load-time characteristic (or pulse shape) is one of the difficult parameters to predict in experiments. A l l the earlier analyses are carried out by assuming a rectangular pressure pulse. To see the effect of pulse shape, analysis of the plate subjected to a triangular pressure pulse (instantaneous rise to maximum pressure) with linearly decreasing pressure is carried out. The triangular pulse has the same duration (15 //sec), and only the pressure intensity is doubled to keep the applied impulse as constant. The load is applied uniformly over the entire plate surface. Chapter 5. Sensitivity Studies 98 The failure modes for the plates under two pulse shapes are shown in Figure 5.16. In this figure, the deflection-to-thickness ratio over a range of impulses are plotted using the Quadratic failure model. From the figure, it is clear that the permanent central deformation of plate in mode I failure range is virtually the same in both cases. This result once again confirms that mode I response can be considered a function of the impulse rather than the peak pressure. However, it should be noted that in all cases the duration of loading was always held constant and higher loads were applied by simply increasing the magnitude of pressure. This result is certainly consistent with the predictions of Youngdahl (1970, 1971) and Fagnan (1996). The threshold impulse to mode II* failure is I*m2, = 0.98 (16.8 Ns) for the rectangular pressure pulse and I*n2, = 1.0 (17.3 Ns) for the triangular pressure pulse. In the current analysis, the loads are assumed to have an instantaneous rise to peak pressure. Fagnan (1996) reported studies on the response of circular plates to triangular (isosceles) pressure pulse. He reported that the threshold impulse to mode II failure of circular plates was almost the same for all pulse shapes. The current result matches well with his observations. Thus, the mode II failure threshold is sensitive only to the applied impulse. The figure clearly indicates that the triangular pressure pulse has a lower threshold to mode III failure than the rectangular pressure pulse. This is due to the fact that the failure occurs before the end of the loading phase and hence it is sensitive to the magnitude of the applied pressure. Thus, the response in mode III failure is somewhat dependent on the pulse shape. Chapter 5. Sensitivity Studies 99 Table 5.1: Comparison of results with grid sizes for a plate in mode II* failure Grid size \ (mm) tin (usee) hear (%) T £ 2x2 9.62 1.16 67.5 50 0.50 0.50 3x3 10.06 2.94 53.5 67 0.54 0.46 4x4 10.28 2.84 50.5 75 0.52 0.48 Table 5.2: Comparison of results with grid sizes for a plate in mode Ha failure Grid V tin tc '-'res T 6 size /h (mm) (//sec) (//sec) 3x3 12.80 4.28 34 138 12.8 0.54 0.46 4x4 12.42 4.90 31.5 117.5 44.8 0.55 0.45 Table 5.3: Comparison of results with time steps for a plate in mode II* failure Time step V £ T (//sec) Vh (mm) (//sec) 0.5 9.53 2.04 62 0.48 0.52 1.0 9.32 1.16 74 0.50 0.50 Chapter 5. Sensitivity Studies 100 Table 5.4: Comparison of results with time steps for a plate in mode Ila failure Time V 4 , tin tc vP £ T Step /h (mm) (//sec) (//sec) (m/sec) (//sec) 0.5 12.43 4.90 31.5 117.5 60.2 0.45 0.55 1.0 12.28 4.72 32 119 58.1 0.45 0.55 Table 5.5: Comparison of results with plate thickness in different modes of failure a) Mode II* h (mm) (Ns) Vh tin (//sec) \ (mm) £ T 1.0 11.9 25.04 58.5 4.46 0.86 0.51 1.6 16.8 13.29 62.5 2.68 0.71 0.71 2.0 19 9.46 60 2.50 0.61 0.80 3.2 8.4 1.91 14 0 0 1.0 b) Mode Ila h (mm) (Ns) V tin (//sec) tc (//sec) '-'res (J) \ (mm) £ T 1.0 13 30.39 32.5 161 10.3 8.0 0.84 0.56 1.6 17.6 14.69 42.5 154.5 13 6.40 0.68 0.73 2.0 19.4 13.66 14.5 108 93 1.52 0.16 0.99 3.2 9.5 1.02 9.5 14.5 34 0 0.01 1.00 Chapter 5. Sensitivity Studies 101 c) Mode lib h (mm) (Ns) V Vh tin (//sec) tc (//sec) '-'res (J) 4p (mm) T 1.0 22 18.69 13.5 14.5 688.8 8.54 0.50 0.88 1.6 22 8.88 13.5 14 406.1 5.42 0.23 0.98 2.0 21 5.45 12 13 299.5 1.88 0.12 0.99 3.2 9.7 0.36 9.0 13.5 36.5 0 0.01 1.00 d) Mode III h (mm) ',,,3 (Ns) Vh tin (//sec) tc (//sec) '-'res (J) \ (mm) s T 1.0 50 4.28 1.5 2.5 4725.0 0.28 1.06 0.02 1.6 40 1.33 1.5 2.5 1881.7 0.0 0.01 1.00 2.0 35 0.47 2.0 2.5 1146.7 0.0 0.01 1.00 3.2 25 0.0 2.0 2.5 362.2 0.0 0.00 1.00 Table 5.6: Cowper-Symonds material constants for calculating dynamic rupture strain SI No D n 1 40.4 5 2 800 5 3 6340 5 Chapter 5. Sensitivity Studies 102 Table 5.7: Comparison of results at the threshold impulse to mode II* failure using dynamic rupture strain D, n ^ dyn-rup Im2* (Ns) V Vh \ (mm) tin (//sec) £ T hear (%) 40.4, 5 0.10 8.0 6.43 0.90 67.5 0.93 0.37 50 800, 5 0.14 10.5 8.52 1.70 62 0.89 0.45 75 6340,5 0.17 12 9.73 2.22 60 0.86 0.52 75 Chapter 5. Sensitivity Studies 103 E E c E 0) u « Q. (A •a 75 i_ C d) O "i | i | r 50 100 150 Time (micro-sec) 200 250 1.0 0.8 Figure 5.1: Temporal variation of central displacement of a plate in mode I failure with grid size Square plate I* = 0.58 A, B = Midpoint on boundary +5 0.6 C "<5 5 0 4 0.2 —\ 0.0 i I 50 100 150 Time (micro-sec) 200 250 Figure 5.2a: Strain ratio Chapter 5. Sensitivity Studies 104 Figure 5.2b: Stress ratio c) Failure function Figure 5.2: Comparison of time histories of strain ratio, stress ratio and failure function of a plate in mode I failure with grid size Chapter 5. Sensitivity Studies 105 Square plate I* = 0.72 LIC 0.0 0.2 0.4 0.6 0.8 1.0 Distance along the centreline Figure 5.3: Comparison of permanent deflection profile of a plate in mode II* failure with grid size Figure 5.4: Comparison of permanent deflection profile of a plate in mode Ha failure with grid size Chapter 5. Sensitivity Studies 1 Figure 5.5: Comparison of central displacement time history of a plate in mode II* failure with time step size 1.0 0.8 Square plate I* = 0.70 o.50 usee 1.00 usee Z: 0.6 *j 0.4 0.2 —\ 0.0 -| i | i | r 20 40 60 Time (micro-sec) Figure 5.6a: Strain ratio 80 100 Chapter 5. Sensitivity Studies 107 1.0 0.8 "-=. 0.6 (A tn CD •fc 0.4 CO 0.2 Square plate I* = 0.70 •i i/ \ i o.50)i sec i.00(i sec 0.0 ~~I 1 1 1 1 T ~ 20 40 60 Time (micro-sec) Figure 5.6b: Stress ratio 80 100 1.0 c o "5 0.6 c 3 «t— 0) 3 0.4 "<5 Square plate I* = 0.70 LIC 0.8 —\ • 0.50 (isec i.00(isec 0.2 0.0 20 40 60 80 Time (micro-sec) c) Failure function Figure 5.6: Variation of strain ratio, stress ratio and failure funtion with time step size 100 Chapter 5. Sensitivity Studies 1 20 ? 15 E c CD E o o ra Q. W 5 Square plate I* = 0.70 LIC 10 0.50 \xsec i.oo usee " 1 i | i | I F 0.2 0.4 0.6 0.8 Distance along the centreline 0.0 1.0 Figure 5.7: Comparison of permanent deflection profile of a plate in mode II* failure with time step size Chapter 5. Sensitivity Studies 1 0.0 0.2 0.4 0.6 0.8 1.0 Distance along the centreline Figure 5.9: Comparison of permanent deflection profile of a plate in mode Ha failure with time step size Chapter 5. Sensitivity Studies 11 CO •_ _3 O *•> CD w 3 Q. E 33 o £ (/) CO 2 H Square plate QIC Mode II* Mode Ila Mode Mb Mode III 20 40 60 a/h 80 100 Figure 5.10: Comparison of threshold impulses to failure for a plate with different thicknesses Impulse (I*) Figure 5.11: Comparison of deflection-to-thickness ratio versus impulse of plate with different thicknesses Chapter 5. Sensitivity Studies 11 1.0 LU >» O) i_ <u c o "43 CD c 0.8 0.6 0.4 —\ Square plate QIC Mode lla Mode lib Mode III CO CD cr. 0.2 o.o 20 40 60 a/h 80 100 Figure 5.12: Comparison of residual kinetic energy of plate with different thicknesses Chapter 5. Sensitivity Studies 1 Figure 5.13: Threshold impulse to different modes of failure for plate with different thicknesses using QIC Chapter 5. Sensitivity Studies 113 6000 4000 CO -+-» (0 I c "re +J CO 2000 Strain-rate Yield stress 1000 800 600 h — 400 200 0 1 2 : Impulse (I*) Figure 5.14: Variation of strain-rate and yield stress with impulse re D_ <A tn CO <A T3 CO '>< o E re c >. G 0.4 0.3 c "re (A 2 0.2 3 Q. 3 0.1 —\ 0.0 Square plate QIC Mode II* threshold Static rupture strain Mode III threshold D,n = 6340, 5 D,n = 800, 5 I D,n = 40.4, 5 1 2 Impulse (I*) Figure 5.15: Dynamic rupture strain and threshold impulse to failure for a plate under explosive load Chapter 5. Sensitivity Studies 1 0 1 2 3 Impulse (I*) Figure 5.16: Plot of deflection-to-thickness ratio versus impulse for a plate under different load-time characterstics Chapter 6 Stiffened Plate Analysis Results 6.1 Introduction The analysis in Chapters 4 and 5 is now extended to the analysis of stiffened plates. Springs are introduced at all possible locations of failure (boundary as well as plate-stiffener interface) to evaluate the influence of shear on the failure of these structures. The equations of motion to be solved in this case are \M'\ [o] ' 10] ["1. i n -K1 [C] [0] [o] [c1. M l M l {"11 H I + jo} (6.1) where is the global spring stiffness matrix; F ^ ^ j j and F^^Jjare the global plate and beam internal force vectors; M p j , [^*], [ C J and [c*] are global mass and damping matrices for plate and beam respectively; and j ^ j , |<?*| and {P} are the global displacement and load vectors respectively. The equations are solved by the implicit Newmark-/? method with Newton-Raphson iteration within each time step as discussed in Chapter 3. The effect of structural damping has been ignored in the analysis, while 115 Chapter 6. Stiffened Plate Analysis Results 116 stiffness proportional damping is introduced to remove the high frequency oscillatory motion of springs. The experimental observation of stiffened plates under explosive loading is presented in section 6.2. The linear elastic as well as large deflection elastic and elastic-plastic responses are investigated to validate the spring model. Both static and dynamic analyses are carried out and the results are discussed in section 6.3. Sections 6.4 and 6.5 are devoted to the response of stiffened plates subjected to blast wave load. 6.2 Experimental Observations Nurick et al (1995) conducted experiments on fully built-in stiffened plates. A fully built-in plate is one in which the plate and the support boundary are integral. This is different from the arrangement of unstiffened plate experiments where the plates were clamped between clamping blocks. A 89x89x1.6 mm square stiffened plate with rectangular stiffeners of four different sizes (2, 4, 5, 9 mm deep and 3 mm wide) were used in the study. The procedure used for the blast experiments was similar to that described in Chapter 4 and the material properties were obtained from uniaxial tensile test of mild steel. Most of the tests reported are in the mode I range. The results indicate that in mode I failure, midpoint deflection increases with the increase in applied impulse. This is true for all the stiffened plates. A limited number of results were reported in mode II failure. A series of stiffened plates in different modes of failure are shown in Figure 6.1. Tearing starts at the middle of the plate boundary parallel to the stiffener (mode IIB) for a plate with 3 x 2 - mm and 3 Chapter 6. Stiffened Plate Analysis Results 117 x 4 - mm stiffeners, while localised tearing of the plate at the stiffener (mode IIS) was observed in all the 3 x 9 - mm stiffened plates as shown in this figure. 6.3 Spring Model Verification: Response without Failure 6.3.1 Static Analysis Results This section presents the results from the static analysis of a 2-bay stiffened plate I shown in Figure 6.2. This example was used by Koko (1990) who provided the results for a clamped boundary condition. Using symmetry, one-half of the structure is modelled by one plate element and one beam element. The relevant beam section properties have to be halved in the computation since the symmetry line divides the beam into two. The results from the linear and nonlinear geometric and material responses are shown in Table 6.1. The linear elastic panel centre and stiffener centre deflections (points A and B respectively in Figure 6.2) at a load of 0.001 N/mm 2 are given in Table 6.1 along with those from a finite strip analysis using 8 strips for one bay of the structure. The panel centre deflection is overestimated by 2.2%, and the model underestimates the strain energy by 3.7%, with respect to the one mode finite strip solution. The effect of different spring stiffnesses on the response of the stiffened plate is shown in Table 6.2. Using stiff springs, the results approach the response of a stiffened plate with clamped boundary condition. The nonlinear elastic and elastic-plastic analyses are carried out under a uniformly distributed load of 0.4 N/mm 2. The results are in excellent agreement with that of the clamped boundary. The spring stiffness of l x l O 1 5 N/m used in the current analysis is Chapter 6. Stiffened Plate Analysis Results 118 more than adequate to simulate the clamped condition of problem. The last column in the table presents the spring force developed at the centre of the plate between the plate and the stiffener in each case. 6.3.2 Dynamic Analysis Results Dynamic analysis of the plate under a step load of 0.3 N/mm 2 is carried out to assess the suitability of the spring model for such analysis. The configuration and material properties of plate II are shown in Figure 6.3. Table 6.3 presents the linear and nonlinear elastic response of this plate along with previously published results (Koko, 1990). The frequency of the structure, peak panel and stiffener centre deflection (points A and B respectively in Figure 6.3) matches very well with that of a rigidly clamped structure. The linear elastic time history of panel centre and stiffener centre deflection are shown in Figure 6.4. The two results are in excellent agreement with the clamped plate results. Next, the nonlinear geometric analysis is carried out and the time history of deflection at the above two locations is shown in Figure 6.5. Once again, the results indicate the predictive capability of model. The spring model proves to be a very versatile model and is used for the failure analysis of stiffened plates. The results are discussed in the next section. Chapter 6. Stiffened Plate Analysis Results 119 6.4 Onset of Failure 6.4.1 Uniform Load A transient analysis is now performed on a one-way stiffened steel square plate with clamped edges. The geometric and material properties of the test plate are as shown below: dimensions of plate = 89 mm x 89 mm x 1.6 mm elastic modulus, E = 1 9 7 GPa tangent modulus, ET = 250 MPa Poisson's ratio, v = 0.3 static yield stress, cr0 = 265 MPa rupture strain, srup = 0.18 static ultimate shear stress, rull = 179 MPa density, p = 7830 Kg/m 3 D = 40.4 n = 5 spring stiffness, ks = 1 x 1015 N/m A series of different stiffener sizes are used in the current study. A l l stiffeners are rectangular in shape with 3 mm width. Four different depths (H), 2 mm, 4 mm, 5 mm and 9 mm are used in the analysis. The plates are designated as 3 x 2 - mm, 3 x 4 - mm, 3 x 5 - mm and 3 x 9 - mm stiffened plate for all future reference, in view of stiffener size. The Chapter 6. Stiffened Plate Analysis Results 120 configuration of the plate and the finite element model used for the analysis are shown in Figure 6.6. To compare with available experimental results, blast wave loading is considered. The pressure loading from the explosive charge is assumed to be uniformly distributed over the plate surface and of rectangular wave form in time. The duration of the loading is assumed to be 15 //sec, equal to the approximate explosive burn time. The pressure loading magnitude is then calculated to correspond to the measured impulses (area under the pressure-time curve). The temporal variation of stiffener centre (point C) displacement of a 3 x 2 - mm stiffened plate with and without damping is shown in Figure 6.7. In the damped analysis, a 20% damping is used to suppress all the high frequency motion of the springs. The results are virtually identical in both cases. The permanent deflection profiles of the plate in the two cases are shown in Figure 6.8. The permanent deformation predicted by the spring model is about 1% less than the deflection predicted by undamped analysis. 6.4.1.1 Model The analysis was carried out over a range of impulses between 5 and 30 Ns for each stiffener case. The analysis was performed using a 2 x 2 grid for the quarter plate model unless stated otherwise, with a time step size of 0.5 //sec. A 20% damping is used to suppress all the high frequency motion of the springs. The time history of stiffener centre deformation (point C) and kinetic energy in the system is shown in Figure 6.9, for a 3 x 2 - mm stiffened plate under 5 Ns impulse. The central deformation increases monotonically and reaches a peak value around 120 -Chapter 6. Stiffened Plate Analysis Results 121 125 //sec. This time to peak displacement is slightly lower than that of an unstiffened plate which occurs around 135 - 140 //sec. The kinetic energy increases until the end of the loading phase, that is 15 //sec and decreases afterwards. A l l the input energy is absorbed by the plate in undergoing deformation. The time to absorb all the energy coincides with the time to peak displacement. This is typical of mode I deformation. The plate vibrates in an elastic manner about the permanent deformation position once energy absorption is complete. The transient deflection profiles of a 3 x 2 - mm stiffened plate are shown in Figure 6.10. The dashed lines in the figure indicate the instantaneous profile of the plate at every 5 //sec interval, between 0 and 100 //sec while, the solid line represents the permanent displacement profile (mode I) of the plate. During the loading phase, the energy is continuously input to the system and simultaneously a portion of it is dissipated into plastic work. The net result is that the kinetic energy steadily increases until the end of loading phase. The profile at the end of loading phase (15 //sec) is indicated in the above figure which shows that the plate has undergone a displacement of 0.4 mm. This value increases substantially to one or two plate thicknesses as the applied load is increased until mode II threshold is reached. The analysis is carried out by enforcing the zero slope condition at the boundary. However, all the figures showing the centreline displacement profile (displacement along CA) are obtained by drawing straight lines through the nodal data points and, hence, show a nonzero slope at the boundary (location-0.0). Chapter 6. Stiffened Plate Analysis Results 122 In Chapter 4, the result from an unstiffened square plate indicated that the plate moves in the transverse direction with almost uniform velocity, which was observed in the flat profile of the plate. In a stiffened plate, observation of transient profile indicates a maximum between the boundary and stiffener. This is due to the fact that initially the stiffeners will move with less velocity because their respectively larger mass/unit area has to be accelerated by the explosive loading. This effect is more pronounced in deep stiffeners ( 3 x 9 - mm) than the thin ( 3 x 2 - mm) ones at high impulses. This velocity difference will lead to relative deformation of the plate between the boundary and stiffeners. As the deformation develops, the velocities become equal and the plate-stiffener configuration deform together. Figure 6.11 shows the time history of the stiffened plate central deformations for plates with different stiffener sizes under an applied impulse of 5 Ns. The figure clearly shows that the stiffened plate responds to large deformation almost linearly with time, reaches a maximum and exhibits a small amount of residual elastic vibration. The time to reach peak displacement is almost the same, thus independent of stiffener sizes. However, the magnitude of peak displacement varies inversely with stiffener size. The predicted mode I failure deflection profiles are shown in Figure 6.12 for all the stiffened plates for the same impulse of 5 Ns. It is seen that the permanent central deformation decreases continuously with the increase in stiffener depth. Also, there is a significant change in the permanent deflection profiles of these plates. The figure also shows the mode I displacement profile of an unstiffened square plate of the same size. The maximum displacement always occurs at the plate centre in an unstiffened plate. Plates with thin stiffeners show the maximum displacement occurring at the stiffener Chapter 6. Stiffened Plate Analysis Results 123 centre. This location shifts to a point between the boundary and the stiffener for plates with thick stiffeners. This transition is more pronounced in 3 x 9 - mm stiffened plate. The maximum deflection of 3 x 2 - mm and 3 x 9 - mm stiffened plates is about 20% and 55% less respectively than the corresponding displacement of an unstiffened square plate made of the same material. Figure 6.13a through 6.13d show the predicted 3-D profiles of all four stiffened plates in mode I failure at 5 Ns impulse. The influence of stiffener size (depth) on the overall deflection of the plate can be seen in this figure clearly. The resisting effect of deep stiffeners leads to a maximum displacement between the plate and the stiffener, as explained before. Several numerical investigations are carried out by modelling the structure with different grid sizes. A comparison of predicted mode I deflection profiles of the plate for the 3 x 2 - mm stiffened plate is shown in Figure 6.14. It is seen that the finer grid provides a slightly sharper curvature at the boundary and the profile matches very well with that of the coarse grid. The predicted permanent central displacement by the fine grid (3 x 3) is only 1.5% more than the coarse grid (2 x 2). Table 6.4a and 6.4b show the central displacement (4.) and maximum strain at the boundary of the plate (point A in Figure 6.6b) with different stiffeners under 5 Ns for the two grid sizes. The convergence of results can be seen from the above two tables. The bending and membrane strains are the same at A , thus independent of stiffener sizes. Figure 6.15 shows the variation of strain along the boundary parallel to the stiffener. It can be seen that the maximum strain occurs at the midpoint on boundary Chapter 6. Stiffened Plate Analysis Results 124 (point A) and the maximum strain value is independent of stiffener size. This can be also be observed by noticing the same rotation of all stiffened plates at the boundary from Figure 6.12, which shows the permanent deflection profiles of the plate. Temporal variations of strain ratio, stress ratio and failure function (e,r,f respectively) for the midpoint on a boundary parallel to a stiffener and for the stiffener centre (points A and C respectively in Figure 6.6b) are shown in Figures 6.16 and 6.17 respectively. The results are for a 3 x 2 - mm stiffened plate under 5 Ns impulse. The strain ratio at A increases monotonically to peak value around 90 /usee and remains more or less constant. The time to peak strain is less than the time to reach peak displacement. This is true with all stiffener sizes. The stress ratio at A increases until 15 //sec and drops when the applied load drops to zero. It reaches another lower maximum when the inertial forces in the system become significant. Thereafter, it drops and oscillates around the zero mean value. The corresponding variation of failure function at A is also shown in Figure 6.16. It reaches a maximum value at the end of loading phase due to high initial shear stress ratio. The failure function drops when the applied load drops to zero. It increases again and reaches the maximum value when the inertial force of the system is substantial. At this stage, the plate has undergone a significant amount of stretching and bending contributing to the failure function via strain ratio. The time to peak failure function occurs around 90 //sec, which also coincides with peak strain. This time to peak failure function changes slightly with the applied impulse but remains the same for all stiffener sizes. Chapter 6. Stiffened Plate Analysis Results 125 Similar variation of these functions with time for the stiffener centre of the above plate (point C) under the same load is shown in Figure 6.17. Both strain and stress ratios have a much smaller magnitude compared to the boundary point. This is true with all stiffener sizes under a uniformly applied load. The strain at C is always less than that of A . The experiments showed failure of the stiffener at the interface of the beam and plate near the centre for a 3 x 9 - mm stiffened plate (Figure 6.1). This is considered to be a shear failure and not a strain dominated failure. The maximum shear stress at C (Figure 6.17) always occurred at the end of the loading phase. During the loading phase, the shear stress at C is of comparable magnitude to that of A (Figure 6.16). This is especially true for a stiffened plate with deep stiffeners. Thus, the failure of the stiffener-plate interface could be the result of high initial shear stress during the loading phase. The time to peak displacement is less than that of an unstiffened plate of the same size. Similarly, the time to peak strain and the second peak of shear stress and the time to peak failure function are all less than the corresponding time for an unstiffened plate. 6.4.1.2 Mode II As the applied impulse is increased, the threshold impulse to cause rupture of stiffened plates is reached. The threshold impulses are 9.5 Ns with LIC and 13.3 Ns with QIC using a 2 x 2 grid for the analysis. The changes in strain ratio and stress ratio as a function of stiffener depth are shown in Figures 6.18a and 6.18b respectively. The figure shows the response of stiffened plates under an impulse of 9.5 Ns, threshold impulse to cause rupture using LIC at A and C. These are the points observed to exhibit failure initiation in Chapter 6. Stiffened Plate Analysis Results 126 experiments. Hence, they are chosen to study the rupture initiation in these plates. The values at these two locations correspond to those obtained before the end of the loading phase. Figure 6.18a shows that although there is some change in the strain ratio with a change in stiffener size, these changes are insignificant. However, the stress ratio corresponding to that time exhibits a much larger change with stiffener size. For the same loading, a 3 x 9 - mm stiffened plate develops 25% more shear force at the interface than a 3 x 2 - mm stiffened plate at the same location (Figure 6.18b). This is the first indication towards the possible failure of the stiffener-plate interface. These values are much below the threshold to cause failure during the loading phase. Figures 6.18c and 6.19 show the corresponding variation of the failure function with stiffener depth at the threshold impulse to failure using Linear and Quadratic Interaction Criteria respectively. The values shown are at the end of the loading phase, to reflect the effect of high initial shear stress on the failure of the stiffener-plate interface. Both figures show that the failure function has a higher value at A than that of C. The failure function at A shows a slight increase with stiffener depth, while the function at C increases monotonically. The gap between them decreases linearly with the increase in stiffener depth. However, these values are well below the threshold to failure. Tables 6.5a and 6.5b present the time to first element failure (tin), strain ratio (s), stress ratio (r) and displacement at the stiffener centre at the time of first failure (zlc,) for all stiffened plates. The applied impulse is just enough to cause mode II failure in all plates. From the table, it is evident that the mode II threshold is independent of stiffener Chapter 6. Stiffened Plate Analysis Results 127 size. Also, the contribution of strain and stress ratio to the failure function which identifies the rupture is almost the same for all sizes of stiffeners. Although the time to failure is also the same, the central displacement at failure decreases monotonically with the stiffener size. The failure at this impulse occurs due to a combination of stress and strain ratios much after the loading phase is over. This is typical of mode II failure which is observed in the unstiffened plates. This strain domination is evident when the strain contribution to reach failure is observed which is characteristic of mode II failure. However, the failure function also has a significant contribution from stress ratio at the threshold impulse (46% of failure function is due to the shear stress ratio at the threshold impulse in stiffened plates, while close to 50% of failure function is due to shear stress ratio in an unstiffened plate when LIC is used for the failure analysis). A strain based criterion would miss this significant effect and thus give a higher threshold to failure (Nurick et al, 1995). The time to failure also coincides with the plate developing very high inertial forces at these times resulting in high shear stress ratios shown in the tables. The results in Table 6.6 were obtained using a 3 x 3 grid under 9.5 Ns of impulse, with LIC to detect rupture. Once again, the numerical results converge when a finer mesh is used to model the stiffened plate. The last column in the two Tables 6.6a and 6.7 shows the mode I displacement (4 „ i ) m a t the p l a t e would have achieved if failure was ignored. Figure 6.20 (a through d) shows the variation of failure function as a function of impulse for different stiffener sizes. The values correspond to the time just before the end of the loading phase using LIC. The failure function at A is consistently higher than that Chapter 6. Stiffened Plate Analysis Results 128 of C at all impulses for all stiffener sizes. That is, failure always starts at the middle of the clamped boundary parallel to the stiffener (point A). Although the difference between the two failure functions reduced significantly with the increase in stiffener size, this was not enough to cause the failure at the stiffener centre as observed in experiments for deep stiffeners ( 3 x 9 - mm). At higher impulses (> 14 Ns), the failure occurs before the loading phase is over (< 15 //sec) due to high values of initial shear stress ratio. This behaviour is also consistent with the results of an unstiffened plate seen in Chapter 4. The predicted stiffener centre displacement is plotted against impulse in Figures 6.21 through 6.24 for 3 x 2 - mm to 3 x 9 - mm stiffened plates, respectively, along with the experimental data. The predicted mode I displacements are slightly higher in the case of 3 x 2 - mm stiffener, but the values are in excellent agreement with experimental data in the case of 3 x 4 - mm and 3 x 5 - mm stiffeners. This is because the analysis for the 3 x 2 - mm case is carried out using the static yield stress of 265 MPa, while the experimental plates had the static yield strength of 302 MPa. Clearly an increase in the yield stress would decrease the permanent displacement predicted by the analysis. Hence, this could be a reason for this slightly higher value. However, the 3 x 9 - mm stiffener in experiments has a much higher central displacement compared to the analysis. The other aspect is that these stiffeners also failed at the interface between the beam and the plate near the centre, which was not predicted by the analysis. 6.4.2 Nonuniform Load From the Figures 6.20 (a through d), it is clear that under the applied uniformly distributed load, the failure always occurred first at the supported boundary parallel to the Chapter 6. Stiffened Plate Analysis Results 129 stiffener. On the other hand, the experiments showed localised tearing at the stiffener-plate interface for a plate with deep stiffener (Figure 6.1). In an attempt to explain this discrepancy, it is worth noting that in the experiments the explosive was laid out on a 12-mm thick polystyrene pad in two concentric square annuli, which were interconnected by two perpendicular strips of explosive called cross-leaders. A short tail of explosive holding the detonator was then attached to the centre of the cross-leaders. The mass of explosive for the tail and the cross-leaders was constant for all the tests. The impulse which was applied to the flat side of the plate was increased by increasing only the mass of the annuli explosive. Since there is an extra mass of explosive along with detonator at the centre of the plate, it was thought this might have introduced a nonuniform load distribution. Therefore, a series of analyses with a nonuniform load distribution over the plate were undertaken. Two different nonuniform load distributions are considered in the analyses. In this, the element next to plate centre is subjected to a higher load magnitude compared to the remaining elements. However, the total impulse applied to the structure is kept the same. These are termed as Nu-2 and Nu-3 respectively, depending on whether the central element is subjected to two or three times the load applied to other elements. Once again, the plate response is computed over a range of impulses for all stiffener sizes. 6.4.2.1 Model Figure 6.25 shows the response of various stiffened plates under an impulse of 5 Ns. The structure is modelled using a 2 x 2 grid, that is 4 plate elements and 2 beam elements for Chapter 6. Stiffened Plate Analysis Results 130 the quarter plate. The element next to the plate centre in this case is subjected to twice the load applied to the other elements (load distribution: Nu-2). The response is typical of mode I failure, where the central deformation increases continuously to reach the peak value. The time to peak displacement is about 125 /usee for all stiffener sizes and is the same for the uniformly distributed load also. However, the magnitudes of displacements are different and they decrease with increasing stiffener depth. Also, the rate at which the plate undergoes deformation is inversely proportional to stiffener depth. The permanent displacement profiles of the plate in mode I failure for the above case are shown in Figure 6.26. The permanent central displacement for each stiffener is higher than the corresponding value obtained with a uniformly distributed load. The relative difference is highest for thick stiffeners and decreases as the depth decreases. However, the maximum displacement for the thick stiffeners (3 x 9 - mm stiffener) still occurs at a point between the stiffener centre and the boundary. The other important feature is that, although the deflection profiles are quite different, they all have same amount of bending rotation. This once again proves that the failure when possible will occur at the same impulse, i f the stress ratio contributions are the same and will occur at the same time for all stiffened plates. The time history of stiffener central displacement for uniform and nonuniform load distribution (Nu-2, Nu-3) is shown in Figure 6.27. The results are for a 3 x 2 - mm stiffened plate under an impulse of 5 Ns. A l l three curves show a monotonic increase in displacement value. The time to peak displacement, however, decreases with nonuniform Chapter 6. Stiffened Plate Analysis Results 131 load distribution due to the higher rate at which the plate deforms. The higher the localised load near the plate centre, the higher the magnitude of displacement also. Figure 6.28 shows the predicted mode I failure displacement profile for all three cases. The permanent central displacements are 4.89 mm for the uniform load and 5.87 and 6.38 mm for the nonuniform load distribution, Nu-2 and Nu-3 respectively. There is a 17% and 24% increase in the permanent central deformation for the load cases Nu-2 and Nu-3 respectively. The element next to the plate centre is subjected to 37.5% and 50% higher load when compared to the uniformly distributed load magnitude. The increase in central deformation does not reflect the same increment as that of the load. The other elements are subjected to a 20% and 33.3% less load respectively when Nu-2 and Nu-3 are applied on a 2 x 2 grid. Also due to the higher localised load, the plate shows a slightly sharper curvature at the boundary. This effect of increased central deformation and curvature at the boundary is true for the other three stiffener sizes also. Figure 6.29 shows the predicted mode I deformation of all four stiffened plates. The applied impulse is 5 Ns and the analysis is carried out by using a 2 x 2 grid for the quarter plate. There is a uniform increase in the permanent central displacement of plates due to nonuniform loading for different stiffener sizes. The magnitude of permanent deformation decreases with increasing stiffener size. Table 6.7 shows the maximum strain at A for all three load distributions. There is a steady increase in the strains, experienced by the plate under nonuniform load. Chapter 6. Stiffened Plate Analysis Results 132 6.4.2.2 Mode II Figures 6.30 and 6.31 shows the variation of failure function as a function of stiffener depth under the nonuniform loading Nu-2 and Nu-3, respectively. Both plots are for a load corresponding to 5 Ns impulse and are analysed using a 2 x 2 grid and the Linear Interaction Criterion. In addition, Figure 6.30 also shows the changes in failure function with stiffener depth for the finer grid (3 x 3). The impulse is typical of an impulse causing mode I failure. The figure is plotted at 14.5 //sec, just before the end of loading phase, to capture the effect of high initial shear stress. The failure function at the midpoint on the boundary (A) in both figures remains almost a constant, thus independent of stiffener size. The same is not true with the failure function at the stiffener centre (C). The localised load tends to increase the shear stress at the stiffener centre. This effect is more pronounced with thick stiffeners ( 3 x 9 - mm stiffener) where the initial shear stress at the centre is more than that at the boundary. The cross-over occurs around 7 mm and 6 mm depth for the coarse and fine grid respectively, with load distribution Nu-2. The threshold moves to the left as seen in Figure 6.31, with higher concentration of load near plate centre. The failure function at the boundary drops from 40% to 36% as load distribution changes from Nu-2 to Nu-3 while the stiffener centre failure function at the same time increases from 20% to 25% for a 3 x 2 - mm stiffened plate. There is a monotonic increment in the value of failure function at the stiffener centre with increase in stiffener depth. The 3 x 9 - mm stiffened plate shows an increment from 45% to 57% in the value Chapter 6. Stiffened Plate Analysis Results 133 of failure function. This increasing tendency is responsible for a local failure of stiffener-plate interface. Even though the failure function at stiffener centre is higher, it is well below the threshold to cause failure under the applied load (5 Ns). If the applied load is increased then failure is identified either at the boundary (A) or at the stiffener centre (C) depending on which of the failure function reaches a value of one first. Next, the failure function is plotted as a function of impulse in Figures 6.32 (a through d) for all four stiffener sizes. The applied load distribution is Nu-2 and the plates are modelled using a 2 x 2 grid for quarter plate. As discussed earlier, the values are plotted just before the loading phase is over to capture the effect of high initial stress ratio. Once again the two possible locations of failure, A and C, are considered in the analysis. The 2, 4 and 5 mm deep stiffeners show that the failure function at the boundary is higher than that of the stiffener centre. The difference between them reduces quite significantly as the stiffener depth increases and also with higher applied impulse. However, only the 9 mm deep stiffener actually shows the stiffener centre having a higher value than that of boundary point. Even though the transition occurs, the magnitude is not enough to cause failure until 13 Ns. At this impulse, the stiffener centre fails earlier than that of the boundary. This effect is also seen when a 3 x 3 grid is used under load distribution Nu-2 for a 3 x 9 - mm stiffener. Thus, one possible explanation for the failure at plate stiffener interface is the effect of high localised loading. The threshold impulse to cause failure at the interface drops when a Nu-3 load distribution is considered. The effect is more pronounced when the load is more localised, and even the threshold moves to the left as discussed earlier. Chapter 6. Stiffened Plate Analysis Results 134 The threshold to cause mode II failure is reached as the applied impulse is increased. The threshold is 13 Ns using QIC for all stiffened plates with load distribution Nu-2. This is slightly lower when compared to the uniform load. This is evident from Figure 6.28 which showed an increase in curvature at the boundary. The threshold impulse to cause mode II failure is, once again, the same for all stiffener sizes, and the failure at boundary (A) is predicted at this impulse. Figures 6.33 through 6.36 shows the mode I failure of stiffened plates under uniform and nonuniform load distribution. In all figures, the permanent displacement of the plate is higher than a corresponding uniform distribution. With the increase in stiffener depth, especially for 4 mm and 5 mm deep stiffeners, the results from nonuniform distribution give an upper bound to experimental results. The importance of load distribution is evident when 3 x 9 - mm stiffened plate results are observed. Since the aim of the study was to find the threshold impulse to failure as well as the failure mode transition, post-failure analysis is not carried out. For the 3 x 9 - mm stiffened plate, the nonuniform load distribution, Nu-3, causes rupture at the stiffener-plate interface. The failure at the interface is predicted for this plate at 10 Ns using LIC. This matches very well with experimental observation. Also, the cross-over in the case of the 3 x 5 - mm stiffened plate occurs around 16 Ns impulse, and is in fact close to the experimentally reported failure (shown by the solid diamond symbol in Figure 6.35). 6.5 Post-Failure Analysis The post-failure analysis of stiffened plates is carried out using the node release algorithm. The technique developed for unstiffened plates is further modified to simulate Chapter 6. Stiffened Plate Analysis Results 135 the progression of rupture in stiffened plates. A l l the results presented in this section are obtained using the 2 x 2 grid to model the quarter plate. At present, there is no experimental evidence to support the failure progression. For illustration purposes, only the results for the 3 x 2 - mm stiffened plate under uniform load is presented. The post-failure analysis of the plate with other stiffener sizes are left for future study. Once sufficient experimental evidence is gathered, they can be used to correlate with the predicted results. Quadratic Interaction Criterion is used to identify failure, as it proved to be a better model than Linear Interaction Criterion. 6.5.1 Mode II* Figure 6.37 shows the time history of central displacement, side pull-in and the kinetic energy of a 3 x 2 - mm stiffened plate under 13.3 Ns impulse. The element side is released once failure is identified at the boundary parallel to the stiffener (point A in Figure 6.6) and the analysis is continued. The remaining energy is absorbed by the plate with no further failure. Thus, a partial failure regime is identified. A plot of the central displacement, side pull-in and the kinetic energy versus time is shown in Figure 6.38. The applied impulse is 14.1 Ns. The failure starts at A (shown by the solid circle in the figure). Further the failure continues on the same side of the stiffened plate (AO in Figure 6.6)and then the boundary adjacent to the stiffener (OB in Figure 6.6) fails (shown by the solid diamond and triangle symbols respectively). The remaining element sides are released and the entire plate is supported by the stiffener end only (point B in Figure 6.6). The input energy is not sufficient to cause complete failure. Figure 6.39 shows the 3-D profile of the deformed plate at the instant when the plate has Chapter 6. Stiffened Plate Analysis Results 136 torn away from all the points along the boundary except at the stiffener. The two views of the deformed plate clearly shows the stiffener end still attached to the boundary. 6.5.2 Mode Ila The temporal variation of the stiffener centre displacement, side pull-in and the kinetic energy of the system is shown in Figure 6.40 for the 3 x 2 - mm stiffened plate under an applied load of 16 Ns. The same trend continues at higher loads with failure always starting at the middle of the boundary parallel to the stiffener (point A), and terminating at B. The solid symbols in the figure are the instances at which each of the failure occurred. Once all the element sides are released, the plate is held by the stiffener end (point B). The reaction force is calculated using the force equilibrium, which is used to estimate the shear stress contribution. Failure is identified at this point when the failure function reaches a value of one (f= 1). At this stage, all the nodes are released and the plate separates from the boundary. Temporal integration is further carried out to account for any deformation occurring during the free-flight of the torn plate until a steady state is reached. Figure 6.41 shows the 3-D profile of the torn plate. The deformed profile indicates that substantial amount of input energy is absorbed by the plate. The plate enters the mode lib and mode III failure regimes, as the applied impulse is increased. The failure is dominated by shear in these two failure modes which is reflected by the insignificant deformation of the plate. Chapter 6. Stiffened Plate Analysis Results 137 6.5.3 Permanent Central Displacement A plot of the stiffener centre displacement versus impulse is shown in Figure 6.42, for the 3 x 2 - mm stiffened plate along with the experimental data (Nurick et al., 1995). The predicted mode II failure always occurred at the boundary and it always started at A . Only one experiment with 3 x 2 - mm stiffened plate showed rupture at the boundary parallel to the stiffener and the results do not offer a comprehensive comparison with the numerical predictions. Experimentally, the mode II failure in 3 x 2 - mm stiffened plates occurred around 14 - 15 Ns impulse. This value compares favorably with the results of 13.3 Ns obtained using the Quadratic Interaction Criterion. The predicted threshold impulse for mode Ila is around 15 Ns. There is a monotonic increase in the stiffener centre displacement with the increase in impulse until the threshold impulse to mode II failure is reached. Similar to the unstiffened plates the predicted central displacement of stifffened plate shows an increasing trend for mode II*, mode Ila and decreasing trend for mode lib. 6.5.4 Side Pull-in and Residual Kinetic Energy Figure 6.43 shows a plot of the side pull-in versus impulse. The amount of side pull-in increases with the impulse and increasing central deformation for mode Ila. Thereafter, this side pull-in decreases with increasing impulse. The variation of the residual kinetic energy with impulse is shown in Figure 6.44. The prediction shows a linear increase in the energy of the system with impulse. Chapter 6. Stiffened Plate Analysis Results 138 6.5.5 Centreline Displacement Profiles The deflection profiles at the time of first element failure for the 3 x 2 - mm stiffened plate are shown in Figure 6.45 for selected cases, using QIC. A l l these failures are predicted to occur at the boundary. The profiles shown in these figures change dramatically with increasing impulse. This is because the failure function gets an increased amount of contribution from stress ratio with increasing impulse. The time to failure also decreases continuously with the higher applied impulse. This means that the structure did not have enough time to deform fully before failure occurred. Chapter 6. Stiffened Plate Analysis Results 139 Table 6.1: Static analysis results of 2-bay stiffened plate I Panel Stiffener Energy Spring Type of Model centre centre (N-m) force at analysis deflection deflection centre (mm) (mm) (N/m) Linear Koko 0.471 0.042 0.335 -elastic with springs1 0.471 0.042 0.335 230.5 Finite strip 0.461 0.052 0.348 -Nonlinear Koko 14.61 8.62 253.79 -elastic with springs1 14.61 8.62 253.83 46935 Large Koko 14.61 11.30 - -deflection elastic-plastic with springs1 14.61 11.30 - 29952 1 Spring stiffness ks — l x l 0 1 5 is used in the analysis Table 6.2: Influence of spring stiffness on the linear elastic response of 2-bay stiffened plate I Spring Panel centre Centre Energy Stiffener Spring stiffness displacement displacement (N-m) centre force (N/m) (mm) (plate end) deflection (N/m) (mm) (mm) 0 0.673 0.860 0.0364 0.0 0.0 103 0.672 0.858 0.0363 0.6x10"6 0.858 106 0.529 0.280 0.0225 0.117 163.4 109 0.475 0.059 0.0171 0.059 225.4 1012 0.471 0.042 0.0167 0.042 230.5 1015 0.471 0.042 0.0167 0.042 230.5 Chapter 6. Stiffened Plate Analysis Results 140 Table 6.3: Dynamic analysis results of 2-bay stiffened plate II Peak panel Peak stiffener Frequency Peak Type of Model centre centre of structure spring analysis deflection (mm) deflection (mm) (Hz) force (N/m) Linear Koko 12.08 1.64 771.45 -elastic with springs1 12.08 1.64 771.40 35465 Nonlinear Koko 3.95 2.23 - -elastic with springs1 3.95 2.24 _ _ 1 Spring stiffness ks = l x l O 1 5 is used in the analysis Table 6.4: Central displacement and maximum strain at midpoint on boundary parallel to stiffener of stiffened plate with different stiffener sizes under 5 Ns impulse a) 2x2 grid H (mm) 4 (mm) Maximum strain at A Cm «* £ 2 4.89 0.008 0.052 0.060 4 4.13 0.007 0.054 0.061 5 3.67 0.007 0.054 0.061 9 1.71 0.008 0.054 0.062 b) 3x3 grid H (mm) 4 (mm) Maximum strain at A em £h £ 2 4.98 0.008 0.052 0.060 4 4.22 0.007 0.052 0.060 5 3.78 0.007 0.053 0.060 9 1.74 0.008 0.054 0.062 Chapter 6. Stiffened Plate Analysis Results 141 Table 6.5: Stress ratio, strain ratio, time to first element failure and stiffener centre displacement at the threshold impulse to mode II failure for all stiffener sizes using 2 x 2 grid a) LIC H An2* T £ hn 4 / 4„> (mm) (Ns) (//sec) (mm) (mm) 2 9.5 0.427 0.574 61.0 4.48 10.32 4 9.5 0.460 0.540 57.5 3.82 9.10 5 9.5 0.469 0.536 56.0 3.49 8.40 9 9.5 0.466 0.537 54.5 2.50 5.20 b) QIC H (mm) An 2* (Ns) 4 , (mm) £ T hn (//sec) 2 13.3 6.64 0.835 0.553 64.0 4 13.3 5.05 0.798 0.604 55.5 5 13.3 4.66 0.796 0.607 54.5 9 13.3 3.51 0.801 0.600 53.0 Table 6.6: Stress ratio, strain ratio, time to first element failure and stiffener centre displacement at the threshold impulse to mode II failure for all stiffener sizes using 3 x 3 grid H (mm) An 2* (Ns) T £ hn (//sec) A, (mm) 4„ i (mm) 2 9.5 0.462 0.544 52.5 3.67 10.55 4 9.5 0.467 0.540 51.5 3.26 9.33 5 9.5 0.465 0.564 51.5 3.08 8.63 9 9.5 0.455 0.546 52.0 2.31 5.42 Chapter 6. Stiffened Plate Analysis Results 142 Table 6.7: Comparison of maximum strain at midpoint on boundary parallel to stiffener under different load distribution Maximum strain at A H Uniform Nu-2 Nu-3 (mm) load 2 0.060 0.071 0.080 4 0.061 0.069 0.076 5 0.061 0.069 0.075 9 0.062 0.070 0.075 Chapter 6. Stiffened Plate Analysis Results Courtesy: International J of Impact Engg 3 x 2 - mm stiffener Mode IIB 143 3 x 4 - mm stiffener Mode IIB 3 x 9 - mm stiffener Mode IIS Figure 6.1: Failure modes of explosively loaded stiffened square plates Chapter 6. Stiffened Plate Analysis Results 144 y,v 2 Panels at 500 mm centres 500 mm x,u f z,w 3 mm Elastic modulus = 71,700 N/mm2 Hardening modulus = 358.5 N/mm2 Yield stress = 284 N/mm2 Poisson's ratio = 0.3 Figure 6.2: Configuration of 2-bay stiffened plate I Chapter 6. Stiffened Plate Analysis Results 145 • y,v 203 mm x,u 1.37 mm IT t= 6.35 mm h « = 1 2 - 7 m m Elastic modulus = 68,900 MPa Density = 2670 kg/m3 Poisson's ratio = 0.3 Figure 6.3: Configuration of 2-bay stiffened plate II Chapter 6. Stiffened Plate Analysis Results 146 0.0 0.5 1.0 1.5 2.0 2.5 Time (msec) Figure 6.4: Comparison of linear elastic response of clamped 2-bay stiffened plate II due to step load 6 —\ -2 Nonlinear elastic analysis Panel centre (Koko) Stiffener centre (Koko) 0 Panel centre (Spring model) A Stiffener centre (Spring model) 0.0 0.5 1.0 1.5 Time (msec) 2.0 2.5 Fig. 6.5: Comparison of nonlinear elastic response of clamped 2-bay stiffened plate II due to step load Chapter 6. Stiffened Plate Analysis Results 147 89 mm 89 mm x,u 1.6 mm 3 mm H (2,4,5,9 mm) a) Configuration of plate 44.5 mm 0 stiffener 8 44.5 mm b) Finite element model of quarter plate (2x2 grid) Figure 6.6: Configuration and finite element model of one-way stiffened plate Chapter 6. Stiffened Plate Analysis Results 148 E £ 6 •»-> c o E a> o _ro 4 -a . w ~ 2 —| a> O Impulse 5 Ns 2 x 2 grid C - Stiffener centre Undamped A 20% damping 50 100 150 Time (micro-sec) 200 250 Figure 6.7: Comparison of mode I displacement time history of 3 x 2 - mm stiffened plate E E, *•> c o E CD U re o. CO Q 2 —\ Impulse 5 Ns 2 x 2 grid Undamped 20% damping 0.0 0.2 0.4 0.6 0.8 Distance from boundary to stiffener centre Figure 6.8: Comparison of permanent deflection profile of 3 x 2 - mm stiffened plate in mode I failure 1.0 Chapter 6. Stiffened Plate Analysis Results 149 Impulse 5 Ns Uniform load 50 100 150 Time (micro-sec) 200 25 >» U) 1_ CD c CO o +3 CO c 250 Figure 6.9: Time history of central displacement and kinetic energy of 3 x 2 - mm stiffened plate in mode I failure E E c 0) E o o re Q 4 H Impulse 5 Ns Uniform load Permanent deflection profile 0.0 0.2 0.4 0.6 0.8 Distance from boundary to stiffener centre Figure 6.10: Transient deflection profiles of 3 x 2 - mm stiffened plate in mode I failure 1.0 Chapter 6. Stiffened Plate Analysis Results 1 E £ 6 c o E <D O _ro 4 -Q . (ft 75 | 2 —| CD O Impulse 5 Ns Uniform load 50 100 150 Time (micro-sec) 200 250 Figure 6.11: Comparison of central displacement time history of stiffened plate for different stiffener sizes in mode I failure E 6 E +J C CD E 4 CD O W Q. (0 Impulse 5 Ns Uniform load 0.0 0.2 0.4 0.6 0.8 Distance from boundary to stiffener centre 1.0 Figure 6.12: Predicted permanent deflection profiles for plate with different stiffeners in mode I failure Chapter 6. Stiffened Plate Analysis Results 1 a) 3 x 2 - mm stiffener b) 3 x 4 - mm stiffener Figure 6.13: 3-D mode I deflection profiles of stiffened plate Chapter 6. Stiffened Plate Analysis Results 152 c) 3 x 5 - mm stiffener d) 3 x 9 - mm stiffener Figure 6.13: 3-D mode I deflection profiles of stiffened plate Chapter 6. Stiffened Plate Analysis Results 1 E E 0) E a> o « a to Q 4 H Impulse 5 Ns Uniform load 3 x 3 grid 2 x 2 grid 0.0 0.2 0.4 0.6 0.8 Distance from boundary to stiffener centre Figure 6.14: Comparison of mode I displacement profile for 3 x 2 - mm stiffened plate with different grid sizes 1.0 0.08 0.06 Impulse 5 Ns Uniform load 0.0 0.2 0.4 0.6 Distance from corner 0.8 1.0 Figure 6.15: Comparison of strain distribution along the boundary parallel to stiffener for stiffened plates Chapter 6. Stiffened Plate Analysis Results 1 o "•*= c 2 .2 to o CO c 2 = (0 <D o = T Q CO 2 CO 1.0 0.8 0.0 -0.2 Impulse 5 Ns 3 x 2 - mm stiffener Uniform load LIC Strain ratio Stress ratio Failure function 50 100 150 Time (micro-sec) 200 250 Figure 6.16: Plot of strain ratio, stress ratio and failure function with time at the midpoint on boundary parallel to stiffener Impulse 5 Ns 3 x 2 - mm stiffener 0 50 100 150 200 250 Time (micro-sec) Figure 6.17: Plot of strain ratio, stress ratio and failure function with time at the stiffener centre Chapter 6. Stiffened Plate Analysis Results 1 1.0 Impulse 9.5 Ns 2x2 grid Uniform load 0.8 —\ A C O •4= 0.6 —| CO 2 0.2 —\ A - Midpoint on boundary parallel to stiffener C - Stiffener centre 0.0 4 6 Stiffener depth (mm) Figure 6.18 a) Strain ratio 10 1.0 0.8 Impulse 9.5 Ns 2x2 grid Uniform load A c O ~ 0.6 2 CO ( A O £ 0.4 0.2 —\ 0.0 A - Midpoint on boundary parallel to stiffener C - Stiffener centre —' 1 ' 1 4 6 Stiffener depth (mm) Figure 6.18 b) Stress ratio 10 Chapter 6. Stiffened Plate Analysis Results 156 1.0 0.8 c o "5 0.6 C a 3 0.4 0.2 0.0 Impulse 9.5 Ns 2x2 grid A C A - Midpoint on boundary parallel to stiffener C - Stiffener centre 1 1 1 1 1—: 1 n p I 0 2 4 6 8 10 Stiffener depth (mm) c) Failure function Figure 6.18: Comparison of failure function and components at the threshold impulse to mode II failure under uniformly distributed load - LIC 1.0 0.8 c o t> 0.6 c 3 0.4 0.2 0.0 Impulse 13.3 Ns 2x2 grid A C A - Midpoint on boundary parallel to stiffener C - Stiffener centre 4 6 Stiffener depth (mm) 10 Figure 6.19: Comparison of failure function at the threshold impulse to mode II failure under uniformly distributed load - QIC Chapter 6. Stiffened Plate Analysis Results 1 1.0 0.8 c o "5 0.6 c 3 4 -O 3 0.4 w 0.2 0.0 Uniform load 2x2 grid LIC A - Midpoint on boundary parallel to stiffener C - Stiffener centre 5 10 Impulse (Ns) a) 3 x 2 - mm stiffener 15 1.0 0.8 c o O 0.6 c 3 <1> 0.4 0.2 0.0 Uniform load 2x2 grid LIC A - Midpoint on boundary parallel to stiffener C - Stiffener centre 0 5 10 15 Impulse (Ns) b) 3 x 4 - mm stiffener Figure 6.20: Comparison of failure function with impulse for plate with different stiffener sizes Chapter 6. Stiffened Plate Analysis Results 1 1.0 0.8 c o t> 0.6 c CD Jj 0.4 're u_ 0.2 0.0 Uniform load 2x2 grid LIC A - Midpoint on boundary parallel to stiffener C - Stiffener centre 5 10 Impulse (Ns) c) 3 x 5 - mm stiffener 15 1.0 0) re 0.8 H c o t> 0.6 c 0.4 0.2 0.0 Uniform load 2x2 grid LIC - - C A - Midpoint on boundary parallel to stiffener C - Stiffener centre 0 5 10 15 Impulse (Ns) d) 3 x 9 - mm stiffener Figure 6.20: Comparison of failure function with impulse for plate with different stiffener sizes Chapter 6. Stiffened Plate Analysis Results 159 20 ? £ 15 c 0) E o o _ro Q . W •5 10 re I • a> O Uniform load 2x2 grid Partial failure • A LIC QIC Experimental data # Mode I Mode I 10 20 30 Impulse (Ns) Figure 6.21: Plot of central displacement versus impulse for 3 x 2 - mm stiffened plate 20 ? £ 15 c CD E o Q . CO Uniform load 2x2 grid Experimental data # Mode I A Mode IIB 10 2 I 5H 0) O Mode I 1 p 10 20 Impulse (Ns) 30 Figure 6.22: Plot of central displacement versus impulse for 3 x 4 - mm stiffened plate Chapter 6. Stiffened Plate Analysis Results 160 0 10 20 30 Impulse (Ns) Figure 6.23: Plot of central displacement versus impulse for 3 x 5 - mm stiffened plate 20 £ 15 Uni form load 2 x 2 gr id 10 —\ •S 5 —\ Exper imenta l data + Mode IIS Mode I 10 20 Impulse (Ns) Figure 6.24: Plot of central displacement versus impulse for 3x9- mm stiffened plate 30 Chapter 6. Stiffened Plate Analysis Results 161 0 50 100 150 200 250 Time (micro-sec) Figure 6.25: Comparison of central displacement time history of stiffened plates under nonuniform load E 6 E 4 —\ Impulse 5 Ns Load: Nu-2 2x2 grid 0.0 0.2 0.4 0.6 0.8 Distance from boundary to stiffener centre 1.0 Figure 6.26: Predicted mode I deflection profile of stiffened plates under nonuniformly distributed load Chapter 6. Stiffened Plate Analysis Results 162 0 50 100 150 200 250 Time (micro-sec) Figure 6.27: Time history of central displacement of 3 x 2 - mm stiffened plate under different load distribution Figure 6.28: Predicted mode I deflection profiles for 3 x 2 - mm stiffened plate under different load distribution Chapter 6. Stiffened Plate Analysis Results 163 E E CO E CO o JS Q . (/) '•B To &_ +•> c CO O 4 — Impulse 5 Ns Nonuniform load 2x2 grid Nu-2 Nu-3 Uniform load 4 6 Stiffener depth (mm) 10 Figure 6.29: Comparison of stiffener centre displacement for plates in mode I failure for different load distribution Chapter 6. Stiffened Plate Analysis Results 164 3 0.4 re UL 0.2 0.0 Impulse 5 Ns Nonuniform load: Nu-2 LIC 2x2 grid 3x3 grid A A - - c c -' " A - Midpoint on boundary parallel to stiffener C - Stiffener centre 1 | i | i i | i 0 2 4 6 8 10 Stiffener depth (mm) Figure 6.30: Variation of failure function with stiffener depth under nonuniformly distributed load (Nu-2) c o o c 3 0) 1_ 3 're 1.0 0.8 —\ 0.6 0.4 0.2 0.0 Impulse 5 Ns 2x2 grid LIC Nonuniform load: Nu-3 A - - C A - Midpoint on boundary parallel to stiffener C - Stiffener centre T" 0 2 4 6 8 10 Stiffener depth (mm) Figure 6.31: Variation of failure function with stiffener size under nonuniformly distributed load (Nu-3) Chapter 6. Stiffened Plate Analysis Results 165 1.0 0.8 c o o 0.6 c 3 CO 0.4 0.2 0.0 2x2 grid LIC Nonuniform load: Nu-2 A - Midpoint on boundary parallel to stiffener C - Stiffener centre T 5 10 Impulse (Ns) a) 3 x 2 - mm stiffener 15 1.0 0.8 c o "S 0.6 c 3 0) 3 0.4 'ro 0.2 H 0.0 2x2 grid LIC Nonuniform load: Nu-2 A c A - Midpoint on boundary parallel to stiffener C - Stiffener centre 5 10 Impulse (Ns) b) 3 x 4 - mm stiffener 15 Figure 6.32: Variation of failure function with impulse for stiffened plates under nonuniformly distributed load Chapter 6. Stiffened Plate Analysis Results 166 1.0 0.8 c o T3 0.6 —| c 3 u-Ci M 0.4 —] 0.2 0.0 2x2 grid LIC Nonuniform load: Nu-2 A - Midpoint on boundary parallel to stiffener C - Stiffener centre 5 10 Impulse (Ns) c) 3 x 5 - mm stiffener 15 1.0 0.8 c o t> 0.6 c 3 O 3 0.4 "(5 0.2 0.0 2x2 grid LIC Nonuniform load: Nu-2 c A - Midpoint on boundary parallel to stiffener C - Stiffener centre 0 5 10 15 Impulse (Ns) d) 3 x 9 - mm stiffener Figure 6.32: Variation of failure function with impulse for stiffened plates under nonuniformly distributed load Chapter 6. Stiffened Plate Analysis Results 167 Figure 6.33: Comparison of stiffener centre displacement versus impulse for 3 x 2 - mm stiffened plate under different load distribution Figure 6.34: Comparison of stiffener centre displacement versus impulse for 3x4- mm stiffened plate under different load distribution Chapter 6. Stiffened Plate Analysis Results 168 20 E E, 15 c o> E a> o iS a. w 75 i_ 4-1 C a> O 2x2 grid QIC Experimental data # Mode I + Mode IIS 10 20 Impulse (Ns) Uniform load Nu-2 Nu-3 30 Figure 6.35: Comparison of stiffener centre displacement versus impulse for 3 x 5 - mm stiffened plate under different load distribution c CO E CO o JS Q. <fl 7 3 20 E £ 15 10 ns CO O 2x2 grid QIC Threshold impulse to failure at plate-stiffener interface (mode IIS) using LIC with Nu-3 Model / / Experimental data + Mode IIS Uniform load Nu-2 Nu-3 10 20 30 Impulse (Ns) Figure 6.36: Comparison of stiffener centre displacement versus impulse for 3 x 9 - mm stiffened plate under different load distribution Chapter 6. Stiffened Plate Analysis Results 169 E E, *•> c o E o u re a CO Q 16 12 4 H 3 x 2 - mm stiffener Impulse 13.3 Ns QIC Central displacement Side pull-in Kinetic energy 50 100 150 Time (micro-sec) 200 200 \— 150 \— 100 50 250 Figure 6.37: Time history of central displacement, side pull-in and kinetic energy of a 3 x 2 - mm stiffened plate in mode II* failure 16 E E *J c E CD o re Q. CO Q 3 x 2 - mm stiffener Impulse 14.1 Ns QIC First element failure parallel to stiffener Second element failure (side parallel to stiffener) a Failure of boundary ^ adjacent to stiffener (OB in Figure 6.6) Central displacement Side pull-in Kinetic energy 50 100 150 Time (micro-sec) 200 200 150 100 \— 50 250 Figure 6.38: Plot of central displacement, side pull-in and kinetic energy versus time for a 3 x 2 - mm stiffened plate in mode II* failure Chapter 6. Stiffened Plate Analysis Results 170 Figure 6.39: 3-D profile of 3 x 2 - mm stiffened plate in mode II* failure (two views of deformed plate, Impulse 14.1 Ns) Chapter 6. Stiffened Plate Analysis Results 171 25 Central displacement Side pull-in Kinetic energy 3 x 2 - mm stiffener Impulse 16 Ns QIC 50 100 150 Time (micro-sec) 200 250 200 150 100 50 >» D5 c CO o CO c 250 Figure 6.40: Temporal variation of central displacement, side pull-in and kinetic energy of 3 x 2 - mm stiffened plate in mode Ila failure Chapter 6. Stiffened Plate Analysis Results 172 Figure 6.41: 3-D profile of 3 x 2 - mm stiffened plate in mode Ila failure (two views of deformed plate, Impulse 16 Ns) Chapter 6. Stiffened Plate Analysis Results 1 0 10 20 30 Impulse (Ns) Figure 6.42: Plot of central displacement versus impulse for a 3 x 2 - mm stiffened plate using QIC Chapter 6. Stiffened Plate Analysis Results 1 E 5 E Q. 3 O •D 55 2 1 H Uniform load 2x2 grid QIC 10 20 Impulse (Ns) Figure 6.43: Plot of side pull-in versus impulse for a 3 x 2 - mm stiffened plate using QIC 30 1200 o 1-a> c CD O 0) c "55 a> or 800 — 400 Unifrom load 2x2 grid QIC 10 20 Impulse (Ns) Figure 6.44: Plot of residual kinetic energy versus impulse for a 3 x 2 - mm stiffened plate using QIC 30 Chapter 6. Stiffened Plate Analysis Results 1 E E 0) E o u J2 Q. b 1 1 1 1 1 1 1 1 r 0.0 0.2 0.4 0.6 0.8 1.0 Distance from boundary to stiffener centre b) QIC Figure 6.45: Deflection profile at the time of first element failure for 3 x 2 - mm stiffened plate under uniform load Chapter 7 Summary, Conclusions and Suggestions 7.1 Summary Super finite elements (beam and plate elements) are used to predict the large deformation, elastic-plastic transient behaviour of unstiffened and stiffened plate structures. The displacement fields of the element are represented by polynomials as well as continuous analytical functions, and the elements have been specially designed so that one element is sufficient to model the deformational response of the entire structure. The von Karman large deflection theory is used to model the geometric nonlinearities. Material nonlinearities are modelled by the von Mises yield criterion and the associated flow rule using a bilinear strain-hardening law. Strain-rate effects are included via Cowper-Symonds relation. The finite element equations are derived using the virtual work principle, and the matrix quantities are evaluated by Gaussian quadrature. Five integration points are used in each in-plane direction, and four points are used through the thickness. Temporal integration is carried out using the Newmark-/? method with Newton-Raphson iteration for solution of the nonlinear equations within each time step. A n interactive failure model is proposed to predict the tearing and rupture of thin steel plates and stiffened plate structures under blast loading. The model accounts for the membrane and bending strains as well as the transverse shear stress experienced by the structure under the applied load. The interaction between the tensile tearing and shearing 176 Chapter 7. Summary, Conclusions and Suggestions 111 mode of failure is considered via an interaction relation between the strain and stress ratios. Two interactive failure criteria are considered, either Linear (LIC) or Quadratic (QIC) based on the way the ratios are added. The bending strain is estimated by assuming that a plastic hinge line develops at the boundary, while the membrane strain is calculated using the finite element prediction of the deformed profile. The total strain, composed of membrane and bending components, is then divided by a specified rupture strain for the material, to obtain the strain ratio. Since the finite element formulation is based on Kirchhoff plate bending theory, the program does not provide any information on shear strains (stresses). In order to achieve a continuous estimation of the shear force and stress along the plate boundary, a series of very stiff springs are introduced there. The use of high stiffness values effectively simulates the clamped boundary condition of the problem. The estimated shear stress is then compared to the ultimate shear strength of the material to form a stress ratio. A node release algorithm is developed to simulate the progression of rupture. When the plate tears completely, it separates from the boundary and flies freely. The post-failure analysis is continued to account for the deformation which continues during the free flight of the torn plate. The new formulation has been applied to predict the failure modes of unstiffened and stiffened square mild steel plates. The verification of the spring supported plate model in linear and nonlinear static as well as dynamic domain confirms the validity of using this approach for modelling plate structures. The predicted failure modes using the above model for blast loaded unstiffened and stiffened plates are presented and compared with available experimental data. The Chapter 7. Summary, Conclusions and Suggestions 178 predicted threshold impulse to mode II failure for unstiffened square plates is in the same range as that of experiments. In fact, the proposed Linear and Quadratic Interaction Criteria for identifying rupture appear to provide a bound on the observed experimental results. The threshold impulse to mode II failure is also predicted quite well for the stiffened plates. Of the two, the Quadratic criterion predicts results which are closer to experiments. The threshold impulse to transverse shear failure mode (mode III) is established with reasonable accuracy. In addition to the failure modes, the model also predicts associated parameters like residual kinetic energy, side pull-in and deformation profiles of the ruptured plates which also compare well with the experimental data. The extensive sensitivity analysis carried out indicates the versatility of the failure model. The simple modelling of the plate structure and the small input data file are indeed very attractive features for the failure analysis of these simple plates. 7.2 Conclusions The proposed interactive failure model provides a satisfactory method for the analysis of unstiffened and stiffened square plates subjected to blast loading. The inclusion of the node release algorithm results in simulating the failure progression, and thus, identifying the partial failure regime. The model captures the interaction effects between the tensile tearing and shearing failure modes and the reasonably accurate predictions instills confidence in the failure models used. The model successfully predicts the threshold to mode II failure in the case of stiffened plates. It also suggests the possible paths of failure progression. For plate with Chapter 7. Summary, Conclusions and Suggestions 179 deep stiffeners, the experimentally observed localised tearing near plate centre, is captured by the model under nonuniform distribution of load. The Quadratic Interaction Criterion predicts consistently better results than the Linear Interaction Criterion as compared to the experimental results. The results clearly indicate the influence of shear on the failure mechanism not only for mode III, but also for mode II. The results confirm the importance of the interaction effects of tensile and bending strain on tearing and shear failure. The rigorous testing of the failure model through an extensive parametric study provides useful guidelines and charts for designing simple plate structures to resist blast loading. The main usefulness of the model is its simplicity and capability to capture the global response with relatively few elements and yet obtain a reasonable overall accuracy in the failure predictions. Even though the model is adequate for identifying the failure at the global or structural level, it lacks the sophistication and detail in modelling the fracture process. The absence of an energy dissipation mechanism will lead to some uncertainties in the predicted failure thresholds. The predicted threshold and range of different failure modes still depends on the input material parameters, especially the rupture strain and ultimate shear strength of the material. Once better understanding and characterization of material behaviour under high rates of loading is possible, the present model offers the scope for incorporating them in the future. Chapter 7. Summary, Conclusions and Suggestions 180 7.3 Suggestions for Future Work 1. Although the present model is adequate to understand the failure modes of unstiffened and stiffened plates, more experimental results would give further confirmation of the phenomenon. Further validation of the model can be made by analysing failure of simple plates made of different materials. 2. In the present work, the plastic hinge length is held constant as there is no theory to predict its value analytically. This will definitely have an effect on the predicted results. Further work can be carried out in establishing the hinge length and some refinements can be made in this regard. 3. The failure model provides the strain and shear stress data in a continuous fashion at all points on the boundary. Thus, it would be worthwhile to conduct research into a possible development of an energy dissipation mechanism, in order to obtain a better description of tearing and rupture of plate structures. 4. The present model can be used to study plates with different stiffener sizes. 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Appendix A Transverse Shear Stress in Plates through Equilibrium Equations The equations of equilibrium when inertial forces are present are given by Z _ A dcrx foxy dx Fx = 0 ; — i + — 2 - + dx dy dz -P He (A.1) + dx dy dz -P ^1 a2 (A.2) dx 'd3 w •+ v-d3 w X-v'Jydx5 dxdy1 (A.3) dy 'd3 w + v-d3 w X-v'Jldy' dx'dy (A.4) foxy dx = —z \-v. d3 w dx2dy (A.5) foxy dy = -z 1-vJ d3 w dxdy2 (A.6) -z d_ dx f d2^ \dt2 j (A.7) ^ v _ _ d_ ^ ~ "dy V dt2 J (A.8) Substituting and rearranging terms we get 190 Appendix A. Transverse Shear Stress in Plates through Equilibrium Equations 191 dz dz l-vzJ\dx' dxdy1 d^w c^w] d W-v2)\dx2dy ' dy3 J + Pdy (*) (A.9) (A. 10) Integrating Equation (A.9) and (A. 10) E d3 w d3 w + 1 - v V \dx3 cxcy C^W d3 w + d .. 2 1 H dx d \-vl)\cxldy dy 3 ' dy v 2 v 2 + a (A.11) (A.12) Apply boundary conditions that the shear stress is zero at top and bottom surface C 3 = - h d3 w d3 w • + • l-v'Jldx3 cxcy1 h2 8 <?w d3 • p — ( w ) dxy ' d \-vl)\dxdy dy dy (A.13) (A.14) Substituting for constants C 3 and C 4 d3 1-v 2 ^ c 3 d3 cxcy2 p d 0 dx <? 0 dy w I ' z 2 2^ 2 8 J (A. 15) In the static case the terms associated with acceleration of the body vanish d3 T., y 1 . 1-v2 dx3 d3 dxdy2 d3 dx2dy dy3 w w I {z2 h_ 2 8 2\ (A. 16) Appendix A. Transverse Shear Stress in Plates through Equilibrium Equations 192 The static shear force/unit length is obtained by integrating the shear stress (Equation A. 16) through the thickness (A. 17) Appendix B Displacement Functions Displacement fields within the plate element are given by 9 M*Y r , -« 1 0 A M T u = Z N"ut + sin 2n^< y _ ] L2{r?)> < > + sin 4 T T £ ' 4 ( 7 ) > MI). A W w 1 5 9 M$V ' v i o ' T v = Z NJv/ + sin 2nn-Li® ' ' V l l > + sin 4TTJ]< • < V 1 4 ' Mb M$. V 1 5 , 16 w = Z tfjVj + r 'w, ' T ' W 8 ' ^2(7) wy, H*(z) Wx • • > < > w1 "4(7). I x<> J > + fa) fa)*, Where u, v, w are the displacements in the rectangular x, y, z coordinate directions, respectively; x and y being in the plane and z normal to it. £ and TJ are the non-dimensional equivalents of x and y, respectively. L are quadratic Lagrange interpolation polynomials, Hj are cubic Hermitian polynomials and (j> is the first vibration mode of a fully clamped beam. The in-plane displacement shape functions, N" and N, are products of the Lagrange polynomials, while NJ are products of the Hermitian 193 Appendix B. Displacement Functions 194 polynomials, y/. represent the variables associated with out-of-plane bending at the corner nodes. Also, w, and v, represent the in-plane nodal variables and it is noted that the extra ones labeled ul0,uu v 1 5 are actually the amplitudes of the trigonometric (Sine) functions used to model the in-plane displacements. Beam Elements The membrane and flexural displacement fields referred to the centroidal axis of a beam in the x-direction are given by 11 = Z2(4 Lffi) + w 4 sin InS, + u5 sin 4TT^ + + e[H{(%), H'2{c;), H'A($)] w = [Htf), H2($; H3(4), H<($] w. wr > + where the primes denote differentiation with respect to x and e is the distance between the centroidal axis of the beam and the mid-plane of the plate. Appendix B. Displacement Functions 195 The effect of torsion and lateral bending in the stiffener beam element has been included in some cases. The rotation, 6 and lateral displacement, v fields are approximated, respectively, by 0 = H2^\ H3($, H4(4)] *2 J + (/)03 v = [Lfc), L2({), 4(3] 3 J where 0X 0X ,.. .. v 3 are the beam nodal rotation, twist or lateral displacement variables. Appendix C Shape Functions 15 15 25 Using the notations u = ^ N " u i > v = X '^"v> ^ w = X 7^^0 ' t h e s h a p e /=i i=i j=i functions for the plate elements are given by N; = L^LM) N; = i^Lfa) N; = £3(^(17) N; = L^)L2{ij) Nl = LffiLfr) N; = L^)L,{rj) iV,"0 = 1 , ( 7 7 ) sin 2n$ JV," = L2(rj) sin 2 K•£ N"2 = 1 3 ( 7 ) 5m InE, N"3 = 1 , (7 ) 5m 4fl£ iV"4 = I 2 (7 ) 5 m 4;r£ 196 Appendix C. Shape Functions 197 N"5 = 4(7/) « « 4 ^ iV," = A7 for / < 9 N;Q = 4(3 2 ^7 A^', = 4 ( £ ) 2TTT7 A^ ,2 = 4(<?) 2 ; z 7 7 N*3 = « n 4 ^ 7 7 tf* = Z2(3 «« 4TT/7 iV* = 4(3 sz/i 4mj N? = H^)H,{rj) N; = H^H^rj) N; = H^)H2{rj) N: = H2{{)H2{«) N: = H^)H,{rj) N; = H<(S)HX(T,) N; = H3{Z)H2(TJ) N? = H^)H2{rj) N: = H3($H3(rj) Nil = Hfc)H3(r,) N;\ = H3($HA(r,) K = HA(S)HA(T,) Appendix C. Shape Functions 1 NW J V | 3 = H^)H3{TJ) = Hfa)H,{ij) NW = HX(Z)H4(tj) J V 1 6 = H2(S)HA(ri) NW = fa)^(v) = fa)Hfa) NW J V 1 9 = Hfa)fa) NW J V 2 0 = Hfa)fa) NW i v 2 , = faH(v) J V 2 2 = faHW AT J V 2 3 = H^)fa) A/" -' v24 = Hfa)fa) J\IW 1\25 = fa) fa) Shape Functions for Beam Element in .x-direction The shape functions for a beam element in the x-direction are given by N;» = N>2" = eHfe) K = eH^) K = Lfa) K = eHfa) Appendix C. Shape Functions 199 = m N'" = e<f>'{{) = sin 2TZE, K = sin 4nE, Nf N* N° = H>® N4B JV5Z Where the superscripts m and B stand for membrane and bending, respectively, e is the distance between the centroidal axis of the beam and mid-plane of the plate and <f> is the first symmetric vibration mode of a clamped beam. For a beam in the ^-direction replace £ by TJ . Appendix D Spring Stiffness Matrix The consistent stiffness matrix of springs in the transverse direction is given by [V]=pf']Tk\N>}k=ftNb]TkM[Ni]dx L I 13Z ILL 2 21L -13L2 35 210 210 420 11Z2 Z 3 13Z2 - 3 Z 3 210 105 420 420 271 13Z2 13Z -11Z 2 210 420 35 210 -13Z 2 - 3 Z 3 - I L L 2 Z 3 420 420 210 105 £, E2 E, E4 i where Ex = J//,(^(<^ o E2 = )H2(fy>(t)d{ 0 1 E3 = \H3(5W)dZ 200 Appendix D. Spring Stiffness Matrix 201 E4 = JHA(fy@dt 0 1 E5 = j > 2 ( £ ) ^ o In the above equation H, Ht are Cubic Hermitian polynomials and <j> is the first symmetric vibration mode of a fully clamped beam. Appendix E Global Force Balance Method The Kirchhoff theory based analysis is not conducive to failure analysis of plates which undergo mode III failure, where transverse shear dominates. Hence, an approximate procedure is used to account for the effect of shear in the failure model. A very simple analytical method is incorporated into the program to determine the shear stress at the support at each time step. The idea is very similar to that used in circular plates (Fagnan, 1996 and Olson et al, 1995) where a uniform shear stress distribution around the clamped boundary is assumed. The shear stress at the boundary is obtained via the reaction force estimated at the boundary through the overall structural equilibrium. The vertical forces including the inertial force are summed up at each time step to determine the reaction force at the boundary. That is, where the first term is the applied load, the second term represents the inertial force of the system and R is the total reaction force. The inertial force of the system using the displacement polynomials of super plate element (Equation 3.7) is given by R = jpdA- j pwdV (E.l) A V (E.2) v v 202 Appendix E. Global Force Balance Method 203 J[iV"]cL4 ph\5p} (E.3) A where p is the material density, h is the plate thickness, and 5 and N are the acceleration vector of nodal variables and shape functions associated with w DOF of plate element. Substituting the above expression for inertial force in Equation (E.l), we get, An average shear stress, ravg, is then calculated by dividing this reaction force by the total sectional area at the clamped boundary. This shear stress is used in the failure model (Equations 3.35 and 3.36) for the prediction of tearing and rupture of plates. Once again, the failure progression is simulated using the node release algorithm, while the post-failure analysis is the same as that explained in Chapter 3. The square plate analysed in Chapter 4 (section 4.4.1) is again used for this study. Figure E . l shows a plot of deflection-to-thickness ratio versus impulse for the experimental results as well as predicted results using the two failure models. The numerical results show the same general trend for the different modes. Overall, the quadratic model appears to correlate more closely with the experimental results than the linear failure model. This method fails to predict the mode II*- partial tearing of the plate. To that extent, the two criteria are conservative. In essence, compared to an actual failure process, the two models do not account for sufficient energy dissipation. This may be a direct result of not incorporating any energy dissipating mechanism into the failure model as tearing of the plate occurs. The threshold impulse for mode Ila (lm2a) is 18.9 Ns for (E.4) Appendix E. Global Force Balance Method 204 QIC and 14.6 Ns for LIC. Similarly, the mode lib threshold (lm2b) is defined at 16 Ns for LIC and 22 Ns for QIC. Mode III is defined as complete tearing of the plate with no significant deformation. The current analysis predicts a small deformation of 1.3mm for a 49 Ns impulse for both models. Thus the predicted threshold lies around 50 Ns impulse. Limitations of the method The results clearly indicate the influence of shear on the failure mechanism not only for mode III, but also for mode II. The results confirm the importance of the interaction effects of tensile and bending strain on tearing and shear failure. One weakness of the analysis is its inability to predict the partial tearing of the plate which was observed in the experiments. In this procedure, the shear stresses required for determining the interactive failure function were assumed to be uniformly distributed over the plate boundary with the magnitude equal to the dynamic reaction force divided by the remaining cross-sectional area. A shortcoming of this averaging technique is that it masks the peaks and troughs of the shear stresses which are vital in the evaluation of the failure functions and detection of critical points in the structure. Appendix E. Global Force Balance Method 205 Figure E.l: Plot of deflection-to-thickness ratio versus impulse for both failure models
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Analysis of the deformation and failure of blast loaded unstiffened and striffened plates Rudrapatna, Nagaraja 1997
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Title | Analysis of the deformation and failure of blast loaded unstiffened and striffened plates |
Creator |
Rudrapatna, Nagaraja |
Date Issued | 1997 |
Description | A numerical investigation into the deformation and failure of clamped unstiffened and stiffened mild steel plates is carried out. Studies indicate that with increasing load intensity, simple plate structures under blast loading exhibit the three general modes of failure: mode I (large inelastic deformation), mode II (tensile tearing and deformation) and mode III (shear rupture). The failure analysis of plate structures subjected to blast loading is still considered a difficult task. These problems are highly nonlinear due to the combined effect of large deformation, material plasticity and high strain-rates. The present work develops a semi-numerical model for failure prediction which is aimed at providing a simple tool for preliminary design/analysis of such structures. Analytical failure models have been incorporated in an existing finite element code which handles the large deformation, elastic-plastic transient behaviour of unstiffened and stiffened plate structures. The finite element formulation employs existing super finite plate and beam elements. An interactive failure model is proposed to predict the tearing and rupture of thin steel plates and stiffened plate structures under blast loading. The model accounts for the membrane and bending strains as well as the transverse shear stress experienced by the structure under the applied load. The interaction between the tensile tearing and shearing mode of failure is considered via an interaction relation between the strain and stress ratios. Two interactive failure criteria are considered, either Linear (LIC) or Quadratic (QIC) based on the way the ratios are added. The bending strain is estimated by assuming that a plastic hinge line develops at the boundary, while the membrane strain is calculated using the finite element prediction of the deformed profile. The total strain, composed of membrane and bending components, is then divided by a specified rupture strain for the material, to obtain the strain ratio. Since the finite element formulation is based on Kirchhoff plate bending theory, there is no direct prediction of shear strains (stresses). In order to achieve a continuous estimation of the shear force and stress along the plate boundary, a series of very stiff springs are introduced there. The use of high stiffness values effectively simulates the clamped boundary condition of the problem. The estimated shear stress is then compared to the ultimate shear strength of the material to form a stress ratio. A node release algorithm is developed to simulate the progression of rupture. The analysis is continued in the post-failure phase to account for the deformation which continues during the free flight of the torn plate. The predicted failure modes using the above model for blast loaded plate structures are presented and compared with previously published experimental data. The Quadratic Interaction Criterion predicts consistently better results than the Linear Interaction Criterion as compared to the experimental results. The results clearly indicate the influence of shear on the failure mechanism not only for mode III, but also for mode II. The results confirm the importance of the interaction effects of tensile and bending strain on tearing and shear failure. |
Extent | 8903330 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-03 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
IsShownAt | 10.14288/1.0050233 |
URI | http://hdl.handle.net/2429/6798 |
Degree |
Doctor of Philosophy - PhD |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1997-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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