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Failure Analysis of Stiffened and Unstiffened Mild Steel Plates

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Failure Analysis of Stiffened and Unstiffened Mild Steel Plates Failure Analysis of Stiffened and Unstiffened Mild Steel Plates Subjected to Blast Loading. by Julien R. Fagnan B.Sc, The University of Alberta, 1982 M. Eng., The University of Alberta, 1984 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF CIVIL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © Julien R. Fagnan, 1996 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 of CNtL /5/JSWggg;/J^ The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract The failure of unstiffened and stiffened mild steel plates subjected to blast loads is investigated. The three failure modes identified in the literature, as the load intensity increases, are; mode I (large ductile deformation), mode II (tensile-tearing) and mode III (transverse shear). The prediction of failure for a large structure subjected to air blasts is a complex problem involving non-linear response in both geometry and material properties. In addition, the effect of high rates of strain on material properties is not well understood. An accurate analysis using finite elements currently requires a very detailed element grid with a large number of elements and the associated volume of input and output data. In the preliminary stages of design this is not practical. The present work develops simplified analysis tools to allow engineering level accuracy for preliminary design of blast loaded plates. The analysis tools consist of an analytical formulation used to predict when material separation has occurred, by either tensile tearing (mode II) or shear failure (mode III), in conjunction with a numerical formulation which models the large inelastic (mode I) behaviour of the plates when subjected to air blasts. This numerical formulation is capable of modelling a large structure with relatively few elements and yet obtain a reasonable overall accuracy but with a resulting loss of detail at the local level. The numerical formulation employed for the square plates employs existing super finite elements. These elements contain all the basic modes of deformation response (mode I) which occur in orthogonally stiffened plates. This allows stiffened plate structures to, be modelled with only one plate element per bay and one beam element per stiffener span, greatly simplifying the input and output data, A regular finite element formulation is also developed to model axisymmetric plates. ' Both these formulations use von Karman large deflection theory and the von Mises yield criterion and associated flow rule to model geometric and material non-linearities respectively. Numerical and temporal integrations are carried out using Gauss quadrature and the Newmark-P method respectively with Newton-Raphson iteration at each time step. The analytical formulation used in conjunction with these numerical models takes advantage of the overall accuracy of the deformed profile of the plates in calculating the in- plane strain at each time step. For mode II failure, the total strain is calculated by estimating the local bending and axial strain in the critical regions and avoiding the less accurate grid dependent local strains. The total strain is then compared to a maximum allowable fracture strain. For the mode III failure of axisymmetric plates, equilibrium is used to calculate the support reaction at the plate boundary at each time step. The calculated shear stress is then compared to a maximum allowable shear stress to determine if shear failure has occurred. The interaction between mode II and III failure is taken into account via an interaction relationship. Once material separation has occurred, post failure analysis continues as the plate translates freely through the air. The results from the computer models for both the axisymmetric and square stiffened and unstiffened plates are compared to experimental data available in the literature. The numerical results show good comparison to the experimental results. iii Table of Contents Abstract List of Tables List of Figures Acknowledgement 1. Introduction 2. Literature Review 2.1 Introduction 2.2. Available Fracture Models 2.2.1 Introduction 2.2.2 Fracture Mechanics 2.2.3 Void Coalescence Mechanisms 2.2.4 Damage Mechanics 2.2.5 Forming Limit Curves 2.2.6 Fracture Limit Strain 2.2.7 Strain Energy Density 2.2.8 Stress Based Failure Curves 2.2.9 Transverse Plastic Shear Hinge Selected Fracture Models 2.3.1 Mode II Failure 2.3.2 Mode III Failure 3. Numerical Formulations 3.1 Introduction iv 3.2 Approximations 14 3.3 Equations of Motion 15 3.4 NAAPFE 16 3.4.1 Introduction 16 3.4.2. Finite Element Discretization 17 3.4.3 Displacement Functions 19 3.4.4 Strain Displacement Relations 21 3.4.5 Continuity and Order of Convergence 22 3.4.6 Constitutive Relations 23 3.4.7 Mass and Damping Matrices 28 3.4.8 Load Vector 29 3.4.9 Stiffness Matrix 32 3.4.10 Numerical Integration 34 3.4.11 Temporal Integration 36 3.4.12 Computer Implementation .37 3!5' NAPSSE 38 3.5.1 Introduction 38 3.5.2 Finite Element Discretization 39 3.5.3 Displacement Functions 42 3.5.3.1 Plate Elements 42 3.5.3.2 Beam Elements 44 3.5.4 Strain Displacement Relations 45 3.5.5 ' Computer Implementation 46 Analytical Failure Formulations 47 4.1 Introduction 47 4.2 Mode II Failure Model 48 V 4.2.1 Introduction 4.2.2 Membrane Strain 4.2.3 Bending Strain 4.2.4 Extension to 2D Problems 4.3 Mode III Failure Model 4.4 Mode II and Mode III Interaction 4.5 Implementation in Axisymmetric Plates 4.5.1 Introduction 4.5.2 Mode II 4.5.2.1 Membrane Strain , 4.5.2.2 Bending Strain 4.5.3 Modem 4.5.4 Interaction • 4.5.5 Post-Failure Analysis 4.6 Implementation in Rectangular Plates 4.6.1 Introduction 4.6.2 Mode II 4.6.2.1 Membrane Strain 4.6.2.2 Bending Strain 4.6.2.3 Biaxial Strain Limit 4.6.3 Modelll Square Plates 5.1 Introduction 5.2 Experiments 5.3 Modelling the Experiment 5.4 Preliminary Studies vi 5.5 Results and Discussion 80 5.5.1 Predictions 80 5.5.2 Comparisons 86 6. Circular Plates 90 6.1 Introduction 90 6.2 Testing of Finite Element Formulation 90 6.2.1' Introduction 90 6.2.2 Static, Linear Elastic Geometry and Material 91 6.2.3 Eigenvalues 95 6.2.4 Dynamic Elastic 97 6.2.5 Static, Non-linear Geometry, Linear.Material 98 6.3 Experiments 99 6.4 Modelling the Experiment 101 6.5 Results and Discussion 102 6.5.1 Predictions 102 6.5.2 Comparisons 122 7: Stiffened Square Plates , 134 7:1 Introduction 134 . 7.2 Experiments 134 7.3 Modelling the Experiment 136 7.4 Results and Discussion 137 ^ 7.4.1 Predictions 137 7.4.2 Comparisons 142 8. Conclusions 148 Bibliography 152 vii List of Tables 5.1 Uniform pressure load for given impulse (square plate). 69 5.2 Preliminary investigation results. 73 5.3 Comparison of membrane strains. 79 5.4 Strains at mode II failure. .85 6.1 Comparisons of exact static results with NAAPFE. 92 6.2 Variation of element forces with grid type: simply supported. 94 6.3 Variation of element forces with grid type: clamped. 96 6.4 Eigenvalues. 97 6.5 Dynamic step load. 98 6.6 Non-linear geometry - linear material. .99 6.7 Residual elastic vibration. 105 7.1 Material properties for stiffened plate specimens. 136 7.2. Predicted mode II failure results (o0 = 265 MPa). 141 viii List of Figures 2.1 Generalised forming limit diagram. 8 2.2 Transverse velocity field. 11 3.1 Discretization of circular plate. 18 3.2 Degrees of freedom for circular plate element. 19 3.3 Bilinear stress-strain relationship. 24 3.4 Strain rate effect. 26 3.5 Loading functions. 31 3.6 Discretization of rectangular stiffened plate. 40 3.7 Degrees of freedom for rectangular plate and beam elements. 41 4.1 Typical centreline deformation profiles for unstiffened square plates. 49 4.2 Generalised stiffener displacement. 64 5.1 Layout of explosives for square plate tests. 68 5.2 Finite element grids for preliminary studies. 70 5.3 Effect of grid size on centreline failure profiles. 72 5.4 Effect of grid size on centreline strain distribution. (20 Ns) 75 5.5 Effect of grid size on centreline strain distribution. (25 Ns) 76 5.6 Effect of grid size on centreline strain distribution. (40 Ns) 77 5.7 Effect of impulse on centreline strain distribution. 78 5.8 •- Time history of midpoint displacement. 81 5.9 Effect of strain rate on centreline deformation profiles. 81 5.10 Effect of impulse on yield stress, strain rate and time to first yield. 82 5.11 Predicted transient centreline deformation profiles. 83 ix 5.12 Predicted final permanent centreline deformation profiles. 84 5.13 Strain perpendicular to boundary at failure. 86 5.1.4 Comparison of experimental and predicted centreline profiles. 87 5.15 Midpoint deflection versus impulse. 88 6.1 Layout of explosives for circular plate tests.
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