RESPONSE OF WIDE FLANGE STEEL COLUMNS SUBJECTED TO CONSTANT AXIAL LOAD AND LATERAL BLAST LOAD

Transcription

1 RESPONSE OF WIDE FLANGE STEEL COLUMNS SUBJECTED TO CONSTANT AXIAL LOAD AND LATERAL BLAST LOAD By Ronald L. Shope Dissertation Submitted to the Faculty of the Virginia Polytechnic Institute and State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY IN CIVIL ENGINEERING Approved by: Raymond H. Plaut, Chair Romesh C. Batra W. Samuel Easterling Kamal B. Rojiani Alfred L. Wicks October 3, 6 Blacksburg, Virginia Keywords: Blast, Steel Column, Axial Load, Finite Element, Residual Stress, Strain Rate Effects, Rotational Spring Supports, Non-Uniform Loads, Biaxial Bending Copyright 6, Ronald L. Shope

2 RESPONSE OF WIDE FLANGE STEEL COLUMNS SUBJECTED TO CONSTANT AXIAL LOAD AND LATERAL BLAST LOAD By Ronald L. Shope Raymond H. Plaut, Committee Chairman (ABSTRACT) The response of wide flange steel columns subjected to constant axial loads and lateral blast loads was examined. The finite element program ABAQUS was used to model W8x4 sections with different slendernesses and boundary conditions. For the response calculations, a constant axial force was first applied to the column and the equilibrium state was determined. Next, a short duration, lateral blast load was applied and the response time history was calculated. Changes in displacement time histories and plastic hinge formations resulting from varying the axial load were examined. The cases studied include single-span and two-span columns. In addition to ideal boundary conditions, columns with linear elastic, rotational supports were also studied. Non-uniform blast loads were considered. Major axis, minor axis, and biaxial bending were investigated. The effects of strain rate and residual stresses were examined. The results for each column configuration are presented as a set of curves showing the critical blast impulse versus axial load. The critical blast impulse is defined as the impulse that either causes the column to collapse or to exceed the limiting deflection criterion. A major goal of this effort was to develop simplified design and analysis methods. To accomplish this, two single-degree-of-freedom approaches that include the effects of the axial load were derived. The first uses a bilinear resistance function that is similar to the one used for beam analysis. This approach provides a rough estimate of the critical impulse and is suitable only for preliminary design or quick vulnerability calculations. The second approach uses a nonlinear resistance function that accounts for the gradual yielding that occurs during the dynamic response. This approach can be easily implemented in a simple computer program or spreadsheet and provides close agreement with the results from the finite element method.

3 ACKNOWLEDGEMENTS I would like to thank my committee chairman and advisor, Dr. Raymond H. Plaut for his instruction and guidance throughout the course of this study. His patience and encouragement are greatly appreciated. I would also like to thank the other members of my committee, Dr. Romesh C. Batra, Dr. W. Samuel Easterling, Dr. Kamal B. Rojiani, and Dr. Alfred L. Wicks, for their valuable participation. I wish to acknowledge the incredible support I received as a Via Doctorial Fellowship recipient. I will always be grateful for the unique opportunities the fellowship provided me. I would like to thank my parents and other family members for their support and frequent words of encouragement. Finally, I would like to thank my wife, Rachel, for her love and support. iii

4 TABLE OF CONTENTS ACKNOWLEDGEMENTS...iii TABLE OF CONTENTS... iv LIST OF FIGURES... x LIST OF TABLES... xxiv Chapter 1 INTRODUCTION... 1 Chapter BACKGROUND BLAST LOADS Explosion Characteristics External Explosions Internal Explosions Blast Loads on Columns STRUCTURAL RESPONSE Strain Rate Effects Residual Stresses Influence of Axial Loads on Lateral Response Effect of Lateral Bracing Loss and Shear Failure ANALYSIS TECHNIQUES Approximate Methods Finite Element Method EXPERIMENTAL WORK CURRENT DESIGN APPROACH Beams iv

5 .5. Columns IMPULSIVE BLAST LOADS... 4 Chapter 3 FINITE ELEMENT ANALYSES APPROACH MODELS RESPONSE Major Axis Bending Minor Axis Bending CRITICAL IMPULSE CURVES CONCLUSIONS Chapter 4 SINGLE-DEGREE-OF-FREEDOM METHOD USING BILINEAR RESISTANCE FUNCTION REVIEW OF SINGLE-DEGREE-OF-FREEDOM RESPONSE SDOF APPROACH FOR BEAMS (NO AXIAL LOAD) Kinetic Energy Resistance Function Resistance Transformation Factors Maximum Allowable Impulse SDOF APPROACH FOR COLUMNS Description Reduced Plastic Moment Capacity Plastic Design Moment Resistance Function v

6 4.3.5 Critical Impulse CONCLUSIONS Chapter 5 SINGLE-DEGREE-OF-FREEDOM METHOD USING NONLINEAR RESISTANCE FUNCTION DESCRIPTION RESISTANCE FUNCTION Assumptions Major Axis Bending, Tension Flange Not Yielded Major Axis Bending, Tension Flange Yielded Strain Calculation Summary of Procedure for Major Axis Bending Minor Axis Bending, Tension Side Not Yielded Minor Axis Bending, Tension Side Yielded Minor Axis Bending, Summary of Procedure COMPARISON OF BILINEAR AND NONLINEAR RESISTANCE FUNCTIONS CRITICAL IMPULSE CONCLUSIONS Chapter 6 NON-UNIFORM BLAST LOADS DESCRIPTION FINITE ELEMENT CALCULATIONS NONLINEAR SDOF ANALYSIS USING BUCKLING MODE SHAPE NONLINEAR SDOF ANALYSIS USING PLASTIC SHAPE CONCLUSIONS vi

7 Chapter 7 ROTATIONAL SPRING SUPPORTS DESCRIPTION DISPLACEMENT CRITERION CRITICAL BUCKLING LOAD NONLINEAR SDOF APPROACH Kinetic Energy Resistance Function CRITICAL IMPULSE CURVES CONCLUSIONS... Chapter 8 BIAXIAL BENDING APPROACH MODEL DYNAMIC ANALYSIS CRITICAL IMPULSE CURVES CONCLUSIONS... 8 Chapter 9 RESIDUAL STRESS AND STRAIN RATE DEPENDENT YIELDING DESCRIPTION RESIDUAL STRESSES Finite Element Model Static Analyses Dynamic Analyses STRAIN RATE EFFECTS vii

8 9.3.1 Finite Element Model Dynamic Analyses COMBINATION OF RESIDUAL STRESS AND STRAIN RATE SIMPLE NONLINEAR SDOF METHOD FOR RESIDUAL STRESS AND STRAIN RATE Assumptions Critical Impulse Curves SDOF METHOD WITH TRANSITION REGION FOR RESIDUAL STRESS AND STRAIN RATE CONCLUSIONS Chapter 1 TWO-SPAN COLUMN APPROACH MODEL STATIC ANALYSIS DYNAMIC ANALYSIS CRITICAL IMPULSE CURVES CONCLUSIONS Chapter 11 SUMMARY AND CONCLUSIONS REFERENCES APPENDIX A: TIME SEQUENCE PLOTS FOR UNIFORMLY LOADED COLUMNS APPENDIX B: TIME SEQUENCE PLOTS FOR NON-UNIFORMLY LOADED COLUMNS viii

9 VITA ix

10 LIST OF FIGURES Figure -1 Ideal Blast Wave (Adapted from TM5-13, 199)... 6 Figure - Surface Burst Wave Form (Adapted from TM5-13, 199)... 8 Figure -3 Positive Phase Shock Wave Parameters for a Spherical TNT Explosion in Free Air at Sea Level (Adapted from TM5-13, 199)... 9 Figure -4 Equivalent Triangular Loads (Adapted from TM5-13, 199) Figure -5 Pressure-Time History from Confined Explosion (Adapted from TM5-13, 199)... 1 Figure -6 Combined Shock and Gas Pressure for an Internal TNT Explosion (Adapted from TM5-13, 199) Figure -7 Interaction of Shock Wave with Rectangular Column Figure -8 Column Loads from Blast (Adapted from TM5-13, 199) Figure -9 Typical Stress-Strain Curve for Low Carbon Steel (Adapted from TM5-13, 199)... Figure -1 Dynamic Increase Factor vs. Strain Rate for Yield Stress of A36 Steel (Adapted from TM5-13, 199)... Figure -11 Critical Buckling Stress vs. Slenderness Ratio for Steel Columns... 5 Figure 3-1 Axial Load Capacity versus Length Figure 3- (a) Column; (b) Load-Time Function; (c) Location of Section Integration Points Figure 3-3 Response of Column When Collapse Governs Figure 3-4 Response of Column When Limiting Deflection Governs Figure 3-5 Stress vs. Time; Major Axis Bending, i = 158 kn-ms/m, P = 1419 kn... 5 Figure 3-6 Strain vs. Time; Major Axis Bending, i = 158 kn-ms/m, P = 1419 kn... 5 Figure 3-7 Strain vs. Displacement; Major Axis Bending, i = 158 kn-ms/m, P = 1419 kn Figure 3-8 Time Sequence Plot; Major Axis Bending, i = 487 kn-ms/m, P = kn... 5 Figure 3-9 Plastic Hinge Formation; Major Axis Bending, i = 487 kn-ms/m, P = kn x

11 Figure 3-1 Time Sequence Plot; Major Axis Bending, i = 158 kn-ms/m, P = 143 kn Figure 3-11 Plastic Hinge Formation; Major Axis Bending, i = 158 kn-ms/m, P = 143 kn Figure 3-1 Displacement vs. Time; Minor Axis Bending Figure 3-13 Stress vs. Time; Minor Axis Bending, i = 14 kn-ms/m, P = 1495 kn Figure 3-14 Strain vs. Time; Minor Axis Bending, i = 14 kn-ms/m, P = 1495 kn Figure 3-15 Strain vs. Displacement; Minor Axis Bending, i = 14 kn-ms/m, P = 1495 kn Figure 3-16 Time Sequence Plot; Minor Axis Bending, i = 44 kn-ms/m, P = kn Figure 3-17 Plastic Hinge Formation; Minor Axis Bending, i = 55 kn-ms/m, P = kn Figure 3-18 Time Sequence Plot; Minor Axis Bending, i = 14 kn-ms/m, P = 1499 kn... 6 Figure 3-19 Plastic Hinge Formation; Minor Axis Bending, i = 14 kn-ms/m, P = 1499 kn Figure 3- Displacement vs. Time Illustrating Critical Impulse... 6 Figure 3-1 Critical Impulse Curves; Major Axis, Simply Supported Figure 3- Critical Impulse Curves; Minor Axis, Simply Supported Figure 3-3 Critical Impulse Curves; Major Axis, Fixed Ends Figure 3-4 Critical Impulse Curves; Minor Axis, Fixed Ends Figure 3-5 Critical Impulse Curves; Major Axis, Propped Cantilever Figure 3-6 Critical Impulse Curves; Minor Axis, Propped Cantilever Figure 4-1 Single-Degree-of-Freedom System Figure 4- Free Body Diagram For Beams Figure 4-3 Propped Cantilever Beam Figure 4-4 Generalized Resistance Function For Beams Figure 4-5 Plastic Bending of Beams, (a) Symmetric Cases, (b) Propped Cantilever xi

12 Figure 4-6 Stress Diagrams for I-Section, Major Axis Bending With Different Magnitudes of Axial Load Figure 4-7 Stress Diagrams for I-Section, Minor Axis Bending With Different Magnitudes of Axial Load Figure 4-8 P vs. M pa for W8x4 Section... 9 Figure 4-9 Free Body Diagram of Column Figure 4-1 Resistance Function for Fixed End Column When Collapse Governs Figure 4-11 Generalized SDOF Resistance Function For Columns When Collapse Governs... 1 Figure 4-1 SDOF Resistance Function for Columns When Deflection Criterion Governs Figure 4-13 SDOF Resistance Function for Columns When Deflection Criterion Is Less Than Elastic Displacement Figure 4-14 Bilinear SDOF Critical Impulse Curves; Major Axis, Simply Supported Figure 4-15 Bilinear SDOF Critical Impulse Curves; Minor Axis, Simply Supported Figure 4-16 Bilinear SDOF Critical Impulse Curves; Major Axis, Fixed Ends Figure 4-17 Bilinear SDOF Critical Impulse Curves; Minor Axis, Fixed Ends Figure 4-18 Bilinear SDOF Critical Impulse Curves; Major Axis, Propped Cantilever Figure 4-19 Bilinear SDOF Critical Impulse Curves; Minor Axis, Propped Cantilever Figure 5-1 Plastic Hinge Formation in Column Figure 5- Stress and Strain Diagrams for Major Axis Bending; No Yielding in Tension Flange Figure 5-3 Stress and Strain Diagrams for Major Axis Bending; Tension Flange Yielded Figure 5-4 Flow Chart for Major Axis Resistance Function Figure 5-5 Stress and Strain Diagrams for Minor Axis Bending; No Yielding on Tension Side... 1 xii

13 Figure 5-6 Stress and Strain Diagrams for Minor Axis Bending; Yielding on Tension Side Figure 5-7 Flow Chart for Minor Axis Resistance Function Figure 5-8 Resistance Functions for Various Magnitudes of Axial Load Figure 5-9 Comparison of Resistance Functions; Major Axis, Simply Supported Figure 5-1 Comparison of Resistance Functions; Minor Axis, Simply Supported Figure 5-11 Comparison of Resistance Functions; Major Axis, Fixed End Figure 5-1 Comparison of Resistance Functions; Minor Axis, Fixed End Figure 5-13 Comparison of Resistance Functions; Major Axis, Propped Cantilever Figure 5-14 Comparison of Resistance Functions; Minor Axis, Propped Cantilever Figure 5-15 Resistance Function; Major Axis, Simply Supported, P = 516 kn Figure 5-16 Resistance Function; Major Axis, Simply Supported, P = 143 kn Figure 5-17 Nonlinear SDOF Critical Impulse Curves; Major Axis, Simply Supported Figure 5-18 Nonlinear SDOF Critical Impulse Curves; Minor Axis, Simply Supported Figure 5-19 Nonlinear SDOF Critical Impulse Curves; Major Axis, Fixed Ends Figure 5- Nonlinear SDOF Critical Impulse Curves; Minor Axis, Fixed Ends Figure 5-1 Nonlinear SDOF Critical Impulse Curves; Major Axis, Propped Cantilever Figure 5- Nonlinear SDOF Critical Impulse Curves; Minor Axis, Propped Cantilever Figure 6-1 Notional Representation of Non-Uniform Blast Load Figure 6- Idealized Representation of Non-Uniform Blast Load Figure 6-3 Critical Impulse Curves; Major Axis, Simply Supported, Non- Uniform Loads Figure 6-4 Critical Impulse Curves; Minor Axis, Simply Supported, Non- Uniform Loads xiii

14 Figure 6-5 Critical Impulse Curves; Major Axis, Fixed Ends, Non-Uniform Loads Figure 6-6 Critical Impulse Curves; Minor Axis, Fixed Ends, Non-Uniform Loads Figure 6-7 Critical Impulse Curves; Major Axis, Propped Cantilever, Non- Uniform Loads Figure 6-8 Critical Impulse Curves; Minor Axis, Propped Cantilever, Non- Uniform Loads Figure 6-9 Plastic Hinge Formation; Minor Axis, Simply Supported, Non- Uniform Loads, Bomb at L/ Figure 6-1 Plastic Hinge Formation; Minor Axis, Simply Supported, Non- Uniform Loads, Bomb at L/ Figure 6-11 Plastic Hinge Formation; Minor Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-1 Plastic Hinge Formation; Minor Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-13 Plastic Hinge Formation; Minor Axis, Propped Cantilever, Non- Uniform Loads, Bomb at L/ Figure 6-14 Plastic Hinge Formation; Minor Axis, Propped Cantilever, Non- Uniform Loads, Bomb at L/ Figure 6-15 Plastic Hinge Formation; Minor Axis, Propped Cantilever, Non- Uniform Loads, Bomb at 3L/ Figure 6-16 Plastic Hinge Formation; Major Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-17 SDOF and Finite Element Critical Impulse Curves; Major Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-18 SDOF and Finite Element Critical Impulse Curves; Minor Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-19 SDOF and Finite Element Critical Impulse Curves; Major Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ xiv

15 Figure 6- SDOF and Finite Element Critical Impulse Curves; Minor Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-1 SDOF and Finite Element Critical Impulse Curves; Major Axis, Propped Cantilever, Non-Uniform Loads, Bomb at L/ Figure 6- SDOF and Finite Element Critical Impulse Curves; Minor Axis, Propped Cantilever, Non-Uniform Loads, Bomb at L/ Figure 6-3 SDOF and Finite Element Critical Impulse Curves; Major Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-4 SDOF and Finite Element Critical Impulse Curves; Minor Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-5 SDOF and Finite Element Critical Impulse Curves; Major Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-6 SDOF and Finite Element Critical Impulse Curves; Minor Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-7 SDOF and Finite Element Critical Impulse Curves; Major Axis, Propped Cantilever, Non-Uniform Loads, Bomb at L/ Figure 6-8 SDOF and Finite Element Critical Impulse Curves; Minor Axis, Propped Cantilever, Non-Uniform Loads, Bomb at L/ Figure 6-9 SDOF and Finite Element Critical Impulse Curves; Major Axis, Propped Cantilever, Non-Uniform Loads, Bomb at 3L/ Figure 6-3 SDOF and Finite Element Critical Impulse Curves; Minor Axis, Propped Cantilever, Non-Uniform Loads, Bomb at 3L/ Figure 6-31 Free Body Diagram of Non-Uniformly Loaded Member Figure 6-3 Plastic Deformed Shape, Simply Supported, Bomb at L/ Figure 6-33 Plastic Shape SDOF Critical Impulse Curves; Major Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-34 Plastic Shape SDOF Critical Impulse Curves; Minor Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-35 Plastic Shape SDOF Critical Impulse Curves; Major Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ xv

16 Figure 6-36 Plastic Shape SDOF Critical Impulse Curves; Minor Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-37 Plastic Shape SDOF Critical Impulse Curves; Major Axis, Propped Cantilever Loads, Bomb at L/ Figure 6-38 Plastic Shape SDOF Critical Impulse Curves; Minor Axis, Propped Cantilever Loads, Bomb at L/ Figure 6-39 Plastic Shape SDOF Critical Impulse Curves; Major Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-4 Plastic Shape SDOF Critical Impulse Curves; Minor Axis, Simply Supported, Non-Uniform Loads, Bomb at L/ Figure 6-41 Plastic Shape SDOF Critical Impulse Curves; Major Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-4 Plastic Shape SDOF Critical Impulse Curves; Minor Axis, Fixed Ends, Non-Uniform Loads, Bomb at L/ Figure 6-43 Plastic Shape SDOF Critical Impulse Curves; Major Axis, Propped Cantilever, Non-Uniform Loads, Bomb at L/ Figure 6-44 Plastic Shape SDOF Critical Impulse Curves; Minor Axis, Propped Cantilever, Non-Uniform Loads, Bomb at L/ Figure 6-45 Plastic Shape SDOF Critical Impulse Curves; Major Axis, Propped Cantilever, Non-Uniform Loads, Bomb at 3L/ Figure 6-46 Plastic Shape SDOF Critical Impulse Curves; Minor Axis, Propped Cantilever, Non-Uniform Loads, Bomb at 3L/ Figure 7-1 Column with Rotation Supports Figure 7- Normalized Buckling Load, q, vs. β Figure 7-3 Column Length Factor, κ, vs. β O Figure 7-4 Kinetic Energy Coefficient, C T, vs. κ Figure 7-5 Resistance Function for a Beam with Rotational Spring Supports Figure 7-6 Critical Impulse Curves; Major Axis, Rotational Spring Support, κ = Figure 7-7 Critical Impulse Curves; Minor Axis, Rotational Spring Support, κ = xvi

17 Figure 7-8 Critical Impulse Curves; Major Axis, Rotational Spring Support, κ = Figure 7-9 Critical Impulse Curves; Minor Axis, Rotational Spring Support, κ = Figure 7-1 Critical Impulse Curves; Major Axis, Rotational Spring Support, κ = Figure 7-11 Critical Impulse Curves; Minor Axis, Rotational Spring Support, κ = Figure 7-1 Comparison of Critical Impulse Curves, Ideal Boundary Conditions vs. Rotational Spring Supports, κ =.85, λ c = Figure 8-1 Total Displacement vs. Time; Simply Supported, Impulse Angle = 1 degrees... 5 Figure 8- x- vs. y-displacement; Simply Supported, Impulse Angle = 1 degrees, i = 669 kn-ms/m, P =... 7 Figure 8-3 x- vs. y-displacement; Simply Supported, Impulse Angle = 1 degrees, i = 1 kn-ms/m, P = 153 kn... 7 Figure 8-4 Total Displacement vs. Time; Simply Supported, Impulse Angle = 45 degrees... 9 Figure 8-5 x- vs. y-displacement; Simply Supported, Impulse Angle = 45 degrees, i = 518 kn-ms/m, P =... 9 Figure 8-6 x- vs. y-displacement; Simply Supported, Impulse Angle = 45 degrees, i = 158 kn-ms/m, P = 1463 kn... 1 Figure 8-7 Total Displacement vs. Time; Simply Supported, Impulse Angle = 8 degrees Figure 8-8 x- vs. y-displacement; Simply Supported, Impulse Angle = 8 degrees, i = 447 kn-ms/m, P = Figure 8-9 x- vs. y-displacement; Simply Supported, Impulse Angle = 8 degrees, i = 14 kn-ms/m, P = 1499 kn... 1 Figure 8-1 Total Displacement vs. Time; Fixed Ends, Impulse Angle = 1 degrees xvii

18 Figure 8-11 x- vs. y-displacement; Fixed Ends, Impulse Angle = 1 degrees, i = 75 kn-ms/m, P = Figure 8-1 x- vs. y-displacement; Fixed Ends, Impulse Angle = 1 degrees, i = 1 kn-ms/m, P = 1716 kn Figure 8-13 Total Displacement vs. Time; Fixed Ends, Impulse Angle = 45 degrees Figure 8-14 x- vs. y-displacement; Fixed Ends, Impulse Angle = 45 degrees, i = 54 kn-ms/m, P = Figure 8-15 x- vs. y-displacement; Fixed Ends, Impulse Angle = 45 degrees, i = 158 kn-ms/m, P = 1485 kn Figure 8-16 Total Displacement vs. Time; Fixed Ends, Impulse Angle = 8 degrees Figure 8-17 x- vs. y-displacement; Fixed Ends, Impulse Angle = 8 degrees, i = 46 kn-ms/m, P = Figure 8-18 x- vs. y-displacement; Fixed Ends, Impulse Angle = 8 degrees, i = 14 kn-ms/m, P = 151 kn Figure 8-19 Total Displacement vs. Time; Propped Cantilever, Impulse Angle = 1 degrees Figure 8- x- vs. y-displacement; Propped Cantilever, Impulse Angle = 1 degrees, i = 65 kn-ms/m, P = Figure 8-1 x- vs. y-displacement; Propped Cantilever, Impulse Angle = 1 degrees, i = 193 kn-ms/m, P = 169 kn Figure 8- Total Displacement vs. Time; Propped Cantilever, Impulse Angle = 45 degrees Figure 8-3 x- vs. y-displacement; Propped Cantilever, Impulse Angle = 45 degrees, i = 48 kn-ms/m, P =... Figure 8-4 x- vs. y-displacement; Propped Cantilever, Impulse Angle = 45 degrees, i = 14 kn-ms/m, P = 153 kn... Figure 8-5 Total Displacement vs. Time; Propped Cantilever, Impulse Angle = 8 degrees... 1 xviii

19 Figure 8-6 x- vs. y-displacement; Propped Cantilever, Impulse Angle = 8 degrees, i = 411 kn-ms/m, P =... 1 Figure 8-7 x- vs. y-displacement; Propped Cantilever, Impulse Angle = 8 degrees, i = 13 kn-ms/m, P = 1566 kn... Figure 8-8 Critical Impulse Curves; Simply Supported, λ c = Figure 8-9 Critical Impulse Curves; Simply Supported, λ c = Figure 8-3 Critical Impulse Curves; Simply Supported, λ c = Figure 8-31 Critical Impulse Curves; Fixed Ends, λ c = Figure 8-3 Critical Impulse Curves; Fixed Ends, λ c = Figure 8-33 Critical Impulse Curves; Fixed Ends, λ c = Figure 8-34 Critical Impulse Curves; Propped Cantilever, λ c = Figure 8-35 Critical Impulse Curves; Propped Cantilever, λ c = Figure 8-36 Critical Impulse Curves; Propped Cantilever, λ c = Figure 9-1 Residual Stress Distribution in W8x4 Column Figure 9- Residual Stress Input for Finite Element... 3 Figure 9-3 Static Results With Residual Stress; Major Axis, Simply Supported Figure 9-4 Static Results With Residual Stress; Minor Axis, Simply Supported Figure 9-5 Critical Impulse Curves with Residual Stress Model ; Major Axis, Simply Supported Figure 9-6 Critical Impulse Curves with Strain Rate Model ; Major Axis, Simply Supported Figure 9-7 Critical Impulse Curves with RS/SR Model; Major Axis, Simply Supported Figure 9-8 Critical Impulse Curves with RS/SR Model; Minor Axis, Simply Supported Figure 9-9 Critical Impulse Curves with RS/SR Model; Major Axis, Fixed Ends... 4 Figure 9-1 Critical Impulse Curves with RS/SR Model; Minor Axis, Fixed Ends... 4 Figure 9-11 Critical Impulse Curves with RS/SR Model; Major Axis, Propped Cantilever Figure 9-1 Critical Impulse Curves with RS/SR Model; Minor Axis, Propped Cantilever xix

20 Figure 9-13 Comparison of Finite Element RS/SR and SDOF With Strain Rate Figure 9-14 Comparison of RS/SR Finite Element and RS/SR SDOF Figure 9-15 Critical Impulse Curves RS/SR SDOF Model; Major Axis, Simply Supported Figure 9-16 Critical Impulse Curves - RS/SR SDOF Model; Minor Axis, Simply Supported Figure 9-17 Critical Impulse Curves - RS/SR SDOF Model; Major Axis, Fixed Ends Figure 9-18 Critical Impulse Curves - RS/SR SDOF Model; Minor Axis, Fixed Ends Figure 9-19 Critical Impulse Curves - RS/SR SDOF Model; Major Axis, Propped Cantilever Figure 9- Critical Impulse Curves - RS/SR SDOF Model; Minor Axis, Propped Cantilever Figure 9-1 Modified Dynamic Yield Stress vs. P/P cr Figure 9- Critical Impulse Curves Modified RS/SR SDOF Model; Major Axis, Simply Supported... 5 Figure 9-3 Critical Impulse Curves Modified RS/SR SDOF Model; Minor Axis, Simply Supported... 5 Figure 9-4 Critical Impulse Curves Modified RS/SR SDOF Model; Major Axis, Fixed Ends Figure 9-5 Critical Impulse Curves Modified RS/SR SDOF Model; Minor Axis, Fixed Ends Figure 9-6 Critical Impulse Curves Modified RS/SR SDOF Model; Major Axis, Propped Cantilever Figure 9-7 Critical Impulse Curves Modified RS/SR SDOF Model; Minor Axis, Propped Cantilever Figure 1-1 Two Span Column Model; (a) Column, (b) Load-Time Function Figure 1- Critical Top Load vs. Length for P = and P = P Figure 1-3 Critical Top Load vs. Length for P = P 1 and P = 3P Figure 1-4 Response of Two-Span Column with P 1 = and P =... 6 xx

21 Figure 1-5 Response of Two-Span Column with P 1 = 45 kn and P = 9 kn... 6 Figure 1-6 Response of Two-Span Column with P 1 = 65 kn and P = 13 kn Figure 1-7 Lateral Displacement for Case Shown in Figure Figure 1-8 Critical Impulse Curves; Two-Span Column (λ c = 1.) Figure 1-9 Comparison of Two-Span Column and Single-Span Column Critical Impulse Curves Figure A-1 Time Sequence Plot; Uniform Load, Simply Supported, Major Axis Bending, i = 487 kn-ms/m, P = kn... 8 Figure A- Time Sequence Plot; Uniform Load, Simply Supported, Major Axis Bending, i = 158 kn-ms/m, P = 143 kn Figure A-3 Time Sequence Plot; Uniform Load, Simply Supported, Minor Axis Bending, i = 44 kn-ms/m, P = kn Figure A-4 Time Sequence Plot; Uniform Load, Simply Supported, Minor Axis Bending, i = 14 kn-ms/m, P = 1499 kn... 9 Figure A-5 Time Sequence Plot; Uniform Load, Fixed Ends, Major Axis Bending, i = 518 kn-ms/m, P = kn Figure A-6 Time Sequence Plot; Uniform Load, Fixed Ends, Major Axis Bending, i = 158 kn-ms/m, P = 1454 kn... 3 Figure A-7 Time Sequence Plot; Uniform Load, Fixed Ends, Minor Axis Bending, i = 466 kn-ms/m, P = kn Figure A-8 Time Sequence Plot; Uniform Load, Fixed Ends, Minor Axis Bending, i = 14 kn-ms/m, P = 151 kn Figure A-9 Time Sequence Plot; Uniform Load, Propped Cantilever, Major Axis Bending, i = kn-ms/m, P = kn Figure A-1 Time Sequence Plot; Uniform Load, Propped Cantilever, Major Axis Bending, i = 14.1 kn-ms/m, P = 1516 kn Figure A-11 Time Sequence Plot; Uniform Load, Propped Cantilever, Minor Axis Bending, i = 4 kn-ms/m, P = kn... 3 Figure A-1 Time Sequence Plot; Uniform Load, Propped Cantilever, Minor Axis Bending, i = 87.6 kn-ms/m, P = 1715 kn xxi

22 Figure B-1 Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Simply Supported, Minor Axis, i = 1.8 kn-ms/m, P = kn Figure B- Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Simply Supported, Minor Axis, i = 35. kn-ms/m, P = 1487 kn Figure B-3 Time Sequence Plot; Non-Uniform Load, Bomb at L/, Simply Supported, Minor Axis, i = 13.5 kn-ms/m, P = kn Figure B-4 Time Sequence Plot; Non-Uniform Load, Bomb at L/, Simply Supported, Minor Axis, i = 6.3 kn-ms/m, P = 1597 kn Figure B-5 Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Fixed Ends, Minor Axis, i = 11. kn-ms/m, P = kn Figure B-6 Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Fixed Ends, Minor Axis, i = 35. kn-ms/m, P = 148 kn Figure B-7 Time Sequence Plot; Non-Uniform Load, Bomb at L/, Fixed Ends, Minor Axis, i = 98.1 kn-ms/m, P = kn Figure B-8 Time Sequence Plot; Non-Uniform Load, Bomb at L/, Fixed Ends, Minor Axis, i = 9.8 kn-ms/m, P = 1516 kn Figure B-9 Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Propped Cantilever, Minor Axis, i = kn-ms/m, P = kn Figure B-1 Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Propped Cantilever, Minor Axis, i = 31.5 kn-ms/m, P = 156 kn Figure B-11 Time Sequence Plot; Non-Uniform Load, Bomb at L/, Propped Cantilever, Minor Axis, i = 9.35 kn-ms/m, P = kn Figure B-1 Time Sequence Plot; Non-Uniform Load, Bomb at L/, Propped Cantilever, Minor Axis, i = 4.51 kn-ms/m, P = 169 kn Figure B-13 Time Sequence Plot; Non-Uniform Load, Bomb at 3L/4, Propped Cantilever, Minor Axis, i = 11.7 kn-ms/m, P = kn Figure B-14 Time Sequence Plot; Non-Uniform Load, Bomb at 3L/4, Propped Cantilever, Minor Axis, i = 8. kn-ms/m, P = 1555 kn Figure B-15 Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Fixed Ends, Major Axis, i = 131. kn-ms/m, P = kn xxii

23 Figure B-16 Time Sequence Plot; Non-Uniform Load, Bomb at L/4, Fixed Ends, Major Axis, i = 35. kn-ms/m, P = 1458 kn xxiii

24 LIST OF TABLES Table -1 Dynamic Increase Factors for Yield Stress of Structural Steels... 1 Table - Dynamic Increase Factors for Ultimate Stress of Structural Steels... 1 Table 4-1 Single-Degree-of Freedom Parameters for Beams Table 4- Equations for Reduced Plastic Moment Capacity, M pa... 9 Table 4-3 Single-Degree-of-Freedom Parameters for Columns... 1 Table 5-1 Coefficients for Column SDOF Table 6-1 Kinetic Energy Simply Supported Table 6- Kinetic Energy Fixed Ends Table 6-3 Kinetic Energy Propped Cantilever Table 6-4 Kinetic Energy Simply Supported, Elastic Mode and Plastic Shape Table 6-5 Kinetic Energy Fixed Ends, Elastic Mode and Plastic Shape Table 6-6 Kinetic Energy Propped Cantilever, Elastic Mode and Plastic Shape Table 9-1 Dynamic Increase Factors for Yield Stress xxiv

25 Chapter 1 INTRODUCTION Disasters such as the terrorist bombings of the U.S. embassies in Nairobi, Kenya and Dar es Salaam, Tanzania in 1998, the Khobar Towers military barracks in Dhahran, Saudi Arabia in 1996, the Murrah Federal Building in Oklahoma City in 1995, and the World Trade Center in New York in 1993 have demonstrated the need for a thorough examination of the behavior of columns subjected to blast loads. At the World Trade Center, a confined explosion in the parking garage destroyed the concrete slabs above and below the explosion. This left several of the main structural columns laterally unbraced for lengths of up to 18 meters (Tarricone, 1993). Although these columns were exposed to an explosion estimated to be the equivalent of 68 kilograms of TNT, collapse did not occur. Sadly, the same cannot be said of the Oklahoma City tragedy. The Oklahoma City bomb was estimated to be the equivalent of 18 kilograms of TNT and was detonated at a short distance of 6 meters from the front of the building. Three columns were destroyed by the blast. This resulted in a catastrophic collapse of roughly one-half of the building s occupied space (FEMA, 1996). It is estimated that 8 percent of the 168 deaths were caused by the building collapse rather than the blast itself (Prendergast, 1995). A better understanding of column behavior is needed before this type of problem can be adequately addressed. This dissertation had two main objectives. First, the response of steel columns subjected to lateral blast loads was investigated using the finite element method. The magnitudes of the blast impulse and axial load, boundary conditions, slenderness, and bending direction (major or minor axis) were considered. Additionally, the inclusion of 1

26 residual stresses and strain rate effects was studied. The purpose of this part of the study was to gain an understanding of how these columns respond to blast loads. Although the finite element method provides the best generalized approach for modeling the physics of the response, its use requires a significant investment in hardware and software. This is especially true if plasticity, strain rate dependent yielding, and residual stress distributions are included, since most of the less expensive commercial codes do not have these features. Perhaps more importantly, obtaining finite element solutions requires a significant investment in time. This precludes its use in response planning during a crisis situation. Also, the uncertainties associated with blast effects can make the use of finite element analysis unwarranted in some applications. Therefore, the second objective was to develop simplified engineering models that predict the critical blast impulse based on the axial load applied to the column at the time of the explosion. These methods are based on the single-degree-of-freedom (SDOF) approach widely used for beam analysis (where no axial load is present). The second chapter presents a review of the existing literature dealing with blast loads and column response. Blast load characteristics and simple methods for predicting blast wave parameters are discussed. Topics relating to structural response are reviewed next. These include strain rate material yielding, residual stresses in wide flange columns, and effects of lateral bracing loss. An overview of current analysis and design techniques is also presented. Chapter 3 describes the finite element modeling approach and presents the results of the baseline study. The baseline model uses an elastic-perfectly plastic material model with no strain rate dependent yielding or residual stresses. The spread of plasticity during

27 the dynamic response is examined for different magnitudes of axial load and different bending directions. The approach used to determine the critical impulse is presented. The chapter concludes by presenting the results as a set of critical impulse curves. These are curves that plot the axial load versus critical impulse for a given bending direction and boundary condition and for a range of slendernesses. Chapter 4 and Chapter 5 present the derivation of two SDOF approaches that include the effects of the axial load in the response. The method in Chapter 4 uses a bilinear resistance function that is somewhat like the one used for beams. This method is based on the assumption that the plastic hinges form suddenly during the dynamic response. The use of this method requires only simple hand calculations. Chapter 5 introduces a nonlinear resistance function that accounts for the gradual yielding in the member during the dynamic response. This method is easily implemented in a simple computer program or spreadsheet. Both chapters conclude with critical impulse curves that are compared with the finite element results. Chapter 6 examines columns subjected to non-uniform blast loads and highlights the contributions of higher modes for unsymmetrical loadings. Two approaches for developing an SDOF approximation are examined. As expected, the SDOF approaches have limitations since the response is multi-modal. However, an SDOF approach is identified that gives reasonable, conservative results suitable for design purposes. Chapter 7 addresses columns that are constrained against lateral motion and have linear elastic rotational spring supports. An SDOF approach for approximating the column response is presented that involves calculating the effective column length factor based on the rotational stiffness of the springs. The results from the SDOF approach, 3

28 which uses the equivalent column length factor, are compared with finite element calculations. Chapter 8 examines columns that experience biaxial bending resulting from blast loads that impact the column at angles not aligned with the primary axes. Midspan displacement paths and plastic zones are presented for various blast wave angles. Guidance on using the SDOF method is also discussed. A study of the effects of residual stresses and strain rate dependent yielding is presented in Chapter 9. The finite element method is used to show how the occurrence of each affects the response individually and in combination. The chapter also presents modifications that can be made to the nonlinear SDOF method which help to account for these effects. In Chapter 1, the finite element method is used to examine two-span (or twostory) columns. A range of floor load to roof load ratios is considered. The results are compared to the single-span cases. Chapter 11 summarizes this research and presents the conclusions. Suggestions are also made for future research in this area. 4

29 Chapter BACKGROUND.1 Blast Loads.1.1 Explosion Characteristics Blast loads are produced from the detonation of explosive materials. Detonation is a chemical reaction which proceeds through the explosive material at a supersonic speed and converts the material into a high pressure gas. This gas expands radially outward, generating a strong blast wave that always travels at a velocity greater than the speed of sound. The fast moving blast wave or shock front generates an almost instantaneous rise in pressure from ambient pressure to a peak incident pressure at any point away from the explosion. The pressure then decays to a value slightly below ambient before returning to ambient pressure. Figure -1, adapted form TM5-13 (199), shows the pressure-time history for an ideal blast wave in air at any point located away from the explosion. The peak pressure and duration at a given point are functions of the type and weight of the explosive material and the distance from the explosion. When the shock wave impacts an unyielding surface such as a building structure, the fast-moving air and gas molecules are stopped abruptly, causing the density and pressure to increase above the values observed in the incident wave. This process of reinforcing and magnifying the blast wave is known as reflection. The reflected pressure is of primary interest because it produces the actual loads on the structure. Baker (1973) reports that the reflected pressure is at least twice as high as the incident pressure and can be more than eight times as high. 5

30 Figure -1 Ideal Blast Wave (Adapted from TM5-13, 199) In determining the characteristics of a blast wave, such as peak pressure and duration, much use is made of the Hopkinson scaling law (Baker, 1973 and Kinney and Graham, 1985). This law states that two explosions will have identical blast wave characteristics at equal scaled distances, Z. The scaled distance is defined as R Z = (-1) 1 W 3 where R = actual distance from the center of the explosive source to the point of interest W = total weight of the explosive material. Publications such as the Department of Defense design manual for accidental explosions (TM5-13, 199) utilize this relationship by presenting curves that give peak pressure, duration, and impulse as a function of Z. 6

31 Explosive materials can exist in all three physical states: solid, liquid, and gas. They include solid high explosives such as symmetrical,4,6 - trinitrotoluene (TNT), liquid chemicals and propellants, and gas fuels such as natural gas. The blast effects of solid high explosives are the most easily predicted. This is primarily because the detonation process in other materials is often incomplete, so only a portion of the explosive material contributes to the explosion. The remaining material is consumed by a fast burning process. Because of this, the majority of blast effects data found in the literature pertains to solid explosives. Among the solid explosives, TNT is most often used because chemically pure TNT is readily available for calibration tests. Most of the literature (TM5-13, 199 and DOE, 199, for example) present methods for determining blast loads on structures based on TNT explosions. If another material is to be considered, the usual approach is to convert the weight of the explosive material to an equivalent weight of TNT. The prediction equations or graphs for TNT can then be used..1. External Explosions TM5-13 (199) divides external explosions into three categories: free air bursts, air bursts, and surface bursts. A free air burst is an explosion that occurs above a structure at an elevation high enough such that the shock wave impacts the structure without amplification from the ground surface. An air burst is similar to a free air burst in that the explosion occurs above the structure; however, the elevation is not high enough to prevent the blast load from being amplified by ground surface reflections. A surface burst is an explosion that occurs at or near the ground surface. It is this type of external explosion that is of primary concern when designing vertical members 7

32 such as walls or columns. The initial shock wave is reflected and magnified immediately after the explosion by the ground s surface. This magnified blast wave travels outward, forming a uniform wave front similar to the one shown in Figure -. Figure - Surface Burst Wave Form (Adapted from TM5-13, 199) The pressure associated with the blast wave is called the side-on or incident pressure. When the wave impacts the structure, it is reflected and magnified by the structure itself. The pressure produced by this additional reflection is called the reflected pressure and this generates the actual loads on the structure. The Department of Defense (TM5-13, 199) and Department of Energy (DOE, 199) manuals as well as books by Baker (1973), Kinney and Graham (1985), and Biggs (1964) all present tables or charts of blast wave parameters for TNT explosions. Figure -3, adapted from TM5-13 (199), shows the blast wave parameters for free air bursts. The figure shows the peak positive reflected pressure as well as the shock front velocity as a function of the scaled distance. Similar charts showing the parameters for the 8

33 negative phase of the free air bursts and the positive and negative phases of surface bursts are also found in the literature. 1, 1, Peak Normal Reflected Pressure, P r (psi) 1, Charge Surface Shock Front Velocity, ft/ms Scaled Unit Reflected Impulse, i r /W 1/3 (psi-ms/lb 1/3 ) Scaled Distance, Z = R / W 1/3 Figure -3 Positive Phase Shock Wave Parameters for a Spherical TNT Explosion in Free Air at Sea Level (Adapted from TM5-13, 199) Two other important parameters usually given in the literature are the incident (side-on) and reflected impulses. These are defined as the areas under the side-on and reflected pressure-time histories, respectively. Baker (1973) notes that these are of considerable importance because the blast parameter graphs are usually derived from experimental data and impulse measurements are much more reliable than pressure measurements. Also, impulse data are important for structural engineers because the duration of the blast load is usually short when compared with the natural period of the structure. When this is the case, the total impulse rather than the shape of the pressure- 9

34 time function governs the structural response. Figure -3 also shows the reflected positive impulse as a function of the scaled distance. Bulmash and Kingery (198) present two methods for producing pressure-time histories for external surface explosions from both conventional and nuclear charges. Both methods utilize a database from 47 hemispherical TNT charges weighing 5-5 tons and 9 nuclear charges in the one-kiloton range. The first method is presented as a set of curves which give pressure as a function of standoff distance for a given elapsed time. The curves are for a charge weight of one-half kiloton of TNT but can be scaled to any charge weight by using Equation (-1). The second method uses polynomial functions to represent the peak incident pressure and shock front arrival time, and exponential functions to represent the waveform decay. In computing the loads on a structure, the pressure-time history for the ideal blast wave shown in Figure -1 is usually represented by the two equivalent triangular loads shown in Figure -4. The peak positive and negative pressures are taken from graphs similar to Figure -3. The duration of each phase is determined by setting the area of each triangle equal to the impulse (positive or negative) also obtained from the graphs. This approach has the advantage of simplifying the load-time functions while keeping the peak pressure and total impulse in agreement with experimental values. 1

35 Figure -4 Equivalent Triangular Loads (Adapted from TM5-13, 199) The outer columns of a structure are the ones most affected by external explosions. This is especially true if the columns are built as an integral part of a wall. The wall will collect a blast load and impart it to the column. Blast loads can also cause a sudden loss of lateral bracing, resulting in an unbraced length much greater than that assumed in the design. Interior columns can also be affected by external explosions, although the loads are less severe. Blast pressures can enter a structure through windows or other openings (Shope and Keenan, 199). These pressures can produce lateral loads on interior columns. Also, interior columns are sometimes part of a lateral force resisting moment frame. In this case, the interior columns will be subjected to bending moments and shear forces resulting from the overall building response. 11

36 .1.3 Internal Explosions The two main types of internal explosions most often discussed in the literature are those caused by either the detonation of solid high explosive materials (TNT, etc.) or the rapid combustion of fuel dispersed within a confined volume. These two types of explosions have pressure-time histories that are quite different. An internal explosion resulting from a high explosive source generates pressure loads that are complex in nature, as shown in Figure -5 (TM5-13, 199). These loads, which act on the face of interior structural elements, consist of a series of reflected shocks followed by a longduration quasi-static pressure (Beshara, 1994). Conversely, an internal explosion resulting from the combustion of dispersed fuel generates a much smoother pressure curve (Kinney and Graham, 1985). These pressures tend to be significantly lower in magnitude and longer in duration than those caused by a detonation. Figure -5 Pressure-Time History from Confined Explosion (Adapted from TM5-13, 199) 1

37 Baker, et al. (1983) divide the loading from an internal high explosive charge into two distinct phases. The first phase consists of the initial short-duration shock with several successive reflected shocks. These loads are influenced by the surrounding walls and will be highly nonuniform throughout the structure. The second phase consists of the longer duration pressure associated with the expanding hot gasses. This pressure is a function of the structure s internal volume and vent area. It is often referred to as the quasi-static or gas pressure and it is distributed more uniformly throughout the structure. The load that each phase imparts on a column will depend on the location of the column within the structure. For example, a free standing interior column that is not built-in to a wall or partition will primarily be affected by the pressures generated during the shock phase. This is because the long-duration gas pressures will flow around the column and equalize on all sides, producing a net zero force. A column that is built-in to a wall or partition, however, will be affected by the load resulting from both phases because the gas pressure will be restricted to acting on one side of the column, producing a net lateral load. TM5-13 (199) presents a method for estimating the shock loads from an internal explosion. This method is based on experimental blast data obtained from explosive tests in steel cubicles. The data was used to compile a series of charts for peak pressure and impulse as a function of the weight and location of the charge, the cubicle height to width and length ratios, the number of reflecting surfaces, and the scaled distance from the charge to the structural element under consideration. The peak pressure and impulse obtained from the charts are used to determine an equivalent triangular load. This method is mainly used for predicting the shock loads on the ceiling and perimeter 13

38 walls of a containment structure; however, an approach is given for surfaces other than entire walls, such as columns. Predicting the actual pressure-time history of the shock phase in complex structures generally requires specialized computer programs. A computer program called SHOCK (NCEL, 1988), developed by the U. S. Naval Civil Engineering Laboratory, is available. This program executes the interpolation procedure and generates the values of peak pressure and impulse given in the TM5-13 charts. The computer code BLASTX (ERDC, 1) is widely used to calculate the loading from incident shock waves combined with the shock waves reflected from surrounding surfaces. As stated earlier, the quasi-static or gas pressure phase is primarily a function of the charge weight, the internal volume of the structure, and the structure s vent area (windows, doors, etc.). A prediction method for this phase is given in TM5-13 (199). The prediction is presented as a graph which gives the peak gas pressure as a function of the charge weight to room volume ratio. Additional graphs are also provided in the manual that give the total gas impulse as a function of the vent area to room volume ratio, the charge weight, and the weight of the vent cover if one is present. The impulse is used to determine the duration of an equivalent triangular load representing the gas pressure. Combining the two phases and using the simplifying assumptions outlined above gives the total pressure-time history shown in Figure -6. The BLASTX code (ERDC, 1) can also be used to calculate the pressures occurring in the gas pressure phase. 14