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 |  موضوع: كتاب Dynamic Models for Structural Plasticity الجمعة 28 مايو 2021, 1:55 am | |
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Dynamic Models for Structural Plasticity
W.J. Stronge and T.X. Yu
With 175 Figures
و المحتوى كما يلي :
Contents
List of Symbols xvii
1 Elastoplastic and Viscoplastic Constitutive
Relations . 1
1.1 Stress Resultants and Generalized Stress -
Deformations and Generalized Strain . 1
1.2 Pure Bending of Rate-Independent Bar . 2
1.2.1 Kinematics of Deformation 2
1.2.2 Elastic Constitutive Equation . 4
1.2.3 Stress Resultants (Axial Force and Bending
Moment) 4
1.2.4 Elastic-Plastic Constitutive Equations . 5
1.2.5 Elastic-Power Law Hardening Constitutive
Equations. ' . 9
1.3 Pure Bending of Rate-Dependent Bar 9
1.3.1 Strain-Rate Dependent Constitutive Equations 9
1.4 Interaction Yield Functions and Associated Plastic
Flow 12
1.4.1 Elastic Limit for Bending and Tension . 12
1.4.2 Fully Plastic Limit Surface for Bending and
Tension in Elastic-Perfectly Plastic Bar 12
1.4.3 Yield and Fully Plastic Stress Condition . 15
1.4.4 Associated Flow Rule for Plastic Deformations. 16
1.4.5 Separated Yield Functions and Separated Plastic
Flow . . . . 18
1.5 Interaction Yield Surfaces Including Shear 19
l.5.1 Tension, Shear and Bending in Rectangular
Cross-Section 19
1.5.2 Tension, Torsion and Bending in Circular
Cross-Section 20
1.6 Elastic Springback . 23
1.6.1 Pure Bending 23
1.6.2 Bending and Tension 25
References . 27xii Contents
2 Principles of Mechanics . 29
2.1 Kinematics . 29
2.1.1 Inertia Properties of Cross-Section 31
2.2 Balance of Forces . 31
2.2.1 Stress Resultants and Generalized Stresses . 31
2.2.2 Equations of Motion ., 32
2.3 Principle of Virtual Velocity 34
2.3.1 Rate of Change for Kinetic Energy of System 35
2.3.2 Rate of Change for Kinetic Energy of
Kinematically Admissible Velocity Field W;C . 35
2.3.3 Extremal Principles for Complete Solution . 36
2.4 Bounds For Rigid-Perfectly Plastic Solids and
Structures . 37
2.4.1 Upper and Lower Bounds on Static Collapse
Force . . . 37
2.4.2 Lower Bound on Dynamic Response Period 38
2.4.3 Upper Bound on Dynamic Response Period 39
2.4.4 Lower Bound on Final Displacement 40
2.4.5 Upper Bound on Final Displacement 41
2.5 Dynamic Modes Of Deformation . 44
2.5.1 Modal Solutions 44
2.5.2 Properties of Modes . 44
2.5.3 Mode Approximations for Structural Response
to Impulsive Loading 47
References . 48
3 Static Deflection 51
3.1 Small Elastic-Plastic Deflections 51
3.1.1 Elastic Deflections 52
3.1.2 Deflection of Elastic-Perfectly Plastic
Cantilever . 54
3.1.3 Deflection of Elastic-Linear Strain Hardening
Cantilever . 55
3.1.4 Residual Deflection After Elastic Unloading 58
3.1.5 Elastoplastic Beam-Columns . 61
3.2 Large Elastic-Plastic Deflections 63
3.2.1 Elastica: Large Elastic Deflection . 63
3.2.2 Plastica: Large Plastic Deflection . 67
References . 72
4 Dynamic Rigid-Plastic Response . 73
4.1 Step Loading 73
4.1.1 Static and Dynamic Loadings . 73
4.1.2 Moderate Dynamic Load (~ < F < 3Fc) . 75
4.1.3 Intense Dynamic Load (F> 3Fc) . 78
4.2 Rectangular Pulse Loading 80
4.2.1 Three Phases in Response of Cantilever 80
4.2.2 Deformed Shape 84
4.2.3 Energy Dissipation 85
4.2.4 Synopsis 87Contents xiii
4.3 Features of Travelling Hinges . 88
4.4 General Pulse Loading . 90
4.4.1 General Considerations 90
4.4.2 Example: Linearly Decaying Pulse 93
4.4.3 Equivalent Replacement of Arbitrary Pulse . 94
4.5 Impact on Cantilever . 96
4.5.1 Problem and Assumptions . 97
4.5.2 Changing Pattern of Deformation . 98
4.5.3 Acceleration, Force and Bending Moment 101
4.5.4 Deformed Shape 103
4.5.5 Energy Dissipation 105
4.5.6 Modal Approximation . 106
References . - 110
5 Second-Order Effects on Dynamic Response . 111
5.1 Strain-Rate Effect . 111
5.1.1 Impulsive Load on Viscoplastic Cantilever . III
5.1.2 Elementary Estimates of the Effect of Strain-Rate
on Final Deformation . 117
5.2 Strain Hardening (Strain Softening) Effect 120
5.2.1 Introduction . 120
5.2.2 Elementary Effect of Strain Hardening on Final
Deformation . 121
5.2.3 Dynamic Analysis of Strain Hardening and
Strain Softening Cantilevers 123
5.3 Effects of Transverse Shear and Rotary Inertia . . . 132
5.3.1 Interface Conditions for Concentrated Mass 132
5.3.2 Shear Deformation Adjacent to Colliding
Particle 134
5.3.3 Shear and Rotary Inertia of Finite Size
Colliding Missile . 139
5.3.4 Shear Rupture due to Impact . 144
5.3.5 Measurements of Energy for Shear Rupture 146
5.4 Effect of Large Deflection 149
5.4.1 General Considerations 149
5.4.2 Large Deflection of Impulsively Loaded
Cantilever . 150
5.4.3 Methods of Approximating Large Deflection
Effects 153
5.4.4 Effect of Centripetal Acceleration on Bending
Moment Distribution 155
5.5 Effect of Elastic Deformation . 157
5.5.1 General Considerations . . 157
5.5.2 Mass-Spring Finite Difference Structural
Model (MS-FD) 158
5.5.3 Timoshenko Beam Finite Element Structural
Model (TB-FE) . , . 163
5.5.4 Dynamic Deformation of Elastic-Plastic
Cantilever from Impact 164xiy Contents
5.5.5 Effect of Elastic Deformation at Root of
Cantilever . 173
5.5.6 Remarks . 181
5.6 Accuracy of Rigid-Plastic Analyses 182
5.6.1 Accuracy of Rigid-Plastic Analysis Estimated
by Single DoF System . 182
5.6.2 Convergence to Dynamic Plastic Mode Studied
by Two DoF System 184
5.6.3 Remarks . 187
References . 188
6 More Complex Configurations . 191
6.1 General Considerations . 192
6.1.1 Extremal Properties of Yield Function at Plastic
Hinge . 192
6.1.2 Differentiability of Arbitrary Functions at
Plastic Hinge . 193
6.1.3 Differentiability of Kinematic Variables at
Plastic Hinge . 194
6.1.4 Differentiability of Generalized Stresses at
Plastic Hinge . 196
6.1.5 Differentiability of Yield Function at Plastic
Hinge . . . 197
6.2 Straight Cantilevers with Smoothly Varying
Cross-Sections 199
6.2.1 Yield Function and Conditions at Plastic
Hinge . . . 199
6.2.2 Suddenly Applied Steady Force at Tip of
Tapered Cantilever: An Example 200
6.3 Oblique Impact on Straight Cantilever 204
6.3.1 Problem and Assumptions . 204
6.3.2 Formulation Based on Single Hinge
Mechanism 205
6.3.3 Solution Based on Single-Hinge Mechanism 212
6.4 Circular Arc Cantilever Subjected to In-plane Step
Force 215
6.4.1 Engineering Background and Assumptions . 215
6.4.2 Radial Force at Tip . 217
6.4.3 Tangential Force at Tip 221
6.4.4 Discussion . 223
6.5 Circular Arc Cantilever Subjected to In-plane Impact . 224
6.5.1 Rigid-Plastic Formulation . 225
6.5.2 Discussion of Solution 228
6.5.3 Modal Approximation . 231
6.6 Circular Arc Cantilever Subjected to Out-of-Plane
Step Force 232
6.6.1 Equations of Motion 233
6.6.2 Solution . 235
6.6.3 Discussion . 240
6.7 Stepped or Bent Cantilever Subjected to Step Force 241Contents
6.7.1 General Considerations .
6.7.2 Stepped Cantilever .
6.7.3 Bent Cantilever .
6.704 Discussion .
6.8 Cantilever with an Initial Crack .
6.8.1 General Considerations
6.8.2 Impact on Cantilever with an Initial Crack .
6.8.3 Crack Stability After Impact
6.804 Numerical Example and Discussion
References
7 Impact Experiments .
7.1 Methods of Applying Dynamic Loads
7.1.1 Missile Impact
7.1.2 Explosive Loading .
7.1.3 Magnetomotive Loading
7.2 Travelling Hinges - Fiction or Fact?
7.3 Elastic Effects on Plastic Deformation
7A Strain Hardening and Strain-Rate Effects
704.1 Mode Approximations
704.2 Transient Analysis Including Rate Effects
7.5 Dynamic Rupture .
7.5.1 Location and Mechanism of Rupture
7.5.2 Measurements of Generalized Strain at
Rupture .
References
Index .
x
Amplification factor for yield stress 269
Associated flow rule 17
Axial force 4, 12, 25, 204
Ballistic pendulum 261
Beam
Euler-Bernoulli 159
ideal sandwich 8, 15, 159
Timoshenko 163
Bending moment 4
at yield 5
fully plastic 7
Blast pulse 261
Body force 32
Bounds 37
dynamic response period 38, 39
final displacement 40, 41
static collapse force 37
Cantilever 51, 73, 260, 262
bent 241, 246
circular arc 215, 224, 232
imperfect 250
stepped 241
subjected to impact 96
sUbjected to general pulse 90
subjected to rectangular pulse 80
subjected to step force 73
tapered 201
with initial crack 251
Centripetal acceleration 156
Characteristic load 56
Circular arc cantilever 215, 224, 232
subjected to in-plane force 215
subjected to in-plane impact 225
subjected to out-of-plane force 232
subjected to radial force 217
subjected to tangential force 221
Circular cross-section
solid 7, 15, 234
thin-walled 15, 234
Compatibility 30
Complete solution 33
Complex configuration 191
Constitutive relation 1
Contact force 32
Convergence theorem 45
Convergence to dynamic mode 184
Correction factor 122
Cowper-Symonds relation 10, 112
Crack stability 254
Curvature 2, 84, 90, 103
Deflection 51, 84
elastic 52
elastic-perfectly plastic 54
large 149
large elastic 63
large elastic-plastic 67
residual 58
Deformed shape 84, 104, 229
Differentiability 193, 198
Double hinge mechanism 241
Drucker's postulate 17
Dynamically admissible stress 33
Dynamic rupture 144, 270
location 270
mechanism 271
pure shear 144
shl~ar 144, 271
shear-flexure 146
transition of mechanism 272
Effective load 95
Elastica 64
Elastic deformation 157,265
at root of cantilever 173
Elastic springback 23, 265
E1astoplastic beam-column 61
Elliptic integral 65
Energy 35
kinetic 35, 104
she'ar rupture 148
Energy ratio 149, 158, 266
Equation of equilibrium 33
Equation of motion 33
Equivalent replacement of arbitrary pulse 94
Euler-Bernoulli beam 159
Extremal principle 36
Final configuration 104, 264
Finite sized colliding missile 139
Free - free beam 266
Hooke's law 4
Ideal sandwich beam 8, 15, 159
Impact 96, 259278
oblique 204
Impulse 81, 91
effective total 95
total 81, 92
Impulsive loading 42
Initial crack 251
J-integral 254
Kinematic variables 194
Kinematically admissible velocity
field (KAVF) 19, 30
Kink 266
Lagrangian reference frame 29
Lee's functional 45
Limit function 13
elastic 13
fully plastic 13
Loading 29
impact 96, 134, 139, 204, 259
impulsive 42, 111, 182
general pulse 90, 261
quasi-static 29, 51
rectangular pulse 80
step force 73, 200, 215, 232, 241
Martin's principle 37
Mass ratio 98
Mass-spring finite difference
(MS-FD) model 158
Material 1
elastic 4
elastic-perfectly plastic 5
rigid-perfectly plastic 16
strain-hardening 5
power-law hardening 9
visco-plastic 9
Moment-curvature relation 6
Modal shape 44, 47, 99
Modal solution 44, 106
initial velocity of 48
Mode 44
best 47
convergence 45
configuration 44
dynamic 44, 186
elastic 186
fundamental 45
primary 45
Neutral axis 3, 13
Neutral surface 3
Normality condition 17
Phase transition 82, 99
Plastic collapse force 37, 52, 56
Plastic hinge 52, 75
combined bending-torsion 235
double 241
effective length of 119, 121, 262-3,267
general features of 88
pure torsion 236
stationary 79, 81, 89, 236
travelling 82, 89, 99, 205, 225, 262
Plastica 68
Power law hardening 9
Principle
maximum plastic dissipation 17
virtual velocity 34
virtual work 34
Pulse duration 80, 91
Pulse loading 80, 90
blast 261
general 90
half-sine 184
linear decaying 93, 184
rectangular 80
Pure bending 2
Quasi-static loading 29
Rate dependency 9
Rate of dissipation of energy 35
Ratio of fully plastic torque to
fully plastic moment 234, 241
Rectangular cross-section 6, 14, 234
Reverse plastic flow 59
Rotary inertia 132, 139
Shape factor 7
for bending 7
for shearing 8
for tension 7
for torsion 8
Shear phase 134
Shear sliding 134, 274
Single DoF system 182
Single hinge mechanism 75, 205
Springback 23, 265
Static admissible stress field 34
Static collapse force 37
Step loading 73
Strain 2
at rupture 272
flexural 272
generalized 2
stretching 272
Strain-hardening 5, 55
effect of 120, 266
Strain-rate 9
effect of 111, 266
Strain-softening 121
Stress 1
generalized 2
resultants 1, 32
safe state of 35
ultimate 264
Tamuzh's principle 37
Tearing modulus 254
Tensile tearing 271
IndexIndex
Timoshenko beam 163
Timoshenko beam finite element
(TB-FE) model 164
Traction 32
Transient phase 98
Transverse shear 132
Two DoF system 184
Virtual velocity 34
Virtual work 34
Virtual work rate 34
Yield condition 5, 15
interactive 12
separated 18
Yield criterion 15
Tresca 15
Von Mises 16
Yield function 15, 192
differentiability of 197
extremal properties of 192
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