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Summary
Summary
Limit and shakedown analysis for structures can provide a very useful tool for design and analysis of engineering structures. "Structural Plasticity - Limit, Shakedown and Dynamic Plastic Analyses of Structure" provides more general solutions of limit and shakedown analysis for structures by using a unified strength theory. A series of solutions of plates from circular, annular plates to rhombus plates and square plates, rotating discs and cylinders, pressure vessels are presented. These results encompass the Tresca-Mohr-Coulomb solution of structure as special cases. The unified solution, which cannot be obtained by using a single criterion, is suitable to more materials and structures.
Maohong Yu is professor of Department of Civil Engineering at Xi'an Jiaotong University, China. He has authored 12 books including "Unified Strength Theory and Its Applications" and "Generalized Plasticity".
Author Notes
Maohong Yu is professor of Department of Civil Engineering at Xi'an Jiaotong University, China. He has authored 12 books including Unified Strength Theory and Its Applications and Generalized Plasticity.
Table of Contents
1 Introduction | p. 1 |
1.1 Background | p. 1 |
1.2 Unification of Yield and Strength Criteria | p. 3 |
1.3 Plastic Limit Analysis | p. 4 |
1.4 Plastic Limit Analysis of Rotating Solids | p. 6 |
1.5 Shakedown Analysis of Structures | p. 7 |
1.6 Plastic Limit Analysis Based on the Unified Strength Theory | p. 8 |
1.7 Summary | p. 9 |
References | p. 9 |
2 Fundamental Concepts of Stress and Srain | p. 16 |
2.1 Stress Components and Invariants | p. 16 |
2.2 Deviatoric Stress Tensor and the Tensor Invariants | p. 18 |
2.3 Principal Shear Stresses | p. 21 |
2.4 Octahedral Shear Stress | p. 21 |
2.5 Strain Components | p. 22 |
2.6 Equations of Equilibrium | p. 24 |
2.7 Generalizes Hooke's Law | p. 24 |
2.8 Compatibility Equations | p. 25 |
2.9 Governing Equations for Plane Stress Problems | p. 26 |
2.10 Governing Equations in Polar Coordinated | p. 27 |
2.11 Bending of Circular Plate | p. 29 |
2.12 Summary | p. 31 |
References | p. 31 |
3 Yield Condition | p. 32 |
3.1 Introduction | p. 32 |
3.2 Conventional Yield Criteria | p. 33 |
3.2.1 Maximum Normal Stress Criterion | p. 33 |
3.2.2 Maximum Shear Stress-based Criteria-Single-Shear Theory | p. 33 |
3.2.3 Octahedral Shear Stress-based Criteria-Three Shear Theory | p. 36 |
3.2.4 Twin- shear Stress-based Criterion-Twin-shear Theory | p. 40 |
3.3 Unified Yield Criterion for Metallic Materials (Non-SD Materials) | p. 44 |
3.4 Unified Strength Theory for SD Materials | p. 45 |
3.4.1 Mechanical Model of the Unified Strength Theory | p. 46 |
3.4.2 Mathematical Modeling of the Unified Strength Theory | p. 47 |
3.4.3 Mathematical Expression of the Unified Strength Theory | p. 48 |
3.4.4 Yield Surfaces and Yield Loci of the Unified Strength Theory | p. 48 |
3.5 Significance of the Unified Strength Theory | p. 52 |
3.6 Unified Strength Theory in the Plane Stress State | p. 52 |
3.6.1 ?1 ? ?2 > 0, ?3 = 0 | p. 53 |
3.6.2 ?1 ? 0,?2 = 0, ?3 | p. 53 |
3.7 Summary | p. 56 |
3.8 Problems | p. 56 |
References | p. 61 |
4 Theorems of Limit Analysis | p. 64 |
4.1 Introduction | p. 64 |
4.2 Perfectly Plastic Solid | p. 66 |
4.3 Power of Dissipation | p. 66 |
4.4 Lower-Bound Theprem | p. 67 |
4.5 Upper-Bound Theorem | p. 68 |
4.6 Fundamental Limit Theorems | p. 68 |
4.7 Important Remarks | p. 69 |
4.7.1 Exact Value of the Limit Load (Complete Solution) | p. 69 |
4.7.2 Elastic-Plastic and Rigid plastic Bodies | p. 69 |
4.7.3 Load-bearing Capacity | p. 70 |
4.7.4 Uniqueness | p. 71 |
References p. 71 | |
5 Plastic Limit Analysis for Simply Supported Circular Plates | p. 74 |
5.1 Introduction | p. 74 |
5.2 Basic Equations of Circular Plate | p. 75 |
5.3 Unified Solutions of Simply Supported Circular Plate for Non-SD Materials | p. 76 |
5.3.1 Uniformly Distributed Load | p. 78 |
5.3.2 Arbitrary Axisymmetrical Load | p. 80 |
5.4 Unified Solutions of Simply Supported Circular Plate for SD Materials | p. 95 |
5.4.1 Partial-uniform Load | p. 97 |
5.4.2 Linearly Distributed Load | p. 102 |
5.5 Summary | p. 107 |
5.6 Problems | p. 108 |
References p. 110 | |
6 Plastic Limit Analysis of Clamped Circular Plates | p. 112 |
6.1 Introduction | p. 112 |
6.2 Unified Solutions of Clamped Circular Plate for Non-SD Materials | p. 112 |
6.2.1 Uniformly Distributed Load | p. 112 |
6.2.2 Arbitrary Loading Radius | p. 117 |
6.2.3 Arbitrary Loading Distribution | p. 122 |
6.3 Unified Solutions of Clamped Circular Plate for SD Materials | p. 127 |
6.4 Summary | p. 133 |
6.5 Problems | p. 134 |
References p. 134 | |
7 Plastic Limit Analysis of Annular Plate | p. 136 |
7.1 Introduction | p. 136 |
7.2 Basic Equations for Annular Plate Based on UYC | p. 137 |
7.2.1 Case (1) | p. 140 |
7.2.2 Case (2) | p. 141 |
7.2.3 Special Case | p. 142 |
7.3 Unified Solutions of Annular Plate for Non-SD Meterials | p. 142 |
7.4 Unified Solutions of Limit Load of Annular Plate for SD Materials | p. 145 |
7.4.1 Unified Strength Theory | p. 145 |
7.4.2 Basic Equations for Annular Plate Based on the UST | p. 146 |
7.4.3 Limit Analysis | p. 147 |
7.4.4 Results and Discussions | p. 149 |
7.5 Summary | p. 150 |
7.6 Problems | p. 152 |
References | p. 152 |
8 Plastic Limit Analyses of Oblique, Rhombic, and Rectangular Plates | p. 154 |
8.1 Introduction | p. 154 |
8.2 Equations for Oblique Plates | p. 156 |
8.2.1 The Equilibrium Equation in Ordinary Coordinate System | p. 156 |
8.2.2 Field of Internal Motion | p. 158 |
8.2.3 Moment Equation Based on the UST | p. 158 |
8.3 Unified Solution of Limit Analysis of Simply Supported Oblique Plates | p. 159 |
8.4 Limit Load of Rhombic Plates | p. 162 |
8.5 Limit Load of Rectangular Plates | p. 163 |
8.6 Unified Limit Load of Square Plates | p. 168 |
8.7 Tabulation of the Limit Load for Oblique, Rhombic and Square Plates | p. 169 |
8.8 Summary | p. 170 |
8.9 Problems | p. 172 |
References | p. 173 |
9 Plastic Limit Analysis of Pressure Vessels | p. 175 |
9.1 Introduction | p. 175 |
9.2 Unified Solution of Limit Pressure of Thin-walled Pressure Vessel | p. 176 |
9.3 Limit Pressure of Thick-walled Hollow Sphere | p. 180 |
9.3.1 Elastic Limit Pressure of Thick-walled Sphere Shell | p. 181 |
9.3.2 Plastic Limit Pressure of Thick-walled Sphere Shell | p. 182 |
9.4 Unified Solution of Elastic Limit Pressure of Thick-walled Cylinder | p. 184 |
9.5 Unified Solution of Plastic Limit Pressure of Thick-walled Cylinder | p. 190 |
9.5.1 Stress Distribution | p. 190 |
9.5.2 Plastic Zone in the Elasto-plastic Range | p. 192 |
9.5.3 Plastic Zone Radius in the Elasto-plastic Range | p. 193 |
9.5.4 Plastic Limit Pressure | p. 193 |
9.6 Summary | p. 198 |
9.7 Problems | p. 200 |
References | p. 202 |
10 Dynamic Plastic Response of Circular Plate | p. 205 |
10.1 Introduction | p. 205 |
10.2 Dynamic Equations and Boundary Conditions of Circular Plate | p. 206 |
10.2.1 First Phase of Motion (0 ? T ? ?) | p. 209 |
10.2.2 Second Phase of Motion (? ? t ? T) | p. 212 |
10.3 Static and Kinetic Admissibility | p. 214 |
10.4 Unified Solution of Dynamic Plastic Response of Circular Plate | p. 216 |
10.5 Special Cases of the Unified Solutions | p. 221 |
10.6 Summary | p. 229 |
References | p. 229 |
11 Limit Angular Speed of Rotating Disc and Cylinder | p. 231 |
11.1 Introduction | p. 231 |
11.2 Elastic Limit of Discs | p. 232 |
11.3 Elasto-plastic Analysis of Discs | p. 233 |
11.4 Elasto-plastic Stress Field of Rotating Disc | p. 236 |
11.5 Solution Procedure and Results | p. 238 |
11.6 Unified Solution of Plastic Limit Analysis of Rotating Cylinder | p. 239 |
11.7 Limit Analysis of a Solid Disc with Variable Thickness | p. 242 |
11.8 Limit Analysis of an Annular Disc with Variable Thickness | p. 246 |
11.8.1 Case (1) (1 > ? ? ?0) | p. 247 |
11.8.2 Case (2) (0 ? ? ? ?0) | p. 248 |
11.9 Special Case of b = 0 | p. 251 |
11.10 Results and Discussion | p. 252 |
11.11 Summary | p. 258 |
11.12 Problems | p. 258 |
References | p. 259 |
12 Projectile Penetration into Semi-infinite Target | p. 261 |
12.1 Introduction | p. 261 |
12.2 Spatial Axisymmetric Form of Unified Strength Theory | p. 262 |
12.3 Fundamental Equations for Concrete Targets | p. 263 |
12.3.1 Conservation Equations | p. 263 |
12.3.2 Relation between Pressure and Bulk Strain | p. 264 |
12.3.3 Failure Criterion Expressed by ?r and ?? | p. 264 |
12.3.4 Interface Conditions | p. 265 |
12.4 Cylindrical Cavity Expansion Analysis | p. 265 |
12.4.1 Elastic Zone (c1t ? r ? cdt, ?1/? ? ? ? 1/?) | p. 266 |
12.4.2 Interface of Elastic-cracked Zones (r = C1t, ? = ?1/? | p. 270 |
12.4.3 Radial Cracked Zone (ct ? r ? C1t, 1 ? ? ? ?1/?) | p. 271 |
12.4.4 Interface of the Plastic and Cracked Zones (r = ct, ? = 1) | p. 273 |
12.4.5 Plastic Zone (vrt ? r ? ct, ? ? ? ? 1) | p. 275 |
12.5 Cavity Expansion Pressure and Velocity | p. 276 |
12.5.1 Incompressible Material | p. 277 |
12.5.2 Compressible Material | p. 279 |
12.6 Penetration Resistance Analysis | p. 284 |
12.7 Analysis and Verification of Penetration Depth | p. 288 |
12.8 Summary | p. 290 |
References | p. 291 |
13 Plastic Analysis of Orthogonal Circular Plate | p. 293 |
13.1 Introduction | p. 293 |
13.2 Orthotropic Yield Criteria | p. 293 |
13.3 General Solutions | p. 296 |
13.4 Simply Supported Orthotropic Circular Plate | p. 301 |
13.4.1 Case I: Point A' Falls on Segment KL | p. 301 |
13.4.2 Case II: Point A' Falls on Segment LA | p. 302 |
13.4.3 Case III: Point A! Falls on Segment AB | p. 302 |
13.4.4 Case IV: Point A' Falls on Segment BC | p. 302 |
13.4.5 Moment, Velocity Fields and Plastic Limit Load | p. 303 |
13.5 Fixed Supported Circular Plate | p. 307 |
13.5.1 Case I: Point A' Falls on Segment KL | p. 307 |
13.5.2 Case II: Point A' Falls on Segment LA | p. 307 |
13.5.3 Case III: Point A' Locates on Segment AB | p. 307 |
13.5.4 Case IV: Point A1 Falls on Segment BC | p. 308 |
13.5.5 Moment Fields, Velocity Fields, and Plastic Limit Load | p. 308 |
13.6 Summary | p. 312 |
References | p. 313 |
14 Unified Limit Analysis of a Wellbore | p. 314 |
14.1 Introduction | p. 314 |
14.2 Unified Strength Theory | p. 315 |
14.3 Equations and Boundary Conditions for the Wellbore | p. 316 |
14.3.1 Strength Analysis for Wellbore | p. 316 |
14.3.2 Pore Pressure Analysis | p. 318 |
14.4 Elastic and Plastic Analysis | p. 318 |
14.4.1 Elastic Phase | p. 328 |
14.4.2 Plastic Limit Pressure | p. 319 |
14.4.3 Elastic-plastic Boundary | p. 320 |
14.4.4 Example | p. 321 |
14.4.5 Limit Depth for Stability of a Shaft | p. 322 |
14.5 Summary | p. 324 |
14.6 Problems | p. 324 |
References | p. 325 |
15 Unified Solution of Shakedown Limit for Thick-walled Cylinder | p. 327 |
15.1 Introduction | p. 327 |
15.2 Shakedown Theorem | p. 329 |
15.2.1 Static Shakedown Theorem (Melan's Theorem) | p. 329 |
15.2.2 Kinematic Shakedown Theorem (Koiter Theorem) | p. 329 |
15.3 Shakedown Analysis for Thick-walled Cylinders | p. 330 |
15.4 Unified Solution of Shakedown Pressure of Thick-walled Cylinders | p. 334 |
15.5 Connection between Shakedown Theorem and Limit Load Theorem | p. 336 |
15.6 Shakedown Pressure of a Thick-walled Spherical Shell | p. 339 |
15.7 Summary | p. 340 |
15.8 Problems | p. 340 |
References | p. 341 |
16 Unified Solution of Shakedown Limit for Circular Plate | p. 344 |
16.1 Introduction | p. 344 |
16.2 Unified Solution of Shakedown Limit for Simply Supported Circular Plate | p. 345 |
16.2.1 Elastic State | p. 345 |
16.2.2 Elastic-plastic State | p. 346 |
16.2.3 Completely Plastic State | p. 348 |
16.2.4 Shakedown Analysis | p. 348 |
16.2.5 Discussion | p. 349 |
16.3 Unified Solution of Shakedown Limit for Clamped Circular Plate | p. 350 |
16.3.1 Elastic State | p. 350 |
16.3.2 Elastic-plastic State | p. 350 |
16.3.3 Completely Plastic State | p. 351 |
16.3.4 Shakedown Analysis | p. 351 |
16.3.5 Discussion | p. 352 |
16.4 Comparison between Shakedown Solution and Limit Results | p. 353 |
16.5 Summary | p. 354 |
16.6 Problems | p. 354 |
References | p. 355 |
17 Shakedown Analysis of Rotating Cylinder and Disc | p. 357 |
17.1 Introduction | p. 357 |
17.2 Elasto-plastic and Shakedown Analyses of Rotating Cylinder and Disc | p. 358 |
17.2.1 Elastic Analyses of Hollow Rotating Circular Bars | p. 358 |
17.2.2 Elasto-plastic Analyses of Hollow Rotating Circular Bars | p. 360 |
17.2.3 Shakedown Analyses of Hollow Rotating Circular Bars | p. 362 |
17.2.3 Elasto-plasticity and Shakedown of Solid Rotating Circular Bars | p. 364 |
17.3 Summary of Elasto-plasticity and Shakedown Analyses of Rotating Circular Bars | p. 366 |
17.4 Elasto-plastic and Shakedown Analyses of Rotating Disc | p. 367 |
17.4.1 Elastic Analyses of Hollow Rotating Discs | p. 368 |
17.4.2 Plastic Limit Analyses of Hollow Rotating Discs | p. 371 |
17.4.3 Shakedown of Hollow Rotating Discs | p. 373 |
17.5 Elasto-plastic and Shakedown Analyses of Solid Rotating Discs | p. 376 |
17.6 Summary of Elastic and Plastic Analyses of Rotating Discs | p. 377 |
17.7 Summary | p. 378 |
References | p. 379 |
Index | p. 381 |