| Preface - Vol. 1 to Vol. 3 |
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xv | |
| Acknowledgments - Vol. 1 to Vol. 3 |
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xxiii | |
| Contents - Vol. 1 to Vol. 3 |
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xxv | |
| About the Authors |
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xxix | |
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1 Stability of Equilibrium of Systems That Have a Finite Number of Degrees of Freedom |
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1 | (84) |
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1.1 Definition of the Equilibrium Stability |
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2 | (11) |
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1.1.1 Lagrange-Dirichlet theorem and Lyapunov theorems |
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5 | (5) |
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10 | (3) |
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1.2 Elastic Systems with a Finite Number of Degrees of Freedom |
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13 | (27) |
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1.2.1 Stability functional - Bolotin functional |
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19 | (1) |
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1.2.2 Linearized models of equilibrium stability problems |
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20 | (4) |
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24 | (8) |
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1.2.4 Example 3 - paradoxes in equilibrium stability problems? |
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32 | (6) |
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1.2.4.1 Non-invariance of critical load with respect to the choice of the system's generalized coordinates |
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38 | (2) |
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1.3 Some General Theorems of the Equilibrium Stability Theory |
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40 | (33) |
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1.3.1 Rayleigh ratio and variational recursive definition of critical loads |
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42 | (3) |
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1.3.2 Decomposition of the original n-dimensional space of generalized displacement vectors into a direct sum of three subspaces |
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45 | (4) |
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1.3.3 Normal coordinates of a system |
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49 | (2) |
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1.3.4 Influence of constraints on the stability of equilibrium of a linearized elastic system |
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51 | (3) |
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1.3.5 Papkovich theorem on convexity of a stability area |
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54 | (7) |
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1.3.5.1 Reservations for the Papkovich theorem |
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61 | (3) |
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1.3.5.2 Application aspect of the Papkovich theorem |
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64 | (4) |
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1.3.6 Geometric stiffness matrix revisited |
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68 | (1) |
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1.3.7 Stability of equilibrium under a non-force action |
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69 | (4) |
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1.4 Characteristic Curve of an Elastic System |
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73 | (9) |
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1.4.1 One-degree-of-freedom system |
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74 | (4) |
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1.4.2 Multiple-degrees-of-freedom system |
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78 | (4) |
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1.5 Final Notes to
Chapter 1 |
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82 | (3) |
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2 Variational Statement of the Problem of Equilibrium Stability for Elastic Bodies |
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85 | (82) |
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2.1 Geometrically Nonlinear Problems of Elasticity |
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85 | (8) |
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2.1.1 Geometric equations |
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85 | (3) |
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2.1.1.1 Varying the stress tensor components |
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88 | (1) |
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2.1.2 Equilibrium equations and static boundary conditions |
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89 | (4) |
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2.2 Stability of Equilibrium of an Elastic Body |
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93 | (14) |
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2.2.1 Linearized models of the equilibrium stability for elastic bodies |
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97 | (4) |
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2.2.2 Mechanical interpretation of particular terms in the stability functional - The concept of an equivalent load |
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101 | (3) |
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2.2.3 Criteria of a system's critical state |
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104 | (3) |
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107 | (4) |
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2.4 Mixed Functionals in the Equilibrium Stability Problems |
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111 | (6) |
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112 | (5) |
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2.5 Timoshenko-Type Functionals |
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117 | (10) |
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2.5.1 Using statically permissible stresses in the equilibrium stability functional |
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123 | (4) |
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2.6 Elastic Systems in Presence of Constraints |
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127 | (15) |
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2.6.1 Elastic systems with a finite number of degrees of freedom |
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127 | (1) |
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128 | (4) |
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2.6.1.2 General case of allowing for constraints by decreasing the problem's dimensionality |
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132 | (4) |
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2.6.1.3 Allowing for constraints by increasing the problem's dimensionality |
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136 | (3) |
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2.6.2 Elastically deformable body with constraints |
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139 | (1) |
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2.6.2.1 Elastic body reinforced by an incompressible thread |
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139 | (3) |
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2.7 Elastic Systems in Presence of Perfectly Rigid Bodies |
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142 | (20) |
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2.7.1 Equilibrium of an elastic system in presence of rigid bodies |
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142 | (4) |
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2.7.2 Kinematic relationships for a perfectly rigid body - Rodrigues formula and its simplifications |
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146 | (3) |
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2.7.3 Work of forces applied to a perfectly rigid body |
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149 | (2) |
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2.7.4 The equilibrium stability functional for a system that contains a perfectly rigid body |
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151 | (4) |
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2.7.5 Geometric stiffness matrix for a perfectly rigid body |
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155 | (2) |
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2.7.5.1 Systems with a finite number of degrees of freedom |
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157 | (1) |
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157 | (2) |
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2.7.6.1 Modeling of springs by compressible bars |
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159 | (3) |
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2.8 Continuous RB-Bodies in Application Models of Elasticity |
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162 | (4) |
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2.9 Final Notes to
Chapter 2 |
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166 | (1) |
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3 Asymptotic Analysis of Post-Critical Behavior |
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167 | (30) |
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3.1 Role Played by Initial Imperfections |
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167 | (16) |
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3.1.1 Stability "in large": Upper and lower critical loads |
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176 | (7) |
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3.2 Systems with Multiple Degrees of Freedom |
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183 | (11) |
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3.2.1 Preliminary analysis |
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185 | (3) |
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3.2.2 Analysis in higher approximations |
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188 | (1) |
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3.2.3 Classification of singular points |
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189 | (2) |
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3.2.4 Quality of equilibrium in singular points |
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191 | (3) |
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3.3 Final Notes to
Chapter 3 |
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194 | (3) |
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4 Stability of Equilibrium of Straight Bars |
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197 | (128) |
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4.1 Stability of Equilibrium of a Compressed Bar |
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198 | (29) |
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4.1.1 Boundary conditions in equilibrium stability analysis of a compressed bar |
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204 | (5) |
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4.1.2 Orthogonality of a bar's buckling modes |
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209 | (2) |
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4.1.3 Initial imperfections |
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211 | (3) |
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4.1.3.1 Analysis with the deformed shape (second-order) |
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214 | (1) |
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4.1.4 Post-critical behavior of a bar in combined bending and compression |
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215 | (6) |
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4.1.5 Equilibrium stability of a Timoshenko bar - allowing for shear deformation |
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221 | (3) |
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4.1.6 Stability of equilibrium of a compressed bar that rests on an elastic bed |
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224 | (3) |
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4.2 Variational Derivation of the Equation of Stability for a Compressed Bar |
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227 | (9) |
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4.2.1 Stability of equilibrium of a compressed bar in the Euler-Bernoulli technical theory of bars |
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227 | (4) |
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4.2.2 Stability of equilibrium of a Timoshenko bar |
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231 | (5) |
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4.3 Stability of Equilibrium of a Compressed Spring |
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236 | (13) |
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4.3.1 Model with two degrees of freedom |
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236 | (2) |
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4.3.2 Discrete-continuous model |
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238 | (4) |
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4.3.2.1 Allowing for shear deformation |
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242 | (1) |
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4.3.3 Model of an equivalent bar |
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243 | (4) |
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4.3.3.1 Allowing for shear |
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247 | (2) |
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4.4 Buckling of a Bar in Tension |
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249 | (6) |
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4.5 Spatial Buckling Modes of a Compressed Bar |
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255 | (5) |
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4.6 Does the Critical Force Depend on the Lateral Load? |
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260 | (6) |
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4.7 Rayleigh Ratio and Timoshenko Formula |
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266 | (7) |
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4.7.1 A Timoshenko-type formula for a Timoshenko bar |
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271 | (2) |
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273 | (26) |
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4.8.1 Bernoulli-Euler bar |
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274 | (6) |
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4.8.2 Stability of equilibrium of a Bernoulli-Euler bar in bending in a three-dimensional space |
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280 | (2) |
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4.8.2.1 Simplifications in the functional (8.28) |
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282 | (1) |
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4.8.2.1.1 Simplification 1 |
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283 | (1) |
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4.8.2.1.2 Simplification 2 |
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283 | (1) |
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4.8.2.1.3 Simplification 3 |
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283 | (1) |
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4.8.2.1.4 Simplification 4 |
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284 | (1) |
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4.8.2.2 Euler equations for the functional (8.32) |
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285 | (1) |
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4.8.2.3 Example 1 - stability of the planar mode of bending of a Bernoulli-Euler bar |
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286 | (2) |
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4.8.2.4 Example 2 - stability of a bar bent in two planes |
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288 | (2) |
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290 | (3) |
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4.8.4 Stability of equilibrium of a flexural Timoshenko bar in a three-dimensional space |
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293 | (4) |
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4.8.4.1 Example 3 - stability of the planar mode of bending of a Timoshenko bar |
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297 | (2) |
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4.9 Stability of Bars in Torsion |
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299 | (23) |
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4.9.1 Integration of the equation set (9.8) |
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305 | (2) |
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4.9.2 Torsion of a bar in absence of a longitudinal force |
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307 | (1) |
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4.9.3 Boundary conditions |
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308 | (2) |
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310 | (1) |
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4.9.4.1 The case of a bar clamped on two ends |
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311 | (2) |
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4.9.4.2 The case of a cantilever bar |
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313 | (1) |
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4.9.4.2.1 Semi-tangential external moment |
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313 | (1) |
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4.9.4.2.2 Moment of dead forces |
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314 | (1) |
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4.9.5 Torsion of a Timoshenko bar |
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315 | (6) |
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4.9.5.1 The case of a cantilever Timoshenko bar |
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321 | (1) |
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4.9.5.1.1 Semi-tangential external moment |
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321 | (1) |
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4.10 Final Notes to
Chapter 4 |
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322 | (3) |
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5 Stability of Equilibrium of Curved Bars |
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325 | (48) |
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5.1 Basic Equations for a Curved Bar in the Linear Model |
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326 | (15) |
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5.1.1 Simplifications of equations for a curved bar with the incompressible axis |
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329 | (3) |
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332 | (6) |
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338 | (3) |
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5.2 Variational Derivation of the Equilibrium Stability Equations for a Curved Bar |
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341 | (3) |
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5.3 Stability of Equilibrium of an Incompressible Curved Bar |
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344 | (12) |
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5.3.1 Stability of an incompressible circular ring under dead radial forces |
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346 | (1) |
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5.3.2 Stability of an incompressible circular arch under hydrostatic pressure |
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347 | (1) |
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5.3.2.1 Circular two-hinged arch |
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348 | (3) |
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5.3.2.2 Circular hinge-free arch |
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351 | (2) |
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5.3.3 Stability of a ring under a polar radial load |
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353 | (1) |
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5.3.4 Stability of arches under a vertical load |
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353 | (3) |
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5.4 Stability of Equilibrium of Flat Arches |
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356 | (17) |
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5.4.1 A model problem - von Mises truss |
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357 | (8) |
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5.4.2 A flat arch under a vertical load |
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365 | (8) |
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6 Stability of Equilibrium of Thin-Walled Bars |
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373 | (62) |
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6.1 Open-Profile Thin-Walled Bar |
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374 | (33) |
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6.1.1 Equilibrium stability functional for a bar in bending and compression |
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374 | (10) |
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6.1.1.1 Boundary conditions |
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384 | (4) |
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6.1.2 Initial state of stress of a bar in bending and compression |
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388 | (3) |
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6.1.2.1 Characteristic equation for critical forces in an eccentrically compressed bar |
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391 | (4) |
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6.1.2.2 Thin-walled bar compressed along the line of shear centers |
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395 | (5) |
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6.1.2.3 Centrally compressed thin-walled bar |
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400 | (2) |
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6.1.2.4 Stability of equilibrium of a thin-walled bar with the non-warped cross-section |
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402 | (5) |
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6.2 Lateral Bending of Thin-Walled Bars |
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407 | (16) |
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6.2.1 Stability of a planar bending mode for a thin-walled bar - The case of pure bending |
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410 | (1) |
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6.2.2 Generalized Prandlt-Michell problem |
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411 | (1) |
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6.2.2.1 Solution based on the Alfutov functional, SA |
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412 | (2) |
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6.2.2.2 Solution based on the correct functional, S |
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414 | (1) |
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6.2.2.3 Comparing two solutions |
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415 | (1) |
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6.2.2.4 Modified problem - problem "B" |
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415 | (1) |
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6.2.3 Generalized Timoshenko problem |
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416 | (3) |
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6.2.4 How to allow for the level of application of the external lateral load |
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419 | (4) |
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6.3 Thin-Walled Bars Considered by the Semi-Shear Theory |
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423 | (12) |
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6.3.1 Stability of equilibrium of an eccentrically compressed bar |
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425 | (3) |
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6.3.1.1 The case of a non-warped cross-section |
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428 | (1) |
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6.3.2 Stability of equilibrium in lateral bending of bars considered by the semi-shear theory |
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429 | (2) |
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6.3.3 A multi-story building as a thin-walled bar |
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431 | (4) |
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7 Conservative External Forces and Moments: Paradoxes and Misbeliefs |
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435 | (84) |
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7.1 Some Cases of Behavior of External Forces |
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437 | (6) |
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443 | (3) |
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7.2.1 The equilibrium stability functional under the action of a hydrostatic load |
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444 | (2) |
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446 | (1) |
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447 | (33) |
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7.4.1 Definition of generalized moments |
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447 | (6) |
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453 | (1) |
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453 | (1) |
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7.4.2 Components of the moment vector and the rotation vector in the Lagrangian and Eulerian coordinate systems |
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454 | (3) |
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7.4.2.1 Semi-Lagrangian coordinates |
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457 | (2) |
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7.4.3 Conditions of conservativeness of the external moment vector |
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459 | (10) |
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7.4.4 General case of a dead force moment |
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469 | (4) |
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7.4.5 Some mechanical realizations of the dead force moments |
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473 | (2) |
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475 | (1) |
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7.4.6 Equilibrium equations that correspond to rotational degrees of freedom of a mechanical system |
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476 | (2) |
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7.4.7 An attempt to introduce a vector of generalized rotations |
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478 | (2) |
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7.5 Stability of Bars in Three-Dimensional Space |
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480 | (30) |
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7.5.1 Back to the problem of stability of a bar in torsion |
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480 | (2) |
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7.5.1.1 Example - stability of equilibrium of an isolated node |
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482 | (2) |
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7.5.2 Back to the problem of stability of the planar mode of bending |
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484 | (1) |
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7.5.2.1 Bernoulli-Euler bars |
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484 | (3) |
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487 | (1) |
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7.5.2.3 Simply supported bar in pure bending |
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488 | (4) |
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7.5.3 Pure bending of a cantilever bar |
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492 | (1) |
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493 | (1) |
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494 | (1) |
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495 | (1) |
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496 | (1) |
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497 | (4) |
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7.5.4 Distributed moment load |
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501 | (1) |
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7.5.4.1 Bernoulli-Euler bar |
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501 | (2) |
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503 | (5) |
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7.5.5 Losing stability in a state of equilibrium without initial stresses |
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508 | (2) |
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7.6 Argyris Paradox and Accompanying Myths |
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510 | (6) |
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7.6.1 Argyris paradox: Myth of a semi-tangential nature of the bending moment |
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510 | (3) |
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7.6.2 Myth of a potential nature of the elasticity force moments |
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513 | (1) |
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7.6.3 Rotation/slope components and derivatives of lateral displacements of a bar's axis |
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513 | (2) |
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7.6.4 Components of the rotation vector as generalized coordinates of the system |
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515 | (1) |
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7.7 Final Notes to
Chapter 7 |
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516 | (3) |
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8 Spatial Curved Bar - Kirchhoff-Klebsch Theory |
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519 | (36) |
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8.1 Basic Knowledge About the Geometry of a Spatial Curve |
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519 | (4) |
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8.2 Curved Bar and Its Geometry |
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523 | (6) |
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8.3 Kinematic Relationships for a Bar |
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529 | (4) |
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8.4 Equations of Equilibrium for a Bar |
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533 | (3) |
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536 | (2) |
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538 | (11) |
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8.6.1 Stability of the planar mode of bending of a curved bar |
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540 | (1) |
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8.6.1.1 A circular arch in pure bending - Timoshenko problem |
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541 | (4) |
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8.6.1.2 A circular ring under a radial load - Nicolai problem |
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545 | (1) |
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8.6.1.2.1 The case of a dead external load |
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545 | (2) |
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547 | (1) |
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548 | (1) |
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8.7 Rectilinear Bar with an Initial Twist |
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549 | (3) |
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8.8 Final Notes to the Kirchhoff-Klebsch Theory |
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552 | (3) |
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555 | (568) |
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Appendix A Grounds for Simplification in the Equilibrium Stability Functional for a Thin-Walled Bar |
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555 | (4) |
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Appendix B Orthogonal Curvilinear Coordinates - Formulas for Strain Components |
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559 | (4) |
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B.1 Orthogonal curvilinear coordinates - General case |
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560 | (2) |
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B.2 Orthogonal curvilinear coordinates produced by a planar curve |
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562 | (1) |
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Appendix C Additions to the Papkovich Theorem |
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563 | (4) |
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C.1 Another justification for the Papkovich theorem |
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563 | (2) |
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C.2 Additional note to the Papkovich theorem |
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565 | (2) |
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Appendix D Qualitative Estimates of Critical Forces |
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567 | (8) |
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D.1 Transformations of the load |
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568 | (5) |
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D.2 Transformations of the stiffness |
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573 | (2) |
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Appendix E Elements of the Catastrophe Theory |
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575 | (28) |
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E.1 Philosophy of the catastrophe theory |
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576 | (3) |
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E.2 Some elementary catastrophes |
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579 | (4) |
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E.3 Effect of initial imperfections |
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583 | (1) |
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E.4 Interaction between buckling modes |
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584 | (7) |
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E.5 Procedure for use of the catastrophe theory |
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591 | (12) |
| References in Vol. 1 |
|
593 | |
| Author Index in Vol. 1 |
|
1 | (4) |
| Subject Index in Vol. 1 |
|
5 | |
| Preface - Vol. 1 to Vol. 3 |
|
xv | |
| Acknowledgments - Vol. 1 to Vol. 3 |
|
xxiii | |
| Contents - Vol. 1 to Vol. 3 |
|
xxv | |
| About the Authors |
|
xxix | |
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9 Stability of Equilibrium of Plates - Kirchhoff-Love and Reissner Plates |
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603 | (68) |
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9.1 Stability of Equilibrium of Kirchhoff-Love Plates |
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604 | (41) |
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9.1.1 Basic relationships in the theory of thin plates |
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604 | (5) |
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9.1.2 Variational derivation of the equilibrium stability equation for the Kirchhoff-Love plates |
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609 | (6) |
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9.1.2.1 Boundary conditions |
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615 | (2) |
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9.1.3 Stability of equilibrium of a cantilever strip |
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617 | (3) |
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620 | (2) |
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9.1.4.1 Stability of equilibrium of a half-strip reinforced by a thread |
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622 | (3) |
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9.1.4.2 Stability of equilibrium of a half-strip without a thread |
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625 | (4) |
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9.1.5 Southwell-Skan problem |
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|
629 | (1) |
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9.1.6 Stability of equilibrium of round plates |
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630 | (3) |
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9.1.6.1 Stability of equilibrium of a round plate under a radial compression by forces on its contour |
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633 | (7) |
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9.1.6.2 Equilibrium stability functional for a plate in polar coordinates |
|
|
640 | (1) |
|
9.1.6.3 Stability of equilibrium of a round plate loaded by a torque - Dean problem |
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640 | (5) |
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9.2 Stability of Equilibrium of Reissner Plates |
|
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645 | (2) |
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9.3 Slender Plates - von Karman Theory |
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647 | (10) |
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9.3.1 Variational statement of the problem |
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653 | (4) |
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9.4 Post-critical Behavior of Slender Plates |
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657 | (10) |
|
9.4.1 Character of post-critical deformation |
|
|
657 | (1) |
|
|
|
658 | (3) |
|
9.4.2 Solution based on von Karman theory |
|
|
661 | (6) |
|
9.5 Final Notes to
Chapter 9 |
|
|
667 | (4) |
|
10 Systems with Unilateral Constraints |
|
|
671 | (40) |
|
10.1 Elements of the Theory of Systems with Unilateral Constraints |
|
|
671 | (10) |
|
|
|
671 | (4) |
|
10.1.2 Limitations for virtual displacements |
|
|
675 | (1) |
|
10.1.3 Equilibrium conditions |
|
|
676 | (5) |
|
10.2 Critical Value of the Load Intensity |
|
|
681 | (9) |
|
10.3 Determining the Upper Critical Load |
|
|
690 | (5) |
|
10.4 Illustrative Examples |
|
|
695 | (8) |
|
10.5 High-rise Building on a Unilateral Elastic Bed |
|
|
703 | (3) |
|
10.6 Possible Destabilization of Systems with Unilateral Constraints |
|
|
706 | (2) |
|
10.7 Final Notes to
Chapter 10 |
|
|
708 | (3) |
|
11 Stability of Equilibrium of Planar Bar Structures |
|
|
711 | (70) |
|
11.1 Planar Bar Structures |
|
|
712 | (49) |
|
11.1.1 General solution of homogeneous equations of equilibrium stability for an individual bar |
|
|
714 | (7) |
|
11.1.2 Stiffness matrix of an individual bar |
|
|
721 | (4) |
|
11.1.2.1 Stiffness matrix of a bar with other methods of its end fixation |
|
|
725 | (4) |
|
11.1.2.2 Some properties of Kornoukhov functions and a procedure for calculation of those |
|
|
729 | (2) |
|
11.1.2.3 Initial stiffness matrix and geometric stiffness matrix for a bar |
|
|
731 | (3) |
|
11.1.3 Criterion for a critical state in a bar structure |
|
|
734 | (6) |
|
11.1.3.1 Do we need higher buckling modes? |
|
|
740 | (2) |
|
11.1.3.2 A qualitative technique for determining critical loads |
|
|
742 | (3) |
|
11.1.4 Example: A paradox in stability problems |
|
|
745 | (7) |
|
|
|
752 | (4) |
|
11.1.6 Stiff inserts at the ends of a bar |
|
|
756 | (4) |
|
11.1.6.1 A possible mistake in the stability analysis in presence of rigid bodies |
|
|
760 | (1) |
|
11.2 Deformed-shape-based Analysis of a Planar Bar Structure |
|
|
761 | (17) |
|
11.2.1 Deformed-shape-based analysis of an individual bar |
|
|
761 | (1) |
|
11.2.1.1 Method of initial parameters |
|
|
762 | (3) |
|
11.2.1.2 Reactions at the ends of a bar caused by lateral actions |
|
|
765 | (4) |
|
11.2.2 Monocycle, quasi-monocycle and polycycle analysis of bar systems |
|
|
769 | (2) |
|
11.2.3 Mohr formula in application to bar structures in combined bending and compression |
|
|
771 | (7) |
|
11.3 Final Notes to
Chapter 11 |
|
|
778 | (3) |
|
12 FEM in Stability Problems |
|
|
781 | (112) |
|
|
|
783 | (10) |
|
12.1.1 Shape functions and a shape function matrix for a finite element |
|
|
784 | (2) |
|
12.1.2 General requirements to shape functions |
|
|
786 | (3) |
|
12.1.3 Comparative analysis of shape functions |
|
|
789 | (3) |
|
12.1.4 General formulas for matrices R0 and G |
|
|
792 | (1) |
|
12.2 Stiffness Matrices of a Bar in Plane |
|
|
793 | (19) |
|
12.2.1 A Bernoulli-Euler bar |
|
|
793 | (2) |
|
|
|
795 | (3) |
|
12.2.2.1 Model I: Linear approximations of displacements and rotations |
|
|
798 | (2) |
|
12.2.2.2 Model II: Coupled approximations of displacements and rotations - Linear-quadratic shape functions |
|
|
800 | (2) |
|
12.2.2.3 Model III: Cubic-quadratic approximations of displacements |
|
|
802 | (3) |
|
12.2.2.4 General representation of matrices R0 and G for a Timoshenko bar - Comparative analysis of three finite element models |
|
|
805 | (2) |
|
|
|
807 | (5) |
|
12.3 Stiffness Matrix of a Spatial Bar |
|
|
812 | (23) |
|
12.3.1 A Bernoulli-Euler bar |
|
|
812 | (7) |
|
|
|
819 | (6) |
|
12.3.3 Stiff inserts at the ends of a bar |
|
|
825 | (8) |
|
12.3.4 Geometric stiffness matrix of a node |
|
|
833 | (2) |
|
12.4 Plate Finite Elements |
|
|
835 | (29) |
|
12.4.1 Plate finite element in application to the Suv functional |
|
|
836 | (2) |
|
12.4.1.1 A rectangular finite element |
|
|
838 | (3) |
|
12.4.2 Finite elements of flexural plate |
|
|
841 | (2) |
|
12.4.2.1 Kirchhoff-Love plate |
|
|
843 | (5) |
|
|
|
848 | (4) |
|
12.4.3 A hybrid FEM approach |
|
|
852 | (4) |
|
|
|
856 | (3) |
|
|
|
859 | (5) |
|
12.5 Perfectly Rigid Bodies as Parts of Discrete Design Models |
|
|
864 | (2) |
|
12.6 FEN Relationships for Geometrically Nonlinear Models |
|
|
866 | (26) |
|
12.6.1 Four floors of geometrically nonlinear models |
|
|
866 | (3) |
|
12.6.2 Decomposition of strains into a sum of linear and quadratic parts: Second-order theory |
|
|
869 | (6) |
|
12.6.3 Matrix-operator form of the full potential energy functional |
|
|
875 | (5) |
|
12.6.4 Equations in increments |
|
|
880 | (6) |
|
12.6.5 Equilibrium stability models |
|
|
886 | (2) |
|
12.6.5.1 Possible simplifications of the equilibrium stability model |
|
|
888 | (1) |
|
12.6.5.2 Stability coefficient |
|
|
889 | (3) |
|
12.7 Final Notes to
Chapter 12 |
|
|
892 | (1) |
|
|
|
893 | (56) |
|
|
|
893 | (3) |
|
13.2 Geometrical Nonlinearity for Truss-type Bars |
|
|
896 | (17) |
|
13.2.1 Geometric equations |
|
|
897 | (4) |
|
13.2.2 Equilibrium equations |
|
|
901 | (3) |
|
|
|
904 | (1) |
|
|
|
904 | (5) |
|
13.2.5 Geometrically nonlinear equations in variations |
|
|
909 | (4) |
|
13.3 Stable Configurations of a Substatic System |
|
|
913 | (9) |
|
13.3.1 Static-kinematic classification |
|
|
914 | (5) |
|
13.3.2 A criterion of selection of instantaneously rigid systems |
|
|
919 | (3) |
|
13.4 Buckling of Nodes Out of a Truss Plane |
|
|
922 | (4) |
|
13.4.1 Unsupported length of diagonals in compression |
|
|
926 | (1) |
|
13.5 Estimation of Forces in Null Bars |
|
|
926 | (2) |
|
13.6 Estimation of the Node Stiffness Effect |
|
|
928 | (6) |
|
|
|
934 | (15) |
|
|
|
934 | (2) |
|
13.7.2 Effect of initial imperfections |
|
|
936 | (3) |
|
13.7.3 Interaction between buckling modes |
|
|
939 | (5) |
|
13.7.4 Spatial compressed lattice bars |
|
|
944 | (1) |
|
13.7.4.1 Tetrahedral bars |
|
|
944 | (4) |
|
|
|
948 | (1) |
|
14 Dynamic Criterion of Stability and Non-Conservative Systems |
|
|
949 | (100) |
|
14.1 Dynamic Analysis of Equilibrium Stability |
|
|
949 | (14) |
|
|
|
949 | (4) |
|
14.1.2 A system with one degree of freedom |
|
|
953 | (1) |
|
|
|
954 | (1) |
|
|
|
955 | (2) |
|
|
|
957 | (1) |
|
14.1.2.4 Combined loading by dead and follower forces |
|
|
958 | (4) |
|
|
|
962 | (1) |
|
14.2 Systems with Multiple Degrees of Freedom |
|
|
963 | (29) |
|
|
|
963 | (6) |
|
14.2.1.1 Conservative system |
|
|
969 | (3) |
|
14.2.1.2 Nonlinear system, general case |
|
|
972 | (1) |
|
14.2.2 System with two degrees of freedom - detailed analysis |
|
|
973 | (4) |
|
14.2.2.1 General analysis of equilibrium stability for a system with two degrees of freedom |
|
|
977 | (1) |
|
14.2.3 Influence of constraints on equilibrium stability of non-conservative systems |
|
|
978 | (3) |
|
14.2.4 Damping and its role in the equilibrium stability |
|
|
981 | (4) |
|
14.2.4.1 Non-conservative external forces and dissipation: Ziegler paradox |
|
|
985 | (7) |
|
|
|
992 | (8) |
|
14.3.1 Tangential external moment - static analysis |
|
|
995 | (1) |
|
14.3.2 Axial external moment - static analysis |
|
|
996 | (1) |
|
14.3.3 Tangential external moment - dynamic analysis |
|
|
996 | (4) |
|
14.4 Continuous Non-conservative Systems |
|
|
1000 | (13) |
|
14.4.1 Variations of external forces under conservative and non-conservative loads |
|
|
1002 | (4) |
|
14.4.2 Discretization of conservative and non-conservative systems |
|
|
1006 | (1) |
|
14.4.2.1 Bubnov-Galyorkin method - general |
|
|
1007 | (3) |
|
14.4.2.2 Bubnov-Galyorkin method using fundamental basis functions |
|
|
1010 | (1) |
|
14.4.2.3 Finite element method for non-conservative problems |
|
|
1011 | (1) |
|
14.4.2.4 Discretization by mass |
|
|
1011 | (2) |
|
|
|
1013 | (8) |
|
14.5.1 A constant-direction force |
|
|
1014 | (2) |
|
|
|
1016 | (2) |
|
14.5.3 A generalized problem |
|
|
1018 | (3) |
|
14.6 Flutter When Fluid Comes Out of Tube |
|
|
1021 | (4) |
|
14.7 Models with a Truncated Number of Inertial Characteristics |
|
|
1025 | (15) |
|
14.7.1 Conservative system |
|
|
1026 | (3) |
|
14.7.2 Non-conservative system |
|
|
1029 | (2) |
|
14.7.3 The effect of truncation by mass on the area of equilibrium stability |
|
|
1031 | (2) |
|
14.7.3.1 Beck's bar with two concentrated masses |
|
|
1033 | (5) |
|
14.7.4 Critics of the dynamic criterion of equilibrium stability |
|
|
1038 | (2) |
|
14.8 On the Application of the Static Approach to Non-conservative Problems |
|
|
1040 | (4) |
|
14.9 Final Notes to
Chapter 14 |
|
|
1044 | (5) |
|
14.9.1 Smith-Herrmann paradox |
|
|
1045 | (1) |
|
14.9.2 Follower force as an "ugly duckling" of mechanics |
|
|
1046 | (3) |
|
15 Post-Critical Deformation |
|
|
1049 | (32) |
|
15.1 Post-critical Behavior of Bars |
|
|
1050 | (7) |
|
15.1.1 Critical state of frame structures |
|
|
1050 | (1) |
|
15.1.2 A bar with its ends resisting axial displacements |
|
|
1051 | (6) |
|
|
|
1057 | (9) |
|
15.2.1 Possibility of a snap-through |
|
|
1057 | (2) |
|
15.2.2 Mixed-method analysis |
|
|
1059 | (3) |
|
15.2.2.1 Beniaminov formula |
|
|
1062 | (3) |
|
|
|
1065 | (1) |
|
15.3 Using the Post-critical Behavior of Plates |
|
|
1066 | (7) |
|
15.3.1 Reduction coefficient |
|
|
1066 | (5) |
|
15.3.2 Post-critical behavior of plates in shear |
|
|
1071 | (2) |
|
15.4 Post-critical Interaction Between Buckling Modes |
|
|
1073 | (7) |
|
15.4.1 Global and local buckling modes of a thin-walled bar |
|
|
1073 | (7) |
|
15.5 Final Notes to
Chapter 15 |
|
|
1080 | (1) |
|
16 Design Models in Stability Problems: Practical Examples |
|
|
1081 | (42) |
|
16.1 Stability of a Multi-story Building: The Effect of Rigidity of Floor Panels |
|
|
1081 | (4) |
|
16.2 Finite Element Modeling of Thin-walled Bars |
|
|
1085 | (4) |
|
16.3 Stability of Masts with Guy Ropes |
|
|
1089 | (6) |
|
16.3.1 Cable elements in design models |
|
|
1089 | (5) |
|
16.3.2 Potential approaches to the solution |
|
|
1094 | (1) |
|
16.4 Energy-based Estimation of Roles of Particular Subsystems |
|
|
1095 | (11) |
|
16.4.1 Restrained and forced buckling |
|
|
1095 | (2) |
|
16.4.2 Energy characteristics |
|
|
1097 | (5) |
|
16.4.3 Modification of a structure |
|
|
1102 | (1) |
|
16.4.4 Calculation of unsupported length |
|
|
1103 | (3) |
|
16.5 Sensitivity of the Critical Load to Changes in the System's Stiffness Values |
|
|
1106 | (6) |
|
16.5.1 Uniform stability and optimization of a structure |
|
|
1106 | (3) |
|
16.5.1.1 Example of an optimization for stability |
|
|
1109 | (3) |
|
16.6 Approximate Estimation of Ferroconcrete Behavior |
|
|
1112 | (11) |
|
16.6.1 Choosing an elasticity modulus value for stability check |
|
|
1113 | (3) |
|
16.6.2 Approximate estimation of creep effects |
|
|
1116 | (2) |
|
16.6.3 An example of design analysis of a real ferroconcrete structure |
|
|
1118 | (5) |
|
|
|
1123 | (380) |
|
Appendix F Jordan Exclusions and Their Use in Structural Mechanics |
|
|
1123 | (14) |
|
|
|
1123 | (3) |
|
F.2 Jordan exclusions with a system's stiffness matrix |
|
|
1126 | (3) |
|
F.3 A finite element's stiffness matrix in case the element is attached to nodes non-rigidly |
|
|
1129 | (5) |
|
F.4 Double Jordan exclusion |
|
|
1134 | (2) |
|
F.5 Jordan transformations in stability problems: A geometric condensation procedure |
|
|
1136 | (1) |
|
Appendix G Asymptotic Analysis of Finite Element Models for a Timoshenko Bar |
|
|
1137 | (6) |
|
Appendix H Generalized Timoshenko problem |
|
|
1143 | (7) |
|
H.1 Exact solution of the problem |
|
|
1145 | (4) |
|
H.2 Solution by the Ritz method |
|
|
1149 | (1) |
|
Appendix I Strong Bending of Bars |
|
|
1150 | (13) |
|
|
|
1151 | (3) |
|
|
|
1154 | (1) |
|
I.3 Equilibrium equations |
|
|
1155 | (1) |
|
I.4 Simplifications on lower floors of geometrical nonlinearity |
|
|
1156 | (2) |
|
I.5 Example: Stability of a bar under a kinematic action |
|
|
1158 | (4) |
|
I.6 Example: Pure bending of a bar |
|
|
1162 | (1) |
|
Appendix J On a Mathematical Model of a Shear Bar in Equilibrium Stability |
|
|
1163 | (26) |
| References in Vol. 2 |
|
1171 | |
| Author Index in Vol. 2 |
|
1 | (6) |
| Subject Index in Vol. 2 |
|
7 | |
| Preface - Vol. 1 to Vol. 3 |
|
xiii | |
| Acknowledgments - Vol. 1 to Vol. 3 |
|
xxi | |
| Contents - Vol. 1 to Vol. 3 |
|
xxiii | |
| About the Authors |
|
xxvii | |
|
17 Stability of Inelastic Systems |
|
|
1189 | (84) |
|
17.1 A Long Way to Modern Concepts |
|
|
1189 | (29) |
|
17.1.1 Stability of bar subjected to axial compression |
|
|
1190 | (1) |
|
17.1.1.1 Concept of reduced-modulus critical stresses |
|
|
1191 | (7) |
|
17.1.1.2 Concept of tangent-modulus critical stresses |
|
|
1198 | (1) |
|
17.1.1.3 Shanley's column |
|
|
1199 | (7) |
|
17.1.1.4 Analysis of buckling process at initial values of load, different from tangent-modulus load |
|
|
1206 | (1) |
|
17.1.2 Influence of initial imperfections |
|
|
1207 | (4) |
|
17.1.3 General remarks to Shanley's theory |
|
|
1211 | (2) |
|
17.1.4 Analysis of behavior of Shanley's column under action of small perturbations |
|
|
1213 | (5) |
|
17.2 Elastoplastic Bar Subjected to Bending and Compression |
|
|
1218 | (10) |
|
17.2.1 Considerations regarding the approach to solution of practical problem |
|
|
1218 | (3) |
|
17.2.2 Role played by lateral load |
|
|
1221 | (4) |
|
17.2.3 Role played by residual stresses |
|
|
1225 | (3) |
|
17.3 Elastoplastic Bar of I-Section |
|
|
1228 | (11) |
|
17.3.1 Elastic core of bar made of ideal elastoplastic material |
|
|
1229 | (1) |
|
17.3.2 Three stages of section's behavior |
|
|
1229 | (2) |
|
17.3.2.1 Section's elastic behavior |
|
|
1231 | (1) |
|
17.3.2.2 Single-sided yield stage |
|
|
1231 | (1) |
|
17.3.2.3 Double-sided yield stage |
|
|
1232 | (1) |
|
17.3.3 Shapes of bar equilibrium |
|
|
1233 | (3) |
|
17.3.3.1 Description of the second shape of equilibrium |
|
|
1236 | (2) |
|
|
|
1238 | (1) |
|
17.4 Bar Systems Method of Two Design Sections |
|
|
1239 | (20) |
|
17.4.1 State equations for bars of system |
|
|
1242 | (1) |
|
17.4.1.1 First and second design sections |
|
|
1243 | (1) |
|
17.4.1.2 State equations for bars of system |
|
|
1244 | (2) |
|
17.4.1.3 System's equilibrium |
|
|
1246 | (1) |
|
17.4.1.4 System state curve and repellence curve |
|
|
1247 | (2) |
|
17.4.1.5 Varied state of equilibrium of bar |
|
|
1249 | (4) |
|
17.4.1.6 Instantaneous stiffness matrix and secant stiffness matrix of bar |
|
|
1253 | (2) |
|
|
|
1255 | (4) |
|
17.5 Semi-empirical and Approximate Analytic Formulas |
|
|
1259 | (6) |
|
17.6 Lateral-Torsional Buckling of Bars Subjected to Bending |
|
|
1265 | (4) |
|
17.7 Final Notes to
Chapter 17 |
|
|
1269 | (4) |
|
|
|
1273 | (26) |
|
18.1 Creep Phenomenon: Necessary General Information |
|
|
1274 | (6) |
|
18.1.1 Linear theory of hereditary creep |
|
|
1276 | (4) |
|
18.2 Simplest Problems of Creep Stability |
|
|
1280 | (14) |
|
|
|
1280 | (1) |
|
18.2.2 Shanley's column under creep conditions |
|
|
1281 | (1) |
|
|
|
1282 | (2) |
|
18.2.2.2 Linear hereditary creep |
|
|
1284 | (2) |
|
18.2.3 Buckling of imperfect viscoelastic bar |
|
|
1286 | (1) |
|
18.2.3.1 Bar as a system with one degree of freedom |
|
|
1286 | (5) |
|
18.2.4 Dynamic analysis of viscoelastic bar's stability |
|
|
1291 | (3) |
|
18.3 Other Approaches and Criteria |
|
|
1294 | (2) |
|
18.4 Final Notes to
Chapter 18 |
|
|
1296 | (3) |
|
|
|
1299 | (66) |
|
19.1 Dynamic Longitudinal Bending |
|
|
1299 | (23) |
|
19.1.1 Dynamic compression of straight bar |
|
|
1300 | (3) |
|
19.1.2 Particular cases and possible simplifications |
|
|
1303 | (2) |
|
19.1.3 Wave process concept - Wave-like behavior of axial force variation in the bar |
|
|
1305 | (1) |
|
|
|
1305 | (5) |
|
|
|
1310 | (4) |
|
19.1.4 Analysis of numerical solution to the problem of impact load effect on the bar |
|
|
1314 | (2) |
|
19.1.5 Sudden application of load: Lavrentiev-Ishlinsky solution |
|
|
1316 | (4) |
|
19.1.6 Elementary approaches to solutions to problems of dynamic longitudinal bending |
|
|
1320 | (2) |
|
19.2 Parametric Resonance |
|
|
1322 | (18) |
|
19.2.1 Equation of parametric system's motion |
|
|
1322 | (1) |
|
19.2.1.1 Some general properties of solutions to Hill equation |
|
|
1323 | (5) |
|
19.2.2 Excitation as per the law of square-sine function |
|
|
1328 | (3) |
|
19.2.3 Classic formulation of the problem |
|
|
1331 | (3) |
|
19.2.3.1 Influence of damping |
|
|
1334 | (6) |
|
19.3 Action of Moving Load |
|
|
1340 | (22) |
|
19.3.1 Action of moving concentrated mass |
|
|
1340 | (2) |
|
19.3.1.1 Willis-Stokes problem |
|
|
1342 | (2) |
|
|
|
1344 | (2) |
|
|
|
1346 | (4) |
|
19.3.2 Motion of infinite load strip |
|
|
1350 | (3) |
|
19.3.3 Traveling bending wave |
|
|
1353 | (5) |
|
19.3.3.1 Generalization of Biderman problem for the case of Timoshenko beam |
|
|
1358 | (4) |
|
19.4 Final Notes to
Chapter 19 |
|
|
1362 | (3) |
|
20 Aerodynamic Instability |
|
|
1365 | (44) |
|
20.1 Vortex Shedding Oscillations |
|
|
1367 | (5) |
|
|
|
1372 | (5) |
|
20.3 Divergence and Flutter |
|
|
1377 | (14) |
|
|
|
1377 | (2) |
|
20.3.1.1 Panel divergence |
|
|
1379 | (3) |
|
|
|
1382 | (6) |
|
|
|
1388 | (2) |
|
|
|
1390 | (1) |
|
20.4 Aeroelastic Vibrations and Instability of Suspension Bridge |
|
|
1391 | (14) |
|
20.4.1 Vertical vibrations of suspension bridges |
|
|
1394 | (5) |
|
20.4.2 Flexural-torsion vibrations |
|
|
1399 | (6) |
|
|
|
1405 | (2) |
|
20.6 Final Notes to
Chapter 20 |
|
|
1407 | (2) |
|
|
|
1409 | (52) |
|
|
|
1409 | (4) |
|
21.2 Challenges Concerning Experimental Technique |
|
|
1413 | (17) |
|
21.2.1 Boundary conditions modeling |
|
|
1415 | (3) |
|
21.2.2 Load transfer control |
|
|
1418 | (4) |
|
21.2.3 Quality of test sample |
|
|
1422 | (4) |
|
21.2.4 Measuring and recording the results |
|
|
1426 | (4) |
|
21.3 Interpretation of Experimental Results |
|
|
1430 | (8) |
|
21.3.1 The Southwell method |
|
|
1431 | (3) |
|
21.3.2 Southwell method modifications |
|
|
1434 | (4) |
|
21.4 Vibration Method of Critical Load Identification |
|
|
1438 | (8) |
|
21.5 Influence of Testing Machine Compliance |
|
|
1446 | (3) |
|
21.6 Description of Certain Experiments |
|
|
1449 | (9) |
|
21.6.1 Roorda disturbance sensitivity experiments |
|
|
1450 | (2) |
|
21.6.2 Verification of paradox results from
Chapter 8 |
|
|
1452 | (3) |
|
21.6.3 Non-conservative compression experiments |
|
|
1455 | (3) |
|
21.7 Final Notes to
Chapter 21 |
|
|
1458 | (3) |
|
22 Stability Check on Design Codes |
|
|
1461 | (42) |
|
22.1 Buckling as a Limit State |
|
|
1462 | (1) |
|
22.2 Stability Safety Factor |
|
|
1463 | (2) |
|
22.3 Traditions of Standardization |
|
|
1465 | (3) |
|
22.4 Effective Length and Stability Analysis |
|
|
1468 | (10) |
|
22.4.1 No general interpretation |
|
|
1468 | (6) |
|
22.4.2 What do design codes recommend? |
|
|
1474 | (4) |
|
22.5 Allowance for Initial Imperfections |
|
|
1478 | (8) |
|
22.6 Code Requirements to General Analysis |
|
|
1486 | (7) |
|
22.6.1 Guidelines for steel structures |
|
|
1487 | (4) |
|
22.6.2 Guidelines for reinforced concrete structures |
|
|
1491 | (2) |
|
22.7 Add-Load on Individual Structural Members |
|
|
1493 | (5) |
|
22.8 More About Second Order Analysis |
|
|
1498 | (5) |
|
|
|
1503 | (18) |
|
Appendix K Concerning Solid Body Finite Rotation Theory |
|
|
1503 | (11) |
|
K.1 Matrix formulation of the Rodrigues formula |
|
|
1503 | (2) |
|
K.2 Finite rotation vector |
|
|
1505 | (1) |
|
|
|
1506 | (1) |
|
|
|
1506 | (3) |
|
K.3 Finite rotations summation formula |
|
|
1509 | (2) |
|
K.4 Finite rotations subtraction formula |
|
|
1511 | (1) |
|
K.5 Elementary work of moment load |
|
|
1512 | (1) |
|
K.5.1 Quod erat demonstrandum |
|
|
1513 | (1) |
|
Appendix L Bar Stability Under Compression by Force with Fixed-Line Action |
|
|
1514 | (7) |
| References in Vol. 3 |
|
1521 | (14) |
| Brief Afterword |
|
1535 | (2) |
| Portrait Gallery |
|
1537 | |
| Author Index in Vol. 3 |
|
1 | (4) |
| Subject Index in Vol. 3 |
|
5 | |
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