Fundamentals of Structural MechanicsThe last few decades have witnessed a dramatic increase in the application of numerical computation to problems in solid and structural mechanics. The burgeoning of computational mechanics opened a pedagogical gap between traditional courses in elementary strength of materials and the finite element method that classical courses on advanced strength of materials and elasticity do not adequately fill. In the past, our ability to formulate theory exceeded our ability to compute. In those days, solid mechanics was for virtuosos. With the advent of the finite element method, our ability to compute has surpassed our ability to formulate theory. As a result, continuum mechanics is no longer the province of the specialist. What an engineer needs to know about mechanics has been forever changed by our capacity to compute. This book attempts to capitalize on the pedagogi cal opportunities implicit in this shift of perspective. It now seems more ap propriate to focus on fundamental principles and formulations than on classical solution techniques. |
Contents
Vectors and Tensors | 1 |
The Geometry of Threedimensional Space | 2 |
Vectors | 3 |
Tensors | 11 |
Vector and Tensor Calculus | 33 |
Integral Theorems | 45 |
Additional Reading | 48 |
Problems | 49 |
Problems | 234 |
The Linear Theory of Beams | 241 |
Equations of Equilibrium | 243 |
The Kinematic Hypothesis | 249 |
Constitutive Relations for Stress Resultants | 252 |
Boundary Conditions | 256 |
The Limitations of Beam Theory | 257 |
The Principle of Virtual Work for Beams | 262 |
The Geometry of Deformation | 57 |
Uniaxial Stretch and Strain | 58 |
The Deformation Map | 62 |
The Stretch of a Curve | 65 |
The Deformation Gradient | 67 |
Strain in Threedimensional Bodies | 68 |
Examples | 69 |
Characterization of Shearing Deformation | 74 |
The Physical Significance of the Components of C | 77 |
Strain in Terms of Displacement | 78 |
Principal Stretches of the Deformation | 79 |
Change of Volume and Area | 84 |
Timedependent motion | 91 |
Additional Reading | 93 |
Problems | 94 |
The Transmission of Force | 103 |
Normal and Shearing Components of the Traction | 109 |
Principal Values of the Stress Tensor | 110 |
Differential Equations of Equilibrium | 112 |
Examples | 115 |
Alternative Representations of Stress | 118 |
Additional Reading | 124 |
Problems | 125 |
Elastic Constitutive Theory | 131 |
Isotropy | 138 |
Definitions of Elastic Moduli | 141 |
Elastic Constitutive Equations for Large Strains | 145 |
Limits to Elasticity | 148 |
Additional Reading | 150 |
Problems | 151 |
Boundary Value Problems in Elasticity | 159 |
Boundary Value Problems of Linear Elasticity | 160 |
A Little Boundary Value Problem | 165 |
Work and Virtual Work | 167 |
The Principle of Virtual Work for the Little Boundary Value Problem | 169 |
Essential and Natural Boundary Conditions | 181 |
The Principle of Virtual Work for 3D Linear Solids | 182 |
Finite Deformation Version of the Principle of Virtual WorkReference Configuration | 186 |
Closure | 188 |
Additional Reading | 189 |
Problems | 190 |
The Ritz Method of Approximation | 193 |
The Ritz Approximation for the Little Boundary Value Problem | 194 |
Orthogonal Ritz Functions | 207 |
The Finite Element Approximation | 216 |
The Ritz Method for Two and Threedimensional Problems | 226 |
Additional Reading | 233 |
The Planar Beam | 266 |
The BernoulliEuler Beam | 273 |
Structural Analysis | 278 |
Additional Reading | 282 |
Problems | 283 |
The Linear Theory of Plates | 293 |
Equations of Equilibrium | 295 |
The Kinematic Hypothesis | 300 |
Constitutive Equations for Resultants | 304 |
Boundary Conditions | 308 |
The Limitations of Plate Theory | 310 |
The Principle of Virtual Work for Plates | 311 |
The KirchhoffLove Plate Equations | 314 |
Additional Reading | 323 |
Problems | 324 |
Energy Principles and Static Stability | 327 |
Virtual Work and Energy Functionals | 330 |
Energy Principles | 341 |
Static Stability and the Energy Criterion | 345 |
Additional Reading | 352 |
Problems | 353 |
Fundamental Concepts in Static Stability | 359 |
Bifurcation of Geometrically Perfect Systems | 361 |
The Effect of Imperfections | 369 |
The Role of Linearized Buckling Analysis | 375 |
Systems with Multiple Degrees of Freedom | 378 |
Additional Reading | 384 |
Problems | 385 |
The Planar Buckling of Beams | 389 |
Derivation of the Nonlinear Planar Beam Theory | 390 |
Eulers Elastica | 397 |
The General Linearized Buckling Theory | 408 |
Ritz and the Linearized Eigenvalue Problem | 415 |
Additional Reading | 421 |
Problems | 423 |
Numerical Computation for Nonlinear Problems | 431 |
Newtons Method | 433 |
Tracing the Equilibrium Path of a Discrete System | 438 |
The Program NEWTON | 444 |
Newtons Method and Virtual Work | 446 |
The Program ELASTICA | 452 |
The Fully Nonlinear Planar Beam | 454 |
The Program NONLINEARBEAM | 462 |
Summary | 469 |
Problems | 470 |
473 | |
Other editions - View all
Common terms and phrases
approximation axial axis base functions base vectors basis beam theory bifurcation diagram body force boundary conditions boundary value problem coefficients components compute Consider constant constitutive equations constraint coordinate critical load cross section curve defined deformation gradient deformation map deformed configuration differential equations directional derivative displacement field displacement map dot product e₁ eigenvalues eigenvectors elastic elastica energy functional equations of equilibrium equilibrium equations equilibrium path essential boundary conditions example expression external virtual Figure Find finite element given imperfection integral Lagrangian little boundary value matrix modulus motion Newton's method nonlinear normal vector orthogonal parameter plane plate polynomial position principle of virtual Ritz method rotation satisfy scalar second derivative shown in Fig simply solution solve stability strain tensor stress resultants stress tensor stretch surface tangent theorem three-dimensional tion traction vector transverse undeformed vector field virtual displacement virtual-work functional zero