Approximate Solution Methods in Engineering Mechanics

Preface. 1. The Role of Approximate Solution Methods in Engineering. 1.1 Introduction. 1.2 Fields of Application. 1.3 Future Progress and Trends. References. Bibliography. 2. Approximate Analysis and Weighted Residuals. 2.1 Introduction. 2.2 Approximation Problem (Trial Functions; Norms or Measures of Error). 2.3 Method of Weighted Residuals (Ordinary Differential Equations). 2.4 Method of Weighted Residuals (Partial Differential Equations). 2.5 Variation Method (Rayleigh-Ritz Method). 2.6 Ritz Method Revisited and Trefftz Method. References. Bibliography. 3. Finite Difference Methods. 3.1 Preliminary Remarks and Concepts. 3.2 Divided Differences and Interpolation Formulas. 3.3 Approximate Expressions for Derivatives. 3.4 Two-Dimensional Harmonic Equation, Biharmonic Equation, and Curved Boundaries. 3.5 Finite Difference Approximation of the Plane Stress Problem. 3.6 Torsion Problem. Appendix 3A: Derivation of Eq. (3.16). Appendix 3B: Derivation of Eq. (3.38). References. Bibliography. 4. The Finite Element Method. 4.1 Introduction. 4.2 Formulation for Plane Elasticity. 4.3 Bilinear Rectangle. 4.4 Linear Isoparametric Quadrilateral. 4.5 Plane Frame Element. 4.6 Closing Remarks. Problems. References. Bibliography. 5. Specialized Methods. 5.1 Introduction. 5.2 Finite Strip Method. 5.3 Formulation of the Finite Strip Method. 5.4 Example of the Finite Strip Method. 5.5 Finite Layer Method. 5.6 Finite Prism Method. 5.7 Applications and Developments of FSM, FLM, and FPM. References. Bibliography. 6. The Boundary Element Method. 6.1 Introduction. 6.2 Integrals in the Boundary Element Method. 6.3 Equations of Elasticity. 6.4 Fundamental or Kelvin's Solution. 6.5 Boundary Element Formulation. 6.6 Displacement and Traction Interpolation. 6.7 Element Contributions. 6.8 Assembly of Boundary Element Matrices. 6.9 Rigid-Body Motion. 6.10 Solution of Boundary Element Equations. 6.11 Displacement at Points in the Interior. 6.12 Body Forces. 6.13 Particular Integral Approach. 6.14 Evaluation of Stresses and Strains. 6.15 Corner Problem in the Boundary Element Method. 6.16 Closing Remarks. References. 7. Meshless Methods of Analysis. 7.1 Introduction. 7.2 Equations of Elasticity. 7.3 Weak Forms of the Governing Equations. 7.4 Moving Least Squares Approximations. 7.5 Characteristics of MLS Approximation. 7.6 MLS Weight Functions. 7.7 Discrete Element-free Galerkin Formulation. 7.8 Numerical Implementation. 7.9 Treatment of Boundary Conditions. 7.10 Other Methods for Meshless Analysis. 7.11 Closing Remarks. References. Author Index. Subject Index.