The Boundary Integral Equatio Method in Axisymmetric Stress Analysis Problems

ISBN-13: 9783540160304 / Angielski / Miękka / 1985 / 213 str.

Adib A. Bakr

Opis

The Boundary Integral Equation (BIE) or the Boundary Element Method is now well established as an efficient and accurate numerical technique for engineering problems. This book presents the application of this technique to axisymmetric engineering problems, where the geometry and applied loads are symmetrical about an axis of rotation. Emphasis is placed on using isoparametric quadratic elements which exhibit excellent modelling capabilities. Efficient numerical integration schemes are also presented in detail. Unlike the Finite Element Method (FEM), the BIE adaptation to axisymmetric problems is not a straightforward modification of the two or three-dimensional formulations. Two approaches can be used; either a purely axisymmetric approach based on assuming a ring of load, or, alternatively, integrating the three-dimensional fundamental solution of a point load around the axis of rotational symmetry. Throughout this ook, both approaches are used and are shown to arrive at identi cal solutions. The book starts with axisymmetric potential problems and extends the formulation to elasticity, thermoelasticity, centrifugal and fracture mechanics problems. The accuracy of the formulation is demonstrated by solving several practical engineering problems and comparing the BIE solution to analytical or other numerical methods such as the FEM. This book provides a foundation for further research into axisymmetric prob lems, such as elastoplasticity, contact, time-dependent and creep prob lems."

Spis treści

1 Introduction and Aims.- 1.1 Introduction.- 1.2 Literature Survey — Axisymmetric Problems.- 1.3 Layout of Notes.- 2 Axisymmetric Potential Problems.- 2.1 Introduction.- 2.2 Analytical Formulation.- 2.2.1 The axisymmetric fundamental solution.- 2.2.2 The boundary integral identity.- 2.2.3 The axisymmetric potential kernels.- 2.2.4 Treatment of the axis of rotational symmetry.- 2.3 Numerical Implementation.- 2.3.1 Isoparametric quadratic elements.- 2.3.2 Numerical integration of the kernels.- 2.3.3 Calculation of the elliptic integrals.- 2.3.4 Solutions at internal points.- 2.3.5 Treatment of non-homogeneous problems.- 2.4 Examples.- 2.4.1 Hollow cylinder.- 2.4.2 Hollow sphere.- 2.4.3 Effect of element curvature.- 2.4.4 Compound sphere.- 2.4.5 Reactor pressure vessel.- 2.4.6 Externally grooved hollow cylinder.- 3 Axisymmetric Elasticity Problems: Formulation.- 3.1 Introduction.- 3.2 Analytical Formulation.- 3.2.1 Basic equations of elasticity.- 3.2.2 Solution of the Navier equations.- 3.2.3 The boundary integral identity.- 3.2.4 Treatment of the axis of rotational symmetry.- 3.2.5 Treatment of non-homogeneous problems.- 3.3 Numerical Implementation.- 3.3.1 Isoparametric quadratic elements.- 3.3.2 Numerical integration of the kernels.- 3.3.3 Surface stresses.- 3.3.4 Solutions at internal points.- 4 Axisymmetric Elasticity Problems: Examples.- 4.1 Introduction.- 4.2 Hollow Cylinder.- 4.3 Hollow Sphere.- 4.4 Thin Sections.- 4.5 Compound Sphere.- 4.6 Spherical Cavity in a Solid Cylinder.- 4.7 Notched Bars.- 4.8 Pressure Vessel with Hemispherical End Closure.- 4.9 Pressure Vessel Clamp.- 4.10 Compression of Rubber Blocks.- 4.11 Externally Grooved Hollow Cylinder.- 4.12 Plain Reducing Socket.- 5 Axisymmetric Thermoelasticity Problems.- 5.1 Introduction.- 5.2 Analytical Formulation.- 5.3 Numerical Implementation.- 5.3.1 Isoparametric quadratic elements.- 5.3.2 Numerical integration of the kernels.- 5.3.3 Solutions at internal points.- 5.4 Examples.- 5.4.1 Hollow cylinder.- 5.4.2 Hollow sphere.- 5.4.3 Compound sphere.- 5.4.4 Comparison with other numerical methods.- 5.4.5 Reactor pressure vessel.- 5.4.6 Externally grooved hollow cylinder.- 6 Axisymmetric Centrifugal Loading Problems.- 6.1 Introduction.- 6.2 Analytical Formulation.- 6.3 Numerical Implementation.- 6.3.1 Isoparametric quadratic elements.- 6.3.2 Numerical integration of the kernels.- 6.4 Examples.- 6.4.1 Rotating disk of uniform thickness.- 6.4.2 Rotating tapered disk.- 6.4.3 Rotating disk of variable thickness.- 7 Axisymmetric Fracture Mechanics Problems.- 7.1 Introduction.- 7.2 Linear Elastic Fracture Mechanics.- 7.3 Numerical Calculation of the Stress Intensity Factor.- 7.3.1 The displacement method.- 7.3.2 The stress method.- 7.3.3 Energy methods.- 7.4 Singularity Elements.- 7.5 Examples.- 7.5.1 Circumferential crack in a round bar.- 7.5.2 Penny-shaped crack in a round bar.- 7.5.3 Internal circumferential crack in a hollow cylinder.- 7.5.4 Flat toroidal crack in a hollow cylinder.- 7.5.5 Pressurised penny-shaped crack in a solid sphere.- 7.5.6 Circumferential cracks in grooved round bars.- 7.5.7 Modelling both faces of the crack.- 8 Conclusions.- References.- Appendix B Numerical Coefficients for the Evaluation of the Elliptical Integrals.- Appendix C Notation for Axisymmetric Vector and Scalar Differentiation.- Appendix D Components of the Traction Kernels.- Appendix E Derivation of the Axisymmetric Displacement Kernels from the Three-Dimensional Fundamental Solution.- Appendix G Differentials of the Displacement and Traction Kernels.- Appendix H The Thermoelastic Kernels.- Appendix I Differentials of the Thermoelastic Kernels.