Finite Element Concepts: A Closed-Form Algebraic Development

Description

This text presents a highly original treatment of the fundamentals of FEM, developed using computer algebra, based on undergraduate-level engineering mathematics and the mechanics of solids. The book is divided into two distinct parts of nine chapters and seven appendices. The first chapter reviews the energy concepts in structural mechanics with bar problems, which is continued in the next chapter for truss analysis using Mathematica programs. The Courant and Clough triangular elements for scalar potentials and linear elasticity are covered in chapters three and four, followed by four-node elements. Chapters five and six describe Taig’s isoparametric interpolants and Iron’s patch test. Rayleigh vector modes, which satisfy point-wise equilibrium, are elaborated on in chapter seven along with successful patch tests in the physical (x,y) Cartesian frame. Chapter eight explains point-wise incompressibility and employs (Moore-Penrose) inversion of rectangular matrices. The final chapter analyzes patch-tests in all directions and introduces five-node elements for linear stresses. Curved boundaries and higher order stresses are addressed in closed algebraic form. Appendices give a short introduction to Mathematica, followed by truss analysis using symbolic codes that could be used in all FEM problems to assemble element matrices and solve for all unknowns. All Mathematica codes for theoretical formulations and graphics are included with extensive numerical examples. Front Matter ....Pages i-xxxvi Finite Element Basics with the Bar Element: Uniaxial Deformations—Interpolants, Stiffness Matrices and Nodal Loads (Gautam Dasgupta)....Pages 1-41 Truss Analysis: A Structural Mechanics Tour of the FEM—Nodal Equilibrium and Compatibility (Gautam Dasgupta)....Pages 43-63 Courant’s Triangular Elements with Linear Interpolants (Gautam Dasgupta)....Pages 65-79 Clough’s Plane Triangular Elements for Linear Elastic Problems: Virtual Work Substituting Variational Principle (Gautam Dasgupta)....Pages 81-102 Taig’s Quadrilateral Elements (Gautam Dasgupta)....Pages 103-120 Irons’ Patch Test (Gautam Dasgupta)....Pages 121-130 Four-Node “Locking-Free” Elements: Capturing Analytical Stresses in Pure Bending: For Two Orthogonal Directions (Gautam Dasgupta)....Pages 131-146 Incompressible Plane Strain Elements: Locking-Free in the x and y Directions (Gautam Dasgupta)....Pages 147-173 Conclusions—Plane Elements: Polynomial Stresses of Degree n (Gautam Dasgupta)....Pages 175-203 Back Matter ....Pages 205-333