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# Finite Element Method Applications in Solids, Structures, and Heat Transfer

By

## Michael R. Gosz

ISBN 9780849334078
Published November 10, 2005 by CRC Press
400 Pages 163 B/W Illustrations

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## Book Description

The finite element method (FEM) is the dominant tool for numerical analysis in engineering, yet many engineers apply it without fully understanding all the principles. Learning the method can be challenging, but Mike Gosz has condensed the basic mathematics, concepts, and applications into a simple and easy-to-understand reference.

Finite Element Method: Applications in Solids, Structures, and Heat Transfer navigates through linear, linear dynamic, and nonlinear finite elements with an emphasis on building confidence and familiarity with the method, not just the procedures. This book demystifies the assumptions made, the boundary conditions chosen, and whether or not proper failure criteria are used. It reviews the basic math underlying FEM, including matrix algebra, the Taylor series expansion and divergence theorem, vectors, tensors, and mechanics of continuous media.

The author discusses applications to problems in solid mechanics, the steady-state heat equation, continuum and structural finite elements, linear transient analysis, small-strain plasticity, and geometrically nonlinear problems. He illustrates the material with 10 case studies, which define the problem, consider appropriate solution strategies, and warn against common pitfalls. Additionally, 35 interactive virtual reality modeling language files are available for download from the CRC Web site.

For anyone first studying FEM or for those who simply wish to deepen their understanding, Finite Element Method: Applications in Solids, Structures, and Heat Transfer is the perfect resource.

INTRODUCTION
MATHEMATICAL PRELIMINARIES
Matrix Algebra
Vectors
Second-Order Tensors
Calculus
Newton's Method
Kinematics of Motion
Problems
ONE-DIMENSIONAL PROBLEMS
The Weak Form
Finite Element Approximations
Plugging in the Trial and Test Functions
Algorithm for Matrix Assembly
One-Dimensional Elasticity
Problems
LINEARIZED THEORY OF ELASTICITY
Cauchy's Law
Principal Stresses
Equilibrium Equation
Small-Strain Tensor
Hooke's Law
Axisymmetric Problems
Weak Form of the Equilibrium Equation
Problems
Derivation of the Steady-State Heat Equation
Fourier's Law
Boundary Conditions
Weak Form of the Steady-State Heat Equation
Problems
CONTINUUM FINITE ELEMENTS
Three-Node Triangle
Four-Node Tetrahedron
Eight-Node Brick
Element Matrices and Vectors
Bending of a Cantilever Beam
Analysis of a Plate with Hole
Thermal Stress Analysis of a Composite Cylinder
Problems
STRUCTURAL FINITE ELEMENTS
Space Truss
Euler-Bernoulli Beams
Mindlin-Reissner Plate Theory
Deflection of a Clamped Plate
Problems
LINEAR TRANSIENT ANALYSIS
Derivation of the Equation of Motion
Semi-Discrete Equations of Motion
Central Difference Method
Trapezoidal Rule
Problems
SMALL-STRAIN PLASTICITY
Basic Concepts
Yield Condition
Flow and Hardening Rules
Derivation of the Elastoplastic Tangent
Finite Element Implementation
One-Dimensional Elastoplastic Deformation of a Bar
Elastoplastic Analysis of a Thick-Walled Cylinder
Problems
TREATMENT OF GEOMETRIC NONLINEARITIES
Large-Deformation Kinematics
Weak Form in the Original Configuration
Linearization of the Weak Form
Snap-Through Buckling of a Truss Structure
Uniaxial Tensile Test of a Rubber Dog-Bone Specimen
Problems
BIBLIOGRAPHY
INDEX

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## Reviews

". . . extremely clearly written, so much so that it could easily be used for self-study . . . achieves an excellent balance between the presentation of fundamental building blocks and the further development and implementation of the theory in the context of specific applications . . . is indeed a very useful addition to the literature on the subject, at the introductory level."

– Batmanathan D. Reddy, in Zentralblatt Math, 2006, Vol. 1095, No. 21