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An Introduction to Nonlinear Finite Element Analysis$
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J. N. Reddy

Print publication date: 2004

Print ISBN-13: 9780198525295

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198525295.001.0001

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Nonlinear Bending of Straight Beams

Nonlinear Bending of Straight Beams

Chapter:
(p.87) 4 Nonlinear Bending of Straight Beams
Source:
An Introduction to Nonlinear Finite Element Analysis
Author(s):

J. N. REDDY

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198525295.003.0004

This chapter considers the application of the finite element method to the analysis of a slightly more complicated one-dimensional nonlinear problem: the bending of straight beams. When the applied loads on the beam are large, the linear load deflection relationship ceases to be valid because the beam develops internal forces that resist deformation, and the magnitude of internal forces increases with the loading as well as the deformation. This nonlinear load deflection response of straight beams is discussed. A nonlinear formulation of straight beams based on assumptions of large transverse displacements, small strains, and small to moderate rotations is presented. Two different theories to model the kinematic behavior of beams are examined: the Euler-Bernoulli beam theory that neglects the transverse shear strain, and the Timoshenko beam theory that accounts for the transverse shear strain in the simplest way. This chapter also discusses weak forms, iterative solutions of nonlinear equations, membrane locking, tangent stiffness matrix, and computer implementation of the problem.

Keywords:   finite element method, computer implementation, nonlinear bending, straight beams, load deflection, strains, kinematic behavior, Euler-Bernoulli beam theory, Timoshenko beam theory, membrane locking

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