A mixed finite element method for beam and frame problems |
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Authors: | R L Taylor F C Filippou A Saritas F Auricchio |
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Affiliation: | (1) Department of Civil and Environmental Engineering, University of California at Berkeley 727 Davis Hall, Berkeley, 94720-1710 California, USA e-mail: rlt@ce.berkeley.edu, US;(2) Dip. Meccanica Strutturale Universita di Pavia 27100 Pavia, Italy, IT |
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Abstract: | In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may
be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross
section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits
consideration in a direct manner of elastic and inelastic behavior with or without shear deformation.
A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu
principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending
moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending
moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement
fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally
consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a
shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements.
The advantages of the approach are illustrated with a few numerical examples.
Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many
years. |
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Keywords: | Inelastic beam Finite elements Mixed method Shear deformation |
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