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Frame elements with mixed formulation for singular section response
Author(s) -
Lee C.L.,
Filippou F. C.
Publication year - 2008
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2531
Subject(s) - stiffness matrix , section (typography) , flexibility method , eigendecomposition of a matrix , finite element method , moore–penrose pseudoinverse , displacement field , singular value decomposition , mathematics , structural engineering , matrix (chemical analysis) , mathematical analysis , eigenvalues and eigenvectors , computer science , geometry , algorithm , engineering , physics , materials science , inverse , quantum mechanics , operating system , composite material
Mixed formulations of frame elements offer significant advantages over more traditional displacement formulations, particularly under large cyclic inelastic deformations including the effects of shear. This paper complements the recent proposal of a consistent variational basis for the mixed formulation of frame elements by supplying the explicit definition of the stress field over the cross‐section. The paper also addresses the numerical stability of the element state determination algorithm in the presence of an ill‐conditioned or even singular section stiffness matrix. The proposed algorithm is based on the eigendecomposition of the section stiffness matrix and uses the Sherman–Morrison–Woodbury formula and the Moore–Penrose pseudoinverse to avoid the inversion of ill‐conditioned matrices in the element state determination. In the extreme case of uniform tension or uniform flexure the section flexibility matrix is split into an elastic and a plastic component before eigendecomposition. With the proposed method the inelastic response of the element under multiple perfectly plastic hinges can be successfully traced. Numerical examples demonstrate the capabilities of the approach. Copyright © 2008 John Wiley & Sons, Ltd.