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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 88
Edited by: B.H.V. Topping and M. Papadrakakis
Paper 146

Elastoplastic Boundary Element Method Formulation for Plates with Geometrical Non-Linearity

L. Waidemam1, W.S. Venturini2 and H.B. Coda2

1Federal University of Technology - Paraná, Campo Mourão, Brazil
2São Carlos School of Engineering, University of São Paulo, São Carlos, Brazil

Full Bibliographic Reference for this paper
L. Waidemam, W.S. Venturini, H.B. Coda, "Elastoplastic Boundary Element Method Formulation for Plates with Geometrical Non-Linearity", in B.H.V. Topping, M. Papadrakakis, (Editors), "Proceedings of the Ninth International Conference on Computational Structures Technology", Civil-Comp Press, Stirlingshire, UK, Paper 146, 2008. doi:10.4203/ccp.88.146
Keywords: boundary element method, material non-linearity, geometrical non-linearity, bending plates.

The direct boundary element method (BEM) applied to analyse plate-bending problems has been successfully used during the last twenty-five years. So far many other works using either direct or indirect formulation have been published demonstrating that the application of the method to plate bending is a reliable numerical tool for engineering problems. Regarding the application of the BEM to geometrically and physically non-linear problems the methods has also demonstrate to be efficient and accurate.

In this paper the BEM formulation for elastoplastic analysis of plates with geometrical non-linearity is reviewed. The adopted boundary integral equations are based on the Kirchhoff's plate theory. The non-linear geometric terms have been incorporated by considering the von Kármán's hypothesis, while the physical non-linear terms are taken into account by applying a correcting term written in terms of stresses. The von Mises criterion with linear isotropic hardening is considered to model the plastic evolution.

Isoparametric linear elements are used to approximate the boundary unknown values and triangular internal cells with linear shape function are used to evaluate the domain value influences. The integral over elements and cells have been performed accurately by adopting a sub-element scheme. Boundary elements and internal cells can be either continuous or discontinuous. In particular for internal cells the discontinuity is used to avoid computing internal forces along the boundary. The internal force fields are always approximated by using only nodes defined inside the domain. The discontinuities of the boundary values are defined when convenient, particularly at corner or at nodes where boundary value discontinuities are expected.

As some products of unknowns are present in the integral terms, we have decided to replace them by auxiliary variables which are approximate over the cells. This scheme reduces the size of the matrices without penalizing the accuracy of the solution. More over the number of integrations over cells are also significantly reduced.

An implicit time marching process is adopted to perform the incremental loading process, while the non-linear algebraic system of equations is solved using a Newton-Raphson procedure for which the consistent tangent operator has been particularly derived for this problem. Numerical examples are presented to demonstrate the validity and the accuracy of the proposed formulation.

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