Keywords: BEM, plate, plasticity, thermal analysis, initial fields, integral equations.

The Boundary Element Method (BEM) has already been recognised as an alternative tool to the domain techniques such as finite difference and finite element methods for the solution of a variety of engineering problems. Apparently, the Bezine [1] and Stern [2] presented pioneering works dealing with a general form of direct integral equations for Kirchhoff´s bending plate problems under elastostatic behaviour. Their integral representations are associated with transverse displacement and normal slope, whose structure is composed by a displacement vector (transverse displacement and normal slope) and by a force vector (Kirchhoff´s shear force, bending moment). In addition, those integrals have additional terms associated with reaction and displacement acting at the corner´s plate.

Many researchers derived formulations for the analyses of bending plate problems under initial fields based on extension of elastic analysis governed by Stern-Bezine´s integral representation. Moshaiov and Vorus [3] studied bending problems under initial fields associated with plastic bending moments. The classical plasticity models are used to represent the yielding surface and plate discretization was made by using constant interpolating functions. Numerical examples were presented for clamped circular and square plates.

Many researches proposed changes in Bezine-Stern model at discretized integral equation level for the elastic analysis of bending plate. Hartmann and Zotemantel [4] discussed a discretization/interpolation process in which transverse displacement is approximated by cubic function written in terms of values of transverse displacement and its tangential gradients. Then, the authors employed special strategies in a finite difference sense to prescribe the natural boundary conditions. Du et al. [5], Song and Mukherjee [6] also approximated transverse displacement by a cubic function, however, displacement and its tangential gradient nodal values were maintained in the discretized equations. Hence, a third boundary integral equation related to tangential slope is required to be written to represent a further quantity in the displacement vector and the remaining boundary quantities are directly written from their corresponding nodal values. In the three-parameter formulations abovementioned, the tangential slopes don't appear at integral equation level and those slopes will be only inserted into the problem when the steps of the discretization process is stablished.

Song and Mukherjee [7] presented a formulation for elastoplastic analysis, obtaining the plastic integral equations by adding the plastic terms to the elastic three-parameter integral representation discussed by Du et al [5]. Numerical examples were presented for different cases of geometry and constraints.

Oliveira Neto and Paiva [8] presented an alternative fashion to Stern- Bezine´s integral representation for elastic analysis of bending plate, applying special mathematical treatments to establish a three-parameter integral equations, resulting in an additional degree of freedom associated with tangential slope in the displacement vector and two quantities in force vector.

The purpose of present work is to extend the three-parameter boundary element elastic model discussed in Oliveira Neto and Paiva to deal with bending problems subjected to initial fields such as thermal and plastic loading. The discretization is done by straight boundary elements and triangular cells for the domain quantities. For modelling the plastic flux, the classical plasticity models are incorporated into three-parameter integral equations. In addition, numerical examples are presented for cases of thermal and mechanical loading.

1

Bezine, G.P. "Boundary integral formulation for plate flexure with arbitrary boundary conditions". Mechanics Research Communications, 5 (4), pp.197-206, 1978. doi:10.1016/0093-6413(78)90033-2

2

Stern, M. "A general boundary integral formulations for the numerical solution of plate bending problems", Int. J. Sol. Struct., Vol. 15, pp. 769-782, 1979. doi:10.1016/0020-7683(79)90003-9

3

Moshaiov, A.; Vorus, W.S. "Elasto-plastic plate bending analysis by the boundary element method with initial plastic moments" Int. J. Solids Structures, Vol.22, n.11, pp.1213-1229, 1986. doi:10.1016/0020-7683(86)90077-6

4

Hartmann, F.; Zotemantel, R. "The direct boundary element method in plate bending" Int. J. Num. Meth. Engng., Vol.23, pp.2049-2069, 1986. doi:10.1002/nme.1620231106

5

Du, Q.; Yao, Z.; Song G. "Solution of some plate bending problems using the boundary element method" Appl. Math. Modelling, Vol.8, Feb., pp.15-22, 1984. doi:10.1016/0307-904X(84)90171-9

6

Song, Guo-Shu; Mukherjee, S. "Boundary element method analysis of bending of elastic plates of arbitrary shape with general boundary conditions". Engng. Analysis, Vol. 3, pp. 36-44, 1986. doi:10.1016/0264-682X(86)90189-9

7

Song, Guo-Shu; Mukherjee, S. "Boundary element method analysis of bending of inelastic plates with general boundary conditions". Computational. Engineering, Vol. 5, pp. 104-112, 1989.

8

Oliveira Neto L, Paiva J.B, "A special BEM for elastostatic analysis of building floor slabs on columns". Computers & Structures 81 (6): pp.359-372, 2003. doi:10.1016/S0045-7949(02)00449-2

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