Keywords: boundary element method, plate-beam interaction, building slabs.

The boundary element method (BEM) was first applied to building slab analysis by Bezine [1], who considered the problem of plates with internal boundary conditions that could be used to simulate rigid columns connected to the plate. Since then, several authors have used the BEM in building slab analysis [2,3]. The major problem in coupling the plate with the beam is that the finite elements used to represent beams and columns need three variables (w, partial w /partial x₁ and partial w /partial x₂) at each node in the plate domain, whereas for nodes located on the boundary of the plate, the BEM employs only two, w and partial w /partial n.

In this article, a new boundary element formulation for the analysis of the plate-beam interaction is presented, in which the plate is modeled by a BEM formulation with three variables at each boundary node [4] and the beam is replaced by its actions on the plate, a distributed load and end forces. Each beam element has three nodes, each with two nodal values, w and partial w /partial x₁, and the transverse displacement of the beam is approximated by a fifth degree polynomial that represents the differential equation analytical solution for the beam under transverse loading with linear variation. After the beam differential equation has been solved, the plate-beam interactions can be written in terms of the beam nodal variables, w and partial w /partial x₁. With this transformation, a final system of equations can be written, involving only the nodal displacements of the beams and plate boundary nodes and the tractions at the boundary of the plate. By imposing the boundary conditions and solving the system of equations, the displacements and tractions on the beams and plate can readily be calculated.

1

Bezine G. "A boundary integral equation method for plate flexure with conditions inside the domain", Int. J. for Numerical Methods in Engineering, 17:1647-1657, 1981. doi:10.1002/nme.1620171106

2

Paiva J.B., Venturini W.S., "Boundary element algorithm for building floor slab analysis", Boundary Element Technology Conference, Adelaide (Austr.), Nov., 1985.

3

Hartley G.A., Abdel-Akher A., Chen P., "Boundary element analysis of thin plates internally bounded by rigid patches", Int. J. for Numerical Methods in Engineering, 35: 1771-1785, 1992. doi:10.1002/nme.1620350904

4

Oliveira Neto L., Paiva J.B. "Cubic approximation for the transverse displacement in BEM for elastic plates analysis", Engineering Analysis with Boundary Elements, v.28, p.869-880, 2004. doi:10.1016/j.enganabound.2003.12.003

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