Keywords: folded plates, thin-walled structures, stiffened structures, boundary element method, plate, flat shell.

The structures constituted by folded plates are largely used as an alternative structural system, mainly, in large structures, for instance, bridges, silos, shear walls, etc. When the bending and membrane states are mobilized in the plate, this one can be called flat shell. Several works have been written about the flat shell problem in the development and improvement of the theories and in the techniques to achieve the solutions of its differential equations and one of these procedures is the numerical methods. One of the most used procedures of numerical techniques applied to solve engineering problems is the so-called Finite Element Method (FEM). For the analysis of folded plates structures the FEM is a well-known formulation Cheung [1], Clough [2], Zienkiewicz [3], etc. Alternatively to the FEM, another technique which have been receiving special attention from the research is the so-called Boundary Element Method (BEM). One of earliest works using this approach applied to the folded plates was written by Palermo Jr. [4] where particular structures in which its longitudinal axe is parallel to one of the axes of the of the global 3D coordinate system were analysed.

In others formulations are described the analysis of some kind of particular folded-plated structures in which the problem is modeled using the boundary element method coupled with other numerical techniques. Tanaka & Bercin [5] proposed a formulation for the analysis of plates stiffened by a generic open cross-section beam. In this approach the problem is divided in regions composed by the plates and the beam. The bending plates is represented by the classical integral equations and for the beam it is applied the thin-walled theory. The influence of the offset of the beams from the plate medium plane is also considered. Galuta & Cheung [6] model girder box bridges using a FEM/BEM combination. In this work the classical plate integral equations are written to the nodal points located over a region represented by the superior flange boundary of the cross-section bridge. The remaining regions of cross-section are modeled using the FEM. The DOFs belonging to the MEF, located over the interface of the common regions, are transformed to the BEM ones.

Differently of all approaches described above, the present formulation is constituted by six integral equations to represent the problem. The bending state is modeled using a integral representation described by Oliveira Neto & Paiva [7] which involves three DOFs to the displacement vector, i.e, transverse displacement, normal and tangential rotations. Besides, the tractions vector is composed of bending moments and the Kirchoff's shear forces. The membrane state is represented by the two classical 2D elastostatics integral equations and by a further equation which includes an outward rotation in the plane of the membrane. Hence, the integral representation of the problem is composed by six integral equations. In general way, the advantage of the present formulation is to model the problem using only integral equations. Besides, the application of the earlier approaches are restricted to the set of particular structures having as common characteristics one longitudinal axis, i.e., "gyrator axis", in which plane that contains it is parallel to the contributing flat shell planes of the folded plate. Beyond this problems, the present formulation can be applied to model structures constituted by generic geometrical configurations of flat shell units in the 3D space.

1

Cheung, Y.K. "Folded plate structures by the finite strip method". Am. Soc. Civ. Eng., vol. 96, 2963-79, 1969.

2

Clough, R.W, Wilson, E.L. "Dynamic finite element analysis of arbitrary thin shells". Comp. & Structures, vol.1, 1971. doi:10.1016/0045-7949(71)90004-6

3

Zienkiewicz, O.C. "The finite element method". 4th ed. Mcgraw Hill, 1991.

4

Palermo Jr., L., Rachid, M., Venturini, W.S. "Analysis of thin walled structures using the boundary element method". Eng. Anal. Bound. Elem., 9, pp.359-363, 1992. doi:10.1016/0955-7997(92)90021-X

5

Tanaka, M., Bercin, A.N. "Static bending analysis of stiffned plates using the boundary element method". Eng. Anal. Bound. Elem., 21, pp.147-154, 1998. doi:10.1016/S0955-7997(98)00002-2

6

Galuta, E.M., Cheung, M.S. "Combined boundary element and finite element analysis of composite box girder bridges". Computers & Structures, v. 57, n.3, pp.427-437, 1995. doi:10.1016/0045-7949(94)00632-D

7

Paiva, J.B., Oliveira Neto, L. "An alternative boundary element formulation for plate bending analysis". In: BETECH 95 , Liege, Belgium, 1995.

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