The analog equation method for large deflection analysis of membranes. A boundary-only solution

Abstract

In this paper the analog equation method (AEM) is applied to nonlinear analysis of elastic membranes with arbitrary shape. In this case the transverse de¯ections inuence the inplane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear. The present formulation, being in terms of the three displacements components, permits the application of geometrical inplane boundary conditions. The membrane is prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with ®ctitious sources under the same boundary conditions. Subsequently, the ®ctitious sources are established using a procedure based on BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method and demonstrate its ef®ciency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of the membranes. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary.

Large deflection analysis of elastic space membranes

“…The discrepancy of the results between AEM and FEM, EFG which appear closer to each other can be explained from the fact that the latter methods use the strain model and the stress-strain relations of the plane elasticity neglecting the curvature of the membrane. This discrepancy was also reported in the non-linear analysis of flat membranes [11] where the AEM results were found more accurate than the FEM ones as compared with existing analytical solutions. The membrane was also analysed under prestress by imposed boundary displacements (ũ =ṽ = 0.01 m) as shown in Figure 8 and subjected to a uniform load.…”

“…The boundary value problem described by Equations (29) and (27) is solved using the analogue equation method (AEM) as developed for the non-linear analysis of flat membranes [11,12]. This method is applied to the problem at hand as follows.…”

“…The unknown sources are established using a procedure based on BEM as it was developed for the non-linear analysis of flat membranes [11,12]. The use of Cartesian co-ordinates simplifies the application of the AEM as it yields simple analogue equations that can be effectively treated on the projected domain.…”

Dynamic analysis of nonlinear membranes by the analog equation method: a boundary-only solution

“…In the latter case existing solutions were available for comparison (Katsikadelis et al, 2001). The employed number of the interior points was not large.…”

The BEM for nonhomogeneous bodies

“…The method presented in this investigation is valid without any limitations. It is based on the concept of the analog equation [4,[6][7][8], which converts the original problem to an equivalent one described by an equation having a simple known fundamental solution, for example, the general second-order equation is replaced by the Poisson equation. Actually, the proposed method introduces an additional unknown domain function, which represents the source of the substitute problem.…”