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The boundary element method (BEM) is applied to the soap bubble problem, that is to the problem of determining the surface that a soap bubble constrained by bounding contours assumes under the action of molecular forces. This is also the shape of a uniformly stretched membrane bounded by one or more nonintersecting curves. As the slopes of the membrane surface are ®nite, their square can not be neglected and the resulting governing equation is non-linear. The problem is solved using the analogue equation method (AEM). According to this method the non-linear membrane is substituted by a linear one subjected to a ®ctitious transverse load. The ®ctitious load is established using the BEM. Numerical examples are presented which illustrate the method and demonstrate its accuracy. This application of the BEM to non-linear problems shows that BEM is a versatile computational method for allpurpose use in engineering analysis. The solution of the problem at hand is very important in engineering, since the soap bubble surface can be used as the best initial form for membrane roofs.

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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.