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In this work, a boundary element formulation to analyse slabs reinforced by beams, combined or not to define a grid sub-system, is proposed. Kirchhoff 's hypothesis is assumed for the plate elements. The beams elements are not required to be displayed over the plate surface, therefore eccentricity effects are taken into account. The formulation is derived by assuming a zoned body where beam elements are introduced by degenerating plate sub-regions. After finding properly a single reciprocity for the whole body, the required integral representations are derived. The integral representations derived for this complex structural element take into account the bending and stretching effects of booth structural elements working together. The standard equilibrium and compatibility conditions along interface are naturally imposed. Moreover, the amount of degrees of freedom required along the interfaces is substantially reduced, leading therefore to small and more accurate algebraic system of linear equations. Several examples are then shown to illustrate the accuracy of the formulation, comparing the obtained results with analytical and other numerical solutions.

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In this work, the plate bending formulation of the boundary element method (BEM) based on the Kirchhoff's hypothesis, is extended to the analysis of stiffened elements usually present in building floor structures. Particular integral representations are derived to take directly into account the interactions between the beams forming grid and surface elements. Equilibrium and compatibility conditions are automatically imposed by the integral equations, which treat this composite structure as a single body. Two possible procedures are shown for dealing with plate domain stiffened by beams. In the first, the beam element is considered as a stiffer region requiring therefore the discretization of two internal lines with two unknowns per node. In the second scheme, the number of degrees of freedom along the interface is reduced by two by assuming that the cross-section motion is defined by three independent components only. IntroductionThe boundary element method (BEM) is already a wellestablished numerical technique to deal with an enormous number of complex engineering problems. Among them, analysis of plate bending problems has proved to offer a particularly adequate field of applications for that technique. The BEM is suitable for evaluating internal force concentrations due to loads distributed over small regions, which very often occur in plate bending analysis. Moreover, BEM can deal with deflections, slopes, moments, and shear forces, approximating them by using the same order polynomials. Thus, shear forces are much better evaluated when compared with other numerical methods; they depend only on the adopted boundary value approximation.The first works discussing the use of direct boundary element formulation, in conjunction with Kirchhoff's theory, are of Bezine [1], Stern [2] and Tottenhan [3]. It is also important to mention some previous studies dealing with plate bending problems in the context of indirect methods [4,5]. These, as well as several other more recent publications, have pointed out the capability of the method for modelling plates in bending, mentioning further accuracy and reliability.In order to use BEM to analyse more complex plates, e.g., stiffened plates of building floor structures, one has to extend the BEM formulations to take into consideration arbitrarily displayed beams, general boundary and internal constraints and several kinds of transversal loads acting over the plate surface or part of it. Along these lines, Song [6], Hartmann and Zotemantel [7] have presented interesting approaches, discussing in detail displacement restrictions at internal points and the use of hermitian interpolations. More recently, Oliveira Neto and Paiva [8] have shown a BEM/FEM for analysing building floor structures.While BEM is strongly recommended for plate bending analysis, in which internal force and displacement fields are always accurately modelled, the natural choice to solve the building floor structures is the BEM/FEM combinations. Boundary elements are recommended to deal with plate elements,...

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ABSTRACT:The objective of this work is to present a simple mathematical model to describe the main features of phase inversion by immersion precipitation. The model was developed for planar geometry and is used to simulate membrane synthesis. Two systems commonly used for membrane preparation were used for simulations: cellulose acetate/acetone/water and polyetherimide/N-methylpyrrolidone/water. The influence of nonsolvent addition to initial polymer casting solutions, solvent addition to the coagulation bath, and geometric variables, such as polymer film thickness, on the final precipitation conditions were studied through simulation and compared to available experimental data. The results are in good agreement with published results. [1][2][3] It is shown that polymer film composition profiles at the moment of precipitation may give important information about the structure and substructure of formed membrane. It is also shown, for both polymeric systems investigated in this work, that the dynamics of the mass transfer process seems to be much more important than the influence of the concentration on the diffusion and thermodynamic partition coefficients, as fair agreement with available experimental data was obtained even when these coefficients were assumed to be constant.