A boundary method using equilibrated basis functions for bending analysis of in-plane heterogeneous thick plates
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O R I G I NA L
Nima Noormohammadi · Bijan Boroomand
A boundary method using equilibrated basis functions for bending analysis of in-plane heterogeneous thick plates
Received: 27 February 2020 / Accepted: 11 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A simple boundary method is developed for the solution of isotropic thick plates with in-plane arbitrarily variable material properties or thickness. Equilibrated basis functions which have proved to be effective in a variety of problems, are adopted for the bending problem of thick plates. The bases are created through a weighted residual approach over a fictitious rectangular domain so as to approximately satisfy the partial differential equation of the problem. This omits the necessity of the bases to analytically satisfy the equilibrium, thus simplifying the application of the method. Boundary conditions are applied to the approximate solution through a collocation technique, which considerably reduces its computational expenses. Mindlin’s first order and Levinson’s third-order shear deformation theories are adopted for the formulation. To accommodate more complicated geometries, a simple domain decomposition approach is also developed. Keywords Equilibrated basis functions · Mindlin · Levinson-Heterogeneous · Mesh-free
1 Introduction Solution of bending problems of plates and shells has always been of high interest due to their extensive usage in industry. Among the vast theories and solution methods so far presented, those considering plate as a two-dimensional structure by properly manipulating the through-thickness deformations and stress resultants, have always been popular due to ease of implementation. Although classic plate theory (CPT) [1] presents good accuracy for thin plate structures, its effectiveness is reduced by increasing the plate thickness. In that case, shear deformation theories lead to better results, of which we name first-order shear deformation theory (FSDT) by Reissner [2] or Mindlin [3], the bending-gradient theory as an extension of Mindlin–Reissner theory by Lebée and Sab [4], higher-order shear deformation theories (HSDT) by Levinson [5], Reddy [6], Krishna Murty [7] and Kant [8], or Zig-Zag theories by Di Sciuva [9]. Refined plate theories have also been developed, by which less deformation components should be manipulated during the solution, see [10] for instance. Unified plate theories lie in between the aforementioned theories, usually implementing elasticity solutions for the through thickness components [11,12]. Apart from the vast efforts to extract analytical solutions in various theories (see for instance [13,14]), numerous researches have also developed numerical techniques to solve the plate problems due to their flexibility and ease of implementation. This includes both mesh-based and mesh-less methods such as the Finite Element Method (FEM) [15–17], Method of Fundamental Solutions (MFS) [18], Boundary Element Method (BEM) [19], Element Free Galerkin (EFG) method [20], Finite
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