BEM for thick plates on unilateral Winkler springs
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A new direct boundary element (BEM) technique is established to analyze plates on tensionless elastic foundation. The soil is modeled as Winkler springs. The considered BEM is based on the formulation of shear deformable plate bending theory according to Reissner. The developed technique is based on coupling a developed plate bending software with iterative process to eliminate tensile stresses underneath the considered plate. Tensile zones are redistributed until the final contact zone of plate is reached. Examples are tested and results are compared to analytical and previously published results to verify the proposed technique.
KeywordsBoundary element method Tensionless Soil-structure interaction Raft
Solving plates on tensionless foundation can be divided into two main categories of solution. First solution is to solve the problem using iterative procedure to consider the miscontact between plate and foundation. Second solution is to embed the contact problem in a system of nonlinear equations and solve it using optimization algorithm. Boundary element method (BEM) is widely used to solve plate on foundation problem. Katsikadelis and Armenakas  presented analysis of thin plates on elastic foundation. A BEM formulation based on shear deformable plates according to Reissner  was derived by Vander Weeën . Rashed et al. [4, 5, 6] extended Vander Weeën formulation to model foundation plates.
Several studies are presented to solve plates on tensionless foundations. Weitsman  presented analysis of tensionless beams, or plates, and their supporting Winkler or Reissner subgrade due to concentrated loads. Celep  presented the behavior of elastic plates of rectangular shape on a tensionless Winkler foundation using auxiliary function. Galerkin’s method is used to reduce the problem to a system of algebraic equations. Li and Dempsey  used an iterative procedure to analyze unbonded contact of a square thin plate under centrally symmetric vertical loading on elastic Winkler or EHS foundation. Sapountzakis and Katsikadelis  presented boundary element solution for unilateral contact problems of thin elastic plates resting on linear or nonlinear subgrade by solving a system of nonlinear algebraic equations. Kexin et al.  presented a BE–LCEM solution for thin free edge plates on elastic half space with unilateral contact. Silva et al.  used finite element method to discretize the plate and foundation then used three alternative optimization linear complementary problems to solve plates on tensionless elastic foundations. T-element analysis of plates on unilateral elastic Winkler type foundation using hybrid-Trefftz finite element algorithm was presented by Jirousek et al. . Xiao  presented a BE–LCEM solution to solve unilateral free edges thick plates. Nonlinear bending behavior of Reissner–Mindlin plates with free edges resting on tensionless elastic foundations using admissible functions was presented by Shen and Yu . Silveira et al.  presented a nonlinear analysis of structural elements under unilateral contact constrains studied by a Ritz approach using a mathematical programming technique. Results of finite element analysis of beam elements on unilateral elastic foundation using special zero thickness element designed for foundation modeling are presented by Torbacki . Buczkowski and Torbacki  presented finite element analysis of plate on layered tensionless foundation. Kongtong and Sukawat  used the method of finite Hankel integral transform techniques for solving the mixed boundary value problem of unilaterally supported rectangular plates loaded by uniformly distributed load.
In this paper, an iterative procedure is developed to solve thick plates on tensionless Winkler foundation. The boundary element formulation is used to extract the stiffness matrix of the plate supported on Winkler springs. The main advantage of the proposed technique is that it merges between the advantage of the boundary element method, modeling plate using integral equation, and the simplicity of finite element method, solving nonlinear equations iteratively in matrix form. One of the advantages of the proposed technique is its practicality. In the formulation of Xiao  although solving of thick plates on tensionless foundations, the results were not accurate near corners as it will be shown in the examples of this paper. Also, this formulation is suitable to be extended for solving plates on Winkler elastic–plastic foundations. Numerical examples are presented to verify efficiency and practicality of the proposed technique.
BEM for plate on Winkler foundation
Proposed iterative procedure
Example 1: clamped supported circular plate on tensionless Winkler foundation
Example 2: free edge square plate on tensionless Winkler foundation
Contact region for different thicknesses in example 2
It can be seen from the results that proposed technique obtained the same contact area that is obtained in  and in FEM. Deflection is in a good agreement with results of Xiao  except at corners, at which some inaccurate results due to singularity problems near boundary elements appear in the formulation of Xiao . It can be seen that the presented formulation results are in good agreement with results from FEM. This demonstrates the advantages of the proposed technique over the formulation of Xiao .
In this paper, solving plates on tensionless Winkler soil is presented using an efficient technique. The plate stiffness matrix is extracted using boundary element formulation. An iterative technique is used to eliminate the tensile stresses.
The simplicity of dealing with the nonlinearity nature of the problem.
Avoiding the stresses concentration zones appears in finite element solutions using the advantages of boundary element formulation.
Suitable to be extended to include solving plates on Winkler elastic–plastic foundations.
Accurate results near corners unlike in the formulation of Xiao  as shown in the examples of this paper.
Solving any geometry and any boundary conditions of the plate as shown in example 1.
This project was supported financially by the Science and Technology Development Fund (STDF), Egypt, Grant no. 14910. The authors would like to acknowledge the support of (STDF).
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