Application of Groebner Bases to Geometrically Nonlinear Analysis of Rectangular Composite Plates Resting on a Pasternak Foundatio

With composite materials becoming increasingly desirable across a wide range of industries due to their high strength to weight ratios and ability to be designed to accommodate specific applications, the need to understand their complex behavior is paramount. These plates often exhibit highly nonlinear and coupled behavior, especially for cases of large deflections, necessitating nonlinear analysis to properly predict the response of the structure. Numerical solutions such as the finite element and finite difference methods are commonly employed to model nonlinear analysis, however these methods have several drawbacks. Numerical solutions estimate solutions at discrete points, resulting in discontinuities between elements depending on the interpolation functions assumed. Moreover, numerical methods cannot provide relationships between the various parameters used in analysis without extensive effort. Purely symbolic analytical solutions are preferred, but expressions for nonlinear analysis only exist for limited situations. It is possible to obtain purely symbolic analytical expressions with the aid of a mathematical method known as Groebner bases.;The primary objective of this thesis is to obtain purely symbolic analytical expressions for rectangular composite plates subjected to a uniformly distributed load that are resting on a Pasternak model soil foundation. A limited amount of literature is available for the Pasternak model, especially for problems involving geometrically nonlinear analysis. This thesis also explores the application of Groebner bases as an aid to solve structural mechanics problems with the hope that it will be applied in other fields of engineering. In the present study, the governing integro-(partial) differential equation was derived using the principle of minimum potential energy. Approximate solutions were assumed based on the Ritz method of approximation for both fully clamped and simply supported boundary conditions. The variational principle was then applied to produce a system of coupled nonlinear multivariate polynomial equations with unknown constant coefficients. Maple 2016 was used to compute the reduced Groebner basis for the system of polynomial equations and then used to symbolically solve for the unknown coefficients. Expressions for deflection were generated for symmetric cross-ply, angle-ply, and quasi-isotropic composite plates. These results were then compared to numerical solutions obtained from the commercial finite element analysis software ANSYS and showed very good agreement. As a result, the deflection expressions can be used to find continuous functions for internal stresses, shear forces, and moments in the plate as well. The results of the current study have shown that the application of Groebner bases provides a valuable alternative to numerical methods, thereby providing incentive for its implementation in other fields of engineering application.