Comparative analysis of numerical methods for calculating flexible foundation plates on an elastic foundation taking into account geometric nonlinearity

Abstract

The article presents the results of a comparative analysis of the efficiency and accuracy of numerical methods in solving problems of structural mechanics, in particular, in calculating flexible foundation plates interacting with an elastic foundation. The relevance of the study is justified by modern trends in the construction of large-span and high-rise buildings, where the use of thin foundation plates leads to significant deflections commensurate with the thickness of the structure itself. In such cases, the classical linear calculation theory yields a significant error, which requires the use of a geometrically nonlinear apparatus based on the system of Theodore von Karman differential equations. The classical one-parameter Winkler model is used as a mathematical model of the interaction of the plate with the soil. The main focus of the work is on comparing two grid methods: the generalized finite difference method (FDM) and the successive approximation method (SAM), the specificity of which lies in the use of piecewise polynomial functions (splines) to construct differentiation matrices. To implement the algorithms, the authors developed computational programs in the MATLAB environment, which allow performing the iterative process of finding a nonlinear solution. During the study, a square foundation plate, rigidly clamped along the contour, under the action of a uniformly distributed load was calculated. A series of calculations were carried out on grids of varying density (from a step of h=1/4 to h=1/64). At the first stage, the obtained results were compared with classical analytical solutions of the nonlinear plate theory without considering the foundation. It has been proven that the successive approximation method (SAM) demonstrates a higher convergence rate and better accuracy on coarse computational grids compared to the classical FDM. In particular, the discrepancy of the maximum deflections for the SAM is less than 3% even at a grid step of h=1/16, while to achieve similar accuracy in the FDM, further grid refinement is required. At the second stage, the behavior of the same plate resting on a continuous elastic foundation was numerically investigated. It was found that taking into account the soil subgrade reaction using the Winkler model leads to a decrease in the maximum deflections of the structure by an average of 15% compared to the plate without a foundation. At the same time, the SAM algorithm maintains its computational stability, ensuring the stabilization of displacements and bending moments already on the h=1/16 grid without the accumulation of errors typical for the FDM in the zones of edge effects. The research results confirm the feasibility of using SAM algorithms in computer-aided design (CAD) software for building structures to increase the reliability of assessing the stress-strain state of foundations in the presence of geometric nonlinearity.