Analytical and numerical study of one-dimensional and two-dimensional stress distribution within an elastic semi-infinite material under the action of an arbitrary rectangular uniform loading

Purpose. The study of stress distribution within an elastic semi-infinite material under the action of an external loading is of great importance in the theory of elasticity. In most cases, the lack of knowledge about the stress distribution within a material can result in incomplete and inappropriate engineering designs, leading to unsatisfactory consequences. The latter include cracks and fractures, created inside the concrete segmental lining in TBM tunneling, as well as indentations that occur in floors due to the lack of pillar design not only in underground mining, but even in civil projects. This study focuses on the one-dimensional and two-dimensional internal stress distribution induced by arbitrary rectangular–square loading, in other words, that applied to an elastic semi-infinite material. Methods. Firstly, this paper uses an analytical method and, subsequently, a numerical method. In the analytical study, the point load equations of Boussinesq and Westergaard are used. Extending these equations to the rectangular loading area, four new equations are introduced. Using the Abaqus finite element software, the numerical study is performed in 3D space. Findings. The results show that the answers from the introduced equations are in high consistency with numerical ones. However, the result of the extended Boussinesq point load equation is closer to the answer obtained by the numerical method. Originality. Four new equations, presented in this paper, describe one-dimensional and two-dimensional stress distribution. Practical implications. The presented equations can provide a simple and convenient way to solve rectangular load problems in many cases such as foundation, civil and mining projects. This study uses initial information on specific segments in the Tabriz Metro line-2 Project.