The article is devoted to the development of mathematical models and new methods for solving contact problems of multilayer elements of structures with an inhomogeneous base, considering their mechanical, structural features, and to the assessment of their internal force factors. A mathematical model was developed and an analytical method was proposed for assessing the internal force factors in multilayer strip slabs on an inhomogeneous base under various loads. The solution of the problems under consideration is based on a series expansion of the reactive pressure of an inhomogeneous base in terms of orthogonal ultra-spherical Gegenbauer polynomials; the solution is reduced to the study of infinite systems of algebraic equations. Their regularity was proved and the corresponding estimates were obtained. The required number of terms of the polynomial in the expansion was established. The analysis of the results obtained made it possible to evaluate the influence of the rigidity characteristics of the filler and the inhomogeneity of the base on the distribution of internal force factors in the slabs.