The methods of the solution of the problems of the planar elasticity theory for the multilayer foundations of the complex structure have been developed. Some internal layers of such foundation can be loaded by the body forces or can have the holes or the cracks. The developed methods are based on the compliance functions technique, the fictitious load method and the discontinuous displacements method. The solution of the initial problems for the multilayer foundation with inhomogeneity in one of the internal layers has been reduced to the solution of two problems: problem, connected with the determination of the stress-strain state of plane, which has the same inhomogeneity as the corresponding layer of the foundation, and problem, connected with the determination of the stress-strain state of the multilayer foundation with the continuous layers, the boundary conditions of which are corrected, taking into the consideration the solution of the first auxiliary problem. The superposition of solutions of both problems makes the boundary conditions of the initial problem completed. The solution of the problem for the multilayer foundation with any finite number of layers has been obtained in an analytical form with the help of one-dimensional Fourier integral transformation. The research of the integral equations, obtained for the problem, connected with the hole and crack in one of the layers has been performed. One has separated the singularity of these equations. The proposed technique of the numerical solution of these equations drives to the system of linear algebraic equations, with unknown values of the fictitious loads (the problem, connected with the hole), or the values of the discontinuous displacements (the problem, connected with the crack) in the discrete system of the points of the contour of the hole or the crack. The coefficients in this system are Fourier integrals. To solve the obtained system, it is necessary to satisfy the boundary conditions at the contour of the hole or the crack. The obtained values are substituted into the analytical expressions for the Fourier transforms and then these expressions are subjected to the inverse Fourier transformation. The new problems for the multilayer foundations of the complex structure have been solved. The effect of the complications in the layers on the value and the distribution of the stresses in multilayer foundation has been researched. The new mechanical relationships have been obtained. Keywords: elastic multilayer foundation, compliance functions, Fourier integral transformation, compliance function technique, hole in the foundation, Kolosov-Mushelishvili complex potential, integral equations, quadrature formulae, fictitious load method, crack, discontinuous displacements method.
