In the modified formulation of problems on contact interaction of thinwalled elements in the form of stringer with massive elastic bodies of various geometric shapes, when the mode of elastic displacements of stringer points is specified in advance, one class of problems that admits an exact solution is considered. These problems are of theoretical and practical interest when studying the rigidity of stringer-elastic foundation systems with subsequent determination of the acting on stringers force factors that ensure a predetermined mode of elastic displacements. The massive elastic bodies are considered as a half-plane, strip, and wedge, under conditions of plane or anti-plane deformation. Solving the considered contact problems introduced to solving Fredholm integral equations of the first kind with a symmetric logarithmic kernel.Their exact (closed-form) solutions are constructed using a unified method based on spectral relations involving Chebyshev polynomials. A numerical analysis of the obtained analytical results is carried out.