On the precise solution of integral equation of one contact problem of the mathematical theory of elasticity

The paper is devoted to the consideration of the problem on contact interaction between a finite length stringer and an elastic layer with a rigidly fastened lower boundary under anti-plain deformation. For the stringer, Melan's model for anti-plain deformation is accepted, which can be described by simplest second order differential equation. Here, as opposed to the traditional formulation of problems on load transmission from stringers to deformable massive solids, the authors adopted a different formulation of the problem, where elastic displacements of the stringer points are given in advance, whereas force factors acting on the stringer and stipulating the given pattern of elastic displacements are determined afterwards. Under such formulation, solving the problem reduces to Fredholm integral equation of the first kind with a symmetrical logarithmic kernel which depends on the difference of the arguments and contains the hyperbolic cotangent. The precise solution of the equation is constructed with the help of spectral relationships, which contain Chebishev polynomials of the first kind with the argument in form of incomplete elliptic function of the first kind. Private case of an absolutely rigid stringer is investigated and dependence is obtained between the rigid displacement of the stringer and axial concentrated force applied at its center