Solving the problems of contact interaction of stringers with massive elastic bodies in a modified formulation

In a generalized plane stress state, the contact interaction of two identical stringers, symmetrically loaded by tangential forces, with an elastic infinite or semi-infinite plate is studied. A related problem of symmetric tangential force transfer from two reinforced strip-shaped stringers to an elastic half-space under antiplane deformation is also considered. Stringers are modeled as a one-dimensional elastic Melan continuum. A modified contact formulation is proposed, assuming equality of axial horizontal displacements of the stringer points and elastic foundations boundary points, unlike the commonly used strain-equality model. In this formulation, solving the problems under consideration is reduced to solving a single Fredholm governing integral equation of the first kind with a symmetric kernel, represented by the sum of its principal part in the form of a symmetric logarithmic function and a regular part in the form of the modulus of the difference of arguments. A numerical-analytical solution of the governing integral equation is constructed by the method of mechanical quadratures using spectral relations for the symmetric logarithmic kernel, containing Chebyshev polynomials of the first kind with the argument of an incomplete elliptic integral of the first kind.