Cylindrical bending of a transversally-isotropic slab partially resting on a rigid foundation

Abstract

In setting up a refined model of bending of transversely isotropic plates of medium thickness, the contact problem of cylindrical bending of the plate, the central area of which is partially in contact with the rigid foundation, and the edges are raised to a certain height by forces distributed along its edge, is solved. The solution of the fourth-order differential equation was obtained, as well as the formulas for the distribution of contact pressure on the lower surface of the interface between the slab and the foundation. The obtained formulas ensure both the exact (in the integral sense) satisfaction of the boundary conditions at the edges of the slab, and the logical (in the physical sense) distribution of the contact pressure along the lower surface of the slab, where it is partially in contact with the rigid foundation. The expression for the contact pressure is obtained due to the exact satisfaction of the boundary condition of equality of zero vertical displacement of the lower surface of the slab in the area of its contact with the rigid foundation. The mentioned vertical displacement is written in the form of a polynomial of the fourth order in the transverse coordinate.

  A comparison with the corresponding results according to the "sliding" theories of B.F. Vlasov and S.A. Ambartsumyan, obtained a partial case of the developed mathematical model of plates of medium thickness when it is assumed that the correction from the deformation of transverse compression is equal to zero. It is shown that this type of solution, where the contact pressure at the boundary of a smooth contact takes certain finite (according to "shear" theories) or infinite (according to the classical Kirchhoff theory) values, does not correspond to the physical content of the problem, as well as to the corresponding solutions of spatial problems of the theory of elasticity and it is not desirable to use them. When determining the size of the contact area, as evidenced by the author's previous research in similar problems for short beams, as well as the corresponding results of other authors, they differ only quantitatively