Exact solution of the mixed boundary-value elasticity problem for an infinite wedge-shaped plate with regard for its proper weight

Abstract

UDC 539.3 We have constructed the exact solution of the mixed boundary-value elasticity problem for an infinite wedge-shaped plate 0 ≤ r <∞ , 0 ≤ϕ ≤ ω, 0 ≤ z ≤ h with regard for the action of its proper weight. We assume that the conditions of sliding attachment are given on the plate faces ϕ= 0 and ϕ=ω , those of the first basic elasticity problem are satisfied on the face z = h , and one of the two types of boundary conditions can be assigned on the face z = 0 . We have obtained a simple elementary solution for the case where only the proper weight acts on the plate. This result has enabled us to consider the plate stressed state depending only on the load assigned in the statement of the problem. For this purpose, we have reduced the formulated boundary-value problems to a vector one-dimensional boundary-value problem with using the appropriate integral transformations with respect to the variables ϕ and r . To obtain its exact solution, we have applied the Hankel vector integral transformation together with the theory of second-order matrix differential equations. 1. Consider an elastic (with Poisson’s ratio μ and shear modulus G ) infinite wedge-shaped plate (with the specific weight of its material γ ), which in a cylindrical coordinate system is defined by