Solution for bending of a polygonal plate with cracks by the linear conjugate method

Abstract

Stress–strain state of a polygonal multiply connected plate under bending is considered. The plate under the action of different force factors (bending moment M and uniformly distributed load of intensity q) is also carried out. Similar problems for a simple configuration plates were solved by different authors. But the bending problem for a plate with different length through cracks is considered first. Using the theory of a complex variable function and analytic functions (Kolosov-Muskheleshvili potential) the problem is solved by the linear conjugate method. Such plates (slabs) are widely used in shipbuilding (hull-deck of ships), in construction (interfloor slabs, balcony flooring slabs) in mechanical engineering, aircraft building, etc. Therewith, at first boundary condition for an entire plate (without cracks) under the action of the applied loads, (M and q) and then a problem on bending of a plate with cracks under the action of loads applied to crack faces and that are determined from the problem on entire plate (using the uniqueness of displacements $v+iv$ and uniqueness of the flexure w), are used. The obtained elliptic integrals F(x) and E(x) are calculated by the methods known from references. As a result for determining integral constants $C_1 ,C_2 ,\ldots ,C_j$, we get a system of equations for the considered cases of applied loads. Some numerical examples are solved.