Conformal mapping function of a complex domain and its application

Abstract

Determination of stress–strain state in an elastic domain of a particular form is considered. In order to determine the stress–strain state for these or other problems of elasticity theory in such a complex domain, the function that maps the given domain S onto the exterior of a unique circle (or onto a half-plane) $\gamma$ is determined first. Then, using the complex variable methods and Kolosov—Muskheleshvili potentials, stress components (of normal and tangential stresses) at characteristic cross-sectional points under the action of applied loads are defined. Some problems of theory of elasticity are considered. Therewith, at first, the boundary value problem of plane theory of elasticity is solved by means of the obtained conformal mapping function. Further, this function is applied to solve the boundary condition of a beam, thus introducing the solution of bending problems of theory of elasticity. This study presents the novelty by introducing a new mapping function (suitable for inversion) which was discovered by the author for the first time in the scientific world. The paper then makes an application of the mapping function to solve a class of elasticity theory problems for such complex domains in much simpler way. The necessity to solve the considered problems is substantiated by a broadening use of such complex elements in different fields of science and engineering (crane girders—for traveling cranes, in multi-story buildings—basis of foundations, concrete and reinforced concrete supports, floorings, etc.), as well as in pipe-line saddles, underground, underwater, ground floorings for pipes, offshore platforms, etc. Therefore, the mapping function presented in this paper has a theoretical and practical significance. The complex elements are presented in compression or in bend. The proposed solution is illustrated by numerical examples.