Systems of Approximating Functions when Using Variational Methods for Calculating Thin-Walled Building Structures

The question of the use of approximating functions in the calculation of thinwalled building structures is investigated and the requirements that they must satisfy are analyzed. A rule is formulated that allows one to distinguish between the principal boundary conditions and natural ones. It is shown that the approximating functions must satisfy the principal boundary conditions, while the natural boundary conditions are included in the equilibrium equations and are satisfied automatically when solving a boundary value problem. The accuracy of their fulfillment depends on the accuracy of the solution of the problem itself. An example shows what errors can result from the use of approximating functions that satisfy the specified boundary conditions, but do not satisfy the completeness conditions. Some systems of functions for which the completeness condition in the energy space has been proven are considered. Using the example of Legendre orthogonal polynomials, a technique is given for forming approximating functions that satisfy the specified boundary conditions and the completeness conditions of a system of functions. The efficiency of using the obtained approximating functions in solving boundary value problems using the Galerkin method is shown.