For the calculation of flat shells by the numerical analytical method of boundary elements

The application of the numerical-analytical boundary elements method (NA BEM) to the calculation of shallow shells is considered. The method is based on the analytical construction of the fundamental system of solutions and the Green’s function for the differential equation of the problem under consideration. The theory of calculation of a shallow shell proposed by V. Z. Vlasov, which for the problem under consideration leads to an eighth-order partial differential equation. The problem of bending a shallow shell is two-dimensional, and in the numerical-analytical boundary elements method, the plate and shell are considered in the form of generalized one-dimensional modules, therefore, the Fourier separation method and the Kantorovich-Vlasov variational method were applied to this equation, which made it possible to obtain ordinary differential equations of the eighth order. It is noted that until recently, the main problem in the subsequent implementation of the algorithm of the numerical-analytical boundary element method was due to the fact that all analytical expressions of the method (fundamental functions, Green’s functions, vectors of external loads) are very cumbersome, and intermediate transformations are associated with determinants of the eighth order. It is proposed to use the direct integration method at the first stage, when, along with the original differential equation, an equivalent system of equations for the unknown shell state vector is considered. In this case, the calculations of some analytic expressions associated with determinants of higher orders can be avoided by using the Jacobi formula. As a result, the calculation of the determinant at an arbitrary point is reduced to its calculation at a zero value of the argument, which leads to a significant simplification of all intermediate transformations and analytical expressions of the numerical-analytical boundary elements method.

On the improved interior regularity of the solution of a fourth order elliptic problem modelling the displacement of a linearly elastic shallow shell lying subject to an obstacle

In this paper we show that the solution of an obstacle problem for linearly elastic shallow shells enjoys higher differentiability properties in the interior of the domain where it is defined.