Подкрепления в вариационно-разностном методе расчета оболочек сложной формы

Рассмотрено введение подкреплений в вариационно-разностном методе расчета оболочек сложной формы. Положение точек срединной поверхности оболочки определено криволинейными ортогональными координатами α, β. Криволинейные ребра расположены вдоль координатных линий. Ребра описаны теорией криволинейных стержней Кирхгофа - Клебша - учтены растяжение, изгиб и кручение ребер. Оболочка описана теорией упругих тонкостенных оболочек Кирхгофа - Лява.

Stiffeners in variational-difference method for calculating shells with complex geometry

We have already considered an introduction of reinforcements in the variational-difference method (VDM) of shells analysis with complex shape. At the moment only ribbed shells of revolution and shallow shells can be calculated with the help of developed analytical and finite-difference methods. Ribbed shells of arbitrary shape can be calculated only using the finite element method (FEM). However there are problems, when using FEM, which are absent in finite- and variational-difference methods: rigid body motion; conforming trial functions; parameterization of a surface; independent stress strain state. In this regard stiffeners are entered in VDM. VDM is based on the Lagrange principle - the principle of minimum total potential energy. Stress-strain state of ribs is described by the Kirchhoff-Clebsch theory of curvilinear bars: tension, bending and torsion of ribs are taken into account. Stress-strain state of shells is described by the Kirchhoff-Love theory of thin elastic shells. A position of points of the middle surface is defined by curvilinear orthogonal coordinates α, β. Curved ribs are situated along coordinate lines. Strain energy of ribs is added into the strain energy to account for ribs. A matrix form of strain energy of ribs is formed similar to a matrix form of the strain energy of the shell. A matrix of geometrical characteristics of a rib is formed from components of matrices of geometric characteristics of a shell. A matrix of mechanical characteristics of a rib contains rib's eccentricity and geometrical characteristics of a rib's section. Derivatives of displacements in the strain vector are replaced with finite-difference relations after the middle surface of a shell gets covered with a grid (grid lines coincide with the coordinate lines of principal curvatures). By this case the total potential energy functional becomes a function of strain nodal displacements. Partial derivatives of unknown nodal displacements are equated to zero in order to minimize the total potential energy. As an example a parabolic-sinusoidal shell with a stiffened hole is analyzed. It is shown that ribs have generally beneficial effect to the zone of the opening: cause a reduction in a modulus of a stress, but an eccentricity affects differently, so material properties and design solutions should be taken into account in an analysis.