Stability of the shell of the minimum surface on a rectangular plan, taking into account geometric nonlinearity under thermal and power loading

Abstract

When developing the classical theory of minimal surface shells, scientists focused on methods that require maximum simplification of the solving equations and elimination of values that do not significantly affect the final results. This reduces the class of minimal surface shells that are studied and excludes some important mechanical and physical effects from view.

With the development of computational complexes, new problems of the theory of shells arise, which must be solved without the involvement of auxiliary hypotheses about the nature of the desired fields along the thickness of the shells of minimal surfaces. This leads to the development of non-classical shell theories, which are classified on the basis of the connection of different methods of constructing systems of equilibrium equations of shell mechanics with methods of boundary value problems of elasticity theory.

When designing minimum surface shells, one has to deal with calculations for the stability of minimum surface shells on different contours, taking into account geometric nonlinearity. Having a small structural mass, a thin-walled spatial shell is an exceptionally rigid structural form.

Geometrically nonlinear problems are those in the theory of elasticity that take into account nonlinearity in the dependence of strains and displacements, while stresses and strains are related linearly. Taking into account nonlinear components of deformations is necessary for the calculation of flexible thin-walled structures.

In general, solving a nonlinear system is reduced to solving a sequence of linear systems. Note that only the right-hand side of the system of equations changes during successive iterations, which allows the stiffness matrix to be factorized only once.

For the first time, a numerical study of a minimum surface shell on a rectangular plan with geometric nonlinearity was performed using the finite element method (FEM), the reliability of the results was verified against theoretical values, the convergence of the finite elements was high, and the dimensions of the finite elements were chosen to be optimal in number and size. The geometric nonlinearity shows a 9% improvement in the actual stresses and displacements from the linear calculation, which is a good material saving and calculation accuracy.