Stability of toroidal shell segments at variation of a deflection angle

The paper presents the study of the stability of panels of steel toroidal thin-walled shell structures with different angles of deviation from the vertical axis. The mathematical model (the model of Timoshenko-Reissner) is geometrically nonlinear and is represented as a functional of the total potential energy of deformation. To reduce the variation problem into solving a system of algebraic equations, the Ritz method was applied with the use of two different types of basis, i.e. trigonometric and polynomial (based on the Legendre polynomials). The process of forming the approximating functions is considered in detail taking into account the symmetry of the toroidal panels. The final system of algebraic equations is nonlinear and solved by Newton method. The solution is made by the Maple 2017. The calculations of segments of toroidal shells are carried out under the action of the external evenly distributed transverse load and the load values with the loss of stability are obtained. The parameter of the large radius was fixed when choosing the variants of constructions for two purposes. The first one is to the covering area of the considered segment of the shell which remained unchanged and the second one is to the small radius which is dependent on the angle of deviation from the vertical axis. In some cases, local stability losses are observed. The effect of the deflection angle from the vertical axis on the values of the stability loss loads and the maximum values of deflections are analyzed. The results obtained for two types of approximation are presented. The calculations showed that both variants of approximation give close results at low loads, but they significantly differ at large loads. The increase in the deflection angle leads to the decrease in value of the critical load, which may be caused by an increase in the surface area of the shell. However, the value of the maximum deflection decreases.