Stability of orthotropic doubly curved shallow shells with a movable hinged fixing of the border

Shell structures are often used in different fields and their studies are important for many applications. To eliminate the stress concentration near the contour, especially at the corner points of the shell, the border of the structure has a fixed movable hinge support. This paper considers double-curved shallow shells, square in a plan, made from orthotropic materials with their border having a fixed movable hinge support. The mathematical model is based on the hypotheses of the theory of Tymoshenko - Reisner shells, which takes into account the transverse shifts and represents the mixed-form equations. In addition, the model takes into account the geometric nonlinearity. To solve the system of differential equations we used the method of Bubnov - Galerkin, that makes it possible to reduce the problem to the solution of a system of nonlinear algebraic equations. The convergence of the method is also shown for the increasing number of terms of approximation. The resulting system is nonlinear and solved by the Newton method. The developed algorithm is implemented in Maple 2017. The proposed algorithm is verified by comparing the calculation results of the test problem with the result obtained by other authors. The combination of the load-deflection curve showed a good consistency of the data. The stability analysis of three variants of shallow shell structures with a double curvature is carried out; each of them is made of four orthotropic materials. The outer uniformly distributed transverse load acts on the shell, the border fixing is hinged-movable. For all the structures studied, the critical buckling load, the maximum value of the deflection, corresponding to this load, and load-deflection curves are given. Conclusions are drawn about the stress-strain state of the shells under consideration.