Nonlinear Deformation of Flexible Shallow Shells of Complex Shape Made of Materials with Different Resistance to Tension and Compression

A new numerical-and-analytical method is developed for solving geometrically and physically nonlinear problems of bending shallow shells of complex shapes made from materials with different resistance to tension and compression. To linearize the initial nonlinear problem, the method of continuous continuation in the parameter associated with the external load was used. For the variational formulation of the linearized problem, a Lagrange functional was constructed, defined at kinematically possible displacement velocities. To find the main unknowns of the problem of nonlinear bending of a hollow shell (displacements, deformations, stresses), the Cauchy problem for a system of ordinary differential equations is formulated. The Cauchy problem was solved by the Runge-Kutta– Merson method with automatic step selection. The initial conditions are found in the solution to the problem of geometrically linear deformation. The right-hand sides of the differential equations at fixed values of the load parameter corresponding to the Runge-Kutta–Merson scheme were obtained from the solution of the variational problem for the Lagrange functional. The variational problems were solved by the Ritz method in combination with the R-function method. The latter makes it possible to present an approximate solution in the form of a formula, which solution structure exactly satisfies all (general structure) or part (partial structure) of the boundary conditions. The problems of nonlinear deformation of a square cylindrical shell and a shell of complex shape with combined fixation conditions are solved. The influence of the direction of external loading, geometric shape, and fixation conditions on the stress-strain state is investigated. It is shown that failure to consider the different behaviors of the material in tension and compression leads to significant errors in calculating the stress-strain state parameters.