In this paper, the problems of ultimate limit load of spherical shell are formulated and solved. An equilibrium finite element developed by the method of Bubnov-Galerkin is suggested. Equilibrium and geometrical equations are created for this element and, based on these equations, the mathematical models of ultimate limit external load optimization problems for the shell structures are constructed. These are nonlinear mathematical programming problems. The methodology is illustrated by the numerical examples. The solution results are obtained for the finite elements of various sizes and show very high accuracy of the suggested element and convergence of the results.

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