An extended shakedown theory on an elastic–plastic spherical shell

The article analyses one of possible optimization methodologies for ideal elastic–plastic structures at shakedown and its application for shallow spherical shells having prescribed geometry and affected by a variable repeated load (VRL) – a system of external forces that may vary independently of each other. The paper accepts that only time-independent upper and lower bounds of variations in external forces are given. The pronounced effect of external forces, i.e. in this context, possible histories of variations in forces, is not examined (the unloading phenomenon of cross-sections is ignored in the course of plastic deformation). The discussed concept of the structure at shakedown refers to the Melan theorem related to statically allowable admissible internal forces. Thus, for the discretization of the spherical shell, with the help of an assumption about small displacements, the equilibrium finite element method based on internal force approximation is applied. The limit axial force of the cross-section is supposed to be constant within the bounds of the finite element, and only the optimal distribution of limit internal forces among elements, according to the selected criterion, is in need of search. The article presents a discrete mathematical model for determining the optimal allocation problem with strength and stiffness requirements. The model conforms to the limit axial force of the shallow spherical shell of the variable repeated load and takes into account ultimate and serviceability limit states of EC3 with corresponding reliability levels. Structural optimization methods refer to extreme energy principles of mechanics and are illustrated with the numerical examples of spherical shell optimization.