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Based on the recent advances of hybrid stress finite elements, a series of alternative stress assumptions for these elements are investigated. Several new element models are proposed by using different concepts for the stress interpolation. Under a unified formulation presented in this paper for Hellinger—Reissner principle based hybrid stress element models, the element series 5β‐family for plane stress and 18β‐family for three‐dimensional problems are discussed. The extra incompatible displacements sometimes also added are not introduced in this unified formulation. A number of popular benchmark elastic problems are examined for both two element families. In each family, the element model presented in this paper using normalized transformed higher order stress trials usually gives better predictions than the others.

A critical assessment of the 4‐node assumed strain element as proposed by Dvorkin and Bathe is made. The element performed excellently in all investigated shell problems which sometimes caused difficulties for other assumed strain techniques. For efficient computation in the non‐linear range, linearization of the virtual work equation is done to yield the consistent tangent stiffness. The shell formulation is done for stress and strain tensors based on local element coordinates. To demonstrate the effectiveness and rapid convergence of the non‐linear formulation, three examples are tested for large displacements.

The present contribution deals with the sensitivity analysis and optimization of structures for path‐dependent structural response. Geometrically as well as materially non‐linear behavior with hardening and softening is taken into account. Prandtl‐Reuss‐plasticity is adopted so that not only the state variables but also their sensitivities are path‐dependent. Because of this the variational direct approach is preferred for the sensitivity analysis. For accuracy reasons the sensitivity analysis has to be consistent with the analysis method evaluating the structural response. The proposed sensitivity analysis as well as its application in structural optimization is demonstrated by several examples.