Equilibrated stress space for nonlinear dimensionally reduced shell models in terms of first-order stress functions

Considering the power series expansion of the three-dimensional variables with respect to the shell thickness coordinate, an equilibrated stress space for the first Piola-Kirchhoff stress vectors is derived in convective curvilinear coordinate system. The infinite series of the two-dimensional translational equilibrium equations are satisfied by introducing two first-order stress function vectors expanded into power series. For the important case of thin shells, the infinite number of two-dimensional equilibrium equations is truncated to obtain a ‘first-order’ model, where the equilibrated stress-space requires three vectorial stress function coefficients only. The formulation presented for thin shells is compared to the nonlinear equilibrium equations of the classical shell theories, written in terms of the first Piola-Kirchhoff stress resultants and stress couples and satisfied by the introduction of three first-order stress function vectors.