Equilibrium-Based Finite-Element Formulation for the Geometrically Exact Analysis of Planar Framed Structures

This paper addresses the development of a hybrid-mixed finite-element formulation for the geometrically exact quasi-static analysis of elastic planar framed structures, modeled using the two-dimensional Reissner beam theory. The proposed formulation relies on a modified principle of complementary energy, which involves, as independent variables, the generalized vectors of stress resultants and displacements and, in addition, a set of Lagrange multipliers used to enforce the stress continuity between elements. The adopted finite-element discretization produces numerical solutions that strongly satisfy the equilibrium differential equations in the elements, as well as the static boundary conditions. It consists, therefore, in a true equilibrium formulation for arbitrarily large displacements and rotations. Furthermore, as it does not suffer from shear locking or any other artificial stiffening phenomena, it may be regarded as an alternative to the standard displacement-based formulation. To validate and assess the accuracy of the proposed formulation, some benchmark problems are analyzed and their solutions are compared with those obtained using the standard two-node displacement-based formulation. Numerical analyses of convergence of the proposed finite-element formulation are also included.