Nonlinear mixed finite element formulations for the analysis of planar curved beams

Abstract

Abstract In the present paper, two mixed type functionals are presented to analyze the geometrical nonlinear bending behavior of slender planar curved beam by Finite Element Method (FEM). Bernoulli-Euler beam theory is taken into consideration within the context of the small strains but large displacements. Non-linear bending of curved beams in-plane has been investigated in both local and Cartesian coordinate systems. The Euler-Lagrange equations of the Functional I and Functional II which give the equilibrium and constitutive equations are in relation with the Green-Lagrange strain tensor and the first-order approximation (consistent linearization) of the Green-Lagrange strain tensor, respectively. The details of this process are given in Appendix. In the formulation of the problem in Cartesian coordinate, in addition to the boundary terms, that expression has been used, also to obtain the strain quantities which should be in the constitutive expressions in-prior to the equilibrium equations for the first time in this work. For the solution of the non-linear equations of the problems “incremental formulation” has been used. The validity, reliability and robustness of the proposed functionals have been presented by solving sample problems found in the literature.