New refined models for curved beams in both linear and nonlinear settings

In this paper, we derive a one-dimensional model for curved linear and nonlinear beams with no assumption on either the shape of the cross-section or the prescribed loads on the boundary of the cross-section. This model contains coupled bending, stretching and torsion effects, which is achieved by truncating the potential energy at the seventh order with respect to small transverse dimensions of the cross-section of the beam based on series expansions. In the linear case, we obtain an estimate for the difference between the exact and approximate stress tensors. Nevertheless, we can only show that for the displacement field, the solution of the approximate problem resulting from the truncation of the displacement field at the sixth order only differs from the exact displacement field by a difference of order of magnitude of the seventh order.