On a nonlinear uniformly valid model for homogeneous beams with double symmetric cross-section

The geometry of beams is such that the two dimensions of the cross-section are of the same order of magnitude and smaller than the third one, the length of the mid-line. A Taylor–Young expansion of the displacement field truncated at fifth order with respect to the transverse dimension is assumed. For beams with two-fold symmetric cross-sections commonly used (e.g., circular, square, rectangular, and elliptical), the two-dimensional beam model is obtained by truncating the potential energy. After giving, the rule for truncation, we define and justify this model by analyzing the Euler–Lagrange equations. Moreover, we can show that this new nonlinear beam model is a uniformly valid beam theory, independent of the orders of magnitude of the applied loads. Indeed, the use of the asymptotic expansion method allows to recover from this nonlinear beam model, five well-known beam models obtained by the rigorous convergence results or the asymptotic expansion method in the literature to the leading order.