Finite element asymptotic post-buckling analysis, being based on fourth-order expansions of the strain energy, requires that nonlinear structural modeling be accurate to same order, at least with respect to the rigid motions of the elements. A corotational description is proposed here as a general tool to satisfy this requirement of objectivity, by referring each element to a local frame which moves (rotates) with the element, so filtering its rigid motion. In this description nonlinearity of the problem derives essentially from the change of reference, from the global fixed frame to the local one, the strain energy being governed by their relative rotations. In finite kinematics, this noticeably complicates the algebra for obtaining exact expressions of its variations. Quite simple, basic expressions for the first four corotational derivatives of the strain energy are provided, allowing the set up of a fourth-order accurate asymptotic analysis starting from standard finite elements based on linear or simplified nonlinear local modelings. The formulation is implemented for the analysis of 3D beam assemblages and several numerical results are presented and discussed showing the effectiveness and robustness of the proposed approach in reproducing the nonlinear equilibrium path in both cases of monomodal and coupled multimodal buckling.