A two-level nonlinear beam analysis method

The nonlinear response of straight, constant cross-section beams is investigated by means of a two-level finite element solution procedure. The higher level model is built starting from the Hellinger–Reissner principle, with the normal stress resultant and moment resultant as additional unknowns field beside the beam reference line position and orientation field. A lower-level nonlinear cross-section problem is defined for each integration point. The two-level models are linked together by the normal stress resultant and moment resultant, in one direction, and by the variation of the complementary strain energy in the other: the cross-section level do deform in such a way that the normal stress resultant and moment resultant are equal to those of the beam model, while the higher level beam model receives the gradient and Hessian of the complementary strain energy with respect to the resultants. The complementary strain energy gradient and Hessian are computed by defining suitable first and second order adjoint problems.