Static and dynamic consistent perturbation analysis for nonlinear inextensible planar frames

• A numerical-perturbation method for analyzing nonlinear planar frames is illustrated.
• The displacement fields are obtained by asymptotic integration of the field equations.
• Calculations are rearranged in a way they can be made by a computer.
• Modeling is carried out in the context of the mixed formulation of elasticity.
• The semi-analytical results are in excellent agreement with FEM analyses.

An asymptotically exact method for static and dynamic analysis of geometrically nonlinear planar frames is illustrated. The method is based on an integration of the nonlinear equations for the beam, carried out via a perturbation method, aiming to express the forces at the ends as series expansion of the displacements at the ends and of the distributed loads. Since the beams are assumed to be inextensible and unshearable, also reactive stresses appear among the unknowns, while compatibility conditions must be appended to the equilibrium equations. The element state-relations are assembled for the frame, and discrete, nonlinear perturbation equations are derived. Examples are worked out and relevant results compared with purely numerical solutions.