Nonlinear Finite Deformation Analysis of Beams and Columns

Abstract

A method for solving the finite‐displacement problem of a curved elastic beam with axial, shear, and flexural deformation subject to distributed and point loads is presented. Within the context of the kinematic assumptions of the Timoshenko theory, a Lagrangian formulation of the problem is developed. In terms of three cross‐sectional stress resultants, three Euler equations of equilibrium for the beam are derived with the aid of a variational principle for finite deformation. Upon linearization to small strains and the adoption of a linear elastic constitutive relation between the stress and strain tensors, it is shown that the problem is reducible to a single second‐order nonlinear ordinary differential equation. Subject to appropriate boundary conditions, the resulting two‐point boundary‐value problem is solved by a finite‐element method. By virtue of a continuation algorithm, accurate solutions of the system of nonlinear equations can be obtained for a variety of bifurcation and buckling problems. Comprehensive results are presented for two cantilever beams as illustrations.