Nonlinear Analysis of Beams and Arches Undergoing Large Rotations

Abstract

A new nonlinear finite element formulation based on total Lagrangian approach is developed by minimizing the total potential using polynomial approximations for displacements. A first-order shear deformation theory is used to model the through-thickness shear distribution. The kinematics used to describe the large displacement and rotation consists of sine and cosine functions. Nonlinear Green strain measures and second Piola-Kirchhoff stress measures are used in the determination of total potential. The approach used in the formulation decomposes the Green strain components into convenient forms for inclusion in the total energy function that is then extended to a nonlinear finite element solution method. The developed beam/arch element contains 11 degrees of freedom. Numerical examples of beam and arch geometries are statically analyzed to observe the characteristics of large displacements and large rotations. Riks and displacement control techniques are used. Results are compared with some of the earlier attempts in using the total Lagrangian formulation to evaluate large rotation problems.