Nonlinear Static Behavior of Slender Beams: Formulation and Solution Algorithms

Accurate analysis of slender beams, such as those found in guyed towers, presents significant challenges due to geometric nonlinearities associated with large deformations. While Timoshenko beam theory improves upon classical models by including shear deformation, it is prone to shear locking in thin beams, rendering it inefficient for slender structures. Conversely, conventional linear analysis fails to account for membrane stretching effects, leading to inaccurate stiffness predictions under large displacements. This study aims to develop a precise and computationally efficient finite element formulation for the nonlinear static analysis of slender beams. The proposed method utilizes the Euler-Bernoulli beam theory integrated with Von Kármán geometric nonlinearity to eliminate shear locking and accurately capture the coupling between bending and axial stretching. Additionally, to navigate the complexities of nonlinear equilibrium paths-particularly at limit points where standard methods diverge―this research implements and evaluates the Modified Riks (Crisfield's) arc-length algorithm alongside conventional Newton-Raphson methods. Numerical verification is performed on an aluminum beam subjected to a constant end moment. The results demonstrate that the nonlinear formulation reveals a distinct stiffening behavior at large deflections, which is completely missed by linear theory. Furthermore, the Modified Riks method is shown to be essential for tracing the complete equilibrium path through points of instability, providing a robust tool for the analysis of complex structural behaviors.

کلیدواژه ها:

Geometric nonlinearity ، Von Kármán beam theory ، Slender beams ، Shear locking ، Modified Riks method ، Finite element analysis