Efficient formulation of a two-noded geometrically exact curved beam element

TL;DR

This paper presents an efficient finite difference-based formulation for curved elastic beams that improves accuracy without increasing degrees of freedom, demonstrated through various numerical examples.

Contribution

The paper extends a 2D geometrically exact beam element formulation to curved beams, introducing a finite difference scheme that enhances accuracy while maintaining constant degrees of freedom.

Findings

High accuracy in modeling curved beams demonstrated

No locking or oscillations observed in numerical examples

Coupling effects between internal forces and deformation variables identified

Abstract

The paper extends the formulation of a 2D geometrically exact beam element proposed in our previous paper [1] to curved elastic beams. This formulation is based on equilibrium equations in their integrated form, combined with the kinematic relations and sectional equations that link the internal forces to sectional deformation variables. The resulting first-order differential equations are approximated by the finite difference scheme and the boundary value problem is converted to an initial value problem using the shooting method. The paper develops the theoretical framework based on the Navier-Bernoulli hypothesis, with a possible extension to shear-flexible beams. Numerical procedures for the evaluation of equivalent nodal forces and of the element tangent stiffness are presented in detail. Unlike standard finite element formulations, the present approach can increase accuracy by refining the integration scheme on the element level while the number of global degrees of freedom is kept constant. The efficiency and accuracy of the developed scheme are documented by seven examples that cover circular and parabolic arches, a spiral-shaped beam, and a spring-like beam with a zig-zag centerline. The proposed formulation does not exhibit any locking. No excessive stiffness is observed for coarse computational grids and the distribution of internal forces is not polluted by any oscillations. It is also shown that a cross effect in the relations between internal forces and deformation variables arises, i.e., the bending moment affects axial stretching and the normal force affects the curvature. This coupling is theoretically explained in the appendix.