On the nonlinear extension-twist coupling of beams

In many applications, beams carry large axial forces, such as centrifugal forces, for instance, and the resulting axial stresses affect their torsional behavior. To capture this important coupling effect within the framework of beam models, a nonlinear theory must be developed that takes into account higher-order strain deformations. The proposed development starts with a three-dimensional elasticity model of beam-like structures. While strain components are assumed to remain small, the strain energy expression is expanded to retain strain components up to cubic order. This expansion of the three-dimensional model leads to material and geometric stiffness matrices. For linear problems, the three-dimensional warping fields induced by each of the six unit sectional strain components can be obtained. These warping fields are used to reduce the three-dimensional elasticity model to one-dimensional beam equations. Byproducts of this reduction process include the sectional stiffness matrix and six nonlinear stiffness matrices representing the geometric stiffness induced by unit sectional strains. It is shown that the nonlinear extension-twist coupling is captured by these geometric stiffness matrices. Comparison of the predictions of the proposed approach with analytical and experimental results demonstrate their effectiveness and accuracy.