Geometrically non-linear formulation of planar flexible multi-beam systems: Stiffness under second order theory

The objective of this study is to develop computational efficient and accurate stress stiffening models that can be used in the dynamical formulation for the analysis of two-dimensional beams. The model which accounts for the tangent stiffness of beam element are derived using second order theory and continuum mechanics approach. Then, three stiffness matrix and two different force models that include different degrees of complexity are presented by expanding it to power series and truncating at quadratic terms. It is shown that the finite-element formulation of multibeam system can be significantly simplified as compared to the classical geometrical nonlinear method developed for the formulation in local element frame. Despite the simplicity of the new models, they account for elastic non-linearity in the strain-displacement relationship, therefore, they lead to more accurate results as compared to the more complex models which does not exactly account for the non-linearity. Numerical results are presented in order to demonstrate the implementation of the new models and test their performances through dynamical analysis of crank-slider mechanism.