Hierarchical multi-layered element of assembled timoshenko beams

A procedure of assembling Timoshenko beam elements into a hierarchical multi-layered element is presented. Using a linear approximation of the tangent displacement u in the thickness direction the traditionally used unknowns w,u, θ are replaced by w, u⁺, u^-, where u⁺ and u^- denote tangent displacements at the top and bottom surfaces of a layer. Then the compatibility conditions at layer interfaces are written and layers are aggregated, leading to a layer-wise model. The hierarchical model is obtained by using a Taylor or Chebyshev expansion of the tangent displacements in the thickness direction. The degree of the expansion can be arbitrarily chosen, and its coefficients are the new independent parameters. A finite element based on the presented procedure has been developed. A layer-wise model, and a single-layer model with an arbitrary order of approximation can be obtained as special cases. Accuracy and versatility of this element are confirmed on several numerical examples.