Finite Element Formulation of Timoshenko Non Prismatic Beam Column Element for Static and Dynamic Analysis of Wind Turbine Towers

In this paper, a numerical formulation is carried out to produce stiffness and consistent mass matrices for two-nodal non-prismatic Timoshenko beam-column elements. The proposed solution is based on the exact shape functions and their derivatives describing the non-uniformity of the element properties. To illustrate the variations in section properties along the tapered element length, the mechanical properties are presented as power functions with tapering indices. The model is applicable for elements with different solid and hollow cross-sections. The Proposed formulation is embedded into a visual basic code to carry out the analysis accompanied with many examples for validating its accuracy and efficiency. The model results are compared with those of commercial software and cited references which showed high accurate results with a small number of elements. Finally, a numerical study using the proposed solution was performed on a wind turbine tower to carry out a free vibration and linear stability analysis.