Keywords: beams of variable cross section, nonlinear analysis, dynamic analysis, flexural-torsional analysis, wind turbine towers, boundary element method.

The objective of this paper is to present a general formulation for the nonlinear flexural-torsional dynamic analysis of beams of arbitrary variable cross section, undergoing moderate large deflections and twisting rotations under general boundary conditions. The beam is subjected to arbitrarily distributed or concentrated transverse loading as well as to twisting and/or axial loading. Four boundary value problems are formulated and solved using the analogue equation method (AEM) [1], a boundary element (BE) based method. Application of the boundary element technique yields a system of nonlinear coupled differential-algebraic equations (DAE) of motion, which are solved iteratively using the Petzold-Gear backward differentiation formula (BDF) [2]. The essential features and novel aspects of the present formulation are summarized as:

1. The cross section is an arbitrarily shaped thin- or thick-walled doubly symmetric one. The formulation does not stand on the assumption of a thin-walled structure.
2. The beam is subjected to arbitrarily distributed or concentrated transverse loads and bending moments in both directions, twisting and warping moments as well as axial loading.
3. The beam is supported by the most general boundary conditions including elastic support or restraint.
4. The effects of rotary and warping inertia are taken into account.
5. The proposed model takes into account all the coupling effects of bending, axial and torsional response of the beam as well as the shortening effect.
6. The proposed method employs a BEM approach for the cross sectional analysis, resulting in line or parabolic elements instead of area elements of the finite element solutions, while a small number of line elements are required to achieve high accuracy.

The model developed is applied to the nonlinear dynamic analysis of a tower of wind turbine structure formed as a steel beam of tubular variable cross section. From the results the significance of geometrical nonlinearity is verified.

1

J.T. Katsikadelis, "The Analog Equation Method. A Boundary - only Integral Equation Method for Nonlinear Static and Dynamic Problems in General Bodies", Theoretical and Applied Mechanics, 27, 13-38, 2002. doi:10.2298/TAM0227013K

2

K.E. Brenan, S.L. Campbell, L.R. Petzold, "Numerical Solution of Initial-value Problems in Differential-Algebraic Equations", North-Holland, Amsterdam, 1989.

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