Dynamics and Buckling of Thin Pre-twisted Beams Under Axial Load and Torque

Free vibration and buckling of pre-twisted beams exhibit interesting coupling phenomena between compression, shears, moments and torque and have been the subject of extensive research due to their importance as models of wind turbines and helicopter rotor blades. The paper investigates the influence of axial compression and torque on the natural vibration of pre-twisted straight beam based on the Euler-Bernoulli theory. The derivation begins with the three-dimensional Green strain tensor. The nonlinear part of the strain tensor is expressed as a product of displacement gradient to derive the strain energy due to initial stresses. The Frenet formulae in differential geometry are employed to treat the pre-twist. The strain energy due to elasticity and the linear kinetic energy are obtained in classical sense. From the variational principle, the governing equations and the associated natural boundary conditions are derived. To the best knowledge of the author, the buckling of pre-twisted beam due to initial torque has not been studied in details. The major contribution of the paper is in the consideration of the influence of initial stresses caused by initial shears, moments and torques for pre-twisted beam-columns by means of the Frenet formulae and second order strains. A number of numerical examples are given. Some particular cases are compared with existing results. It is noted that the first mode increases together with the rate of twist but the second decreases seeming to close the first two modes together. The gaps close monotonically as the rate of twist increases for natural frequencies and buckling compressions.