Nonlinear nonuniform torsional vibrations of bars by the boundary element method

Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations

“…In order to establish the equations of motion, the principle of virtual work under a Total Lagrangian formulation is employed as this is accomplished in [35]. Performing the decomposition of shear strains into primary and secondary parts, as it is described 12 for shear stresses in eqns (3b-c, 4, 5), the contribution of shear stresses in the virtual work of internal forces can be written as …”

“…The required algebra to reach the above expressions starting from those of eqns (9) It is also convenient to define the axial stress resultant N and the warping moment w M arising from normal stresses as in [35]. Substituting the stress components given in eqns (3), the strain ones given in eqns (2) and the displacement ones given in eqns (1) to the principle of virtual work, the governing partial differential equations of motion of the bar are obtained after some algebra as…”

“…[33] analyzed flexural-flexural-torsional free vibrations of twisted rotating beams employing nonlinear normal modes. Rozmarynowski and Szymczak [34] and Sapountzakis and Tsipiras [35] focus to the problem of nonlinear torsional vibrations. Nevertheless, dynamic analysis of bars at a torsional post-buckled state, as this is presented for the static case in [36], is not performed.…”

Bars under Torsional Loading: A Generalized Beam Theory Approach

“…In all of the aforementioned research efforts the evaluation of the warping shear stress distribution is not discussed. Finally, Sapountzakis and Tsipiras in [111,112] developed a boundary element solution for the nonuniform torsional vibration problem of bars of arbitrary doubly symmetric constant cross section taking into account the effect of geometrical nonlinearity. In these last research efforts the bar is subjected to arbitrarily distributed or concentrated conservative dynamic twisting and warping moments along its length, while its edges are supported by the most general torsional and axial boundary conditions.…”

“…From the examined example problems presented in [111,112] it is concluded that (i) the geometrical nonlinearity leads to coupling between the torsional and axial equilibrium equations and alters the modeshapes of vibration;…”