Bending And Buckling Of Timoshenko Beam:Effect Of Transition Parameter

Based on a recently-developed unified model where a transition parameter characterizing the direction of the axial force during the deformation is introduced,the static bending and buckling problems of beams are investigated.The mechanical behavior of Timoshenko beams varies with the values of transition parameter characterizing the direction of the axial force.Based on the model,the following work has been done:(1)The static bending problem of the beams is studied.First,the governing equations of the static bending problem of Timoshenko beams considering transition parameter are obtained.The general solutions of the governing equations are obtained by means of Laplace transform and the inverse transform,and closed-form expressions of deflections and rotation angles of the Hinged-Hinged beam,Clamped-Clamped beam,Clamped-Hinged beam and Clamped-Free beam are obtained.The validity of the analytical and numerical solutions is mutually verified by means of finite element analysis.In addition,numerical calculations are carried out to quantify the effect of the transition parameter on the static bending of the beams,and that reveals the effect is closely related to the boundary conditions of the beams.(2)The buckling problem of the beam is studied.First,the governing equation of the buckling problem of Timoshenko beams considering transition parameter is presented and the general solution is obtained.The corresponding characteristic equations of Hinged-Hinged beam,Clamped-Clamped beam,Clamped-Hinged beam and Clamped-Free beam are obtained according to the boundary conditions,from which the critical buckling loads may be figured out in an analytical or numerical way.It is highlighted that the inherent relations of the critical buckling loads corresponding to the Timoshenko beams and Euler-Bernoulli beams are established.Comparing with the results existing in the literature,the present solutions are verified.Numerical calculations are presented to quantify the effects of the transition parameter on critical buckling loads of Timoshenko beams with various cross sections,including square,rectangle and circle.