Defect Of Timoshenko Beam Theory And Modification Of Its Motion Equations

Since the Timoshenko beam theory(TBT)with two generalized displacements was firstly presented by Timoshenko in 1922,it was widely used both in research and engineering,especially in the analysis of the transverse vibration of non-slender beams.Meantime,a long standing topic of debate on Timoshenko beam theory(TBT) raised as a counterpart,which focus on the problem of the two frequency spectra.The formation cause of two frequency spectra for Timoshenko beam theory (TBT)is firstly pointed out:it is Timoshenko who omitted the angular acceleration caused by sheafing deformation,which made the fourth order term in time arise in the single governing differential equation of the transverse displacement,so that TBT has two frequency spectra.Based on the theorem of moment of momentum,the complete angular acceleration is used to derive the moment equilibrium equation of beam element and the motion equation of TBT is modified,which is called OTB.Then the wave characters are studied in details.Consequently,OTB model has only one sequence of phase velocity,one sequence of group velocity,and one sequence of natural frequency,that means OTB has only one frequency spectrum.Through analysis,it is found that for both OTB and TBT,their moment equilibrium equation is only suitable for rigid element.But,the beam element is actually deformable.In order to consider this fact,a deformation coefficientηis introduced into the equation,so that the motion equation of TBT is modified again. The beam with such a deformation coefficient is named modified Timoshenko beam (MTB).If the motion equation of MTB is used,the beam not only has one frequency spectrum,but also has high accuracy when a suitable deformation coefficient is chosen,so that the problems while utilizing the Timoshenko beam model in complicated structures can be avoided.Taken a simple supported beam as example in this paper,the expression of deformation coefficientηis derived by using the first frequency formula of TBT. In this case,MTB not only has one frequency spectrum,but also has the same accuracy with the first frequency spectrum of TBT.This expression can also be applied to study the general characters of the coefficient,which will be useful for determining a concise and precise deformation coefficient.Based on the principle of Lagrange's equations,this thesis derives a set of new consistent stiffness,and mass matrix formulations for MTB element.It has included the rotary inertia correction caused by the shear deformation of the beam for the mass matrix.A numerical example presented by N.G..Stephen is re-examined using both TBT and MTB theory respectively with reference to exact plane stress elastodynamic theory for the discussion of TBT's second spectrum.Agreement is excellent for MTB FEM when a suitable deformation coefficient is chosen.The dynamic tests show that numerical results agree well.