Free Vibration Of An Axially Loaded Timoshenko Beam With The Transition Parameter

In this thesis,the free vibration of an axially loaded Timoshenko beam with the transition parameter is investigated,and we develop shear coefficient with the effects of size and transition parameter for nanobeams.We aims to investigate free vibrations of an axially loaded Timoshenko beam in a systematic manner.A recently-developed unified model is adopted,where a transition parameter characterizing the direction of the axial force during the deformation of the beam is introduced.By setting the values of the transition parameter,the direction of axial force can vary within the range between normal to the deformed cross section and tangential to the neutral axis of the beam arbitrarily.The equations of motion with the transition parameter are obtained by using element balance method.The governing equations of free vibration of axial loaded Timoshenko beams with various boundary conditions are obtained by virtue of the method of separation of variable.Considering the influence of cut-off frequency,the form of the general solution is provided.With the help of various boundary conditions,frequency equations of beams are obtained.The results available in the literature and numerical calculations are carried out to discuss the validity of the present solutions and to quantify the effects of the transition parameter,the level of axial force,height-to-span ratio and parameters of cross section.Significant effects of transition parameter are revealed on the free vibration of the axially loaded Timoshenko beams.The transition parameter is observed to have great influences on natural frequencies of the axially loaded Timoshenko beams,of which the shear deformation accounts for a large proportion among the total deformation,especially for the clamped beams with thin-walled cross section.Moreover,the height-to-span ratio,the level of axial force and mode orders also influence the effects of the transition coefficient on the natural frequencies.For the study of the shear coefficient of Timoshenko beam theory,the nanobeam is modeled as a bulk material plus an elastic surface layer in this paper.The equations of motion of free vibration are obtained by using the element balance method and the Hellinger-Reissner variational principle,respectively.The corresponding governing equations was obtained by virtue of the method of separation of variable.The terms of the different governing equations obtained by the element balance method and the Hellinger-Reissner variational principle should be identical.Then the shear coefficient of the nanobeam containing both the influence of the transition parameter and the surface parameters is developed.The shear coefficients for a variety of cross sections are derived,including the rectangular cross section,circular cross section and hollow circular cross section.The model of based-surface-layer nanobeam is degenerated into the macro-scale Timoshenko beam model,readily by ignoring the parameters of surface.The results available in the literature are presented to verify the validity of present solutions.Numerical examples show that material parameters of the surface and the transition parameter will influence the values of the shear coefficient.With the decrease of height of nanobeams or the increase of the absolute value of surface elastic modulus and surface residual stress,the effect of surface energy on shear coefficient will increase.Meanwhile,the influences of the transition parameter on the shear coefficient are increased with the height-to-span ratio and the level of axial force.