Buckling Analysis of Non-Local Timoshenko Beams by Using Fourier Series

Abstract

In this study, buckling analysis of a nano sized beam has been performed by using Timoshenko beam theory and Eringen’s nonlocal elasticity theory. Timoshenko beam theory takes into account not only bending moment but also shear force. Therefore, it gives more accurate outcomes than Euler Bernoulli beam theory. Moreover, Eringen’s nonlocal elasticity theory takes into account the small scale effect. Thus, these two theories are utilized in this study. The vertical displacement function is chosen as a Fourier sine series.  Similarly, the rotation function is chosen as a Fourier cosine series. These functions are enforced by Stokes’ transformation, and higher order derivatives of them are obtained. These derivatives are written in the governing equations for the buckling of nonlocal Timoshenko beams. Hence Fourier coefficients are acquired.  Subsequently boundary condition of established beam model is identified with Timoshenko beam and Eringen’s nonlocal elasticity theories, and the linear equations are obtained.  A coefficients matrix is created by utilizing these linear systems of equations. When determinant of this coefficient matrix is calculated, the critical buckling loads are acquired. Finally, achieved outcomes are compared with other studies in the literature.  Calculated results are also presented in a series of figures and tables

Keywords

• Timoshenko beam theory,Eringen’s nonlocal elasticity theory,elastic buckling analysis,Stokes’ transformation

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