A Timoshenko beam element based on the modified couple stress theory

2014 ◽  
Vol 79 ◽  
pp. 75-83 ◽  
Author(s):  
M.H. Kahrobaiyan ◽  
M. Asghari ◽  
M.T. Ahmadian
Author(s):  
M. H. Kahrobaiyan ◽  
M. Khajehpour ◽  
M. T. Ahmadian

In this paper, the modified couple stress theory is employed to develop a size-dependent beam element able to predict the size-dependency observed in microbeams. The stiffness matrix is obtained for the aforementioned beam element. As an example, the deflection of a microcantilever is evaluated using the proposed beam elements and the results of the finite element method are compared to the analytical results obtained by the classical beam theory. The maximum deflection of the beam is depicted versus the ratio of the beam thickness to the material length scale parameter, the parameter appearing in non-classical continuum theories. The results show that when the characteristic size of the beam (thickness, diameter, etc) is small, like the beam used in MEMS and NEMS, the difference between the results of the current model and those obtained by the classical beam theory is significant but it diminishes as the characteristic size increases.


2016 ◽  
Vol 24 (3) ◽  
pp. 527-548 ◽  
Author(s):  
RA Jafari-Talookolaei ◽  
M Abedi ◽  
M Şimşek ◽  
M Attar

In this study, the free and forced vibration analysis of a micro scale Timoshenko beam resting on a Pasternak elastic foundation and subjected to a moving micro particle is presented. Based on the modified couple stress theory and employing Hamilton’s principle, the governing equations along with the boundary conditions are derived. A semi-analytical solution is obtained for the free vibration of the problem by expressing the dynamic lateral displacement and cross-section rotation in terms of the series of Legendre polynomials and extremizing the objective functional of the problem with respect to the unknown displacements and Lagrange multipliers. Correspondingly, the computed eigenvalue information of the system is utilized in the modal expansion technique to obtain the transient dynamic response. For comparison purposes, the free vibration frequencies of the micro beam and the dynamic deflections using the classical Timoshenko beam theory are compared with previously published studies and very good agreements have been observed. Furthermore, more numerical examples for natural frequencies and dynamic deflection of the beam are presented and the effects of some parameters, such as the material length scale parameter, the velocity of micro particle, the Pasternak elastic foundation parameters, shear deformation effects and boundary conditions are examined.


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