Static stability and free vibration characteristics of a micro laminated beam under varying axial load using modified couple stress theory and Ritz method

A model based on a modified couple stress theory for the free vibration and buckling analyses of beams is presented. The model also incorporates the Poisson's effect and allows the analysis of Timoshenko beams with any arbitrary end boundary condition. The natural frequencies and buckling loads are computed using the Ritz method. Parametric studies show that, while the natural frequencies and the buckling loads increase monotonically with the increase of the material length scale, they present a minimum in certain values of the Poisson's ratio. A study relating the classical elasticity and the couple stress strain energies is also presented. By establishing this relation, explicit formulas to obtain the natural frequencies and buckling loads, in which the couple stress and Poisson's effects are accounted for, in terms of the buckling loads of the classical elasticity are found. These formulas, which are valid when the shear strain and stress are zero, allow an expedite computation of natural frequencies and buckling loads of beams with couple stress and Poisson's effect.

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Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

Journal of Vibration and Control ◽

10.1177/10775463211048272 ◽

2021 ◽

pp. 107754632110482

Author(s):

J Ranjan Banerjee ◽

Stanislav O Papkov ◽

Thuc P Vo ◽

Isaac Elishakoff

Keyword(s):

Free Vibration ◽

Couple Stress ◽

Scale Parameter ◽

Couple Stress Theory ◽

Dynamic Stiffness ◽

Modified Couple Stress Theory ◽

Length Scale Parameter ◽

Length Scale ◽

Stress Theory ◽

Modified Couple Stress

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

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