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Bending responses are the important characteristics of structures. In this paper, the bending solution of the thin and thick beams which are elastically restrained against rotation and translation are presented using various theories. Hence, accurate and direct modeling technique is offered for modeling of the thin and thick beams. The effect of the values of the span-to-depth ratio and type of the beam supports are assessed to state accurate comparison of various theories. Finally, the numerical examples are shown in order to present the evaluation of the efficiency and simplicity of the various theories. The results of the theories are compared with the results of the finite element method (ABAQUS). Based on the results, using the Timoshenko beam theory, the obtained values are in good agreement with the Finite Element modeling for the values of the span-to-depth ratio (L/h) less than 3. On the other hands, due to ignoring the shear deformation effect, the Euler-Bernoulli theory underestimates the deflection of the moderately deep beams (L/h=5).

In this study, the dynamic stability of an embedded viscoelastic composite cylindrical shell reinforced by boron nitride nanotubes (BNNTs) is investigated. The composite cylindrical shell is coated by a viscoelastic piezomagnetic layer and subjected to combined magneto-electro-mechanical loads. The composite polymer matrix and the coating layer are made of polyvinylidene fluoride (PVDF) and iron oxide (CoFe2O4), respectively. The composite cylindrical shell conveys pulsating fluid flows, which results in harmonic oscillations. The equivalent characteristics of composite are determined using micro-electro-mechanical models. Considering the magneto-electro-mechanical coupling, motion equations are obtained using Hamilton’s principle. Results show the influences of fluid velocity, geometrical parameters of shell, viscoelastic foundation, orientation angle and percentage of BNNTs on the resonance frequency and stability of a PVDF-coated nanocomposite shell. The result of this study may be used for the design of rotating machines, hydraulic systems and motors.

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