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A four-variable shear and normal deformable quasi-3D beam model to analyze the free and forced vibrations of FG-GPLRC beams under moving load

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Abstract

This paper mainly focuses on the dynamic response of composite beams reinforced with graphene platelets subjected to a moving load. The governing equations of motion of the beam are obtained by using a quasi-3D theory which takes the through-the-thickness shear strains and thickness stretching effects into account. By employing the Halpin–Tsai micro-mechanical rule, which captures the dimensions of the reinforcement, the elastic modulus of reinforced composite is evaluated. A moving load with constant magnitude and velocity, which moves on the free surface of the beam is assumed. In order to discretize the governing equations and obtain a set of ordinary time-dependent equations, the Navier solution which is proper for simply-supported boundary conditions is used. Since the governing equations are time-dependent, a Newmark algorithm is used to acquire the temporal evolution of displacements in FG-GPLRC beams. To be assured of the accuracy of results, simple cases are compared with other data in the open literature, initially. After that, novel results are presented to investigate different factors regarding natural frequencies and dynamic deflection of composite beams reinforced with graphene platelets. It is indicated that the adopted theory has sufficient accuracy to estimate the natural frequencies and dynamic response of arbitrary thick FG-GPLRC beams.

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Authors and Affiliations

  1. Mechanical Engineering Department, Shahrekord University, Shahrekord, Iran

    P. Jafari

  2. Faculty of Engineering, Shahrekord University, Shahrekord, Iran

    Y. Kiani

  3. Nanotechnology Research Institute, Shahrekord University, Shahrekord, Iran

    Y. Kiani

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Appendix

Appendix

The elements of mass and stiffness matrices associated to Eq. (29) are

$$M_{11} = I_{1}$$
$$M_{12} = - \alpha_{m} I_{2}$$
$$M_{13} = I_{4}$$
$$M_{14} = 0$$
$$M_{22} = I_{1} + I_{3} \alpha_{m}^{2}$$
$$M_{23} = - \alpha_{m} I_{5}$$
$$M_{24} = I_{7}$$
$$M_{33} = I_{6}$$
$$M_{34} = 0$$
$$M_{44} = I_{8}$$
$$K_{11} = A_{11} \alpha_{m}^{2}$$
$$K_{12} = - B_{11} \alpha_{m}^{3}$$
$$K_{13} = C_{11} \alpha_{m}^{2}$$
$$K_{14} = - D_{13} \alpha_{m}$$
$$K_{22} = E_{11} \alpha_{m}^{4}$$
$$K_{23} = - F_{11} \alpha_{m}^{3}$$
$$K_{24} = G_{13} \alpha_{m}^{2}$$
$$K_{33} = H_{11} \alpha_{m}^{2} + L_{55}$$
$$K_{34} = \left( {L_{55} - J_{13} } \right)\alpha_{m}$$
$$K_{44} = L_{55} \alpha_{m}^{2} + P_{33}$$

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Jafari, P., Kiani, Y. A four-variable shear and normal deformable quasi-3D beam model to analyze the free and forced vibrations of FG-GPLRC beams under moving load. Acta Mech 233, 2797–2814 (2022). https://doi.org/10.1007/s00707-022-03256-w

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