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Vibration and Buckling Analysis of Elastically Supported Bi-directional FGM Mindlin Circular Plates Having Variable Thickness

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Abstract

In the present study, the effect of non-linear thickness variation and that of Winkler’s foundation is investigated on radially symmetric vibrations of bi-directional FGM circular plates subjected to uniform in-plane peripheral loading. It is considered that the elastic modulus, as well as density of the FGM, varies in two mutually perpendicular directions (i.e., radial and transverse). This variation follows the power law. However, the variation in thickness of FGM plate varies along the radial direction and follows quadratic variation. Hamilton’s principle is used to obtain the coupled differential equations governing the motion. The approximate solutions of these equations for simply supported and clamped edge conditions are obtained by an elegant numerical technique: the Harmonic differential quadrature method. The effect of density parameter, gradient index, heterogeneity parameter, taper parameters, and thickness parameter of the plate is analyzed on the vibration characteristics for the first two modes of vibration for different values of foundation parameter together with in-plane peripheral loading parameter. Furthermore, critical buckling loads in compression by taking the frequencies zero are evaluated for both plates. The validity of the current approach is established by comparing the frequency parameter with true values and with the results of other authors.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The authors are thankful for the constructive comments from the reviewers to improve the quality of the article.

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Authors and Affiliations

  1. Department of Mathematics, Jaypee Institute of Information Technology, Noida, India

    Neha Ahlawat

  2. Department of Mathematics (Applied Sciences), School of Engineering and Technology, H N B Garhwal University (A Central University), Srinagar, Uttarakhand, India

    Rahul Saini

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Appendix

Appendix

$$X = \left[ {\begin{array}{*{20}c} {W_1 } & {W_2 } & {W_3 } & \ldots & {W_m } & {\phi_1 } & {\phi_2 } & {\phi_3 } & \ldots & {\phi_m } \\ \end{array} } \right]^{\prime} , F^C = \left[ {\begin{array}{*{20}c} {A_{ij}^{\left( {11} \right)} } & {A_{ij}^{\left( {12} \right)} } \\ {A_{ij}^{\left( {21} \right)} } & {A_{ij}^{\left( {22} \right)} } \\ {A_{R1} } & {A_{R2} } \\ \end{array} } \right]_{\left( {2m - 2} \right) \times 2m} ,$$
$${\text{where }}A_{ij}^{\left( {11} \right)} = U_{5,i} C_{ij}^{\left( 1 \right)} ,{ }i = 2,{ }3,{ }4,{ } \ldots ,\left( {m - 1} \right),{ }j = 1,{ }2,{ }3, \ldots ,{ }m,$$
$$A_{ij}^{\left( {12} \right)} = \left\{ {\begin{array}{*{20}c} {U_{1,i} C_{ij}^{\left( 2 \right)} + U_{2,i} C_{ij}^{\left( 1 \right)} , if i \ne j,} \\ {U_{1,i} C_{ij}^{\left( 2 \right)} + U_{2,i} C_{ij}^{\left( 1 \right)} + U_{3,i} + k_1 U_{4,i} \lambda^2 , if i = j,} \\ \end{array} i = 2, 3, 4, \ldots ,\left( {m - 1} \right), j = 1, 2, 3, \ldots ,m} \right.,$$
$$A_{ij}^{\left( {21} \right)} = \left\{ {\begin{array}{*{20}c} {V_{1,i} C_{ij}^{\left( 2 \right)} + V_{2,i} C_{ij}^{\left( 1 \right)} , if i \ne j,} \\ {V_{1,i} C_{ij}^{\left( 2 \right)} + V_{2,i} C_{ij}^{\left( 1 \right)} + V_{3,i} \lambda^2 + V_{4,i} , if i = j,} \\ \end{array} i = 2, 3, 4, \ldots ,\left( {m - 1} \right), j = 1, 2, 3, \ldots ,m} \right.,$$
$$A_{ij}^{\left( {22} \right)} = \left\{ {\begin{array}{*{20}c} {V_{5,i} C_{ij}^{\left( 1 \right)} , if i \ne j,} \\ {V_{5,i} C_{ij}^{\left( 1 \right)} + V_{6,i} , if i = j,} \\ \end{array} i = 2, 3, 4, \ldots ,\left( {m - 1} \right), j = 1, 2, 3, \ldots ,m} \right.,$$
$${\text{Regularity conditions }}A_{R1} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & \ldots & 0 & 0 \\ {C_{n1}^{\left( 1 \right)} } & {C_{n2}^{\left( 1 \right)} } & {C_{n3}^{\left( 1 \right)} } & \ldots & {C_{nn}^{\left( 1 \right)} } & {C_{nn}^{\left( 1 \right)} } \\ \end{array} } \right]_{2 \times m} ,{ }A_{R2} = \left[ {\begin{array}{*{20}c} {{ }1} & 0 & 0 & \ldots & 0 \\ {{ }1} & 0 & 0 & \ldots & 0 \\ \end{array} } \right]_{2 \times m} ,$$
$$F^S = \left[ {\begin{array}{*{20}c} { 0} & 0 & 0 & \ldots & 1 & 0 & 0 & 0 & \ldots & 0 & 0 \\ { 0} & 0 & 0 & \ldots & 0 & {C_{n1}^{\left( 1 \right)} } & {C_{n2}^{\left( 1 \right)} } & {C_{n3}^{\left( 1 \right)} } & \ldots & {C_{nn}^{\left( 1 \right)} } & {C_{nn}^{\left( 1 \right)} + \nu } \\ \end{array} } \right]_{2 \times 2m} ,$$
$$F^C = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & \ldots & 1 & 0 & 0 & 0 & \ldots & 0 \\ 0 & 0 & 0 & \ldots & 0 & 0 & 0 & 0 & \ldots & 1 \\ \end{array} } \right]_{2 \times 2m} .$$

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Ahlawat, N., Saini, R. Vibration and Buckling Analysis of Elastically Supported Bi-directional FGM Mindlin Circular Plates Having Variable Thickness. J. Vib. Eng. Technol. 12, 513–532 (2024). https://doi.org/10.1007/s42417-023-00856-1

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