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Fluid–structure–soil interaction effects on the free vibrations of functionally graded sandwich plates

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Abstract

This study investigates the effects of fluid–structure and soil–structure interaction on the free vibration response of functionally graded sandwich plates. To this aim, an exemplary problem is analyzed, whereas a metal/ceramic sandwich plate is placed at the bottom of a tank filled in with fluid. Two cases are considered: (i) soft core, i.e., a sandwich plate with metal core and ceramic skins, and (ii) hard core, i.e., a sandwich plate with ceramic core and metal skins. In both cases, the skins are modelled as suitable functionally graded materials (FGMs). The soil is modelled as a Pasternak foundation. The free vibration analysis is carried out according to the extended higher order sandwich plate theory (EHSAPT). The fluid is assumed to be inviscid, incompressible, and irrotational. Hamilton’s principle is exploited to deduce the governing equations and the corresponding boundary conditions. The Rayleigh–Ritz method with two-variable orthogonal polynomials is used to compute the natural frequencies of the sandwich plate. The adopted approach is first validated through comparison with results published in the literature. Then, the effects are studied of several parameters on the dynamic response of the system.

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Acknowledgement

The first and second authors acknowledge the funding support of Babol Noshirvani University of Technology through Grant program No. BNUT/964113035/97.

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Correspondence to Ramazan-Ali Jafari-Talookolaei.

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Appendices

Appendix A

The eleven governing equations of motion (three equations for the top face sheet, five equations for the core, and three equations for the bottom face sheet) for the FG sandwich plate resting on a Pasternak foundation in contact with fluid are as follows:

For the top face sheet:

$$\begin{gathered} N_{xx,x}^{t} + N_{xy,y}^{t} + \frac{2}{{f_{c}^{2} }}M_{2xx,x}^{c} + \frac{4}{{f_{c}^{3} }}M_{3xx,x}^{c} + \frac{2}{{f_{c}^{2} }}M_{2xy,y}^{c} + \frac{4}{{f_{c}^{3} }}M_{3xy,y}^{c} + \frac{4}{{f_{c}^{2} }}M_{Q1xz}^{c} + \frac{12}{{f_{c}^{3} }}M_{Q2xz}^{c} - \hfill \\ I_{0}^{t} u_{0,tt}^{t} + I_{1}^{t} w_{0,xtt}^{t} - \frac{2}{{f_{c}^{2} }}I_{2}^{c} u_{0,tt} - \frac{4}{{f_{c}^{3} }}I_{3}^{c} u_{0,tt} - \frac{2}{{f_{c}^{2} }}I_{3}^{c} u_{1,tt} - \frac{4}{{f_{c}^{3} }}I_{4}^{c} u_{1,tt} - \frac{2}{{f_{c}^{2} }}I_{4}^{c} u_{2,tt} - \frac{4}{{f_{c}^{3} }}I_{5}^{c} u_{2,tt} - \hfill \\ \frac{2}{{f_{c}^{2} }}I_{5}^{c} u_{3,tt} - \frac{4}{{f_{c}^{3} }}I_{6}^{c} u_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a1)
$$\begin{gathered} N_{yy,y}^{t} + N_{xy,x}^{t} + \frac{2}{{f_{c}^{2} }}M_{2yy,y}^{c} + \frac{4}{{f_{c}^{3} }}M_{3yy,y}^{c} + \frac{2}{{f_{c}^{2} }}M_{2xy,x}^{c} + \frac{4}{{f_{c}^{3} }}M_{3xy,x}^{c} + \frac{4}{{f_{c}^{2} }}M_{Q1yz}^{c} + \frac{12}{{f_{c}^{3} }}M_{Q2yz}^{c} - \hfill \\ I_{0}^{t} v_{0,tt}^{t} + I_{1}^{t} w_{0,ytt}^{t} - \frac{2}{{f_{c}^{2} }}I_{2}^{c} v_{0,tt} - \frac{4}{{f_{c}^{3} }}I_{3}^{c} v_{0,tt} - \frac{2}{{f_{c}^{2} }}I_{3}^{c} v_{1,tt} - \frac{4}{{f_{c}^{3} }}I_{4}^{c} v_{1,tt} - \frac{2}{{f_{c}^{2} }}I_{4}^{c} v_{2,tt} - \frac{4}{{f_{c}^{3} }}I_{5}^{c} v_{2,tt} - \hfill \\ \frac{2}{{f_{c}^{2} }}I_{5}^{c} v_{3,tt} - \frac{4}{{f_{c}^{3} }}I_{6}^{c} v_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a2)
$$\begin{gathered} - M_{xx,xx}^{t} - M_{yy,yy}^{t} - 2M_{xy,xy}^{t} + \frac{{f_{t} }}{{f_{c}^{2} }}M_{2xx,xx}^{c} + \frac{{f_{t} }}{{f_{c}^{3} }}M_{3xx,xx}^{c} + \frac{{f_{t} }}{{f_{c}^{2} }}M_{2yy,yy}^{c} + \frac{{f_{t} }}{{f_{c}^{3} }}M_{3yy,yy}^{c} + \frac{1}{{f_{c} }}R_{z}^{c} + \hfill \\ \frac{4}{{f_{c}^{2} }}M_{z}^{c} + \frac{{f_{t} }}{{f_{c}^{2} }}M_{2xy,xy}^{c} + \frac{{2f_{t} }}{{f_{c}^{3} }}M_{3xy,xy}^{c} + \frac{{f_{t} }}{{f_{c}^{2} }}M_{2xy,xy}^{c} + \frac{{2f_{t} }}{{f_{c}^{3} }}M_{3xy,xy}^{c} + \frac{{2f_{t} }}{{f_{c}^{3} }}M_{Q1xz,x}^{c} + \frac{{6f_{t} }}{{f_{c}^{3} }}M_{Q2xz,y}^{c} + \hfill \\ \frac{1}{{f_{c} }}M_{Q1xz,x}^{c} + \frac{2}{{f_{c}^{2} }}M_{Q2xz,x}^{c} + \frac{{2f_{t} }}{{f_{c}^{2} }}M_{Qnyz,y}^{c} + \frac{{6f_{t} }}{{f_{c}^{3} }}M_{Q2yz,y}^{c} + \frac{1}{{f_{c} }}M_{Q1yz,y}^{c} + \frac{2}{{f_{c}^{2} }}M_{Q2yz,y}^{c} - I_{0}^{t} w_{0,tt}^{t} + \hfill \\ I_{1}^{t} u_{0,xtt}^{t} + I_{1}^{t} v_{0,ytt}^{t} - I_{2}^{t} w_{0,xxtt}^{t} - I_{2}^{t} w_{0,yytt}^{t} - \frac{{f_{t} }}{{f_{c}^{2} }}I_{2}^{c} u_{0,xtt} - \frac{{2f_{t} }}{{f_{c}^{3} }}I_{3}^{c} u_{0,xtt} - \frac{{f_{t} }}{{f_{c}^{2} }}I_{3}^{c} u_{1,xtt} - \hfill \\ \frac{{2f_{t} }}{{f_{c}^{3} }}I_{4}^{c} u_{1,xtt} - \frac{{f_{t} }}{{f_{c}^{2} }}I_{4}^{c} u_{2,xtt} - \frac{{2f_{t} }}{{f_{c}^{3} }}I_{5}^{c} u_{2,xtt} - \frac{{f_{t} }}{{f_{c}^{2} }}I_{5}^{c} u_{3,xtt} - \frac{{2f_{t} }}{{f_{c}^{3} }}I_{6}^{c} u_{3,xtt} - \frac{{f_{t} }}{{f_{c}^{2} }}I_{2}^{c} v_{0,y} - \frac{{2f_{t} }}{{f_{c}^{3} }}I_{3}^{c} v_{0,y} - \hfill \\ \frac{{f_{t} }}{{f_{c}^{2} }}I_{3}^{c} v_{1,y} - \frac{{2f_{t} }}{{f_{c}^{3} }}I_{4}^{c} v_{1,y} - \frac{{f_{t} }}{{f_{c}^{2} }}I_{4}^{c} v_{2,ytt} - \frac{{2f_{t} }}{{f_{c}^{3} }}I_{5}^{c} v_{2,ytt} - \frac{{f_{t} }}{{f_{c}^{2} }}I_{5}^{c} v_{3,ytt} - \frac{{2f_{t} }}{{f_{c}^{3} }}I_{6}^{c} v_{3,ytt} - \frac{1}{{f_{c} }}I_{1}^{c} w_{0,tt} - \hfill \\ \frac{2}{{f_{c}^{2} }}I_{2}^{c} w_{0,tt} - \frac{1}{{f_{c} }}I_{2}^{c} w_{1,tt} - \frac{2}{{f_{c}^{2} }}I_{3}^{c} w_{1,tt} - \frac{1}{{f_{c} }}I_{3}^{c} w_{2,tt} - \frac{2}{{f_{c}^{2} }}I_{4}^{c} w_{2,tt} - \frac{1}{2}\rho_{f} \dot{\phi } = 0 \hfill \\ \end{gathered}$$
(a3)

For the core:

$$\begin{gathered} N_{xx,x}^{c} - \frac{4}{{f_{c}^{2} }}M_{2xx,x}^{c} + N_{xy,y}^{c} - \frac{4}{{f_{c}^{2} }}M_{2xy,y}^{c} - \frac{8}{{f_{c}^{2} }}M_{Q1xz}^{c} - I_{0}^{c} u_{0,tt} + \frac{4}{{f_{c}^{2} }}I_{2}^{c} u_{0,tt} - I_{1}^{c} u_{1,tt} + \frac{4}{{f_{c}^{2} }}I_{3}^{c} u_{1,tt} - \hfill \\ I_{2}^{c} u_{2,tt} + \frac{4}{{f_{c}^{2} }}I_{4}^{c} u_{2,tt} - I_{3}^{c} u_{3,tt} + \frac{4}{{f_{c}^{2} }}I_{5}^{c} u_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a4)
$$\begin{gathered} M_{1xx,x}^{c} - \frac{4}{{f_{c}^{2} }}M_{3xx,x}^{c} + M_{1xy,y}^{c} - \frac{4}{{f_{c}^{2} }}M_{3xy,y}^{c} + Q_{xz}^{c} - \frac{12}{{f_{c}^{2} }}M_{Q2xz}^{c} - I_{1}^{c} u_{0,tt} + \frac{4}{{f_{c}^{2} }}I_{3}^{c} u_{0,tt} - I_{2}^{c} u_{1,tt} \hfill \\ + \frac{4}{{f_{c}^{2} }}I_{4}^{c} u_{1,tt} - I_{3}^{c} u_{2,tt} + \frac{4}{{f_{c}^{2} }}I_{5}^{c} u_{2,tt} - I_{4}^{c} u_{3,tt} + \frac{4}{{f_{c}^{2} }}I_{6}^{c} u_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a5)
$$\begin{gathered} N_{yy,y}^{c} - \frac{4}{{f_{c}^{2} }}M_{2yy,y}^{c} + N_{xy,x}^{c} - \frac{4}{{f_{c}^{2} }}M_{2xy,x}^{c} - \frac{8}{{f_{c}^{2} }}M_{Q1yz}^{c} - I_{0}^{c} v_{0,tt} + \frac{4}{{f_{c}^{2} }}I_{2}^{c} v_{0,tt} - I_{1}^{c} v_{1,tt} + \frac{4}{{f_{c}^{2} }}I_{3}^{c} v_{1,tt} - \hfill \\ I_{2}^{c} v_{2,tt} + \frac{4}{{f_{c}^{2} }}I_{4}^{c} v_{2,tt} - I_{3}^{c} v_{3,tt} + \frac{4}{{f_{c}^{2} }}I_{5}^{c} v_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a6)
$$\begin{gathered} M_{1yy,y}^{c} - \frac{4}{{f_{c}^{2} }}M_{3yy,y}^{c} + M_{1xy,y}^{c} - \frac{4}{{f_{c}^{2} }}M_{3xy,x}^{c} + Q_{yz}^{c} - \frac{12}{{f_{c}^{2} }}M_{Q2xz}^{c} - I_{1}^{c} v_{0,tt} + \frac{4}{{f_{c}^{2} }}I_{3}^{c} v_{0,tt} - I_{2}^{c} v_{1,tt} + \hfill \\ \frac{4}{{f_{c}^{2} }}I_{4}^{c} v_{1,tt} - I_{3}^{c} v_{2,tt} + \frac{4}{{f_{c}^{2} }}I_{5}^{c} v_{2,tt} - I_{4}^{c} v_{3,tt} + \frac{4}{{f_{c}^{2} }}I_{6}^{c} v_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a7)
$$Q_{xz,x}^{c} - \frac{8}{{f_{c}^{2} }}M_{z}^{c} - \frac{4}{{f_{c}^{2} }}M_{Q2xz,x}^{c} + Q_{yz,y}^{c} - \frac{4}{{f_{c}^{2} }}M_{Q2yz,y}^{c} - I_{0}^{c} w_{0,tt} + \frac{4}{{f_{c}^{2} }}I_{2}^{c} w_{0,tt} - I_{1}^{c} w_{1,tt} + \frac{4}{{f_{c}^{2} }}I_{3}^{c} w_{1,tt} - I_{2}^{c} w_{2,tt} - \frac{4}{{f_{c}^{2} }}I_{4}^{c} w_{2,tt} = 0$$
(a8)

For the bottom face sheet:

$$\begin{gathered} N_{xx,x}^{b} + N_{xy,y}^{b} + \frac{2}{{f_{c}^{2} }}M_{2xx,x}^{c} - \frac{4}{{f_{c}^{3} }}M_{3xx,x}^{c} + \frac{2}{{f_{c}^{2} }}M_{2xy,y}^{c} - \frac{4}{{f_{c}^{3} }}M_{3xy,y}^{c} + \frac{4}{{f_{c}^{2} }}M_{Q1xz}^{c} - \frac{12}{{f_{c}^{3} }}M_{Q2xz}^{c} - \hfill \\ I_{0}^{b} u_{0,tt}^{b} + I_{1}^{b} w_{0,xtt}^{b} - \frac{2}{{f_{c}^{2} }}I_{2}^{c} u_{0,tt} + \frac{4}{{f_{c}^{3} }}I_{3}^{c} u_{0,tt} - \frac{2}{{f_{c}^{2} }}I_{3}^{c} u_{1,tt} + \frac{4}{{f_{c}^{3} }}I_{4}^{c} u_{1,tt} - \frac{2}{{f_{c}^{2} }}I_{4}^{c} u_{2,tt} + \frac{4}{{f_{c}^{3} }}I_{5}^{c} u_{2,tt} - \hfill \\ \frac{2}{{f_{c}^{2} }}I_{5}^{c} u_{3,tt} + \frac{4}{{f_{c}^{3} }}I_{6}^{c} u_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a9)
$$\begin{gathered} N_{yy,y}^{b} + N_{xy,x}^{b} + \frac{2}{{f_{c}^{2} }}M_{2yy,y}^{c} - \frac{4}{{f_{c}^{3} }}M_{3yy,y}^{c} + \frac{2}{{f_{c}^{2} }}M_{2xy,x}^{c} - \frac{4}{{f_{c}^{3} }}M_{3xy,x}^{c} + \frac{4}{{f_{c}^{2} }}M_{Q1yz}^{c} - \frac{12}{{f_{c}^{3} }}M_{Q2yz}^{c} - \hfill \\ I_{0}^{b} b_{0,tt}^{b} + I_{1}^{b} w_{0,ytt}^{b} - \frac{2}{{f_{c}^{2} }}I_{2}^{c} v_{0,tt} + \frac{4}{{f_{c}^{3} }}I_{3}^{c} v_{0,tt} - \frac{2}{{f_{c}^{2} }}I_{3}^{c} v_{1,tt} + \frac{4}{{f_{c}^{3} }}I_{4}^{c} v_{1,tt} - \frac{2}{{f_{c}^{2} }}I_{4}^{c} v_{2,tt} + \frac{4}{{f_{c}^{3} }}I_{5}^{c} v_{2,tt} - \hfill \\ \frac{2}{{f_{c}^{2} }}I_{5}^{c} v_{3,tt} + \frac{4}{{f_{c}^{3} }}I_{6}^{c} v_{3,tt} = 0 \hfill \\ \end{gathered}$$
(a10)
$$\begin{gathered} - M_{xx,xx}^{b} - M_{yy,yy}^{b} - 2M_{xy,xy}^{b} - \frac{{f_{b} }}{{f_{c}^{2} }}M_{2xx,xx}^{c} + \frac{{f_{b} }}{{f_{c}^{3} }}M_{3xx,xx}^{c} - \frac{{f_{b} }}{{f_{c}^{2} }}M_{2yy,yy}^{c} + \frac{{f_{b} }}{{f_{c}^{3} }}M_{3yy,yy}^{c} - \frac{1}{{f_{c} }}R_{z}^{c} + \hfill \\ \frac{4}{{f_{c}^{2} }}M_{z}^{c} - \frac{{f_{b} }}{{f_{c}^{2} }}M_{2xy,xy}^{c} + \frac{{2f_{b} }}{{f_{c}^{3} }}M_{3xy,xy}^{c} - \frac{{f_{b} }}{{f_{c}^{2} }}M_{2xy,xy}^{c} + \frac{{2f_{b} }}{{f_{c}^{3} }}M_{3xy,xy}^{c} - \frac{{2f_{b} }}{{f_{c}^{2} }}M_{Q1xz,x}^{c} + \frac{{6f_{b} }}{{f_{c}^{3} }}M_{Q2xz,y}^{c} - \hfill \\ \frac{1}{{f_{c} }}M_{Q1xz,x}^{c} + \frac{2}{{f_{c}^{2} }}M_{Q2xz,x}^{c} - \frac{{2f_{t} }}{{f_{c}^{2} }}M_{Q1yz,y}^{c} + \frac{{6f_{b} }}{{f_{c}^{3} }}M_{Q2yz,y}^{c} - \frac{1}{{f_{c} }}M_{Q1yz,y}^{c} + \frac{2}{{f_{c}^{2} }}M_{Q2yz,y}^{c} - I_{0}^{b} w_{0,tt}^{b} + \hfill \\ I_{1}^{b} u_{0,xtt}^{b} + I_{1}^{b} v_{0,ytt}^{b} - I_{2}^{b} w_{0,xxtt}^{b} - I_{2}^{b} w_{0,yytt}^{b} + \frac{{f_{b} }}{{f_{c}^{2} }}I_{2}^{c} u_{0,xtt} - \frac{{2f_{b} }}{{f_{c}^{3} }}I_{3}^{c} u_{0,xtt} + \frac{{f_{b} }}{{f_{c}^{2} }}I_{3}^{c} u_{1,xtt} - \hfill \\ \frac{{2f_{b} }}{{f_{c}^{3} }}I_{4}^{c} u_{1,xtt} + \frac{{f_{b} }}{{f_{c}^{2} }}I_{4}^{c} u_{2,xtt} - \frac{{2f_{b} }}{{f_{c}^{3} }}I_{5}^{c} u_{2,xtt} + \frac{{f_{b} }}{{f_{c}^{2} }}I_{5}^{c} u_{3,xtt} - \frac{{2f_{b} }}{{f_{c}^{3} }}I_{6}^{c} u_{3,xtt} + \frac{{f_{b} }}{{f_{c}^{2} }}I_{2}^{c} v_{0,ytt} - \frac{{2f_{b} }}{{f_{c}^{3} }}I_{3}^{c} v_{0,ytt} + \hfill \\ \frac{{f_{b} }}{{f_{c}^{2} }}I_{3}^{c} v_{1,ytt} - \frac{{2f_{b} }}{{f_{c}^{3} }}I_{4}^{c} v_{1,ytt} + \frac{{f_{b} }}{{f_{c}^{2} }}I_{4}^{c} v_{2,ytt} - \frac{{2f_{b} }}{{f_{c}^{3} }}I_{5}^{c} v_{2,ytt} + \frac{{f_{b} }}{{f_{c}^{2} }}I_{5}^{c} v_{3,ytt} - \frac{{2f_{b} }}{{f_{c}^{3} }}I_{6}^{c} v_{3,ytt} + \frac{1}{{f_{c} }}I_{1}^{c} w_{0,tt} - \hfill \\ \frac{2}{{f_{c}^{2} }}I_{2}^{c} w_{0,tt} + \frac{1}{{f_{c} }}I_{2}^{c} w_{1,tt} - \frac{2}{{f_{c}^{2} }}I_{3}^{c} w_{1,tt} + \frac{1}{{f_{c} }}I_{3}^{c} w_{2,tt} - \frac{2}{{f_{c}^{2} }}I_{4}^{c} w_{2,tt} + k_{w} w_{0}^{b} + k_{g} w_{0,xx}^{b} + k_{g} w_{0,yy}^{b} = 0 \hfill \\ \end{gathered}$$
(a11)

Moreover, the corresponding boundary conditions are as follows:

At \(x = 0\) and \(x = a\):

$$\begin{gathered} N_{{xx}}^{i} = 0~\,{\text{or}}\,~u_{0}^{i} = 0~~;~~M_{{xx}}^{i} = 0~\,{\text{or}}\,w_{{0,x}}^{i} = 0~~;~~N_{{xy}}^{i} = 0\,{\text{or}}\,~v_{0}^{i} = 0 \hfill \\ M_{{xy}}^{i} = 0~\,{\text{or}}\,~w_{{0,y}}^{i} = 0~~;~~Q_{{xz}}^{i} = 0~\,{\text{or}}\,~w_{0}^{i} = 0~~;~~M_{{Q1xz}}^{c} = 0~\,{\text{or}}\,w_{1} = 0 \hfill \\ M_{{Q2xz}}^{c} = 0~\,{\text{or}}\,w_{2} = 0~~;~~Q_{{xz}}^{c} = 0~\,{\text{or}}\,w_{0} = 0~~;~~N_{{xx}}^{c} = 0~\,{\text{or}}\,u_{0} = 0 \hfill \\ M_{{xx}}^{c} = 0~\,{\text{or}}\,u_{1} = 0~~;~~M_{{2xx}}^{c} = 0\,{\text{or}}\,u_{2} = 0~~;~~M_{{3xx}}^{c} = 0\,{\text{or}}\,u_{3} = 0 \hfill \\ N_{{xy}}^{c} = 0~\,{\text{or}}\,v_{0} = 0~~;~~M_{{xy}}^{c} = 0\,{\text{or}}\,v_{1} = 0~~;~~M_{{2xy}}^{c} = 0\,{\text{or}}\,v_{2} = 0 \hfill \\ M_{{3xy}}^{c} = 0~\,{\text{or}}\,v_{3} = 0 \hfill \\ \end{gathered}$$
(a12)

At \(y = 0\) and \(y = a\):

$$\begin{gathered} N_{{yy}}^{i} = 0\,{\text{or}}\,v_{0}^{i} = 0~~;~~M_{{yy}}^{i} = 0~\,{\text{or}}\,w_{{0,y}}^{i} = 0~~;~~N_{{xy}}^{i} = 0\,{\text{or}}\,u_{0}^{i} = 0; \hfill \\ M_{{xy}}^{i} = 0~\,{\text{or}}\,w_{{0,y}}^{i} = 0~~;~~Q_{{yz}}^{i} = 0~\,{\text{or}}\,w_{0}^{i} = 0~~;~~M_{{Q1yz}}^{c} = 0~\,{\text{or}}\,w_{1} = 0~; \hfill \\ M_{{Q2yz}}^{c} = 0~\,{\text{or}}\,w_{2} = 0~~;~~Q_{{yz}}^{c} = 0~\,{\text{or}}\,w_{0} = 0~~;~~N_{{xy}}^{c} = 0\,{\text{or}}\,u_{0} = 0~~; \hfill \\ M_{{xy}}^{c} = 0\,{\text{or}}\,u_{1} = 0~~;~~M_{{2xy}}^{c} = 0~\,{\text{or}}\,u_{2} = 0~~;~~M_{{3xy}}^{c} = 0\,{\text{or}}\,u_{3} = 0~~; \hfill \\ N_{{yy}}^{c} = 0~\,{\text{or}}\,v_{0} = 0~~;~~M_{{yy}}^{c} = 0\,{\text{or}}\,v_{1} = 0~~;~~M_{{2yy}}^{c} = 0\,{\text{or}}\,v_{2} = 0~~; \hfill \\ M_{{3yy}}^{c} = 0\,{\text{or}}\,v_{3} = 0~; \hfill \\ \end{gathered}$$
(a13)

Appendix B

The base functions \(\left( {\lambda_{i}^{k} \left( {k = u^{t} , v^{t} , w^{t} , \ldots , v_{1} } \right)} \right)\) can be considered as follows:

a)

$$\lambda_{1}^{{w^{l} }} = x^{{\gamma_{1} }} y^{{\gamma_{2} }} \left( {x - a} \right)^{{\gamma_{1} }} \left( {y - b} \right)^{{\gamma_{2} }}$$
(b1)

where \(l=t,b,c\) and

$$\gamma_{1,2} = \left\{ {\begin{array}{ll} 0 & {\text{if edge is free}} \\ 1 & {\text{if edge is simply - suppoted}} \\ 2 & {\text{if edge is clampedif edge is clamped}} \\ \end{array} } \right.$$

b)

$$\lambda_{1}^{{u^{l} }} = x^{{\beta_{1} }} y^{{\beta_{2} }} \left( {x - a} \right)^{{\beta_{1} }} \left( {y - b} \right)^{{\beta_{2} }}$$
(b2)

and

$$\lambda_{1}^{{u^{l} }} = \lambda_{1}^{{x^{k} }} = \lambda_{1}^{{u_{1}^{c} }}$$
(b3)

where \(l=t,b,c, k=t,b\) and

$$\beta_{1,2} = \left\{ {\begin{array}{ll} 0 & {{\text{if edge is free or simply}} - {\text{suported in }}y - {\text{direction}}} \\ 1 & {{\text{if edge is simply}} - {\text{suported or clamped in}} x - {\text{direction }}} \\ \end{array} } \right.$$

c)

$$\lambda_{1}^{{v^{l} }} = x^{{\xi_{1} }} y^{{\xi_{2} }} \left( {x - a} \right)^{{\xi_{1} }} \left( {y - b} \right)^{{\xi_{2} }}$$
(b4)

and

$$\lambda_{1}^{{v^{l} }} = \lambda_{1}^{{y^{k} }} = \lambda_{1}^{{v_{1}^{c} }}$$
(b5)

where \(l=t,b,c, k=t,b\) and:

$$\xi_{1,2} = \left\{ {\begin{array}{ll} 0 & {{\text{if edge is free or simply}} - {\text{suported in}} x - {\text{direction}}} \\ 1 & {{\text{if edge is simply}} - {\text{suported or clamped in}} y - {\text{direction}}} \\ \end{array} } \right.$$

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Ramian, A., Jafari-Talookolaei, RA., Valvo, P.S. et al. Fluid–structure–soil interaction effects on the free vibrations of functionally graded sandwich plates. Engineering with Computers 38 (Suppl 3), 1901–1921 (2022). https://doi.org/10.1007/s00366-021-01348-0

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