*2.4. Governing Equations*

Hamilton's principle is used herein to derive the equations of motion. In the absence of external forces, the principle can be stated in the analytical form as [32]:

$$\int\_{t\_1}^{t\_2} (\delta lI - \delta K) dt = 0 \tag{11}$$

where δ*U* is the variation of the strain energy, δ*K* is the variation of the kinetic energy, *t*<sup>1</sup> and *t*<sup>2</sup> are arbitrary time variables. The strain energy of the plate can be calculated as:

$$\begin{array}{ll} II &= \frac{1}{2} \int \int \int \left( \sigma\_{xx} \epsilon\_{xx} + \sigma\_{yy} \epsilon\_{yy} + \tau\_{xy} \gamma\_{xy} + \tau\_{xz} \gamma\_{xz} + \tau\_{yz} \gamma\_{yz} \right) \Big( 1 + \frac{z}{R\_c} \Big) \Big( 1 + \frac{z}{R\_g} \Big) dx dy dx \\ &= \frac{1}{2} \int \int \Big( \mathbf{N}\_{xx} \epsilon\_{xx}^{0} + \mathbf{N}\_{yy} \epsilon\_{yy}^{0} + \mathbf{N}\_{xy} \gamma\_{xy}^{0} + \mathbf{M}\_{yx} \gamma\_{yx}^{0} + \mathbf{M}\_{yy}^{b} \kappa\_{yy}^{b} + \mathbf{M}\_{xy}^{b} \kappa\_{yy}^{b} + \mathbf{M}\_{yx}^{b} \kappa\_{yx}^{b} + \\ & M\_{xx}^{a} \kappa\_{xx}^{s} + M\_{yy}^{a} \kappa\_{yy}^{s} + M\_{yx}^{a} \kappa\_{xy}^{s} + M\_{yx}^{a} \kappa\_{yx}^{s} + Q\_{yy} \gamma\_{yx}^{s} + Q\_{xy} \gamma\_{yx}^{s} \Big) dx dy \end{array} \tag{12}$$

where stress resultants (*N, M* and *Q*) are defined by:

⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ *Nxx Nxy Qxs* ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ <sup>=</sup> *<sup>n</sup> k*=1 *zk*+<sup>1</sup> *zk* - 1 + *<sup>z</sup> Ry* ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ σ*k xx* σ*k xy* τ*k xz* ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ *dz*; ⎧ ⎪⎪⎪⎨ ⎪⎪⎪⎩ *Nyy Nyx Qys* ⎫ ⎪⎪⎪⎬ ⎪⎪⎪⎭ <sup>=</sup> *<sup>n</sup> k*=1 *zk*+<sup>1</sup> *zk* 1 + *<sup>z</sup> Rx* ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ σ*k yy* σ*k yx* τ*k yz* ⎫ ⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎭ *dz*; *M<sup>b</sup> xx M<sup>b</sup> xy* <sup>=</sup> *<sup>n</sup> k*=1 *zk*+<sup>1</sup> *zk* - 1 + *<sup>z</sup> Ry* σ*<sup>k</sup> xx* σ*k xy zdz*; *Mb yy M<sup>b</sup> yx* <sup>=</sup> *<sup>n</sup> k*=1 *zk*+<sup>1</sup> *zk* 1 + *<sup>z</sup> Rx* σ*<sup>k</sup> yy* σ*k yx zdz*; *M<sup>s</sup> xx M<sup>s</sup> xy* <sup>=</sup> *<sup>n</sup> k*=1 *zk*+<sup>1</sup> *zk* - 1 + *<sup>z</sup> Ry* σ*<sup>k</sup> xx* σ*k xy f*(*z*)*dz*; *M<sup>s</sup> yy M<sup>s</sup> yx* <sup>=</sup> *<sup>n</sup> k*=1 *zk*+<sup>1</sup> *zk* 1 + *<sup>z</sup> Rx* σ*<sup>k</sup> yy* σ*k yx f*(*z*)*dz*. (13)

Based on the constitutive relations (8), strain-displacement relation (6) and displacement field (4), the force and moment resultants can be rewritten in terms of displacement components as:

*Nxx Nyy Nxy Nyx M<sup>b</sup> xx Mb yy M<sup>b</sup> xy M<sup>b</sup> yx M<sup>s</sup> xx Ms yy M<sup>s</sup> xy M<sup>s</sup> yx* ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *A*<sup>11</sup> *A*<sup>12</sup> *A*<sup>16</sup> *A*<sup>16</sup> *B*<sup>11</sup> *B*<sup>12</sup> *B*<sup>16</sup> *B*<sup>16</sup> *B s* <sup>11</sup> *Bs* <sup>12</sup> *B s* <sup>16</sup> *Bs* 16 *A*<sup>12</sup> *A*ˆ <sup>22</sup> *A*<sup>26</sup> *A*ˆ <sup>26</sup> *B*<sup>12</sup> *B*ˆ <sup>22</sup> *B*<sup>26</sup> *B*ˆ <sup>26</sup> *B<sup>s</sup>* <sup>12</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* <sup>22</sup> *Bs* <sup>26</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* 26 *A*<sup>16</sup> *A*<sup>26</sup> *A*<sup>66</sup> *A*<sup>66</sup> *B*<sup>16</sup> *B*<sup>26</sup> *B*<sup>66</sup> *B*<sup>66</sup> *B s* <sup>16</sup> *Bs* <sup>26</sup> *<sup>B</sup>*<sup>66</sup> *Bs* 66 *A*<sup>16</sup> *A*ˆ <sup>26</sup> *A*<sup>66</sup> *A*ˆ <sup>66</sup> *B*<sup>16</sup> *B*ˆ <sup>26</sup> *B*<sup>66</sup> *B*ˆ <sup>66</sup> *B<sup>s</sup>* <sup>16</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* <sup>26</sup> *Bs* <sup>66</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* 66 *<sup>B</sup>*<sup>11</sup> *<sup>B</sup>*<sup>12</sup> *<sup>B</sup>*<sup>16</sup> *<sup>B</sup>*<sup>16</sup> *<sup>D</sup>*<sup>11</sup> *<sup>D</sup>*<sup>12</sup> *<sup>D</sup>*<sup>16</sup> *<sup>D</sup>*<sup>16</sup> *<sup>D</sup><sup>s</sup>* <sup>11</sup> *Ds* <sup>12</sup> *Ds* <sup>16</sup> *D<sup>s</sup>* 16 *B*<sup>12</sup> *B*ˆ <sup>22</sup> *B*<sup>26</sup> *B*ˆ <sup>26</sup> *D*<sup>12</sup> *D*ˆ <sup>22</sup> *D*<sup>26</sup> *D*ˆ <sup>26</sup> *D<sup>s</sup>* <sup>12</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* <sup>22</sup> *Ds* <sup>26</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* 26 *<sup>B</sup>*<sup>16</sup> *<sup>B</sup>*<sup>26</sup> *<sup>B</sup>*<sup>66</sup> *<sup>B</sup>*<sup>66</sup> *<sup>D</sup>*<sup>16</sup> *<sup>D</sup>*<sup>26</sup> *<sup>D</sup>*<sup>66</sup> *<sup>D</sup>*<sup>66</sup> *<sup>D</sup><sup>s</sup>* <sup>16</sup> *Ds* <sup>26</sup> *Ds* <sup>66</sup> *D<sup>s</sup>* 66 *B*<sup>16</sup> *B*ˆ <sup>26</sup> *B*<sup>66</sup> *B*ˆ <sup>66</sup> *D*<sup>16</sup> *D*ˆ <sup>26</sup> *D*<sup>66</sup> *D*ˆ <sup>66</sup> *D<sup>s</sup>* <sup>16</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* <sup>26</sup> *Ds* <sup>66</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* 66 *B s* <sup>11</sup> *Bs* <sup>12</sup> *B s* <sup>16</sup> *B<sup>s</sup>* <sup>16</sup> *<sup>D</sup><sup>s</sup>* <sup>11</sup> *Ds* <sup>12</sup> *Ds* <sup>16</sup> *Ds* <sup>16</sup> *E s* <sup>11</sup> *Es* <sup>12</sup> *E s* <sup>16</sup> *Es* 16 *Bs* <sup>12</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* <sup>22</sup> *Bs* <sup>26</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* <sup>26</sup> *<sup>D</sup><sup>s</sup>* <sup>12</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* <sup>22</sup> *Ds* <sup>26</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* <sup>26</sup> *<sup>E</sup><sup>s</sup>* <sup>12</sup> *<sup>E</sup>*ˆ*<sup>s</sup>* <sup>22</sup> *Es* <sup>26</sup> *<sup>E</sup>*ˆ*<sup>s</sup>* 26 *B s* <sup>16</sup> *Bs* <sup>26</sup> *<sup>B</sup>*<sup>66</sup> *<sup>B</sup><sup>s</sup>* <sup>66</sup> *<sup>D</sup><sup>s</sup>* <sup>16</sup> *Ds* <sup>26</sup> *Ds* <sup>66</sup> *Ds* <sup>66</sup> *E s* <sup>16</sup> *Es* <sup>26</sup> *E s* <sup>66</sup> *Es* 26 *Bs* <sup>16</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* <sup>26</sup> *Bs* <sup>66</sup> *<sup>B</sup>*ˆ*<sup>s</sup>* <sup>66</sup> *<sup>D</sup><sup>s</sup>* <sup>16</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* <sup>26</sup> *Ds* <sup>66</sup> *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup>* <sup>66</sup> *<sup>E</sup><sup>s</sup>* <sup>16</sup> *<sup>E</sup>*ˆ*<sup>s</sup>* <sup>26</sup> *Es* <sup>26</sup> *<sup>E</sup>*ˆ*<sup>s</sup>* 66 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ε0 *xx* ε0 *yy* γ0 *xy* γ0 *yx* κ*b xx* κ*b yy* κ*b xy* κ*b yx* κ*s xx* κ*s yy* κ*s xy* κ*s yx* ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ (14)

$$
\left\{ \begin{array}{c} Q\_{ys} \\ Q\_{\rm xs} \end{array} \right\} = \left[ \begin{array}{c} \mathring{A}\_{44}^{s} \\ A\_{45}^{s} \end{array} \right] \left\{ \begin{array}{c} \mathcal{V}\_{yz}^{s} \\ \mathcal{V}\_{\rm xx}^{s} \end{array} \right\} \tag{15}
$$

*J. Compos. Sci.* **2019**, *3*, 104

in which:

 *Aij*, *Bij*, *Dij*, *Bs ij*, *Ds ij*, *<sup>A</sup><sup>s</sup> ij* <sup>=</sup> *<sup>N</sup>* 1 *<sup>z</sup>*+<sup>1</sup> *zk <sup>Q</sup>*(*k*) *ij* 1, *z*, *z*2, *f*(*z*), *z f*(*z*), *g*2(*z*) *dz*; *Aij*, *Bij*, *Dij*, *B s ij*, *Ds ij*, *<sup>A</sup><sup>s</sup> ij* = *Aijx*, *Bijx*, *Dijx*, *B<sup>s</sup> ijx*, *<sup>D</sup><sup>s</sup> ijx*, *As ijx* + *Bijx*,*Dijx*,*Eijx*,*Ds ijx*,*E<sup>s</sup> ijx*,*AAs ijx Ry* ; *Aijx*, *Bijx*, *Dijx*, *Eijx*, *B<sup>s</sup> ijx*, *Ds ijx*, *Es ijx*, *As ijx*, *AA<sup>s</sup> ijx* = *N* 1 *<sup>z</sup>*+<sup>1</sup> *zk <sup>Q</sup>*(*k*) *ij* {1,*z*,*z*2,*z*3, *<sup>f</sup>*(*z*),*z f*(*z*),*z*<sup>2</sup> *<sup>f</sup>*(*z*),*g*(*z*),*zg*(*z*)} <sup>1</sup>+*z*/*Rx dz A*ˆ *ij*, *B*ˆ*ij*, *D*ˆ *ij*, *B*ˆ*<sup>s</sup> ij*, *<sup>D</sup>*<sup>ˆ</sup> *<sup>s</sup> ij*, *<sup>A</sup>*ˆ*<sup>s</sup> ij* = *Aijy*, *Bijy*, *Dijy*, *Bs ijy*, *Ds ijy*, *<sup>A</sup><sup>s</sup> ijy* + *Bijy*,*Dijy*,*Eijy*,*Ds ijy*,*Es ijy*,*AA<sup>s</sup> ijy Rx Aijy*, *Bijy*, *Dijy*, *Eijy*, *B<sup>s</sup> ijy*, *Ds ijy*, *<sup>E</sup><sup>s</sup> ijy*, *<sup>A</sup><sup>s</sup> ijy*, *AAs ijy* <sup>=</sup> *<sup>N</sup>* 1 *<sup>z</sup>*+<sup>1</sup> *zk <sup>Q</sup>*(*k*) *ij* {1,*z*,*z*2,*z*3, *<sup>f</sup>*(*z*),*z f*(*z*),*z*<sup>2</sup> *<sup>f</sup>*(*z*),*g*(*z*),*zg*(*z*)} <sup>1</sup>+*z*/*Ry dz* (16)

The variation of the kinetic energy of the panel can be written as:

$$\begin{array}{ll} K &= \frac{1}{2} \int\_{0}^{a} \int\_{0}^{b} \int\_{-h/2}^{\cdot} \rho(z) \Big(\dot{\bar{u}}^{2} + \dot{\bar{v}}^{2} + \dot{\bar{w}}^{2}\Big) \Big(1 + \frac{z}{K\_{\text{x}}}\Big) \Big(1 + \frac{z}{K\_{\text{y}}}\Big) dz dy dx \\ &= \frac{1}{2} \int\_{0}^{a} \int\_{0}^{b} \Big(\Big(\tilde{l}\_{0}\dot{\boldsymbol{u}}\_{0} + \tilde{l}\_{2}\dot{\boldsymbol{\phi}}\_{\text{x}b}^{2} + \overline{K}\_{1}\dot{\boldsymbol{\phi}}\_{\text{x\text{x}}}^{2} + 2\overline{l}\_{1}\dot{\boldsymbol{u}}\_{0}\dot{\boldsymbol{\phi}}\_{\text{x\text{y}}} + 2\overline{l}\_{1}\dot{\boldsymbol{u}}\_{0}\dot{\boldsymbol{\phi}}\_{\text{x\text{x}}} + 2\overline{l}\_{2}\dot{\boldsymbol{\phi}}\_{\text{x\text{y}}}\dot{\boldsymbol{\phi}}\_{\text{x\text{x}}} + \overline{l}\_{0}\dot{\boldsymbol{\phi}}\_{0}^{2} + \overline{l}\_{2}\dot{\boldsymbol{\phi}}\_{\text{y\text{}}}^{2} \\ &+ \overline{K}\_{1}\dot{\boldsymbol{\phi}}\_{\text{y\text{x}}}^{2} + 2\overline{l}\_{1}\dot{\boldsymbol{v}}\_{0}\dot{\boldsymbol{\phi}}\_{\text{y\text{}}} + 2\overline{l}\_{1}\dot{\boldsymbol{v}}\_{0}\dot{\boldsymbol{\phi}}\_{\text{y\text{x}}} + 2\overline{l}\_{2}\dot{\boldsymbol{\phi}}\_{\text{y\text{d}}}\dot{\boldsymbol{\phi}}\_{\text{y\text{e}}} + \overline{l}\_{0}(\dot{\boldsymbol{w}}\_{\text{$$

where:

$$\phi\_{\rm xb} = \left(\frac{u\_0}{R\_\mathbf{x}} - \frac{\partial w\_b}{\partial \mathbf{x}}\right); \phi\_{\rm xs} = -\frac{\partial w\_s}{\partial \mathbf{x}}; \phi\_{\rm yb} = \left(\frac{v\_0}{R\_\mathbf{y}} - \frac{\partial w\_b}{\partial \mathbf{y}}\right); \phi\_{\rm ys} = -\frac{\partial w\_s}{\partial \mathbf{y}} \tag{18}$$

and ρ(*z*) is the mass density, and the mass moments of inertia *Ii* (i = 0, 1, 2) are defined as [30,33]:

$$\begin{aligned} \tilde{I}\_i &= I\_i + I\_{i+1} \Big( \frac{1}{\mathcal{K}\_x} + \frac{1}{\mathcal{K}\_y} \Big) + \frac{I\_{i+2}}{\mathcal{K}\_x \mathcal{R}\_y}; \\ \{I\_0, I\_1, I\_2, I\_3\} &= \sum\_{k=1}^N \int\_{z\_k}^{z\_{i+1}} \rho(z) \{1, z, z^2, z^3\} dz; \\ \tilde{I}\_i &= f(z) \mathbb{I}\_{i-1}; \ \overline{\mathcal{K}}\_1 = f^2(z) \mathbb{I}\_0 \end{aligned} \tag{19}$$

Substituting the expressions of *U* and *K* from Equation (12) and Equation (17) into Equation (11), and by performing some mathematical manipulations, the equations of motion of the shell panel are obtained as follows:

$$\begin{aligned} 0 = -\int\_{\dot{A}} \begin{bmatrix} \frac{\partial N\_{xx}}{\partial x} + \frac{\partial N\_{yx}}{\partial y} + \frac{Q\_{ab}}{R\_x} - \tilde{I}\_0 \ddot{u}\_0 - \tilde{I}\_1 \left(\frac{\ddot{u}\_0}{R\_x} - \frac{\partial \ddot{w}\_b}{\partial x}\right) + \tilde{I}\_1 \frac{\partial \ddot{u}\_s}{\partial x} \\ \frac{\partial N\_{yy}}{\partial y} + \frac{\partial N\_{yx}}{\partial x} + \frac{Q\_{ab}}{R\_y} - \tilde{I}\_0 \ddot{w}\_0 - \tilde{I}\_1 \left(\frac{\ddot{u}\_0}{R\_y} - \frac{\partial \ddot{w}\_b}{\partial y}\right) + \tilde{I}\_1 \frac{\partial \ddot{w}\_b}{\partial y} \Big| \delta w\_0 \\ \left[ -\frac{\partial N\_{xx}}{\partial x} - \frac{\partial N\_{yy}}{R\_y} + \frac{\partial Q\_{ab}}{\partial x} + \frac{\partial Q\_{ab}}{\partial y} - \tilde{I}\_0 (\ddot{w}\_b + \ddot{w}\_s) \right] \delta w\_b \\ -\frac{\partial N\_{xx}}{\partial x} - \frac{\partial N\_{yy}}{R\_y} + \frac{\partial Q\_{ab}}{\partial x} + \frac{\partial Q\_{ab}}{\partial y} - \tilde{I}\_0 (\ddot{w}\_b + \ddot{w}\_s) \Big| \delta w\_s \\ \end{bmatrix} dA \tag{20} \\ + \int\_0^b \|\Gamma\_x\|\_0^a dy + \int\_0^a \|\Gamma\_y\|\_0^b dx \end{aligned} \tag{21}$$

*J. Compos. Sci.* **2019**, *3*, 104

where:

$$\begin{cases} Q\_{xb} = \frac{\partial M^b\_{xx}}{\partial x} + \frac{\partial M^b\_{yx}}{\partial y} - \left(\tilde{I}\_1 + \frac{\tilde{I}\_2}{R\_x}\right)\ddot{u}\_0 + \tilde{I}\_2\frac{\partial \dot{w}\_b}{\partial x} + \tilde{I}\_2\frac{\partial \dot{w}\_s}{\partial x} \\\\ Q\_{yb} = \frac{\partial M^b\_{yy}}{\partial y} + \frac{\partial M^b\_{xy}}{\partial x} - \left(\tilde{I}\_1 + \frac{\tilde{I}\_2}{R\_y}\right)\ddot{v}\_0 + \tilde{I}\_2\frac{\partial \dot{w}\_b}{\partial y} + \tilde{I}\_2\frac{\partial \dot{w}\_s}{\partial y} \\\\ \overline{Q}\_{xs} = \frac{\partial M^b\_{xx}}{\partial x} + \frac{\partial M^b\_{yx}}{\partial y} + Q\_{xs} - \left(\tilde{I}\_1 + \frac{\tilde{I}\_2}{R\_x}\right)\ddot{u}\_0 + \tilde{I}\_2\frac{\partial \dot{w}\_b}{\partial x} + \overline{K}\_1\frac{\partial \dot{w}\_s}{\partial x} \\\\ \overline{Q}\_{ys} = \frac{\partial M^b\_{yy}}{\partial y} + \frac{\partial M^b\_{xy}}{\partial x} + Q\_{ys} - \left(\tilde{I}\_1 + \frac{\tilde{I}\_2}{R\_y}\right)\ddot{v}\_0 + \tilde{I}\_2\frac{\partial \dot{w}\_b}{\partial x} + \overline{K}\_1\frac{\partial \dot{w}\_s}{\partial y} \end{cases} \tag{21}$$

and Γ*x*, Γ*<sup>y</sup>* are boundary expressions:

$$\begin{aligned} \Gamma\_{\mathbf{x}} &= \overline{N}\_{\mathbf{x}\mathbf{x}} \delta u\_0 + \overline{N}\_{\mathbf{x}\mathbf{y}} \delta v\_0 + Q\_{\mathbf{z}\mathbf{b}} \delta w\_\mathbf{b} + \overline{Q}\_{\mathbf{x}\mathbf{x}} \delta w\_\mathbf{s} + M^b\_{\mathbf{x}\mathbf{x}} \delta \tilde{\phi}\_{\mathbf{z}\mathbf{b}} + M^b\_{\mathbf{x}\mathbf{y}} \delta \tilde{\phi}\_{\mathbf{y}\mathbf{b}} + M^\mathbf{x}\_{\mathbf{x}\mathbf{x}} \delta \phi\_{\mathbf{x}\mathbf{s}} + M^\mathbf{x}\_{\mathbf{y}\mathbf{y}} \delta \phi\_{\mathbf{y}\mathbf{s}} \\\\ \Gamma\_{\mathbf{y}} &= \overline{N}\_{\mathbf{y}\mathbf{y}} \delta v\_0 + \overline{N}\_{\mathbf{y}\mathbf{x}} \delta u\_0 + Q\_{\mathbf{y}\mathbf{b}} \delta w\_\mathbf{b} + \overline{Q}\_{\mathbf{y}\mathbf{s}} \delta w\_\mathbf{s} + M^b\_{\mathbf{y}\mathbf{y}} \delta \tilde{\phi}\_{\mathbf{y}\mathbf{b}} + M^\mathbf{y}\_\mathbf{y} \delta \phi\_{\mathbf{z}\mathbf{b}} + M^\mathbf{s}\_{\mathbf{y}\mathbf{y}} \delta \phi\_{\mathbf{y}\mathbf{s}} + M^\mathbf{s}\_{\mathbf{y}\mathbf{x}} \delta \phi\_{\mathbf{z}\mathbf{s}} \end{aligned} \tag{22}$$

in which:

$$\begin{aligned} \overline{N}\_{\text{XX}} &= \left( \mathcal{N}\_{\text{xx}} - \frac{\mathcal{M}\_{\text{xx}}^{b}}{\mathcal{R}\_{\text{x}}} \right); \ \overline{N}\_{\text{xy}} = \left( \mathcal{N}\_{\text{xy}} - \frac{\mathcal{M}\_{\text{xy}}^{b}}{\mathcal{R}\_{\text{y}}} \right); \ \overline{N}\_{\text{yy}} = \left( \mathcal{N}\_{\text{yy}} - \frac{\mathcal{M}\_{\text{yy}}^{b}}{\mathcal{R}\_{\text{y}}} \right); \ \overline{N}\_{\text{yx}} = \left( \mathcal{N}\_{\text{yx}} - \frac{\mathcal{M}\_{\text{yx}}^{b}}{\mathcal{R}\_{\text{x}}} \right); \\\\ \overline{\phi}\_{\text{xb}} = -\frac{\partial w\_{\text{b}}}{\partial x}; \ \phi\_{\text{xc}} = -\frac{\partial w\_{\text{b}}}{\partial x}; \ \overline{\phi}\_{\text{y}b} = -\frac{\partial w\_{\text{b}}}{\partial y}; \ \phi\_{\text{ys}} = -\frac{\partial w\_{\text{s}}}{\partial y} \end{aligned} \tag{23}$$

By setting the coefficients of the virtual displacements δ*u*0, δ*v*0, δ*wb*, δ*ws* to zeros, the governing equations are obtained as follows:

$$\begin{aligned} \delta u\_0 &: \frac{\partial N\_{\rm tx}}{\partial x} + \frac{\partial N\_{\rm yx}}{\partial y} + \frac{Q\_{\rm sh}}{R\_x} = \tilde{I}\_0 \ddot{u}\_0 + \tilde{I}\_1 \left(\frac{\ddot{u}\_0}{R\_x} - \frac{\partial \ddot{u}\_0}{\partial y}\right) - \tilde{I}\_1 \frac{\partial \ddot{u}\_s}{\partial x} \\\\ \delta v\_0 &: \frac{\partial N\_{\rm yx}}{\partial y} + \frac{\partial N\_{\rm yx}}{\partial x} + \frac{Q\_{\rm ab}}{R\_y} = \tilde{I}\_0 \ddot{v}\_0 + \tilde{I}\_1 \left(\frac{\ddot{v}\_0}{R\_y} - \frac{\partial \ddot{v}\_0}{\partial y}\right) - \tilde{I}\_1 \frac{\partial \ddot{u}\_0}{\partial y} \\\\ \delta w\_b &: \frac{\partial N\_{\rm xx}}{R\_x} + \frac{\partial N\_{\rm yy}}{R\_y} - \frac{\partial Q\_{\rm ab}}{\partial x} - \frac{\partial Q\_{\rm ab}}{\partial y} = -\tilde{I}\_0 (\ddot{w}\_b + \ddot{w}\_b) \end{aligned} \tag{24}$$
 
$$\delta w\_s : \frac{\partial N\_{\rm xx}}{\partial x} + \frac{\partial N\_{\rm yy}}{R\_y} - \frac{\partial Q\_{\rm ax}}{\partial x} - \frac{\partial Q\_{\rm av}}{\partial y} = -\tilde{I}\_0 (\ddot{w}\_b + \ddot{w}\_s)$$
