For derivation of bending and shear stiffness matrix the ordinary finite element technique is used. Finite element deflection is expressed as product of deflection shape functions and nodal displacements

$$w\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}=\u3008{\Phi}_{k}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}\u3009\left\{\delta \right\},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}k=1,\text{\hspace{0.17em}}2\dots 12,$$(14)

where according to Figure 2

$$\mathrm{\{}\delta \mathrm{\}}=\mathrm{\{}{\mathrm{\{}\delta \mathrm{\}}}_{l}\mathrm{\}},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{}\delta {\text{}}}_{l}=\left\{\begin{array}{c}{w}_{l}\\ {\varphi}_{l}\\ {\psi}_{l}\end{array}\right\},\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}l=1,\text{\hspace{0.17em}}2,\text{\hspace{0.17em}}3,\text{\hspace{0.17em}}4.$$(15)

In a similar way one can write for rotation angles

$$\begin{array}{l}{\psi}_{x}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}=\u3008{\Psi}_{xk}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}\u3009\left\{\delta \right\},\\ {\psi}_{y}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}=\u3008{\Psi}_{yk}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}\u3009\left\{\delta \right\},\end{array}$$(16)

and shear angles

$$\begin{array}{l}{\gamma}_{x}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}=\u3008{\Gamma}_{xk}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}\u3009\left\{\delta \right\},\\ {\gamma}_{y}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}=\u3008{\Gamma}_{yk}\mathrm{(}\xi ,\text{\hspace{0.17em}}\eta \mathrm{)}\u3009\left\{\delta \right\}.\end{array}$$(17)

According to (3) the finite element bending curvature and warping can be presented as vector

$$\mathrm{\{}\kappa \mathrm{\}}=\left\{\begin{array}{c}\frac{\partial {\psi}_{x}}{\partial x}\\ \frac{\partial {\psi}_{y}}{\partial y}\\ \frac{\partial {\psi}_{x}}{\partial y}+\frac{\partial {\psi}_{y}}{\partial x}\end{array}\right\}$$(18)

By substituting (12) into (16), and taking (18) into account, yields

$$\left\{\kappa \right\}=-{[L]}_{b}\left\{\delta \right\},$$(19)

where

$${[L]}_{b}=\left[\begin{array}{c}\frac{1}{{a}^{2}}\u3008{\mathrm{(}{{X}^{\u2033}}_{ib}{Y}_{j}\mathrm{)}}_{k}\u3009\\ \frac{1}{{b}^{2}}\u3008{\mathrm{(}{X}_{i}{{Y}^{\u2033}}_{jb}\mathrm{)}}_{k}\u3009\\ \frac{1}{ab}\u3008{\mathrm{(}{{X}^{\prime}}_{ib}{{Y}^{\prime}}_{j}\mathrm{)}}_{k}+{\mathrm{(}{{X}^{\prime}}_{i}{{Y}^{\prime}}_{jb}\mathrm{)}}_{k}\u3009\end{array}\right].$$(20)

By using a general formulation of stiffness matrix from the finite element method based on variational principle [21], one can write for bending stiffness matrix

$${[K]}_{b}=ab{\displaystyle \underset{0}{\overset{1}{\int}}{\displaystyle \underset{0}{\overset{1}{\int}}{[L]}_{b}^{T}{[D]}_{b}{[L]}_{b}\text{d}\xi \text{d}\eta}},$$(21)

where

$${[D]}_{b}=D\left[\begin{array}{ccc}1& \nu & 0\\ \nu & 1& 0\\ 0& 0& \frac{1-\nu}{2}\end{array}\right]$$(22)

Elements of matrix [*K*]_{b} , after multiplication of the subintegral matrices, can be presented in the form

$$\begin{array}{l}{K}_{kl}^{b}=\frac{D}{ab}{\displaystyle \underset{0}{\overset{1}{\int}}{\displaystyle \underset{0}{\overset{1}{\int}}\{{p}_{k}\left[{\left(\frac{b}{a}\right)}^{2}{p}_{l}+\nu {q}_{l}\right]+}}\\ {q}_{k}\left[{\mathrm{(}\frac{a}{b}\mathrm{)}}^{2}{q}_{l}+\nu {p}_{l}\right]+\frac{1-\nu}{2}{r}_{k}{r}_{l}\}\text{d}\xi \text{d}\eta ,\end{array}$$(23)

where, according to (20)

$$\begin{array}{rl}& {p}_{k}=({{X}^{\u2033}}_{ib}{Y}_{j}{)}_{k},{q}_{k}=({X}_{i}{{Y}^{\u2033}}_{jb}{)}_{k},\\ & {r}_{k}=({{X}^{\prime}}_{ib}{{Y}^{\prime}}_{j}{)}_{k}+({{X}^{\prime}}_{i}{{Y}^{\prime}}_{jb}{)}_{k}.\end{array}$$(24)

Related to the shear stiffness the shear strain vector according to (5) reads

$$\left\{\gamma \right\}=\left\{\begin{array}{c}{\gamma}_{x}\\ {\gamma}_{y}\end{array}\right\}.$$(25)

By substituting (13) into (17), and then into (25), yields

$$\left\{\gamma \right\}={[L]}_{s}\left\{\delta \right\},$$(26)

where

$${[L]}_{s}=\left[\begin{array}{c}\frac{1}{a}\u3008{\mathrm{(}{{X}^{\prime}}_{is}{Y}_{j}\mathrm{)}}_{k}\u3009\\ \frac{1}{b}\u3008{\mathrm{(}{X}_{i}{{Y}^{\prime}}_{js}\mathrm{)}}_{k}\u3009\end{array}\right]$$(27)

Analogously to (21) one can write for the shear stiffness matrix

$${[K]}_{s}=ab{\displaystyle \underset{0}{\overset{1}{\int}}{\displaystyle \underset{0}{\overset{1}{\int}}{[L]}_{s}^{T}{[D]}_{s}{[L]}_{s}\text{d}\xi \text{d}\eta}},$$(28)

where

$${[D]}_{s}=S\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$$(29)

is matrix of the plate shear rigidity. Since [*D*]_{s} includes unit matrix, Eq. (28) is reduced to the form

$${[K]}_{s}=Sab{\displaystyle \underset{0}{\overset{1}{\int}}{\displaystyle \underset{0}{\overset{1}{\int}}{[L]}_{s}^{T}{[L]}_{s}\text{d}\xi \text{d}\eta}}$$(30)

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