# Rectangular Plate Bending Element

This derivation follows the procedure given in Chapter 12 of "A First Course in the Finite Element Method, 4th Edition" by Daryl L. Logan. Logan's method has been improved upon to allow for orthotropic behavior. Closed form solutions for stiffness and load matrices are provided.

We'll start by importing a few Python libraries that are useful for symbolic math, and initializing "pretty" printing.


```python
from sympy import symbols, Matrix, diff, integrate, simplify, factor, latex, init_printing
from IPython.display import display, Math
init_printing()
```

The plate width will be defined as $2b$, and the height will be $2c$ to be consistent with Figure 12-1. We'll set up some Sympy symbols to represent $b$ and $c$.


```python
b, c = symbols('b, c')
```

The plate is defined by four nodes specified in counter-clockwise order: i, j, m, and n. The local x-axis runs from node i toward node j, and the local y-axis runs from node i toward node n. Next we'll define the element's local displacement vector, $[d]$, at each node. There are 3 degrees of freedom at each node: $w$, $\theta_x$, and $\theta_y$.


```python
wi, theta_xi, theta_yi = symbols('w_i, theta_x_i, theta_yi')
wj, theta_xj, theta_yj = symbols('w_j, theta_xj, theta_yj')
wm, theta_xm, theta_ym = symbols('w_m, theta_xm, theta_ym')
wn, theta_xn, theta_yn = symbols('w_n, theta_xn, theta_yn')
d = Matrix([wi, theta_xi, theta_yi, wj, theta_xj, theta_yj, wm, theta_xm, theta_ym, wn, theta_xn, theta_yn])
display(Math('[d] = ' + latex(d)))
```


$\displaystyle [d] = \left[\begin{matrix}w_{i}\\\theta_{x i}\\\theta_{yi}\\w_{j}\\\theta_{xj}\\\theta_{yj}\\w_{m}\\\theta_{xm}\\\theta_{ym}\\w_{n}\\\theta_{xn}\\\theta_{yn}\end{matrix}\right]$


A 12-term polynomial displacement function will be assumed to define the out-of-plane displacement, w, at any point (x, y) in the plate's local coordinate system. The rotations about each axis are derivatives of this displacement:


```python
a1, a2, a3, a4, a5, a6, a7, a8, a9, a10, a11, a12 = symbols('a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_10, a_11, a_12')
x, y, w, theta_x, theta_y = symbols('x, y, w, theta_x, theta_y')

w = a1 + a2*x + a3*y + a4*x**2 + a5*x*y + a6*y**2 + a7*x**3 + a8*x**2*y + a9*x*y**2 + a10*y**3 + a11*x**3*y + a12*x*y**3
theta_x = diff(w, y)
theta_y = -diff(w, x)

display(Math('w = ' + latex(w)))
display(Math('\\theta_x = ' + latex(theta_x)))
display(Math('\\theta_y = ' + latex(theta_y)))
```


$\displaystyle w = a_{1} + a_{10} y^{3} + a_{11} x^{3} y + a_{12} x y^{3} + a_{2} x + a_{3} y + a_{4} x^{2} + a_{5} x y + a_{6} y^{2} + a_{7} x^{3} + a_{8} x^{2} y + a_{9} x y^{2}$



$\displaystyle \theta_x = 3 a_{10} y^{2} + a_{11} x^{3} + 3 a_{12} x y^{2} + a_{3} + a_{5} x + 2 a_{6} y + a_{8} x^{2} + 2 a_{9} x y$



$\displaystyle \theta_y = - 3 a_{11} x^{2} y - a_{12} y^{3} - a_{2} - 2 a_{4} x - a_{5} y - 3 a_{7} x^{2} - 2 a_{8} x y - a_{9} y^{2}$


The negative sign on $\frac{dw}{dx}$ is required to be consistent with the right hand rule. These equations can be rewritten in matrix form as follows:

$[\psi] = [P][a]$

where $[\psi]$ is shorthand for  $\begin{bmatrix} w \\ \theta_x \\ \theta_y \end{bmatrix}$ and $[P]$ is defined as follows:


```python
P = Matrix([[1, x, y, x**2, x*y, y**2, x**3, x**2*y, x*y**2, y**3, x**3*y, x*y**3],
            [0, 0, 1, 0, x, 2*y, 0, x**2, 2*x*y, 3*y**2, x**3, 3*x*y**2],
            [0, -1, 0, -2*x, -y, 0, -3*x**2, -2*x*y, -y**2, 0, -3*x**2*y, -y**3]])

display(Math('P = ' + latex(P)))
```


$\displaystyle P = \left[\begin{array}{cccccccccccc}1 & x & y & x^{2} & x y & y^{2} & x^{3} & x^{2} y & x y^{2} & y^{3} & x^{3} y & x y^{3}\\0 & 0 & 1 & 0 & x & 2 y & 0 & x^{2} & 2 x y & 3 y^{2} & x^{3} & 3 x y^{2}\\0 & -1 & 0 & - 2 x & - y & 0 & - 3 x^{2} & - 2 x y & - y^{2} & 0 & - 3 x^{2} y & - y^{3}\end{array}\right]$


This general equation for $[P]$ will be evaluated at each node to give us a larger set of equations:

$[d] = [C][a]$

where $[C]$ is merely $[P]$ evaluated at each node, and $[d]$ is correpsondingly $[\psi]$ at each node. Knowing that the plate width is $2b$ and the plate height is $2c$, we can obtain the matrix $[C]$.


```python
C = Matrix([P, P, P, P])
C[0:3, 0:12] = C[0:3, 0:12].subs(x, 0).subs(y, 0)      # i-node @ x = 0, y = 0
C[3:6, 0:12] = C[3:6, 0:12].subs(x, 2*b).subs(y, 0)    # j-node @ x = 2b, y = 0
C[6:9, 0:12] = C[6:9, 0:12].subs(x, 2*b).subs(y, 2*c)  # m-node @ x = 2b, y = 2c
C[9:12, 0:12] = C[9:12, 0:12].subs(x, 0).subs(y, 2*c)  # n-node @ x = 0, y = 2c
display(Math('[C] = ' + latex(C)))
print(C)
```


$\displaystyle [C] = \left[\begin{array}{cccccccccccc}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\1 & 2 b & 0 & 4 b^{2} & 0 & 0 & 8 b^{3} & 0 & 0 & 0 & 0 & 0\\0 & 0 & 1 & 0 & 2 b & 0 & 0 & 4 b^{2} & 0 & 0 & 8 b^{3} & 0\\0 & -1 & 0 & - 4 b & 0 & 0 & - 12 b^{2} & 0 & 0 & 0 & 0 & 0\\1 & 2 b & 2 c & 4 b^{2} & 4 b c & 4 c^{2} & 8 b^{3} & 8 b^{2} c & 8 b c^{2} & 8 c^{3} & 16 b^{3} c & 16 b c^{3}\\0 & 0 & 1 & 0 & 2 b & 4 c & 0 & 4 b^{2} & 8 b c & 12 c^{2} & 8 b^{3} & 24 b c^{2}\\0 & -1 & 0 & - 4 b & - 2 c & 0 & - 12 b^{2} & - 8 b c & - 4 c^{2} & 0 & - 24 b^{2} c & - 8 c^{3}\\1 & 0 & 2 c & 0 & 0 & 4 c^{2} & 0 & 0 & 0 & 8 c^{3} & 0 & 0\\0 & 0 & 1 & 0 & 0 & 4 c & 0 & 0 & 0 & 12 c^{2} & 0 & 0\\0 & -1 & 0 & 0 & - 2 c & 0 & 0 & 0 & - 4 c^{2} & 0 & 0 & - 8 c^{3}\end{array}\right]$


    Matrix([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [1, 2*b, 0, 4*b**2, 0, 0, 8*b**3, 0, 0, 0, 0, 0], [0, 0, 1, 0, 2*b, 0, 0, 4*b**2, 0, 0, 8*b**3, 0], [0, -1, 0, -4*b, 0, 0, -12*b**2, 0, 0, 0, 0, 0], [1, 2*b, 2*c, 4*b**2, 4*b*c, 4*c**2, 8*b**3, 8*b**2*c, 8*b*c**2, 8*c**3, 16*b**3*c, 16*b*c**3], [0, 0, 1, 0, 2*b, 4*c, 0, 4*b**2, 8*b*c, 12*c**2, 8*b**3, 24*b*c**2], [0, -1, 0, -4*b, -2*c, 0, -12*b**2, -8*b*c, -4*c**2, 0, -24*b**2*c, -8*c**3], [1, 0, 2*c, 0, 0, 4*c**2, 0, 0, 0, 8*c**3, 0, 0], [0, 0, 1, 0, 0, 4*c, 0, 0, 0, 12*c**2, 0, 0], [0, -1, 0, 0, -2*c, 0, 0, 0, -4*c**2, 0, 0, -8*c**3]])


An important matrix that we will come back to later is the shape function matrix $[N]$, defined as:

$[N] = [P][C]^{-1}$

The closed form solution of $[N]$ for a rectangular plate is:


```python
N = P*C.inv()
display(Math('[N] = ' + latex(simplify(N))))
```


$\displaystyle [N] = \left[\begin{array}{cccccccccccc}1 - \frac{3 y^{2}}{4 c^{2}} + \frac{y^{3}}{4 c^{3}} - \frac{x y}{4 b c} + \frac{3 x y^{2}}{8 b c^{2}} - \frac{x y^{3}}{8 b c^{3}} - \frac{3 x^{2}}{4 b^{2}} + \frac{3 x^{2} y}{8 b^{2} c} + \frac{x^{3}}{4 b^{3}} - \frac{x^{3} y}{8 b^{3} c} & y - \frac{y^{2}}{c} + \frac{y^{3}}{4 c^{2}} - \frac{x y}{2 b} + \frac{x y^{2}}{2 b c} - \frac{x y^{3}}{8 b c^{2}} & - x + \frac{x y}{2 c} + \frac{x^{2}}{b} - \frac{x^{2} y}{2 b c} - \frac{x^{3}}{4 b^{2}} + \frac{x^{3} y}{8 b^{2} c} & \frac{x \left(2 b^{2} c^{2} y - 3 b^{2} c y^{2} + b^{2} y^{3} + 6 b c^{3} x - 3 b c^{2} x y - 2 c^{3} x^{2} + c^{2} x^{2} y\right)}{8 b^{3} c^{3}} & \frac{x y \left(4 c^{2} - 4 c y + y^{2}\right)}{8 b c^{2}} & \frac{x^{2} \left(4 b c - 2 b y - 2 c x + x y\right)}{8 b^{2} c} & \frac{x y \left(- 2 b^{2} c^{2} + 3 b^{2} c y - b^{2} y^{2} + 3 b c^{2} x - c^{2} x^{2}\right)}{8 b^{3} c^{3}} & \frac{x y^{2} \left(- 2 c + y\right)}{8 b c^{2}} & \frac{x^{2} y \left(2 b - x\right)}{8 b^{2} c} & \frac{y \left(6 b^{3} c y - 2 b^{3} y^{2} + 2 b^{2} c^{2} x - 3 b^{2} c x y + b^{2} x y^{2} - 3 b c^{2} x^{2} + c^{2} x^{3}\right)}{8 b^{3} c^{3}} & \frac{y^{2} \left(- 4 b c + 2 b y + 2 c x - x y\right)}{8 b c^{2}} & \frac{x y \left(- 4 b^{2} + 4 b x - x^{2}\right)}{8 b^{2} c}\\\frac{- 12 b^{3} c y + 6 b^{3} y^{2} - 2 b^{2} c^{2} x + 6 b^{2} c x y - 3 b^{2} x y^{2} + 3 b c^{2} x^{2} - c^{2} x^{3}}{8 b^{3} c^{3}} & 1 - \frac{2 y}{c} + \frac{3 y^{2}}{4 c^{2}} - \frac{x}{2 b} + \frac{x y}{b c} - \frac{3 x y^{2}}{8 b c^{2}} & \frac{x \left(4 b^{2} - 4 b x + x^{2}\right)}{8 b^{2} c} & \frac{x \left(2 b^{2} c^{2} - 6 b^{2} c y + 3 b^{2} y^{2} - 3 b c^{2} x + c^{2} x^{2}\right)}{8 b^{3} c^{3}} & \frac{x \left(4 c^{2} - 8 c y + 3 y^{2}\right)}{8 b c^{2}} & \frac{x^{2} \left(- 2 b + x\right)}{8 b^{2} c} & \frac{x \left(- 2 b^{2} c^{2} + 6 b^{2} c y - 3 b^{2} y^{2} + 3 b c^{2} x - c^{2} x^{2}\right)}{8 b^{3} c^{3}} & \frac{x y \left(- 4 c + 3 y\right)}{8 b c^{2}} & \frac{x^{2} \left(2 b - x\right)}{8 b^{2} c} & \frac{12 b^{3} c y - 6 b^{3} y^{2} + 2 b^{2} c^{2} x - 6 b^{2} c x y + 3 b^{2} x y^{2} - 3 b c^{2} x^{2} + c^{2} x^{3}}{8 b^{3} c^{3}} & \frac{y \left(- 8 b c + 6 b y + 4 c x - 3 x y\right)}{8 b c^{2}} & \frac{x \left(- 4 b^{2} + 4 b x - x^{2}\right)}{8 b^{2} c}\\\frac{2 b^{2} c^{2} y - 3 b^{2} c y^{2} + b^{2} y^{3} + 12 b c^{3} x - 6 b c^{2} x y - 6 c^{3} x^{2} + 3 c^{2} x^{2} y}{8 b^{3} c^{3}} & \frac{y \left(4 c^{2} - 4 c y + y^{2}\right)}{8 b c^{2}} & 1 - \frac{y}{2 c} - \frac{2 x}{b} + \frac{x y}{b c} + \frac{3 x^{2}}{4 b^{2}} - \frac{3 x^{2} y}{8 b^{2} c} & \frac{- 2 b^{2} c^{2} y + 3 b^{2} c y^{2} - b^{2} y^{3} - 12 b c^{3} x + 6 b c^{2} x y + 6 c^{3} x^{2} - 3 c^{2} x^{2} y}{8 b^{3} c^{3}} & \frac{y \left(- 4 c^{2} + 4 c y - y^{2}\right)}{8 b c^{2}} & \frac{x \left(- 8 b c + 4 b y + 6 c x - 3 x y\right)}{8 b^{2} c} & \frac{y \left(2 b^{2} c^{2} - 3 b^{2} c y + b^{2} y^{2} - 6 b c^{2} x + 3 c^{2} x^{2}\right)}{8 b^{3} c^{3}} & \frac{y^{2} \left(2 c - y\right)}{8 b c^{2}} & \frac{x y \left(- 4 b + 3 x\right)}{8 b^{2} c} & \frac{y \left(- 2 b^{2} c^{2} + 3 b^{2} c y - b^{2} y^{2} + 6 b c^{2} x - 3 c^{2} x^{2}\right)}{8 b^{3} c^{3}} & \frac{y^{2} \left(- 2 c + y\right)}{8 b c^{2}} & \frac{y \left(4 b^{2} - 8 b x + 3 x^{2}\right)}{8 b^{2} c}\end{array}\right]$


We can now solve for the $[a]$ matrix in terms of the nodal displacements:


```python
a = simplify(C.inv()*d)
display(Math('[a] = ' + latex(a)))
```


$\displaystyle [a] = \left[\begin{matrix}w_{i}\\- \theta_{yi}\\\theta_{x i}\\\frac{2 b \left(2 \theta_{yi} + \theta_{yj}\right) - 3 w_{i} + 3 w_{j}}{4 b^{2}}\\\frac{2 b \left(\theta_{yi} - \theta_{yn}\right) + 2 c \left(- \theta_{x i} + \theta_{xj}\right) - w_{i} + w_{j} - w_{m} + w_{n}}{4 b c}\\\frac{- 2 c \left(2 \theta_{x i} + \theta_{xn}\right) - 3 w_{i} + 3 w_{n}}{4 c^{2}}\\\frac{- b \left(\theta_{yi} + \theta_{yj}\right) + w_{i} - w_{j}}{4 b^{3}}\\\frac{2 b \left(- 2 \theta_{yi} - \theta_{yj} + \theta_{ym} + 2 \theta_{yn}\right) + 3 w_{i} - 3 w_{j} + 3 w_{m} - 3 w_{n}}{8 b^{2} c}\\\frac{2 c \left(2 \theta_{x i} - 2 \theta_{xj} - \theta_{xm} + \theta_{xn}\right) + 3 w_{i} - 3 w_{j} + 3 w_{m} - 3 w_{n}}{8 b c^{2}}\\\frac{c \left(\theta_{x i} + \theta_{xn}\right) + w_{i} - w_{n}}{4 c^{3}}\\\frac{b \left(\theta_{yi} + \theta_{yj} - \theta_{ym} - \theta_{yn}\right) - w_{i} + w_{j} - w_{m} + w_{n}}{8 b^{3} c}\\\frac{c \left(- \theta_{x i} + \theta_{xj} + \theta_{xm} - \theta_{xn}\right) - w_{i} + w_{j} - w_{m} + w_{n}}{8 b c^{3}}\end{matrix}\right]$


The next step is to define the curvature matrix:

$[\kappa] = \begin{bmatrix} -\frac{d^2w}{dx^2} \\ -\frac{d^2w}{dy^2} \\ -\frac{2d^2w}{dxdy} \end{bmatrix} = [Q][a]$

It should be recognized that $w/[a]$ is simply the first row of our $[P]$ matrix. Evaluating the derivatives in this expression gives $[Q]$ as follows:


```python
Q = Matrix([-diff(diff(P[0, :], x), x),
            -diff(diff(P[0, :], y), y),
            -2*diff(diff(P[0, :], x), y)])
display(Math('[Q] = ' + latex(Q)))
print(Q)
```


$\displaystyle [Q] = \left[\begin{array}{cccccccccccc}0 & 0 & 0 & -2 & 0 & 0 & - 6 x & - 2 y & 0 & 0 & - 6 x y & 0\\0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & - 2 x & - 6 y & 0 & - 6 x y\\0 & 0 & 0 & 0 & -2 & 0 & 0 & - 4 x & - 4 y & 0 & - 6 x^{2} & - 6 y^{2}\end{array}\right]$


    Matrix([[0, 0, 0, -2, 0, 0, -6*x, -2*y, 0, 0, -6*x*y, 0], [0, 0, 0, 0, 0, -2, 0, 0, -2*x, -6*y, 0, -6*x*y], [0, 0, 0, 0, -2, 0, 0, -4*x, -4*y, 0, -6*x**2, -6*y**2]])


With $[Q]$ in hand we can now solve for the $[B]$ matrix which is essential for formulating the stiffness matrix $[k]$


```python
B = simplify(Q*C.inv())
display(Math('[B] = ' + latex(B)))
```


$\displaystyle [B] = \left[\begin{array}{cccccccccccc}\frac{3 \left(2 b c - b y - 2 c x + x y\right)}{4 b^{3} c} & 0 & \frac{- 8 b c + 4 b y + 6 c x - 3 x y}{4 b^{2} c} & \frac{3 \left(- 2 b c + b y + 2 c x - x y\right)}{4 b^{3} c} & 0 & \frac{- 4 b c + 2 b y + 6 c x - 3 x y}{4 b^{2} c} & \frac{3 y \left(- b + x\right)}{4 b^{3} c} & 0 & \frac{y \left(- 2 b + 3 x\right)}{4 b^{2} c} & \frac{3 y \left(b - x\right)}{4 b^{3} c} & 0 & \frac{y \left(- 4 b + 3 x\right)}{4 b^{2} c}\\\frac{3 \left(2 b c - 2 b y - c x + x y\right)}{4 b c^{3}} & \frac{8 b c - 6 b y - 4 c x + 3 x y}{4 b c^{2}} & 0 & \frac{3 x \left(c - y\right)}{4 b c^{3}} & \frac{x \left(4 c - 3 y\right)}{4 b c^{2}} & 0 & \frac{3 x \left(- c + y\right)}{4 b c^{3}} & \frac{x \left(2 c - 3 y\right)}{4 b c^{2}} & 0 & \frac{3 \left(- 2 b c + 2 b y + c x - x y\right)}{4 b c^{3}} & \frac{4 b c - 6 b y - 2 c x + 3 x y}{4 b c^{2}} & 0\\\frac{2 b^{2} c^{2} - 6 b^{2} c y + 3 b^{2} y^{2} - 6 b c^{2} x + 3 c^{2} x^{2}}{4 b^{3} c^{3}} & \frac{c^{2} - 2 c y + \frac{3 y^{2}}{4}}{b c^{2}} & \frac{- b^{2} + 2 b x - \frac{3 x^{2}}{4}}{b^{2} c} & \frac{- 2 b^{2} c^{2} + 6 b^{2} c y - 3 b^{2} y^{2} + 6 b c^{2} x - 3 c^{2} x^{2}}{4 b^{3} c^{3}} & \frac{- c^{2} + 2 c y - \frac{3 y^{2}}{4}}{b c^{2}} & \frac{x \left(4 b - 3 x\right)}{4 b^{2} c} & \frac{2 b^{2} c^{2} - 6 b^{2} c y + 3 b^{2} y^{2} - 6 b c^{2} x + 3 c^{2} x^{2}}{4 b^{3} c^{3}} & \frac{y \left(4 c - 3 y\right)}{4 b c^{2}} & \frac{x \left(- 4 b + 3 x\right)}{4 b^{2} c} & \frac{- 2 b^{2} c^{2} + 6 b^{2} c y - 3 b^{2} y^{2} + 6 b c^{2} x - 3 c^{2} x^{2}}{4 b^{3} c^{3}} & \frac{y \left(- 4 c + 3 y\right)}{4 b c^{2}} & \frac{b^{2} - 2 b x + \frac{3 x^{2}}{4}}{b^{2} c}\end{array}\right]$


Now we form the constitutive matrix for orthotropic materials, [D]. This matrix is analagous to the flexural stiffness of a beam EI.


```python
Ex, Ey, nu_xy, nu_yx, G, t = symbols('E_x, E_y, \\nu_{xy}, \\nu_{yx}, G, t')

D = 1/(1 - nu_xy*nu_yx)*Matrix([[   Ex,    nu_yx*Ex,          0         ],
                                [nu_xy*Ey,    Ey,             0         ],
                                [   0,        0,     (1 - nu_xy*nu_yx)*G]])

display(Math('[D] = \\frac{t^3}{12}' + latex(D)))

print(t**3/12*D)
```


$\displaystyle [D] = \frac{t^3}{12}\left[\begin{matrix}\frac{E_{x}}{- \nu_{xy} \nu_{yx} + 1} & \frac{E_{x} \nu_{yx}}{- \nu_{xy} \nu_{yx} + 1} & 0\\\frac{E_{y} \nu_{xy}}{- \nu_{xy} \nu_{yx} + 1} & \frac{E_{y}}{- \nu_{xy} \nu_{yx} + 1} & 0\\0 & 0 & G\end{matrix}\right]$


    Matrix([[E_x*t**3/(12*(-\nu_{xy}*\nu_{yx} + 1)), E_x*\nu_{yx}*t**3/(12*(-\nu_{xy}*\nu_{yx} + 1)), 0], [E_y*\nu_{xy}*t**3/(12*(-\nu_{xy}*\nu_{yx} + 1)), E_y*t**3/(12*(-\nu_{xy}*\nu_{yx} + 1)), 0], [0, 0, G*t**3/12]])


Now we can calculate the stiffness matrix:

$[k] = \int_0^{2c} \int_0^{2b} [B]^T[D][B] dx dy$


```python
k = simplify(integrate(integrate(B.T*D*B, (x, 0, 2*b)), (y, 0, 2*c)))

display(Math('[k] = \\frac{t^3}{12}' + latex(k)))

# Uncomment the line below for a version that can be copied/pasted
# Be sure to multiply by t**3/12
print(k)
```


$\displaystyle [k] = \frac{t^3}{12}\left[\begin{array}{cccccccccccc}\frac{- \frac{E_{x} \nu_{yx} b^{2} c^{2}}{4} - E_{x} c^{4} - \frac{E_{y} \nu_{xy} b^{2} c^{2}}{4} - E_{y} b^{4} + \frac{7 G \nu_{xy} \nu_{yx} b^{2} c^{2}}{5} - \frac{7 G b^{2} c^{2}}{5}}{b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} c^{2}}{2} - E_{y} b^{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{x} c^{2} + \frac{E_{y} \nu_{xy} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} + \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} \nu_{yx} b^{2} c^{2} + 20 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} - 10 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} \nu_{yx} c^{2}}{2} - \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} c^{2} - G \nu_{xy} \nu_{yx} b^{2} + G b^{2}}{5 b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{x} \nu_{yx} b^{2} c^{2} + 10 E_{x} c^{4} - 5 E_{y} \nu_{xy} b^{2} c^{2} + 10 E_{y} b^{4} + 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} - 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} \nu_{yx} b^{2} c^{2} - 10 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} + 20 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{y} b^{2} + G \nu_{xy} \nu_{yx} c^{2} - G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} c^{2}}{2} - \frac{E_{y} \nu_{xy} b^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)}\\\frac{- \frac{E_{y} \nu_{xy} c^{2}}{2} - E_{y} b^{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{4 \left(- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{y} \nu_{xy}}{\nu_{xy} \nu_{yx} - 1} & \frac{\frac{E_{y} \nu_{xy} c^{2}}{2} - \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - 4 G \nu_{xy} \nu_{yx} c^{2} + 4 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{\frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0\\\frac{\frac{E_{x} \nu_{yx} b^{2}}{2} + E_{x} c^{2} - \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} + \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{x} \nu_{yx}}{\nu_{xy} \nu_{yx} - 1} & \frac{4 \left(- 5 E_{x} c^{2} + 2 G \nu_{xy} \nu_{yx} b^{2} - 2 G b^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{x} c^{2} + G \nu_{xy} \nu_{yx} b^{2} - G b^{2}}{5 b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{2 \left(- 5 E_{x} c^{2} - G \nu_{xy} \nu_{yx} b^{2} + G b^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- 5 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} b^{2}}{2} + \frac{E_{x} c^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} - \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & - \frac{10 E_{x} c^{2} + 8 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)}\\\frac{5 E_{x} \nu_{yx} b^{2} c^{2} + 20 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} - 10 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} \nu_{yx} c^{2}}{2} - \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{x} c^{2} + G \nu_{xy} \nu_{yx} b^{2} - G b^{2}}{5 b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} b^{2} c^{2}}{4} - E_{x} c^{4} - \frac{E_{y} \nu_{xy} b^{2} c^{2}}{4} - E_{y} b^{4} + \frac{7 G \nu_{xy} \nu_{yx} b^{2} c^{2}}{5} - \frac{7 G b^{2} c^{2}}{5}}{b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} c^{2}}{2} - E_{y} b^{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- E_{x} c^{2} - \frac{E_{y} \nu_{xy} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} - \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} \nu_{yx} b^{2} c^{2} - 10 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} + 20 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{y} b^{2} + G \nu_{xy} \nu_{yx} c^{2} - G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} c^{2}}{2} + \frac{E_{y} \nu_{xy} b^{2}}{2} - \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{x} \nu_{yx} b^{2} c^{2} + 10 E_{x} c^{4} - 5 E_{y} \nu_{xy} b^{2} c^{2} + 10 E_{y} b^{4} + 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} - 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)}\\\frac{\frac{E_{y} \nu_{xy} c^{2}}{2} - \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - 4 G \nu_{xy} \nu_{yx} c^{2} + 4 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- \frac{E_{y} \nu_{xy} c^{2}}{2} - E_{y} b^{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{4 \left(- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{E_{y} \nu_{xy}}{\nu_{xy} \nu_{yx} - 1} & \frac{5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{\frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0\\\frac{5 E_{x} c^{2} - G \nu_{xy} \nu_{yx} b^{2} + G b^{2}}{5 b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{2 \left(- 5 E_{x} c^{2} - G \nu_{xy} \nu_{yx} b^{2} + G b^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} b^{2}}{2} - E_{x} c^{2} + \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} - \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{E_{x} \nu_{yx}}{\nu_{xy} \nu_{yx} - 1} & \frac{4 \left(- 5 E_{x} c^{2} + 2 G \nu_{xy} \nu_{yx} b^{2} - 2 G b^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} \nu_{yx} b^{2}}{2} - \frac{E_{x} c^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} + \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & - \frac{10 E_{x} c^{2} + 8 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- 5 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)}\\\frac{- 5 E_{x} \nu_{yx} b^{2} c^{2} + 10 E_{x} c^{4} - 5 E_{y} \nu_{xy} b^{2} c^{2} + 10 E_{y} b^{4} + 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} - 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} \nu_{yx} b^{2} c^{2} - 10 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} + 20 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{x} c^{2} - 25 E_{y} \nu_{xy} b^{2} + 2 b^{2} \left(15 E_{y} \nu_{xy} - G \nu_{xy} \nu_{yx} + G\right)}{10 b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} b^{2} c^{2}}{4} - E_{x} c^{4} - \frac{E_{y} \nu_{xy} b^{2} c^{2}}{4} - E_{y} b^{4} + \frac{7 G \nu_{xy} \nu_{yx} b^{2} c^{2}}{5} - \frac{7 G b^{2} c^{2}}{5}}{b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} \nu_{yx} c^{2}}{2} + E_{y} b^{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- E_{x} c^{2} - \frac{E_{y} \nu_{xy} b^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} \nu_{yx} b^{2} c^{2} + 20 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} - 10 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} c^{2}}{2} + \frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- E_{x} c^{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)}\\\frac{- \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- 5 E_{y} b^{2} + G \nu_{xy} \nu_{yx} c^{2} - G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{\frac{E_{y} \nu_{xy} c^{2}}{2} + E_{y} b^{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{4 \left(- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{y} \nu_{xy}}{\nu_{xy} \nu_{yx} - 1} & \frac{- \frac{E_{y} \nu_{xy} c^{2}}{2} + \frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - 4 G \nu_{xy} \nu_{yx} c^{2} + 4 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0\\\frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- 5 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} \nu_{yx} b^{2}}{2} - \frac{E_{x} c^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} + \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & - \frac{10 E_{x} c^{2} + 8 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} b^{2}}{2} - E_{x} c^{2} + \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} - \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{x} \nu_{yx}}{\nu_{xy} \nu_{yx} - 1} & \frac{4 \left(- 5 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{x} c^{2} - \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & - \frac{10 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)}\\\frac{5 E_{x} \nu_{yx} b^{2} c^{2} - 10 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} + 20 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} c^{2} + 25 E_{y} \nu_{xy} b^{2} - 2 b^{2} \left(15 E_{y} \nu_{xy} - G \nu_{xy} \nu_{yx} + G\right)}{10 b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{x} \nu_{yx} b^{2} c^{2} + 10 E_{x} c^{4} - 5 E_{y} \nu_{xy} b^{2} c^{2} + 10 E_{y} b^{4} + 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} - 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{5 E_{x} \nu_{yx} b^{2} c^{2} + 20 E_{x} c^{4} + 5 E_{y} \nu_{xy} b^{2} c^{2} - 10 E_{y} b^{4} - 28 G \nu_{xy} \nu_{yx} b^{2} c^{2} + 28 G b^{2} c^{2}}{20 b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} c^{2}}{2} + \frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{x} c^{2} - \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- \frac{E_{x} \nu_{yx} b^{2} c^{2}}{4} - E_{x} c^{4} - \frac{E_{y} \nu_{xy} b^{2} c^{2}}{4} - E_{y} b^{4} + \frac{7 G \nu_{xy} \nu_{yx} b^{2} c^{2}}{5} - \frac{7 G b^{2} c^{2}}{5}}{b^{3} c^{3} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} \nu_{yx} c^{2}}{2} + E_{y} b^{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{E_{x} c^{2} + \frac{E_{y} \nu_{xy} b^{2}}{2} - \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)}\\\frac{- 5 E_{y} b^{2} + G \nu_{xy} \nu_{yx} c^{2} - G c^{2}}{5 b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - G \nu_{xy} \nu_{yx} c^{2} + G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- \frac{E_{y} b^{2}}{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- \frac{E_{y} \nu_{xy} c^{2}}{2} + \frac{E_{y} b^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} - \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{2 \left(- 5 E_{y} b^{2} - 4 G \nu_{xy} \nu_{yx} c^{2} + 4 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{\frac{E_{y} \nu_{xy} c^{2}}{2} + E_{y} b^{2} - \frac{G \nu_{xy} \nu_{yx} c^{2}}{5} + \frac{G c^{2}}{5}}{b c^{2} \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{4 \left(- 5 E_{y} b^{2} + 2 G \nu_{xy} \nu_{yx} c^{2} - 2 G c^{2}\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{E_{y} \nu_{xy}}{\nu_{xy} \nu_{yx} - 1}\\\frac{- \frac{E_{x} \nu_{yx} b^{2}}{2} + \frac{E_{x} c^{2}}{2} + \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} - \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & - \frac{10 E_{x} c^{2} + 8 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{\frac{E_{x} c^{2}}{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & \frac{- 5 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{- E_{x} c^{2} + \frac{G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & 0 & - \frac{10 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)} & \frac{\frac{E_{x} \nu_{yx} b^{2}}{2} + E_{x} c^{2} - \frac{G \nu_{xy} \nu_{yx} b^{2}}{5} + \frac{G b^{2}}{5}}{b^{2} c \left(\nu_{xy} \nu_{yx} - 1\right)} & - \frac{E_{x} \nu_{yx}}{\nu_{xy} \nu_{yx} - 1} & \frac{4 \left(- 5 E_{x} c^{2} + 2 G b^{2} \left(\nu_{xy} \nu_{yx} - 1\right)\right)}{15 b c \left(\nu_{xy} \nu_{yx} - 1\right)}\end{array}\right]$


    Matrix([[(-E_x*\nu_{yx}*b**2*c**2/4 - E_x*c**4 - E_y*\nu_{xy}*b**2*c**2/4 - E_y*b**4 + 7*G*\nu_{xy}*\nu_{yx}*b**2*c**2/5 - 7*G*b**2*c**2/5)/(b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*c**2/2 - E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2 + E_y*\nu_{xy}*b**2/2 - G*\nu_{xy}*\nu_{yx}*b**2/5 + G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*\nu_{yx}*b**2*c**2 + 20*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 - 10*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (E_x*\nu_{yx}*c**2/2 - E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*c**2 - G*\nu_{xy}*\nu_{yx}*b**2 + G*b**2)/(5*b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_x*\nu_{yx}*b**2*c**2 + 10*E_x*c**4 - 5*E_y*\nu_{xy}*b**2*c**2 + 10*E_y*b**4 + 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 - 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*\nu_{yx}*b**2*c**2 - 10*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 + 20*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2 - G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2/2 - E_y*\nu_{xy}*b**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1))], [(-E_y*\nu_{xy}*c**2/2 - E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 4*(-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), E_y*\nu_{xy}/(\nu_{xy}*\nu_{yx} - 1), (E_y*\nu_{xy}*c**2/2 - E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - 4*G*\nu_{xy}*\nu_{yx}*c**2 + 4*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0], [(E_x*\nu_{yx}*b**2/2 + E_x*c**2 - G*\nu_{xy}*\nu_{yx}*b**2/5 + G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), E_x*\nu_{yx}/(\nu_{xy}*\nu_{yx} - 1), 4*(-5*E_x*c**2 + 2*G*\nu_{xy}*\nu_{yx}*b**2 - 2*G*b**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_x*c**2 + G*\nu_{xy}*\nu_{yx}*b**2 - G*b**2)/(5*b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, 2*(-5*E_x*c**2 - G*\nu_{xy}*\nu_{yx}*b**2 + G*b**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), -(E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-5*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*b**2/2 + E_x*c**2/2 + G*\nu_{xy}*\nu_{yx}*b**2/5 - G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, -(10*E_x*c**2 + 8*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1))], [(5*E_x*\nu_{yx}*b**2*c**2 + 20*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 - 10*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (E_x*\nu_{yx}*c**2/2 - E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_x*c**2 + G*\nu_{xy}*\nu_{yx}*b**2 - G*b**2)/(5*b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*b**2*c**2/4 - E_x*c**4 - E_y*\nu_{xy}*b**2*c**2/4 - E_y*b**4 + 7*G*\nu_{xy}*\nu_{yx}*b**2*c**2/5 - 7*G*b**2*c**2/5)/(b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*c**2/2 - E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*c**2 - E_y*\nu_{xy}*b**2/2 + G*\nu_{xy}*\nu_{yx}*b**2/5 - G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*\nu_{yx}*b**2*c**2 - 10*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 + 20*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2 - G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*c**2/2 + E_y*\nu_{xy}*b**2/2 - G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_x*\nu_{yx}*b**2*c**2 + 10*E_x*c**4 - 5*E_y*\nu_{xy}*b**2*c**2 + 10*E_y*b**4 + 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 - 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), -(E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1))], [(E_y*\nu_{xy}*c**2/2 - E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - 4*G*\nu_{xy}*\nu_{yx}*c**2 + 4*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-E_y*\nu_{xy}*c**2/2 - E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 4*(-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), -E_y*\nu_{xy}/(\nu_{xy}*\nu_{yx} - 1), (5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0], [(5*E_x*c**2 - G*\nu_{xy}*\nu_{yx}*b**2 + G*b**2)/(5*b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, 2*(-5*E_x*c**2 - G*\nu_{xy}*\nu_{yx}*b**2 + G*b**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*b**2/2 - E_x*c**2 + G*\nu_{xy}*\nu_{yx}*b**2/5 - G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), -E_x*\nu_{yx}/(\nu_{xy}*\nu_{yx} - 1), 4*(-5*E_x*c**2 + 2*G*\nu_{xy}*\nu_{yx}*b**2 - 2*G*b**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (E_x*\nu_{yx}*b**2/2 - E_x*c**2/2 - G*\nu_{xy}*\nu_{yx}*b**2/5 + G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, -(10*E_x*c**2 + 8*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-5*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1))], [(-5*E_x*\nu_{yx}*b**2*c**2 + 10*E_x*c**4 - 5*E_y*\nu_{xy}*b**2*c**2 + 10*E_y*b**4 + 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 - 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), -(E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*\nu_{yx}*b**2*c**2 - 10*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 + 20*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_x*c**2 - 25*E_y*\nu_{xy}*b**2 + 2*b**2*(15*E_y*\nu_{xy} - G*\nu_{xy}*\nu_{yx} + G))/(10*b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*b**2*c**2/4 - E_x*c**4 - E_y*\nu_{xy}*b**2*c**2/4 - E_y*b**4 + 7*G*\nu_{xy}*\nu_{yx}*b**2*c**2/5 - 7*G*b**2*c**2/5)/(b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (E_x*\nu_{yx}*c**2/2 + E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*c**2 - E_y*\nu_{xy}*b**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*\nu_{yx}*b**2*c**2 + 20*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 - 10*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*c**2/2 + E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*c**2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1))], [(-E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-5*E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2 - G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (E_y*\nu_{xy}*c**2/2 + E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 4*(-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), E_y*\nu_{xy}/(\nu_{xy}*\nu_{yx} - 1), (-E_y*\nu_{xy}*c**2/2 + E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - 4*G*\nu_{xy}*\nu_{yx}*c**2 + 4*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0], [(E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-5*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (E_x*\nu_{yx}*b**2/2 - E_x*c**2/2 - G*\nu_{xy}*\nu_{yx}*b**2/5 + G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, -(10*E_x*c**2 + 8*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*b**2/2 - E_x*c**2 + G*\nu_{xy}*\nu_{yx}*b**2/5 - G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), E_x*\nu_{yx}/(\nu_{xy}*\nu_{yx} - 1), 4*(-5*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2 - G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, -(10*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1))], [(5*E_x*\nu_{yx}*b**2*c**2 - 10*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 + 20*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*c**2 + 25*E_y*\nu_{xy}*b**2 - 2*b**2*(15*E_y*\nu_{xy} - G*\nu_{xy}*\nu_{yx} + G))/(10*b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_x*\nu_{yx}*b**2*c**2 + 10*E_x*c**4 - 5*E_y*\nu_{xy}*b**2*c**2 + 10*E_y*b**4 + 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 - 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (5*E_x*\nu_{yx}*b**2*c**2 + 20*E_x*c**4 + 5*E_y*\nu_{xy}*b**2*c**2 - 10*E_y*b**4 - 28*G*\nu_{xy}*\nu_{yx}*b**2*c**2 + 28*G*b**2*c**2)/(20*b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*c**2/2 + E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2 - G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*\nu_{yx}*b**2*c**2/4 - E_x*c**4 - E_y*\nu_{xy}*b**2*c**2/4 - E_y*b**4 + 7*G*\nu_{xy}*\nu_{yx}*b**2*c**2/5 - 7*G*b**2*c**2/5)/(b**3*c**3*(\nu_{xy}*\nu_{yx} - 1)), (E_x*\nu_{yx}*c**2/2 + E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (E_x*c**2 + E_y*\nu_{xy}*b**2/2 - G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1))], [(-5*E_y*b**2 + G*\nu_{xy}*\nu_{yx}*c**2 - G*c**2)/(5*b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2 + G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-E_y*b**2/2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), (-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-E_y*\nu_{xy}*c**2/2 + E_y*b**2/2 + G*\nu_{xy}*\nu_{yx}*c**2/5 - G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 2*(-5*E_y*b**2 - 4*G*\nu_{xy}*\nu_{yx}*c**2 + 4*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (E_y*\nu_{xy}*c**2/2 + E_y*b**2 - G*\nu_{xy}*\nu_{yx}*c**2/5 + G*c**2/5)/(b*c**2*(\nu_{xy}*\nu_{yx} - 1)), 4*(-5*E_y*b**2 + 2*G*\nu_{xy}*\nu_{yx}*c**2 - 2*G*c**2)/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), -E_y*\nu_{xy}/(\nu_{xy}*\nu_{yx} - 1)], [(-E_x*\nu_{yx}*b**2/2 + E_x*c**2/2 + G*\nu_{xy}*\nu_{yx}*b**2/5 - G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, -(10*E_x*c**2 + 8*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), -(E_x*c**2/2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, (-5*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (-E_x*c**2 + G*b**2*(\nu_{xy}*\nu_{yx} - 1)/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), 0, -(10*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1)), (E_x*\nu_{yx}*b**2/2 + E_x*c**2 - G*\nu_{xy}*\nu_{yx}*b**2/5 + G*b**2/5)/(b**2*c*(\nu_{xy}*\nu_{yx} - 1)), -E_x*\nu_{yx}/(\nu_{xy}*\nu_{yx} - 1), 4*(-5*E_x*c**2 + 2*G*b**2*(\nu_{xy}*\nu_{yx} - 1))/(15*b*c*(\nu_{xy}*\nu_{yx} - 1))]])


The surface force matrix $[F_s]$ can be obtained from the shape function matrix. Since we're interested in the surface force matrix for uniform pressures in the direction of w,


```python
q = symbols('q')

Fs = integrate(integrate(N[0, :].T*q, (x, 0, 2*b)), (y, 0, 2*c))

display(Math('[F_s] = 4qcb' + latex(Fs/(4*q*c*b))))

# Uncomment the line below for a version that can be copied/pasted
print(Fs)
```


$\displaystyle [F_s] = 4qcb\left[\begin{matrix}\frac{1}{4}\\\frac{c}{12}\\- \frac{b}{12}\\\frac{1}{4}\\\frac{c}{12}\\\frac{b}{12}\\\frac{1}{4}\\- \frac{c}{12}\\\frac{b}{12}\\\frac{1}{4}\\- \frac{c}{12}\\- \frac{b}{12}\end{matrix}\right]$


    Matrix([[b*c*q], [b*c**2*q/3], [-b**2*c*q/3], [b*c*q], [b*c**2*q/3], [b**2*c*q/3], [b*c*q], [-b*c**2*q/3], [b**2*c*q/3], [b*c*q], [-b*c**2*q/3], [-b**2*c*q/3]])


## Membrane Action

### Shape Functions


```python
from sympy import factor, symbols
r, s = symbols('r, s')
h1 = factor(1/4*(1-r)*(1-s))
h2 = factor(1/4*(1+r)*(1-s))
h3 = factor(1/4*(1+r)*(1+s))
h4 = factor(1/4*(1-r)*(1+s))
```


```python
x1, y1, x2, y2, x3, y3, x4, y4 = symbols('x_1, y_1, x_2, y_2, x_3, y_3, x_4, y_4')
x = h1*x1 + h2*x2 + h3*x3 + h4*x4
y = h1*y1 + h2*y2 + h3*y3 + h4*y4
```


```python
J = Matrix([[diff(x, r), diff(y, r)],
            [diff(x, s), diff(y, s)]])

display(Math('J = (1/4)' + latex(factor(J*4))))
print(J*4)
```


$\displaystyle J = (1/4)\left[\begin{matrix}x_{1} \left(s - 1\right) - 1.0 x_{2} \left(s - 1\right) + x_{3} \left(s + 1\right) - 1.0 x_{4} \left(s + 1\right) & y_{1} \left(s - 1\right) - 1.0 y_{2} \left(s - 1\right) + y_{3} \left(s + 1\right) - 1.0 y_{4} \left(s + 1\right)\\x_{1} \left(r - 1\right) - 1.0 x_{2} \left(r + 1\right) + x_{3} \left(r + 1\right) - 1.0 x_{4} \left(r - 1\right) & y_{1} \left(r - 1\right) - 1.0 y_{2} \left(r + 1\right) + y_{3} \left(r + 1\right) - 1.0 y_{4} \left(r - 1\right)\end{matrix}\right]$


    Matrix([[x_1*(s - 1) - 1.0*x_2*(s - 1) + x_3*(s + 1) - 1.0*x_4*(s + 1), y_1*(s - 1) - 1.0*y_2*(s - 1) + y_3*(s + 1) - 1.0*y_4*(s + 1)], [x_1*(r - 1) - 1.0*x_2*(r + 1) + x_3*(r + 1) - 1.0*x_4*(r - 1), y_1*(r - 1) - 1.0*y_2*(r + 1) + y_3*(r + 1) - 1.0*y_4*(r - 1)]])



```python
print(4*diff(h1, r))
print(4*diff(h2, r))
print(4*diff(h3, r))
print(4*diff(h4, r))
```

    s - 1.0
    1.0 - 1.0*s
    s + 1.0
    -1.0*s - 1.0



```python

```

43ba8e52d1d712d4a7bfa4f53734fe5025b510c2
ipynb
Jupyter Notebook
Derivations/Rectangular Plate Element.ipynb
maderero/PyNite
20fd90c126e3eb0487d541b86bd5a057208af780
MIT
Derivations/Rectangular Plate Element.ipynb
maderero/PyNite
20fd90c126e3eb0487d541b86bd5a057208af780
MIT
Derivations/Rectangular Plate Element.ipynb
maderero/PyNite
20fd90c126e3eb0487d541b86bd5a057208af780
MIT
Qwen/Qwen-72B
1. YES
2. YES