Question p.6.14: The following interpolation formula is suggested as a displa...
The following interpolation formula is suggested as a displacement function for deriving the stiffness of a plane stress rectangular element of uniform thickness t shown in Fig. P.6.14.
Form the strain matrix and obtain the stiffness coefficients K_{11} and K_{12} in terms of the material constants c, d and e defined below.
In the elasticity matrix [D]
D_{11}=D_{22}=c \quad D_{12}=d \quad D_{33}=e \quad \text { and } \quad D_{13}=D_{23}=0
For a = 1, b = 2
u=\frac{1}{8}\left[(1-x)(2-y) u_1+(1+x)(2-y) u_2+(1+x)(2+y) u_3+(1-x)(2+y) u_4\right]Similarly for v
Then
In matrix form
\left[\begin{array}{c}\frac{\partial u}{\partial x} \\\frac{\partial v}{\partial y} \\\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\end{array}\right]
=\frac{1}{8}\left[\begin{array}{cccccccc}-(2-y) & 0 & (2-y) & 0 & (2+y) & 0 & -(2+y) & 0 \\0 & -(1-x) & 0 & -(1+x) & 0 & (1+x) & 0 & (1-x) \\-(1-x) & -(2-y) & -(1+x) & (2-y) & (1+x) & (2+y) & (1-x) & -(2+y)\end{array}\right] \left\{\begin{array}{l}u_1 \\v_1 \\u_2 \\v_2 \\u_3 \\v_3 \\u_4 \\v_4\end{array}\right\}
Also
D=\left[\begin{array}{lll}c & d & 0 \\d & c & 0 \\0 & 0 & e\end{array}\right]Then
[D][B]
Then
[B]^{\mathrm{T}}[D][B]=\frac{1}{64} \left[\begin{array}{ccc}-(2-y) & 0 & -(1-x) \\0 & -(1-x) & -(2-y) \\& \vdots & \\& \vdots & \\& \vdots & \\& \vdots & \\& \vdots & \\& \vdots &\end{array}\right] \left[\begin{array}{lllll}-c(2-y) & -d(1-x) & \ldots & \ldots & \ldots \\-d(2-y) & -c(1-x) & \ldots & \ldots & \ldots \\-e(1-x) & -e(2-y) & \ldots & \ldots&\ldots\end{array}\right]
Therefore
K_{11}=\frac{t}{64} \int_{-2}^2 \int_{-1}^1\left[c(2-y)^2+e(1-x)^2\right] \mathrm{d} x \mathrm{~d} ywhich gives K_{11}=\frac{t}{6}(4 c+e)
K_{12}=\frac{t}{64} \int_{-2}^2 \int_{-1}^1[d(2-y)(1-x)+e(1-x)(2-y)] \mathrm{d} x \mathrm{~d} ywhich gives K_{12}=\frac{t}{4}(d+e).