Thin plate: nodal displacements and stresses

The plate is a single CST in plane stress. Units: lb and in.

Assumptions

• Node coordinates (in): 1 (0, 0), 2 (0, 10), 3 (20, 10). Nodes are taken in anticlockwise order 1-3-2 ($i=1,\ j=3,\ m=2$) so that the area is positive.
• Node 1 is pinned ($u_1=v_1=0$); node 2 is on a roller that stops horizontal movement ($u_2=0$). Free dofs: $v_2,\ u_3,\ v_3$.
• Loads at node 3: $F_{3x}=+2000$ lb, $F_{3y}=-2000$ lb.
• $E=30\times10^6$ psi, $\nu=0.3$, $t=0.3$ in.

Step 1: Geometry

$$A=\tfrac12(20)(10)=100\ \text{in}^2$$

$b_i=y_j-y_m=0,\ b_j=y_m-y_i=10,\ b_m=y_i-y_j=-10$; $c_i=x_m-x_j=-20,\ c_j=x_i-x_m=0,\ c_m=x_j-x_i=20$.

$$[B]=\frac{1}{200}\begin{bmatrix}0&0&10&0&-10&0\\0&-20&0&0&0&20\\-20&0&0&10&20&-10\end{bmatrix}$$

(columns in the order $u_1,v_1,u_3,v_3,u_2,v_2$).

Step 2: Elasticity matrix

$$[D]=\frac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}=\begin{bmatrix} 3.2967\times10^{7} & 9890110 & 0 \\ 9890110 & 3.2967\times10^{7} & 0 \\ 0 & 0 & 1.1538\times10^{7} \end{bmatrix}\ \text{psi}$$

Step 3: Stiffness and reduced equations

$[k]=tA[B]^T[D][B]$ is assembled in global dof order $(u_1,v_1,u_2,v_2,u_3,v_3)$. Removing the rows and columns of $u_1,v_1,u_2$:

$$\begin{bmatrix} 1.0755\times10^{7} & 1483516 & -865385 \\ 1483516 & 2472527 & 0 \\ -865385 & 0 & 865385 \end{bmatrix}\begin{bmatrix}v_2\\u_3\\v_3\end{bmatrix}=\begin{bmatrix}0\\2000\\-2000\end{bmatrix}$$

Step 4: Nodal displacements

$$v_2=-3.556\times10^{-4}\ \text{in},\quad u_3=1.022\times10^{-3}\ \text{in},\quad v_3=-2.667\times10^{-3}\ \text{in}$$

Reactions (check): $R_{1x}=4000$, $R_{1y}=2000$, $R_{2x}=-6000$ lb. Then $\Sigma F_x = 4000-6000+2000=0$ and $\Sigma F_y = 2000-2000=0$.

Step 5: Element stresses, $\{\sigma\}=[D][B]\{u\}$

$$\sigma_x=1333.3\ \text{psi},\quad \sigma_y=-666.7\ \text{psi},\quad \tau_{xy}=-1333.3\ \text{psi}$$

Principal stresses: $\sigma_1=2000.0$ psi, $\sigma_2=-1333.3$ psi, at $\theta_p=-26.57^\circ$ from the $x$-axis.

Answer: $v_2=-3.556\times10^{-4}$ in, $u_3=1.022\times10^{-3}$ in, $v_3=-2.667\times10^{-3}$ in; $\sigma_x=1333.3$, $\sigma_y=-666.7$, $\tau_{xy}=-1333.3$ psi.