CST plate: stresses at centroid

A CST has constant strain, so the stresses at the centroid equal the stresses everywhere in the plate. The plate is modelled as one CST in plane stress (units N and mm).

Assumptions

• The load "10 kN/mm$^2$" is read as a line load $w = 10$ kN/m $= 10$ N/mm along the 400 mm left edge, acting horizontally (normal to the edge, in $+x$). Total load $= 10\times400 = 4000$ N, shared equally by the end nodes of the edge.
• Nodes (anticlockwise): 1 (0, 0) top-left, 2 (0, -400) bottom-left, 3 (300, 0) top-right. The top edge 1-3 is supported, so $u_1=v_1=u_3=v_3=0$.
• $E=2\times10^5$ N/mm$^2$, $\nu=0.3$, $t=10$ mm.

Step 1: Geometry

$$A=\tfrac12(300)(400)=60000\ \text{mm}^2$$

$b_1=y_2-y_3=-400,\ b_2=y_3-y_1=0,\ b_3=y_1-y_2=400$; $c_1=x_3-x_2=300,\ c_2=x_1-x_3=-300,\ c_3=x_2-x_1=0$.

$$[B]=\frac{1}{120000}\begin{bmatrix} -400&0&0&0&400&0\\ 0&300&0&-300&0&0\\ 300&-400&-300&0&0&400\end{bmatrix}$$

Step 2: [D] for plane stress

$$[D]=\frac{E}{1-\nu^2}\begin{bmatrix}1&0.3&0\\0.3&1&0\\0&0&0.35\end{bmatrix}=\begin{bmatrix} 219780 & 65934 & 0 \\ 65934 & 219780 & 0 \\ 0 & 0 & 76923 \end{bmatrix}\ \text{N/mm}^2$$

Step 3: Stiffness of the free node 2

$$[k]_{22}=tA[B]^T[D][B]\Big|_{u_2,v_2}=\begin{bmatrix} 288462 & 0 \\ 0 & 824176 \end{bmatrix}\ \text{N/mm}$$

Step 4: Displacement

Load at node 2: $F_{2x}=4000/2=2000$ N, $F_{2y}=0$.

$$u_2=\frac{2000}{288461.5}=0.00693\ \text{mm},\qquad v_2=0$$

Step 5: Stresses

$$\{\varepsilon\}=[B]\{u\}=\begin{bmatrix}0\\0\\-1.733\times10^{-5}\end{bmatrix},\qquad \{\sigma\}=[D]\{\varepsilon\}=\begin{bmatrix}0\\0\\-1.333\end{bmatrix}\ \text{N/mm}^2$$

Check: the element force at node 2 in the $x$ direction, from $tA[B]^T\{\sigma\}$, is $\tfrac{t}{2}(b_2\sigma_x+c_2\tau_{xy})=5\times(-300)\times(-1.333)=2000$ N, which balances the applied $F_{2x}$.

Answer: at the centroid $\sigma_x=0$, $\sigma_y=0$, $\tau_{xy}=-1.333$ N/mm$^2$, i.e. the plate is in pure shear.