CST plate: stresses and strains at centroid

The plate is modelled by one CST element ABC in plane stress ($t=20$ mm, $E=2.1\times10^5$ N/mm$^2$, $\nu=0.3$). Node numbers: A = 1, B = 2, C = 3.

Step 1: Geometry

• Node A (1): $(0,\ 0)$ mm
• Node B (2): $(4000,\ 0)$ mm
• Node C (3): $(0,\ 3000)$ mm

Area: $A=\tfrac12[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]=6\times10^{6}\ \text{mm}^2$

$b_1=y_2-y_3=-3000,\ b_2=y_3-y_1=3000,\ b_3=y_1-y_2=0$ mm

$c_1=x_3-x_2=-4000,\ c_2=x_1-x_3=0,\ c_3=x_2-x_1=4000$ mm

Step 2: [B] and [D] matrices (plane stress)

$$[B]=\frac{1}{2A}\begin{bmatrix}-3000 & 0 & 3000 & 0 & 0 & 0 \\ 0 & -4000 & 0 & 0 & 0 & 4000 \\ -4000 & -3000 & 0 & 3000 & 4000 & 0\end{bmatrix},\quad 2A=1.2\times10^{7}\ \text{mm}^2$$ $$[D]=\frac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}=10^{5}\begin{bmatrix}2.308 & 0.692 & 0 \\ 0.692 & 2.308 & 0 \\ 0 & 0 & 0.808\end{bmatrix}\ \text{N/mm}^2$$

Step 3: Element stiffness matrix

$$[k]=t\,A\,[B]^T[D][B]= 10^{6}\begin{bmatrix}2.808 & 1.500 & -1.731 & -0.808 & -1.077 & -0.692 \\ 1.500 & 3.683 & -0.692 & -0.606 & -0.808 & -3.077 \\ -1.731 & -0.692 & 1.731 & 0 & 0 & 0.692 \\ -0.808 & -0.606 & 0 & 0.606 & 0.808 & 0 \\ -1.077 & -0.808 & 0 & 0.808 & 1.077 & 0 \\ -0.692 & -3.077 & 0.692 & 0 & 0 & 3.077\end{bmatrix}\ \text{N/mm}$$

(DOF order: $u_1,v_1,u_2,v_2,u_3,v_3$.)

Step 4: Load vector

The 50 kN upward load ($+y$) and 70 kN horizontal load ($+x$) act at B (node 2): $F_{x2}=70\,000$ N, $F_{y2}=50\,000$ N. Self-weight is ignored (as stated). Coordinates are taken in mm with A at the origin.

$$\{F\}=\begin{Bmatrix}0 \\ 0 \\ 70\,000 \\ 50\,000 \\ 0 \\ 0\end{Bmatrix}\ \text{N}$$

Boundary conditions and solution

A and C are on the supported left edge, so $u_1=v_1=u_3=v_3=0$. Only $u_2,v_2$ are unknown; use rows and columns 3 and 4 of $[k]$.

$$10^{6}\begin{bmatrix}1.731 & 0 \\ 0 & 0.606\end{bmatrix}\begin{Bmatrix}u_2 \\ v_2\end{Bmatrix}=\begin{Bmatrix}70\,000 \\ 50\,000\end{Bmatrix}$$ $$\begin{aligned} u_2&=0.04044\ \text{mm} \\ v_2&=0.08254\ \text{mm} \end{aligned}$$

Strains and stresses (constant over the element, so equal at the centroid)

Centroid: $(1333.33,\ 1000.00)$ mm.

$$\{\varepsilon\}=[B]\{d\}=\begin{Bmatrix}1.011\times10^{-5} \\ 0 \\ 2.063\times10^{-5}\end{Bmatrix}$$ $$\{\sigma\}=[D]\{\varepsilon\}=\begin{Bmatrix}2.333 \\ 0.7 \\ 1.667\end{Bmatrix}\ \text{MPa}$$

Answer: $u_B=0.0404$ mm, $v_B=0.0825$ mm. Strains at the centroid: $\varepsilon_x=1.01\times10^{-5}$, $\varepsilon_y=0$, $\gamma_{xy}=2.06\times10^{-5}$. Stresses at the centroid: $\sigma_x=2.333$, $\sigma_y=0.700$, $\tau_{xy}=1.667$ N/mm$^2$.

Check: reactions from $[k]\{d\}$ at A and C are $(-136.7,-78.0)$ kN and $(66.7,28.0)$ kN; with the loads at B, $\Sigma F_x=0$ and $\Sigma F_y=0$.