CST: stiffness, nodal displacement, stresses

Data: plane stress, one CST element, $t=10$ mm, $E=210\times10^3$ MPa. From $G=E/[2(1+\nu)]=105\times10^3$ MPa, $\nu=E/(2G)-1=0$. Nodes: 1 = top left $(0,0)$, 2 = bottom of left edge $(0,-300)$, 3 = top right $(400,0)$. Units: N, mm.

Step 1: Geometry

• Node 1 (top left): $(0,\ 0)$ mm
• Node 2 (bottom left): $(0,\ -300)$ mm
• Node 3 (top right): $(400,\ 0)$ mm

Area: $A=\tfrac12[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]=60000\ \text{mm}^2$

$b_1=y_2-y_3=-300,\ b_2=y_3-y_1=0,\ b_3=y_1-y_2=300$ mm

$c_1=x_3-x_2=400,\ c_2=x_1-x_3=-400,\ c_3=x_2-x_1=0$ mm

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

$$[B]=\frac{1}{2A}\begin{bmatrix}-300 & 0 & 0 & 0 & 300 & 0 \\ 0 & 400 & 0 & -400 & 0 & 0 \\ 400 & -300 & -400 & 0 & 0 & 300\end{bmatrix},\quad 2A=120000\ \text{mm}^2$$ $$[D]=\frac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}=10^{5}\begin{bmatrix}2.100 & 0 & 0 \\ 0 & 2.100 & 0 \\ 0 & 0 & 1.050\end{bmatrix}\ \text{N/mm}^2$$

Step 3: Element stiffness matrix

$$[k]=t\,A\,[B]^T[D][B]= 10^{6}\begin{bmatrix}1.488 & -0.525 & -0.700 & 0 & -0.787 & 0.525 \\ -0.525 & 1.794 & 0.525 & -1.400 & 0 & -0.394 \\ -0.700 & 0.525 & 0.700 & 0 & 0 & -0.525 \\ 0 & -1.400 & 0 & 1.400 & 0 & 0 \\ -0.787 & 0 & 0 & 0 & 0.787 & 0 \\ 0.525 & -0.394 & -0.525 & 0 & 0 & 0.394\end{bmatrix}\ \text{N/mm}$$

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

Step 4: Load vector

Top edge load $25$ kN/m $=25$ N/mm over $400$ mm $=10\,000$ N, shared: $5000$ N at node 1 (absorbed by the support) and $5000$ N at node 3. Self-weight $W=78.5\times10^{-6}\times10\times60000=47.1$ N, $15.70$ N downward at each node. Load on the free node 3: $F_{y3}=-(5000+15.70)=-5015.70$ N.

$$\{F\}=\begin{Bmatrix}0 \\ 0 \\ 0 \\ 0 \\ 0 \\ -5016\end{Bmatrix}\ \text{N}$$

Boundary conditions and solution

Nodes 1 and 2 lie on the supported left edge ($u=v=0$). Free DOFs: $u_3,v_3$ (rows/columns 5 and 6).

$$10^{5}\begin{bmatrix}7.875 & 0 \\ 0 & 3.938\end{bmatrix}\begin{Bmatrix}u_3 \\ v_3\end{Bmatrix}=\begin{Bmatrix}0 \\ -5016\end{Bmatrix}$$ $$\begin{aligned} u_3&=0\ \text{mm} \\ v_3&=-0.01274\ \text{mm} \end{aligned}$$

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

Centroid: $(133.33,\ -100.00)$ mm.

$$\{\varepsilon\}=[B]\{d\}=\begin{Bmatrix}0 \\ 0 \\ -3.185\times10^{-5}\end{Bmatrix}$$ $$\{\sigma\}=[D]\{\varepsilon\}=\begin{Bmatrix}0 \\ 0 \\ -3.344\end{Bmatrix}\ \text{MPa}$$

Answer: (a) element stiffness matrix as given in Step 3 (N/mm); (b) $u_3=0$ mm, $v_3=-0.0127$ mm (downward), others zero; (c) at the centroid $\varepsilon_x=0$, $\varepsilon_y=0$, $\gamma_{xy}=-3.18\times10^{-5}$ and $\sigma_x=0.000$, $\sigma_y=0.000$, $\tau_{xy}=-3.344$ N/mm$^2$.