CST steel plate: stiffness, load vector, displacements

Assumptions: plane stress, one CST element. Nodes: 1 = bottom of left edge $(0,-500)$, 2 = top right $(500,0)$, 3 = top left $(0,0)$ (counter-clockwise). $E=200\times10^3$ MPa, $\nu=0.3$, $t=10$ mm, unit weight $78.5\times10^{-6}$ N/mm$^3$. Units: N, mm.

Step 1: Geometry

• Node 1 (bottom of left edge): $(0,\ -500)$ mm
• Node 2 (top right): $(500,\ 0)$ mm
• Node 3 (top left): $(0,\ 0)$ mm

Area: $A=\tfrac12[x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)]=125000\ \text{mm}^2$

$b_1=y_2-y_3=0,\ b_2=y_3-y_1=500,\ b_3=y_1-y_2=-500$ mm

$c_1=x_3-x_2=-500,\ c_2=x_1-x_3=0,\ c_3=x_2-x_1=500$ mm

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

$$[B]=\frac{1}{2A}\begin{bmatrix}0 & 0 & 500 & 0 & -500 & 0 \\ 0 & -500 & 0 & 0 & 0 & 500 \\ -500 & 0 & 0 & 500 & 500 & -500\end{bmatrix},\quad 2A=250000\ \text{mm}^2$$ $$[D]=\frac{E}{1-\nu^2}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\0&0&\frac{1-\nu}{2}\end{bmatrix}=10^{5}\begin{bmatrix}2.198 & 0.659 & 0 \\ 0.659 & 2.198 & 0 \\ 0 & 0 & 0.769\end{bmatrix}\ \text{N/mm}^2$$

Step 3: Element stiffness matrix

$$[k]=t\,A\,[B]^T[D][B]= 10^{6}\begin{bmatrix}0.385 & 0 & 0 & -0.385 & -0.385 & 0.385 \\ 0 & 1.099 & -0.330 & 0 & 0.330 & -1.099 \\ 0 & -0.330 & 1.099 & 0 & -1.099 & 0.330 \\ -0.385 & 0 & 0 & 0.385 & 0.385 & -0.385 \\ -0.385 & 0.330 & -1.099 & 0.385 & 1.484 & -0.714 \\ 0.385 & -1.099 & 0.330 & -0.385 & -0.714 & 1.484\end{bmatrix}\ \text{N/mm}$$

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

Step 4: Load vector

Loads: (i) self-weight $W=\gamma tA=78.5\times10^{-6}\times10\times125000=98.125$ N, shared equally: $W/3=32.708$ N downward at each node. (ii) The 5 kN/m$^2$ = 0.005 N/mm$^2$ traction on the left edge 1-3 (length 500 mm, acting horizontally toward $+x$, assumed): total $0.005\times10\times500=25$ N, half (12.5 N) to each end node.

$$\{F\}=\begin{Bmatrix}12.5 \\ -32.71 \\ 0 \\ -32.71 \\ 12.5 \\ -32.71\end{Bmatrix}\ \text{N}$$

Boundary conditions and solution

The top edge is fixed, so nodes 2 and 3 have zero displacement. The only free node is node 1 ($u_1,v_1$); use rows and columns 1 and 2 of $[k]$.

$$10^{6}\begin{bmatrix}0.385 & 0 \\ 0 & 1.099\end{bmatrix}\begin{Bmatrix}u_1 \\ v_1\end{Bmatrix}=\begin{Bmatrix}12.5 \\ -32.71\end{Bmatrix}$$ $$\begin{aligned} u_1&=3.25\times10^{-5}\ \text{mm} \\ v_1&=-2.976\times10^{-5}\ \text{mm} \end{aligned}$$

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

Centroid: $(166.67,\ -166.67)$ mm.

$$\{\varepsilon\}=[B]\{d\}=\begin{Bmatrix}0 \\ 5.953\times10^{-8} \\ -6.5\times10^{-8}\end{Bmatrix}$$ $$\{\sigma\}=[D]\{\varepsilon\}=\begin{Bmatrix}0.003925 \\ 0.01308 \\ -0.005\end{Bmatrix}\ \text{MPa}$$

Answer: $[k]$ as above (units N/mm); nodal loads $\{F\}$ as above (only node 1 matters: $F_{x1}=12.5$ N, $F_{y1}=-32.71$ N); $u_1=3.25\times10^{-5}$ mm, $v_1=-2.98\times10^{-5}$ mm. The plate is very stiff, so the displacements are extremely small.