Truss element stiffness matrices

The stiffness of a 2D truss element is $[k]=\dfrac{AE}{L}\begin{bmatrix}c^2&cs&-c^2&-cs\\cs&s^2&-cs&-s^2\\-c^2&-cs&c^2&cs\\-cs&-s^2&cs&s^2\end{bmatrix}$ with $c=\dfrac{x_j-x_i}{L}$, $s=\dfrac{y_j-y_i}{L}$.

Geometry and assumptions

Coordinates in mm: node 1 (0, 0), node 2 (500, 0), node 3 (500, 400), node 4 (0, 400). Element 1 is the diagonal 1-3; element 2 is the member 3-4, as drawn. $AE=300\times2.1\times10^5=6.3\times10^7$ N. Global dofs are $(u_n, v_n)$ at each node $n$.

Element 1 (nodes 1 to 3)

$$L=\sqrt{500^2+400^2}=640.31\ \text{mm},\quad c=\frac{500}{L}=0.7809,\quad s=\frac{400}{L}=0.6247,\quad \frac{AE}{L}=98389.47\ \text{N/mm}$$

Rows and columns correspond to $(u_1,\ v_1,\ u_3,\ v_3)$:

$$[k]_1=\begin{bmatrix} 59993.58 & 47994.86 & -59993.58 & -47994.86 \\ 47994.86 & 38395.89 & -47994.86 & -38395.89 \\ -59993.58 & -47994.86 & 59993.58 & 47994.86 \\ -47994.86 & -38395.89 & 47994.86 & 38395.89 \end{bmatrix}\ \text{N/mm}$$

Element 2 (nodes 3 to 4)

The member is horizontal: $L=500$ mm, $c=-1$ (from node 3 to node 4), $s=0$, $AE/L=126000$ N/mm. Only $c^2=1$ and $cs=0$ appear in the matrix, so the sign of $c$ does not matter. Rows and columns correspond to $(u_3,\ v_3,\ u_4,\ v_4)$:

$$[k]_2=\begin{bmatrix} 126000 & 0 & -126000 & 0 \\ 0 & 0 & 0 & 0 \\ -126000 & 0 & 126000 & 0 \\ 0 & 0 & 0 & 0 \end{bmatrix}\ \text{N/mm}$$

Checks

• Both matrices are symmetric and every row sums to zero.
• For element 1, $c^2+s^2=0.6098+0.3902=1$.
• A horizontal bar has no stiffness in the $v$ direction, so the $v$ rows of $[k]_2$ are zero.

Answer: $[k]_1$ and $[k]_2$ as above, with $AE/L=98389.47$ N/mm and $126000$ N/mm respectively.