Bar stiffness matrix and 2D truss element

A bar (rod) element carries only axial force. It has two nodes and one dof per node in its local axis; a 2D truss element is the same bar placed at an angle, with two dofs per node.

Part 1: Stiffness matrix of the bar element

   1 o--------------------o 2  ---> x
     u1   (A, E, L)       u2

Assume a linear displacement field $u(x)=\alpha_1+\alpha_2x$. Applying $u(0)=u_1$ and $u(L)=u_2$:

$$u(x)=N_1u_1+N_2u_2,\qquad N_1=1-\frac{x}{L},\quad N_2=\frac{x}{L}$$

Strain:

$$\varepsilon=\frac{du}{dx}=\begin{bmatrix}-\frac1L&\frac1L\end{bmatrix}\begin{bmatrix}u_1\\u_2\end{bmatrix}=[B]\{u\}$$

Stress $\sigma=E\varepsilon$. The stiffness matrix from the strain energy (or the principle of minimum potential energy):

$$[k]=\int_0^L[B]^TE[B]A\,dx=AE\int_0^L\begin{bmatrix}\frac1{L^2}&-\frac1{L^2}\\-\frac1{L^2}&\frac1{L^2}\end{bmatrix}dx$$ $$[k]=\frac{AE}{L}\begin{bmatrix}1&-1\\-1&1\end{bmatrix}$$

Direct check: a unit displacement $u_1=1,\ u_2=0$ needs force $AE/L$ at node 1 and $-AE/L$ at node 2 (equilibrium). The matrix is symmetric and singular (rigid-body motion is possible until supports are added).

Part 2: Rotating the bar - 2D truss element

                 v2 ^  u2 (global)
                    |/
   y ^           2 o
     |          /  
     |        / L      theta = angle of the bar
     |      /          from global x
     |    o 1
     |__theta_______> x

Let the bar make angle $\theta$ with the global $x$-axis, $c=\cos\theta=\dfrac{x_2-x_1}{L}$, $s=\sin\theta=\dfrac{y_2-y_1}{L}$. Global dofs: $\{d\}=[u_1\ v_1\ u_2\ v_2]^T$; local axial dofs: $\{d'\}=[u_1'\ u_2']^T$.

The local axial displacement at a node is the projection of the global displacement on the bar axis:

$$\begin{bmatrix}u_1'\\u_2'\end{bmatrix}=\underbrace{\begin{bmatrix}c&s&0&0\\0&0&c&s\end{bmatrix}}_{[T]}\begin{bmatrix}u_1\\v_1\\u_2\\v_2\end{bmatrix}$$

The strain energy $\tfrac12\{d'\}^T[k']\{d'\}=\tfrac12\{d\}^T[T]^T[k'][T]\{d\}$ must equal $\tfrac12\{d\}^T[k]\{d\}$, so

$$[k]=[T]^T[k'][T]$$ $$[k]=\frac{AE}{L}\begin{bmatrix}c^2&cs&-c^2&-cs\\cs&s^2&-cs&-s^2\\-c^2&-cs&c^2&cs\\-cs&-s^2&cs&s^2\end{bmatrix}$$

Properties

• Symmetric; every row sums to zero (equilibrium of a free element).
• Singular: three rigid-body modes in the plane (two translations and one rotation).
• $\theta=0$: only $u$ terms remain (horizontal bar). $\theta=90^\circ$: only $v$ terms remain (vertical bar).
• After finding the nodal displacements, the bar force is $F=\dfrac{AE}{L}\,(u_2'-u_1')=\dfrac{AE}{L}\begin{bmatrix}-c&-s&c&s\end{bmatrix}\{d\}$.