# Bar And Rod Elements

The bar element stiffness is

$\mathbf{K} = \frac{AE}{l}\begin{bmatrix} 1 & 0 & 0 & - 1 & 0 & 0 \\ & 0 & 0 & 0 & 0 & 0 \\ & & 0 & 0 & 0 & 0 \\ & & & 1 & 0 & 0 \\ & & & & 0 & 0 \\ & & & & & 0 \\ \end{bmatrix}$

The mass matrix is

$\mathbf{M} = \rho Al\begin{bmatrix} \frac{1}{3} & 0 & 0 & \frac{1}{6} & 0 & 0 \\ & \frac{1}{3} & 0 & 0 & \frac{1}{6} & 0 \\ & & \frac{1}{3} & 0 & 0 & \frac{1}{6} \\ & & & \frac{1}{3} & 0 & 0 \\ & & & & \frac{1}{3} & 0 \\ & & & & & \frac{1}{3} \\ \end{bmatrix}$

And the geometric stiffness is

$\mathbf{K}_{g} = \frac{F_{x}}{l}\begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ & 1 & 0 & 0 & - 1 & 0 \\ & & 1 & 0 & 0 & - 1 \\ & & & 0 & 0 & 0 \\ & & & & 1 & 0 \\ & & & & & 1 \\ \end{bmatrix}$

The rod element stiffness is

$\mathbf{K} = \begin{bmatrix} \frac{AE}{l} & 0 & 0 & 0 & 0 & 0 & - \frac{AE}{l} & 0 & 0 & 0 & 0 & 0 \\ & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ & & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ & & & \frac{GJ}{l} & 0 & 0 & 0 & 0 & 0 & - \frac{GJ}{l} & 0 & 0 \\ & & & & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ & & & & & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ & & & & & & \frac{AE}{l} & 0 & 0 & 0 & 0 & 0 \\ & & & & & & & 0 & 0 & 0 & 0 & 0 \\ & & & & & & & & 0 & 0 & 0 & 0 \\ & & & & & & & & & \frac{GJ}{l} & 0 & 0 \\ & & & & & & & & & & 0 & 0 \\ & & & & & & & & & & & 0 \\ \end{bmatrix}$