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Bar And Rod Elements

The bar element stiffness is

K=AEl[100−100000000000100000]\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

M=ρAl[130016001300160130016130013013]\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

Kg=Fxl[000000100−10100−1000101]\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

K=[AEl00000−AEl00000000000000000000000000GJl00000−GJl00000000000000000AEl00000000000000GJl00000]\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}