====== Trave di Eulero-Bernoulli ======


===== Matrice di rigidezza locale =====

I coefficienti della matrice sono tutti nulli tranne

$$k_{l,1,1} = \frac{E \, A}{l}$$
$$k_{l,1,7} = k_{l,7,1} = -\frac{E \, A}{l} $$

$$k_{l,2,2} = \frac{12 E \, J_{zz}}{l^3}$$
$$k_{l,2,6} = k_{l,6,2} = \frac{6 E \, J_{zz}}{l^2} $$
$$k_{l,2,8} = k_{l,8,5} = - \frac{12 E \, J_{zz}}{l^3}$$
$$k_{l,2,12} = k_{l,12,2} = \frac{6 E \, J_{zz}}{l^2} $$

$$k_{l,3,3} = \frac{12 E \, J_{yy}}{l^3}$$
$$k_{l,3,5} = k_{l,5,3} = - \frac{6 E \, J_{yy}}{l^2} $$
$$k_{l,3,9} = k_{l,9,3} = - \frac{12 E \, J_{yy}}{l^3}$$
$$k_{l,3,11} = k_{l,11,3} = - \frac{6 E \, J_{yy}}{l^2} $$

$$k_{l,4,4} = \frac{G \, J_t}{l}$$
$$k_{l,4,10} = k_{l,10,4} = - \frac{G \, J_t}{l}$$

$$k_{l,5,5} = \frac{4 E \, J_{yy}}{l}$$
$$k_{l,5,9} = k_{l,9,5} = \frac{6 E \, J_{yy}}{l^2} $$
$$k_{l,5,11} = k_{l,11,5} = \frac{2 E \, J_{yy}}{l} $$

$$k_{l,6,6} = \frac{4 E \, J_{zz}}{l}$$
$$k_{l,6,8} = k_{l,8,6} = - \frac{6 E \, J_{zz}}{l^2} $$
$$k_{l,6,12} = k_{l,12,6} = \frac{2 E \, J_{zz}}{l} $$

$$k_{l,7,7} = \frac{E \, A}{l}$$

$$k_{l,8,8} = \frac{12 E \, J_{zz}}{l^3}$$
$$k_{l,8,12} = k_{l,12,8} = - \frac{6 E \, J_{zz}}{l^2} $$

$$k_{l,9,9} = \frac{12 E \, J_{yy}}{l^3}$$
$$k_{l,9,11} = k_{l,11,9} = \frac{6 E \, J_{yy}}{l^2} $$

$$k_{l,10,10} = \frac{G \, J_t}{l}$$

$$k_{l,11,11} = \frac{4 E \, J_{yy}}{l}$$

$$k_{l,12,12} = \frac{4 E \, J_{zz}}{l}$$

La matrice di rigidezza locale è allora uguale a

$$\begin{bmatrix}
\frac{E \, A}{l} & 0 & 0 & 0 & 0 & 0 & - \frac{E \, A}{l} & 0 & 0 & 0 & 0 & 0 \\\\
0 & \frac{12 E \, J_{zz}}{l^3} & 0& 0 & 0 & \frac{6 E \, J_{zz}}{l^2} & 0 & -\frac{12 E \, J_{zz}}{l^3} & 0 & 0 & 0 & \frac{6 E \, J_{zz}}{l^2} \\\\
0 & 0 & \frac{12 E \, J_{yy}}{l^3 }& 0 & - \frac{6 E \, J_{yy}}{l^2} & 0 & 0 & 0 & -\frac{12 E \, J_{yy}}{l^3)} & 0 & -\frac{6 E \, J_{yy}}{l^2} & 0 \\\\
0 & 0 & 0& \frac{G \, J_{t}}{l} & 0 & 0 & 0 & 0 & 0 & - \frac{G \, J_{t}}{l} & 0 & 0 \\\\
0 & 0 & - \frac{6 E \, J_{yy}}{l^2}& 0 & \frac{4 E \, J_{yy}}{l} & 0 & 0 & 0 & \frac{6 \, E \, J_{yy}}{l^2} & 0 & \frac{2 E \, J_{yy}}{l} & 0 \\\\
0 & \frac{6 E \, J_{zz}}{l^2} & 0& 0 & 0 & \frac{4 E \, J_{zz}}{l} & 0 & - \frac{6 \, E \, J_{zz}}{l^2} & 0 & 0 & 0 & \frac{2 E \, J_{zz}}{l} \\\\
- \frac{E \, A}{l} & 0 & 0 & 0 & 0 & 0 & \frac{E \, A}{l} & 0 & 0 & 0 & 0 & 0 \\\\
0 & -\frac{12 E \, J_{zz}}{l^3} & 0& 0 & 0 & - \frac{6 \, E \, J_{zz}}{l^2} & 0 & \frac{12 E \, J_{zz}}{l^3 } & 0 & 0 & 0 & - \frac{6 E \, J_{zz}}{l^2} \\\\
0 & 0 & -\frac{12 E \, J_{yy}}{l^3}& 0 & \frac{6 \, E \, J_{yy}}{l^2} & 0 & 0 & 0 & \frac{12 E \, J_{yy}}{l^3} & 0 & \frac{6 E \, J_{yy}}{l^2} & 0 \\\\
0 & 0 & 0& - \frac{G \, J_{t}}{l} & 0 & 0 & 0 & 0 & 0 & \frac{G \, J_{t}}{l} & 0 & 0 \\\\
0 & 0 & -\frac{6 E \, J_{yy}}{l^2}& 0 & \frac{2 E \, J_{yy}}{l} & 0 & 0 & 0 & \frac{6 E \, J_{yy}}{l^2} & 0 & \frac{4 E \, J_{yy}}{l} & 0 \\\\
0 & \frac{6 E \, J_{zz}}{l^2} & 0& 0 & 0 & \frac{2 E \, J_{zz}}{l} & 0 & - \frac{6 E \, J_{zz}}{l^2} & 0 & 0 & 0 & \frac{4 E \, J_{zz}}{l} \\\\
\end{bmatrix}$$