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Bar element

The bar element only bears axial strains with a multi-linear axial stiffness. A point P of the structure in the global reference is located by its position on the mean line :

\[ \overrightarrow {OP} = \vec {x_o}(s) \]

Then comes the axial strain, \(\varepsilon = \left \lVert\frac {\partial \vec {x_o}}{\partial s} \right \rVert -1\), and the axial force, \(\vec N_x = EA \varepsilon \vec n\) , with \(\vec n = \frac {\vec {x_0}}{\lVert \vec{x_o} \rVert}\).

Finally, the internal efforts contribution of a bar element is given by :

\[ G_{internal}^{bar}(\vec x, \delta\vec{x}) = <\vec{N_x}.\delta\vec{x_o}> \]

Let \(A_{\rho}\) be the lineic mass diagonal matrix. Then the inertia contribution writes :

\[ G_{internal}^{bar} = < A_{\rho} \ddot{\vec {x_o}}.\delta\vec{x_o}> \]