5. Stiffness matrix

The stiffness matrix \(K\) of the element is derived from Fréchet from \(F\) in the direction of displacement \(\deltau\) of the nodes:

\(\deltaf =K\deltau\).

\(K\) is calculated by the following classical formula, used extensively in [bib3], [bib4] and [bib5]:

\(\deltaf =\underset{\varepsilon \to 0}{\text{lim}}\frac{\delta }{\delta \varepsilon }F(u+\varepsilon \deltau )\). Eq 5-1

The details of the calculations are given in [§An1] and the final expression for \(K\) is as follows:

\(K=\left[\begin{array}{ccc}{K}_{\text{11}}+\frac{N}{{l}_{1}}{I}_{3}& {K}_{\text{12}}& -{K}_{\text{11}}-{K}_{\text{12}}-\frac{N}{{l}_{1}}{I}_{3}\\ {K}_{\text{12}}^{T}& {K}_{\text{22}}+\frac{N}{{l}_{2}}{I}_{3}& -{K}_{\text{22}}-{K}_{\text{12}}^{T}-\frac{N}{{l}_{2}}{I}_{3}\\ -{K}_{\text{11}}-{K}_{\text{12}}^{T}-\frac{N}{{l}_{1}}{I}_{3}& -{K}_{\text{22}}-{K}_{\text{12}}-\frac{N}{{l}_{2}}{I}_{3}& {K}_{\text{11}}+{K}_{\text{22}}+{K}_{\text{12}}+{K}_{\text{12}}^{T}+(\frac{1}{{l}_{1}}+\frac{1}{{l}_{2}})N{I}_{3}\end{array}\right]\) eq 5-2

\(N\) is given by [éq 3-4] and [éq 3-3];

\(\begin{array}{}{K}_{\text{11}}=(\frac{\text{EA}}{{l}_{0}}-\frac{N}{{l}_{1}})\frac{1}{{l}_{1}^{2}}{l}_{1}{l}_{1}^{T};\\ {K}_{\text{12}}=\frac{\text{EA}}{{l}_{0}{l}_{1}{l}_{2}}{l}_{1}{l}_{2}^{T};\\ {K}_{\text{22}}=(\frac{\text{EA}}{{l}_{0}}-\frac{N}{{l}_{2}})\frac{1}{{l}_{2}^{2}}{l}_{2}{l}_{2}^{T}\text{.}\end{array}\)

\(K\) is symmetric, because of the symmetry of \({K}_{\text{11}}\) and \({K}_{\text{22}}\) and the global block symmetry.

But \(K\) depends on the movements of \({N}_{1},{N}_{2}\text{et}{N}_{3}\) via \({l}_{1},{l}_{2}\text{et}N\): the finite cable-pulley element is therefore a non-linear element.