Stiffness matrix
===================

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

:math:`\deltaf =K\deltau`.

:math:`K` is calculated by the following classical formula, used extensively in [:ref:`bib3 <bib3>`], [:ref:`bib4 <bib4>`] and [:ref:`bib5 <bib5>`]:


.. _RefEquation 5-1:

:math:`\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 [:ref:`§An1 <§An1>`] and the final expression for :math:`K` is as follows:

:math:`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

:math:`N` is given by [:ref:`éq 3-4 <éq 3-4>`] and [:ref:`éq 3-3 <éq 3-3>`];

:math:`\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}`

:math:`K` is **symmetric**, because of the symmetry of :math:`{K}_{\text{11}}` and :math:`{K}_{\text{22}}` and the global block symmetry.

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