6. Digital realization by finite elements#
We denote by the lower index \(h\) the matrices discretized in finite elements. If \(v\) is a vector defined on the cable (position, displacement, acceleration,…) we have at the current point of a finite element with \(i,j,\dots\) nodes:
\(v=\left[L\right]{v}_{e}\)
\({v}_{e}\) being the vector made up of the components of \(v\) at the nodes. Likewise:
\({\left[\frac{\partial }{\partial {s}_{o}}1\right]}_{h}v=\left[L\text{'}\right]{v}_{e}\text{.}\)
According to [eq]:
\({B}_{h}={(x+u)}_{e}^{T}L{\text{'}}^{T}\text{.}L\text{'}\)
The internal forces \({F}_{\text{int}}^{e}\) of a finite element \(e\) of structure are the forces that must be exerted in its nodes to maintain it in its current deformed configuration. According to the virtual work theorem for continuous environments, the work of these point forces is equal to the work of the constraints in the element, that is to say to \({W}_{\text{int}}\), for any virtual field of displacement. We therefore have, according to [eq]:
\({F}_{\text{int}}^{e}={\int }_{{s}_{1}}^{{s}_{2}}N{B}_{h}^{T}\text{ds}={\int }_{{s}_{1}}^{{s}_{2}}NL{\text{'}}^{T}L\text{'}\mathrm{ds}{(x+u)}_{e}\).
On the other hand, the inertial forces distributed in the element are replaced by point forces at the nodes \({F}_{\text{iner}}^{e}\) such that their work is equal to that of the real inertia forces for any virtual field of displacement. According to [eq], we therefore have:
\({F}_{\text{iner}}^{e}=-{\int }_{{s}_{1}}^{{s}_{2}}{L}^{T}\rho AL\mathrm{ds}\mathrm{.}\ddot{{u}_{e}}\)
Likewise, distributed external forces are replaced by concentrated nodal forces \({F}_{\text{ext}}^{e}\) equivalent to the meaning of virtual work. The differential of the virtual work of the internal forces of a finite cable element is written, according to [eq]:
, \({D}_{h}{W}_{\text{int}}(u,\delta u)\text{.}\Delta u={(\delta {u}_{e})}^{T}({K}_{M}+{K}_{G})\Delta {u}_{e}\)
with:
\(\begin{array}{c}{K}_{M}={\int }_{{s}_{1}}^{{s}_{2}}{B}_{h}^{T}{E}_{a}A{B}_{h}\text{ds}\\ {K}_{G}={\int }_{{s}_{1}}^{{s}_{2}}L{\text{'}}^{T}NL\text{'}\text{ds}\end{array}\)
\({K}_{M}\) and \({K}_{G}\) are called material and geometric stiffness matrices for the element. \(({K}_{M}+{K}_{G})\Delta {u}_{e}\) is the main part of the \(\Delta {F}_{\text{int}}\) variation in internal forces at the nodes due to the correction of \(\Delta {u}_{e}\) displacements. The differential of the virtual work of the inertial forces is deduced from [eq]:
\({D}_{h}{W}_{\text{iner}}(\ddot{u},\delta u)\text{.}\Delta u=-{(\delta {u}_{e})}^{T}M\Delta {\ddot{u}}_{e}\)
with:
\(M={\int }_{{s}_{1}}^{{s}_{2}}{L}^{T}\rho AL\text{ds}\)
\(M\) is the mass matrix of the element. \(-M\Delta {\ddot{u}}_{e}\) is the \(\Delta {F}_{\text{iner}}\) variation in inertia forces at the nodes due to the acceleration correction \(\Delta {\ddot{u}}_{e}\).