1. Reference problem#
1.1. Geometry#
Figure 1.1-1 : model studied
It is a concrete cylinder with a height of \(H\mathrm{=}10m\) and an average radius of \(R\mathrm{=}20.5m\).
The thickness of the veil is \(e\mathrm{=}1m\).
The cables each describe a circle in a horizontal plane, and thus travel along the length of the structure. All cables are anchored on the same line. The 10 cables are spaced \(1m\) between \(z\mathrm{=}\mathrm{-}4.5m\) and \(z\mathrm{=}4.5m\).
The cross-sectional area of each cable is \({S}_{a}\mathrm{=}7.923{10}^{\mathrm{-}3}{m}^{2}\).
1.2. Material properties#
1.2.1. Material: concrete constituting the veil#
Elastic properties:
Young’s module |
\({E}_{b}={3.10}^{10}Pa\) |
Poisson’s Ratio |
\({\nu }_{b}\mathrm{=}\mathrm{0,2}\) |
1.2.2. Material: steel constituting the cables#
Elastic properties:
Young’s module |
\({E}_{a}\mathrm{=}\mathrm{1,915}{\text{.}10}^{11}Pa\) |
Poisson’s Ratio |
\({\nu }_{a}\mathrm{=}\mathrm{0,3}\) |
Characteristic parameters for estimating voltage losses: |
|
Elastic limit stress of steel |
\({f}_{\mathit{prg}}\mathrm{=}\mathrm{1,814}\mathrm{.}{10}^{9}\mathit{Pa}\) |
Coefficient of friction |
\(\mu \mathrm{=}\mathrm{0,17}\) |
Online loss coefficient |
\(k\mathrm{=}\frac{\mathrm{0,0015}}{\mathrm{0,17}}{m}^{\mathrm{-}1}\) |
Relaxation of steel at 1000 hours |
\({\rho }_{1000}\mathrm{=}\mathrm{2,5}\text{\%}\) |
1.3. Loading#
A normal force of traction is applied to both ends of each cable. The value of the stress applied in the steel is \({\sigma }_{0}\mathrm{=}\mathrm{0,8}{F}_{\mathit{prg}}\mathrm{=}\mathrm{1487,48}\mathit{MPa}\) or a tension \({F}_{0}\mathrm{=}\mathrm{11,785}{.10}^{6}N\).
To assess the voltage losses in the vicinity of the anchorages, a setback at anchors \(\Delta \mathrm{=}{8.10}^{\mathrm{-}3}m\) is taken into account.
For models A and B, the characteristics are evaluated after 65 years, i.e. \(t\mathrm{=}569790h\) which corresponds to the parameter NBH_RELAX.
For modeling C, only losses due to friction and anchor recoil are modelled. Then, it is assumed that a break in the cable 5 takes place at the curvilinear abscissa \(s=\mathrm{64,39}m\) and that the tension is re-anchored linearly over a distance of 20 m (on either side of the break point). The profiles before and after breakage are shown in the following figure.
Figure 1.3-1: Tension profile in cable 5, before and after breakage