2. Benchmark solution#

2.1. Curvilinear abscissa and cumulative angular deviation#

The cables each describe a path in the form of a semicircle in a horizontal plane. As a result, the curvilinear abscissa \(s\) and the cumulative angular deviation \(\alpha\) are expressed very simply:

\(\mathrm{\{}\begin{array}{c}s\mathrm{=}{R}_{c}\theta \\ \alpha \mathrm{=}\theta \end{array}\)

where \({R}_{c}\) refers to the radius of the semicircle described by the cable, and \(\theta\) the azimuth in cylindrical coordinates.

The reference values for the tests are estimated using these expressions.

2.2. Normal effort in the cables#

We consider a cable describing a horizontal semicircle with radius \({R}_{c}\). Note \({\theta }_{0}\) the azimuth marking the end of the zone where the recoil tension losses apply to the first anchor; \(\pi \mathrm{-}{\theta }_{0}\) marks the start of the zone where the recoil tension losses apply to the second anchor.

Taking into account the previous expressions of the curvilinear abscissa and the cumulative angular deviation, the tension profile along the cable can be parameterized by azimuth \(\theta\), outside the areas where recoil losses apply to anchors:

\(F(\theta )\mathrm{=}\mathrm{-}{F}_{0}({x}_{\mathit{flu}}+{x}_{\mathit{ret}})\)

\(+{F}_{0}\left[1+r(j)\mathrm{\times }\frac{5}{100}{\rho }_{1000}\mathrm{\times }{\mu }_{0}\right]\mathrm{exp}(\mathrm{-}(f+\varphi {R}_{c})\theta )\)

\(\mathrm{-}{F}_{0}\mathrm{\times }r(j)\mathrm{\times }\frac{5}{100}{\rho }_{1000}\mathrm{\times }\frac{{F}_{0}}{{S}_{a}{f}_{\mathit{prg}}}\mathrm{\times }\mathrm{exp}(\mathrm{-}2(f+\varphi {R}_{c})\theta )\) on the \(\left[{\theta }_{0};\frac{\pi }{2}\right]\) interval

\(F(\theta )\mathrm{=}\mathrm{-}{F}_{0}({x}_{\mathit{flu}}+{x}_{\mathit{ret}})\)

\(+{F}_{0}\left[1+r(j)\mathrm{\times }\frac{5}{100}{\rho }_{1000}\mathrm{\times }{\mu }_{0}\right]\mathrm{exp}(\mathrm{-}(f+\varphi {R}_{c})(\pi \mathrm{-}\theta ))\)

\(\mathrm{-}{F}_{0}\mathrm{\times }r(j)\mathrm{\times }\frac{5}{100}{\rho }_{1000}\mathrm{\times }\frac{{F}_{0}}{{S}_{a}{f}_{\mathit{prg}}}\mathrm{\times }\mathrm{exp}(\mathrm{-}2(f+\varphi {R}_{c})(\pi \mathrm{-}\theta ))\) on the \(\left[\frac{\pi }{2};\pi \mathrm{-}{\theta }_{0}\right]\) interval

The reference values for the tests are estimated using these expressions, which define a tension profile that is symmetric with respect to the central node.

2.3. Projection index#

The projection of a node belonging to one of the cables onto a mesh of the concrete veil gives rise to the assignment of a projection index \(\mathit{IPROJ}\) in accordance with the following rule:

Projection on a triangle mesh of \(\mathit{N1}\), \(\mathit{N2}\) and \(\mathit{N3}\) vertex nodes:

\(\mathit{IPROJ}\mathrm{=}0\)	if the projected point is inside the triangle;
\(\mathit{IPROJ}\mathrm{=}\mathrm{11,}12\) or \(13\)	if the projected point belongs respectively to the edge \(\mathrm{[}\mathit{N1};\mathit{N2}\mathrm{]}\), \([\mathrm{N2};\mathrm{N3}]\), or \([\mathrm{N3};\mathrm{N1}]\);
\(\mathit{IPROJ}\mathrm{=}2\)	if the projected point coincides with a vertex node.

Projection on a quadrangle mesh of \(\mathit{N1},\mathit{N2}\mathit{N3}\) and \(\mathrm{N4}\) vertex nodes:

\(\mathit{IPROJ}\mathrm{=}0\)	if the projected point is inside the quadrangle;
\(\mathit{IPROJ}\mathrm{=}\mathrm{11,}\mathrm{12,}13\) or \(14\)	if the projected point belongs respectively to the edge \([\mathrm{N1};\mathrm{N2}]\), \([\mathrm{N2};\mathrm{N3}]\), \([\mathrm{N3};\mathrm{N4}]\) or \([\mathrm{N4};\mathrm{N1}]\);
\(\mathrm{IPROJ}=2\)	if the projected point coincides with a vertex node.

The reference values for the tests are estimated by predicting the places where the cables will be projected, taking into account their location in relation to the veil and the arrangement of the meshes on this one.

2.4. Eccentricity#

The eccentricity of a node belonging to one of the cables is defined as the distance from this node to the mesh of the concrete veil over which it is projected.

Viewed in a horizontal plane, the mesh trace is a chord on the semicircle with radius \(R\). Note \(\alpha\) the angular sector covered by the mesh. The cable node, noted \(\mathit{NC}\), is located on the semicircle with radius \({R}_{c}\). Its relative position with respect to the mesh is marked by azimuth \(\beta\).

Vector \((\mathrm{cos}\frac{\alpha }{2};\mathrm{sin}\frac{\alpha }{2})\) is normal to the string, which passes through point \((R;0)\).

So the equation of the string is \(\mathrm{cos}\frac{\alpha }{2}x+\mathrm{sin}\frac{\alpha }{2}y\mathrm{-}R\mathrm{cos}\frac{\alpha }{2}\mathrm{=}0\)

The distance from a point to a line, in the plane, is given by:

\(d\mathrm{=}\frac{∣a{x}_{0}+b{x}_{0}+c∣}{\sqrt{{a}^{2}+{b}^{2}}}\)

where \(({x}_{\mathrm{0 }};{y}_{0})\) is the coordinates of the point and \(\mathit{ax}+\mathit{by}+c\mathrm{=}0\) is the equation of the line.

Node \(\mathit{NC}\) belonging to the cable has coordinates \(({R}_{c}\mathrm{cos}\beta ;{R}_{c}\mathrm{sin}\beta )\). Its eccentricity in relation to the mesh of the veil on which it is projected is therefore worth:

\({\mathit{ex}}_{c}\mathrm{=}∣{R}_{c}\mathrm{cos}(\frac{\alpha }{2}\mathrm{-}\beta )\mathrm{-}R\mathrm{cos}\frac{\alpha }{2}∣\)

The reference values for the tests are estimated using this expression.