3. A and B modeling

3.1. Characteristics of the models

The concrete veil is represented by elements DKT for modeling A and by elements Q4GG for modeling B, supported in both cases by quadrangle cells with 4 nodes: there are 10 cells on a vertical generator and 32 on a horizontal semicircle. With this layout, the meshes have dimensions close to those of a square with a side of \(1m\).

All of the following are common to both A and B models.

A thickness \(e\mathrm{=}\mathrm{0,6}m\) is assigned to all the meshes of the veil, as well as a concrete material for which the behaviors ELAS and BPEL_BETON are defined: the parameters take the values given previously in paragraph [§ 2.2.1].

Each of the cables is represented by 128 MECA_BARRE elements, supported by 2-node segment meshes. On a horizontal semicircle, there are therefore 4 times more meshes on a cable than on the concrete web.

A cross-sectional area \({S}_{a}\mathrm{=}\mathrm{1,5}{.10}^{\mathrm{-}4}{m}^{2}\) is assigned to all cable meshes, as well as a steel material for which the behaviors ELAS and BPEL_ACIER are defined: the parameters take the values given earlier in paragraph [§ 2.2.2].

Voltage \({F}_{0}\mathrm{=}{2.10}^{5}N\) is applied to the two end nodes of each cable. The value of this tension is consistent with the values of cross section and elastic limit stress, for stranded pretension cables. The evaluation of tension losses due to relaxation and recoil at the anchors is carried out in accordance with the rules of BPEL; the parameters take the values given earlier in paragraph [§ 2.2.3].

Since the same tension is applied to both ends of each cable, and since the same setback is imposed on the anchors, the tension profiles obtained must be symmetric with respect to the middle nodes of the cables.

Taking into account the geometric characteristics and the mesh, the nodes of cables No. 1 and No. 2 project onto the summit nodes and the edges of the meshes of the concrete web. For the nodes of these two cables, the projection indices obtained must be in accordance with the following sequence: 2 for the first node, then \(13-13-13-2\) until the last node.

The nodes of cables No. 3 and No. 4 are projected on the edges and inside the meshes of the concrete veil. For the nodes of these two cables, the projection indices obtained must be in accordance with the following sequence: 14 for the first node, then \(0-0-0-12\) to the last node.

The rule for assigning the projection index is defined earlier in paragraph [§ 3.3].

Given the characteristics of the mesh, the eccentricities of the cable nodes are evaluated using the expression in paragraph [§ 3.4] with \(\alpha \mathrm{=}\frac{\pi }{32}\) and \(\beta \mathrm{=}\mathrm{0,}\frac{\alpha }{4},\frac{\alpha }{2},\frac{3\alpha }{4}\).

3.2. Tested sizes and results

3.2.1. Curvilinear abscissa

The tested component is ABS_CURV.

Identification (node)	Reference type	Reference value [\(m\) ]	Tolerance ( \(\text{\%}\) )
NC001032	“ANALYTIQUE”	7,608545.100	0.10%
NC001033	“ANALYTIQUE”	7,853982.100	0.10%
NC001034	“ANALYTIQUE”	8,099419.100	0.10%
NC001064	“ANALYTIQUE”	1,546253.101	0.10%
NC001065	“ANALYTIQUE”	1,570796.101	0.10%
NC001066	“ANALYTIQUE”	1,595340.101	0.10%
NC001096	“ANALYTIQUE”	2,331651.101	0.10%
NC001097	“ANALYTIQUE”	2,356194.101	0.10%
NC001098	“ANALYTIQUE”	2,380738.101	0.10%
NC002032	“ANALYTIQUE”	7,608545.100	0.10%
NC002033	“ANALYTIQUE”	7,853982.100	0.10%
NC002034	“ANALYTIQUE”	8,099419.100	0.10%
NC002064	“ANALYTIQUE”	1,546253.101	0.10%
NC002065	“ANALYTIQUE”	1,570796.101	0.10%
NC002066	“ANALYTIQUE”	1,595340.101	0.10%
NC002096	“ANALYTIQUE”	2,331651.101	0.10%
NC002097	“ANALYTIQUE”	2,356194.101	0.10%
NC002098	“ANALYTIQUE”	2,380738.101	0.10%
NC003032	“ANALYTIQUE”	7,646587.100	0.10%
NC003033	“ANALYTIQUE”	7,893252.100	0.10%
NC003034	“ANALYTIQUE”	8,139916.100	0.10%
NC003064	“ANALYTIQUE”	1,553984.101	0.10%
NC003065	“ANALYTIQUE”	1,578650.101	0.10%
NC003066	“ANALYTIQUE”	1,603317.101	0.10%
NC003096	“ANALYTIQUE”	2,343309.101	0.10%
NC003097	“ANALYTIQUE”	2,367975.101	0.10%
NC003098	“ANALYTIQUE”	2,392642.101	0.10%
NC004032	“ANALYTIQUE”	7,684630.100	0.10%
NC004033	“ANALYTIQUE”	7,932521.100	0.10%
NC004034	“ANALYTIQUE”	8,180413.100	0.10%
NC004064	“ANALYTIQUE”	1,561715.101	0.10%
NC004065	“ANALYTIQUE”	1,586504.101	0.10%
NC004066	“ANALYTIQUE”	1,611293.101	0.10%
NC004096	“ANALYTIQUE”	2,354967.101	0.10%
NC004097	“ANALYTIQUE”	2,379756.101	0.10%
NC004098	“ANALYTIQUE”	2,404546.101	0.10%

3.2.2. Cumulative angular deviation

The tested component is ALPHA.

Identification (node)	Reference type	Reference value [\(\mathrm{rad}\) ]	Tolerance ( \(\text{\%}\) )
NC001032	“ANALYTIQUE”	7,608545.10—1	1.00%
NC001033	“ANALYTIQUE”	7,853982.10—1	1.00%
NC001034	“ANALYTIQUE”	8,099419.10—1	1.00%
NC001064	“ANALYTIQUE”	1,546253.100	1.00%
NC001065	“ANALYTIQUE”	1,570796.100	1.00%
NC001066	“ANALYTIQUE”	1,595340.100	1.00%
NC001096	“ANALYTIQUE”	2,331651.100	1.00%
NC001097	“ANALYTIQUE”	2,356194.100	1.00%
NC001098	“ANALYTIQUE”	2,380738.100	1.00%
NC002032	“ANALYTIQUE”	7,608545.10—1	1.00%
NC002033	“ANALYTIQUE”	7,853982.10—1	1.00%
NC002034	“ANALYTIQUE”	8,099419.10—1	1.00%
NC002064	“ANALYTIQUE”	1,546253.100	1.00%
NC002065	“ANALYTIQUE”	1,570796.100	1.00%
NC002066	“ANALYTIQUE”	1,595340.100	1.00%
NC002096	“ANALYTIQUE”	2,331651.100	1.00%
NC002097	“ANALYTIQUE”	2,356194.100	1.00%
NC002098	“ANALYTIQUE”	2,380738.100	1.00%
NC003032	“ANALYTIQUE”	7,608545.10—1	1.00%
NC003033	“ANALYTIQUE”	7,853982.10—1	1.00%
NC003034	“ANALYTIQUE”	8,099419.10—1	1.00%
NC003064	“ANALYTIQUE”	1,546253.100	1.00%
NC003065	“ANALYTIQUE”	1,570796.100	1.00%
NC003066	“ANALYTIQUE”	1,595340.100	1.00%
NC003096	“ANALYTIQUE”	2,331651.100	1.00%
NC003097	“ANALYTIQUE”	2,356194.100	1.00%
NC003098	“ANALYTIQUE”	2,380738.100	1.00%
NC004032	“ANALYTIQUE”	7,608545.10—1	1.00%
NC004033	“ANALYTIQUE”	7,853982.10—1	1.00%
NC004034	“ANALYTIQUE”	8,099419.10—1	1.00%
NC004064	“ANALYTIQUE”	1,546253.100	1.00%
NC004065	“ANALYTIQUE”	1,570796.100	1.00%
NC004066	“ANALYTIQUE”	1,595340.100	1.00%
NC004096	“ANALYTIQUE”	2,331651.100	1.00%
NC004097	“ANALYTIQUE”	2,356194.100	1.00%
NC004098	“ANALYTIQUE”	2,380738.100	1.00%

3.2.3. Normal effort in the cables

The tested component is TENSION.

Identification (node)	Reference type	Reference value [\(N\) ]	Tolerance **** (%) **
NC001032	“ANALYTIQUE”	1,334446.105	0.50%
NC001033	“ANALYTIQUE”	1,325720.105	0.50%
NC001034	“ANALYTIQUE”	1,317036.105	0.50%
NC001064	“ANALYTIQUE”	1,076002.105	0.50%
NC001065	“ANALYTIQUE”	1,068586.105	0.50%
NC001066	“ANALYTIQUE”	1,076002.105	0.50%
NC001096	“ANALYTIQUE”	1,317036.105	0.50%
NC001097	“ANALYTIQUE”	1,325720.105	0.50%
NC001098	“ANALYTIQUE”	1,334446.105	0.50%
NC002032	“ANALYTIQUE”	1,334446.105	0.50%
NC002033	“ANALYTIQUE”	1,325720.105	0.50%
NC002034	“ANALYTIQUE”	1,317036.105	0.50%
NC002064	“ANALYTIQUE”	1,076002.105	0.50%
NC002065	“ANALYTIQUE”	1,068586.105	0.50%
NC002066	“ANALYTIQUE”	1,076002.105	0.50%
NC002096	“ANALYTIQUE”	1,317036.105	0.50%
NC002097	“ANALYTIQUE”	1,325720.105	0.50%
NC002098	“ANALYTIQUE”	1,334446.105	0.50%
NC003032	“ANALYTIQUE”	1,334270.105	0.50%
NC003033	“ANALYTIQUE”	1,325538.105	0.50%
NC003034	“ANALYTIQUE”	1,316850.105	0.50%
NC003064	“ANALYTIQUE”	1,075696.105	0.50%
NC003065	“ANALYTIQUE”	1,068278.105	0.50%
NC003066	“ANALYTIQUE”	1,075696.105	0.50%
NC003096	“ANALYTIQUE”	1,316850.105	0.50%
NC003097	“ANALYTIQUE”	1,325538.105	0.50%
NC003098	“ANALYTIQUE”	1,334270.105	0.50%
NC004032	“ANALYTIQUE”	1,334093.105	0.50%
NC004033	“ANALYTIQUE”	1,325356.105	0.50%
NC004034	“ANALYTIQUE”	1,316664.105	0.50%
NC004064	“ANALYTIQUE”	1,075391.105	0.50%
NC004065	“ANALYTIQUE”	1,067969.105	0.50%
NC004066	“ANALYTIQUE”	1,075391.105	0.50%
NC004096	“ANALYTIQUE”	1,316664.105	0.50%
NC004097	“ANALYTIQUE”	1,325356.105	0.50%
NC004098	“ANALYTIQUE”	1,334093.105	0.50%

3.2.4. Projection index

The tested component is INDICE_PROJECTION.

Identification (node)	Reference type	Reference value	Tolerance
NC001032	“ANALYTIQUE”	13	1.00E-003
NC001033	“ANALYTIQUE”	2	1.00E-003
NC001034	“ANALYTIQUE”	13	1.00E-003
NC001064	“ANALYTIQUE”	13	1.00E-003
NC001065	“ANALYTIQUE”	2	1.00E-003
NC001066	“ANALYTIQUE”	13	1.00E-003
NC001096	“ANALYTIQUE”	13	1.00E-003
NC001097	“ANALYTIQUE”	2	1.00E-003
NC001098	“ANALYTIQUE”	13	1.00E-003
NC002032	“ANALYTIQUE”	0	1.00E-003
NC002033	“ANALYTIQUE”	12	1.00E-003
NC002034	“ANALYTIQUE”	0	1.00E-003
NC002064	“ANALYTIQUE”	0	1.00E-003
NC002065	“ANALYTIQUE”	12	1.00E-003
NC002066	“ANALYTIQUE”	0	1.00E-003
NC002096	“ANALYTIQUE”	0	1.00E-003
NC002097	“ANALYTIQUE”	12	1.00E-003
NC002098	“ANALYTIQUE”	0	1.00E-003
NC003032	“ANALYTIQUE”	13	1.00E-003
NC003033	“ANALYTIQUE”	2	1.00E-003
NC003034	“ANALYTIQUE”	13	1.00E-003
NC003064	“ANALYTIQUE”	13	1.00E-003
NC003065	“ANALYTIQUE”	2	1.00E-003
NC003066	“ANALYTIQUE”	13	1.00E-003
NC003096	“ANALYTIQUE”	13	1.00E-003
NC003097	“ANALYTIQUE”	2	1.00E-003
NC003098	“ANALYTIQUE”	13	1.00E-003
NC004032	“ANALYTIQUE”	0	1.00E-003
NC004033	“ANALYTIQUE”	12	1.00E-003
NC004034	“ANALYTIQUE”	0	1.00E-003
NC004064	“ANALYTIQUE”	0	1.00E-003
NC004065	“ANALYTIQUE”	12	1.00E-003
NC004066	“ANALYTIQUE”	0	1.00E-003
NC004096	“ANALYTIQUE”	0	1.00E-003
NC004097	“ANALYTIQUE”	12	1.00E-003
NC004098	“ANALYTIQUE”	0	1.00E-003

3.2.5. Eccentricity

The tested component is EXCENTRICITE.

Identification (node)	Reference type	Reference value [\(m\) ]	Tolerance
NC001032	“ANALYTIQUE”	9,033625.10—3	0.10%
NC001033	“ANALYTIQUE”	0	1.00E-003
NC001034	“ANALYTIQUE”	9,033625.10—3	0.10%
NC001064	“ANALYTIQUE”	9,033625.10—3	0.10%
NC001065	“ANALYTIQUE”	0	1.00E-003
NC001066	“ANALYTIQUE”	9,033625.10—3	0.10%
NC001096	“ANALYTIQUE”	9,033625.10—3	0.10%
NC001097	“ANALYTIQUE”	0	1.00E-003
NC001098	“ANALYTIQUE”	9,033625.10—3	0.10%
NC002032	“ANALYTIQUE”	9,033625.10—3	0.10%
NC002033	“ANALYTIQUE”	0	1.00E-003
NC002034	“ANALYTIQUE”	9,033625.10—3	0.10%
NC002064	“ANALYTIQUE”	9,033625.10—3	0.10%
NC002065	“ANALYTIQUE”	0	1.00E-003
NC002066	“ANALYTIQUE”	9,033625.10—3	0.10%
NC002096	“ANALYTIQUE”	9,033625.10—3	0.10%
NC002097	“ANALYTIQUE”	0	1.00E-003
NC002098	“ANALYTIQUE”	9,033625.10—3	0.10%
NC003032	“ANALYTIQUE”	5,901857.10—2	0.10%
NC003033	“ANALYTIQUE”	5.10—2	0.10%
NC003034	“ANALYTIQUE”	5,901857.10—2	0.10%
NC003064	“ANALYTIQUE”	5,901857.10—2	0.10%
NC003065	“ANALYTIQUE”	5.10—2	0.10%
NC003066	“ANALYTIQUE”	5,901857.10—2	0.10%
NC003096	“ANALYTIQUE”	5,901857.10—2	0.10%
NC003097	“ANALYTIQUE”	5.10—2	0.10%
NC003098	“ANALYTIQUE”	5,901857.10—2	0.10%
NC004032	“ANALYTIQUE”	1,090035.10—1	0.10%
NC004033	“ANALYTIQUE”	10—1	0.10%
NC004034	“ANALYTIQUE”	1,090035.10—1	0.10%
NC004064	“ANALYTIQUE”	1,090035.10—1	0.10%
NC004065	“ANALYTIQUE”	10—1	0.10%
NC004066	“ANALYTIQUE”	1,090035.10—1	0.10%
NC004096	“ANALYTIQUE”	1,090035.10—1	0.10%
NC004097	“ANALYTIQUE”	10—1	0.10%
NC004098	“ANALYTIQUE”	1,090035.10—1	0.10%

3.3. notes

Relative differences for the cumulative angular deviation of an order of magnitude are obtained that are much higher than those obtained for the curvilinear abscissa, which is calculated with very good precision.

The cause is inherent in the cubic spline interpolation method used to interpolate cable paths. This method consists in interpolating the trajectory on each sub-interval by a 3rd order polynomial, guaranteeing the continuity of the first derivatives and the second derivatives across the sub-intervals. The interpolation of a circular trajectory by third-order polynomials is in itself an approximation, the effects of which are more sensitive on the second derivatives than on the first derivatives.

This is why the precision obtained for the curvilinear abscissa, calculated using the first derivatives of the trajectory, is much better than that obtained for the cumulative angular deviation, calculated using cross products between the first derivatives and the second derivatives. To improve the precision of the cumulative angular deviation, the mesh would have to be significantly refined: this approach is not valid a prima facie, because it would lead to very significant additional calculation costs for a low gain in precision.

In fact, tension profiles that are symmetric with respect to the middle nodes of the cables are obtained. The deviations by default on the curvilinear abscissa and the cumulative angular deviation induce differences by excess on the calculated voltage values, but the precision remains satisfactory: this validates the calculation method.

The projection indices obtained are in accordance with those expected, and the eccentricities are calculated with very good precision: this validates the projection operations.