3. Modeling A#
3.1. Characteristics of modeling#
The concrete beam is represented by 2080 MECA_HEXA8 elements, supported by as many hexahedral meshes with 8 knots. The 5 cables are represented using 20 MECA_BARRE elements each supported by as many 2-node segments. The figure below shows the mesh of the beam.
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Figure 3.1-a: Mesh Description
The concrete material is defined by the behaviors ELAS and BPEL_BETON: the characteristic parameters of this relationship are set to 0 because we do not want to include the tension losses due to the shrinkage and creep of the concrete.
The steel material for cables is defined by the behaviors ELAS and BPEL_ACIER. The non-zero values for BPEL_ACIER relate to friction (\(\mathrm{FROT}\text{\_}\mathrm{LINE}=\mathrm{1,5}{10}^{-3}{m}^{-1}\)) and the elastic limit since a zero value is illegal (\(\mathrm{SY}=\mathrm{1,94}{10}^{11}\mathrm{Pa}\)). The cable radius is \(\mathrm{2,8209}{10}^{-2}m\).
The blocked degrees of freedom are as follows:
\(\mathrm{DZ}\) for the underside
\(\mathrm{DX}\) and \(\mathrm{DY}\) for points located on the axis of symmetry of the beam
\(\mathrm{DX}\) for point \((0.50.0.)\) and \(\mathrm{DY}\) for point \((–0.50.0.)\)
Voltage \({F}_{0}=\mathrm{3,75}{10}^{6}N\) is applied to the lower nodes of cables 1, 2, 3 and 4 and to both ends for cable 5.
Charging is done in 6 steps:
\(t=50\text{et}100s\), the first two steps in elasticity with the IMPLEX method: this is used to test the convergence of the method in the elastic domain in a single time step
\(t=150s\), taking into account gravity: STAT_NON_LINE with boundary conditions, gravity and kinematic relationships for all cables. Since the cables do not contribute to the rigidity of the model, they are assigned a law of behavior “SANS” (zero constraint),
\(t=300s\), tension of cables 1 and 2: CALC_PRECONT with the boundary conditions, cables 1 and 2 being the cables to be tensioned, cables 3, 4 and 5 being inactive,
\(t=450s\), tension of cables 3 and 4: CALC_PRECONT with the boundary conditions, the kinematic relationships for cables 1 and 2, cables 3 and 4 being the cables to be put under tension, cable 5 being inactive,
\(t=600s\), tension of cable 5: CALC_PRECONT with the boundary conditions, the kinematic relationships for cables 1, 2, 3 and 4, cable 5 being the cable to be tensioned.
3.2. Calculation steps#
The main calculation steps correspond to the functionalities that we want to validate:
operator DEFI_MATERIAU: definition of the BPEL_BETON behavior relationships with the values by default and BPEL_ACIER, in the case of loss by linear friction,
operator DEFI_CABLE_BP: determination of a tension profile along the prestress cable, taking into account linear friction losses and anchor recoil losses; calculation of the coefficients of the kinematic relationships between the degrees of freedom of the cable nodes and the degrees of freedom of the cable nodes and the degrees of freedom of the « neighboring » nodes of the concrete beam, in the case of a beam modelled by elements \(\mathrm{3D}\),
operator AFFE_CHAR_MECA: definition of a RELA_CINE_BP load (RELA_CINE =” OUI “),
operator STAT_NON_LINE, option COMPORTEMENT: calculation of the equilibrium state taking into account the load of type RELA_CINE_BP, in the case of a beam modelled by elements \(\mathrm{3D}\),
law of behavior SANS,
macro command CALC_PRECONT with a non-virgin initial state, whether or not there are inactive cables, whether or not there are one or more cables to be energized.
In addition, we test the calculation with Newton’s exact iterative method and the approximate method IMPLEX. In this test case, elasticity is maintained; the results obtained by the two resolution methods must therefore be identical.
3.3. Model A results#
The value extracted from field SIEF_ELNO is compared to the reference value obtained with CASTEM
and this for the various characteristic moments (voltage not zero) and for nodes equally distributed along the cables.
The component that the test focuses on is the voltage in cables \(N\).
The tests are carried out twice, once when the problem is solved implicitly (METHODE =” NEWTON “) and once when the problem is solved with the IMPLEX method (METHODE =” IMPLEX”). In both cases, the tolerances are identical.
3.3.1. Voltage in cable 1#
\(T=150s\)
non-regression tests for method IMPLEX for the global value of stresses and displacements.
\(T=300s\)
Identification (knot/mesh) |
Reference type |
Reference value ( \(N\) ) |
Tolerance ( \(\text{\%}\) ) |
N1 - M5655 |
“SOURCE_EXTERNE” |
3,648.106N |
0.10% |
N6 - M5660 |
“SOURCE_EXTERNE” |
3,675.106N |
0.10% |
N11 - M5664 |
“SOURCE_EXTERNE” |
3,693.106N |
0.10% |
N16 - M5670 |
“SOURCE_EXTERNE” |
3,667.106N |
0.10% |
N101 - M5674 |
“SOURCE_EXTERNE” |
3,640.106N |
0.10% |
\(T=450s\)
Identification (knot/mesh) |
Reference type |
Reference value ( \(N\) ) |
Tolerance ( \(\text{\%}\) ) |
N1 - M5655 |
“SOURCE_EXTERNE” |
3,561.106N |
1.00% |
N6 - M5660 |
“SOURCE_EXTERNE” |
3,588.106N |
1.00% |
N11 - M5664 |
“SOURCE_EXTERNE” |
3,628.106N |
1.00% |
N16 - M5670 |
“SOURCE_EXTERNE” |
3,645.106N |
1.00% |
N101 - M5674 |
“SOURCE_EXTERNE” |
3,629.106N |
1.00% |
\(T=600s\)
Identification (knot/mesh) |
Reference type |
Reference value ( \(N\) ) |
Tolerance ( \(\text{\%}\) ) |
N1 - M5655 |
“SOURCE_EXTERNE” |
3,519.106N |
1.00% |
N6 - M5660 |
“SOURCE_EXTERNE” |
3,546.106N |
1.00% |
N11 - M5664 |
“SOURCE_EXTERNE” |
3,597.106N |
1.00% |
N16 - M5670 |
“SOURCE_EXTERNE” |
3,635.106N |
1.00% |
N101 - M5674 |
“SOURCE_EXTERNE” |
3,614.106N |
1.00% |
3.3.2. Voltage in cable 3#
\(T=450s\)
Identification (knot/mesh) |
Reference type |
Reference value ( \(N\) ) |
Tolerance ( \(\text{\%}\) ) |
N41 - M5695 |
“SOURCE_EXTERNE” |
3,647.106N |
0.10% |
N46 - M5700 |
“SOURCE_EXTERNE” |
3,675.106N |
0.10% |
N51 - M5705 |
“SOURCE_EXTERNE” |
3,695.106N |
0.10% |
N56 - M5710 |
“SOURCE_EXTERNE” |
3,667.106N |
0.10% |
N103 - M5714 |
“SOURCE_EXTERNE” |
3,640.106N |
0.10% |
\(T=600s\)
Identification (knot/mesh) |
Reference type |
Reference value ( \(N\) ) |
Tolerance ( \(\text{\%}\) ) |
N41 - M5695 |
“SOURCE_EXTERNE” |
3,6075.106N |
1.00% |
N46 - M5700 |
“SOURCE_EXTERNE” |
3,6346.106N |
1.00% |
N51 - M5705 |
“SOURCE_EXTERNE” |
3,6720.106N |
1.00% |
N56 - M5710 |
“SOURCE_EXTERNE” |
3,6529.106N |
1.00% |
N103 - M5714 |
“SOURCE_EXTERNE” |
3,6241.106N |
1.00% |
3.3.3. Voltage in cable 5#
\(T=600s\)
Identification (knot/mesh) |
Reference type |
Reference value ( \(\text{N}\) ) |
Tolerance ( \(\text{\%}\) ) |
N81 - M5735 |
“SOURCE_EXTERNE” |
3,647.106N |
0.10% |
N86 - M5740 |
“SOURCE_EXTERNE” |
3,674.106N |
0.10% |
N91 - M5745 |
“SOURCE_EXTERNE” |
3,695.106N |
0.10% |
N96 - M5750 |
“SOURCE_EXTERNE” |
3,674.106N |
0.10% |
N105 - M5754 |
“SOURCE_EXTERNE” |
3,647.106N |
0.10% |