3. Modeling A#
3.1. Characteristics of modeling#
The figure below gives a simplified representation of the mesh of the beam.
The concrete beam is represented by 20 elements of type DKT, supported by as many quadrangle meshes with 4 knots.
A thickness \(p\mathrm{=}\mathrm{0,2}m\) is assigned to them, as well as a concrete material for which the behaviors ELAS (Young’s modulus \({E}_{b}\mathrm{=}{3.10}^{10}\mathit{Pa}\)) and BPEL_BETON are defined: the characteristic parameters of this relationship are set to 0 because the tension losses along the prestress cable are neglected.
The degrees of freedom \(\mathit{DX}\), \(\mathit{DY}\), \(\mathit{DZ}\), and \(\mathit{DRY}\) of node \(\mathit{NB001001}\) are blocked.
The cable is represented by 20 MECA_BARRE elements, supported by as many 2-node segment meshes. The left and right endpoints are nodes \(\mathit{NC001001}\) and \(\mathit{NC001021}\) respectively.
An area of right cross section \({S}_{a}\mathrm{=}\mathrm{1,5}{.10}^{\mathrm{-}4}{m}^{2}\) is assigned to the elements, as well as a steel material for which the behaviors ELAS (Young’s modulus \({E}_{a}\mathrm{=}\mathrm{2,1}{.10}^{11}\mathit{Pa}\)) and BPEL_ACIER are defined: the characteristic parameters of this relationship are set to 0 (tension losses neglected), with the exception of the elastic limit stress for which the value of \({f}_{\mathit{prg}}\mathrm{=}\mathrm{1,77}{.10}^{9}\mathit{Pa}\) is chosen.
Node \(\mathit{NC001001}\)”s degrees of freedom \(\mathit{DX}\), \(\mathit{DY}\), and \(\mathit{DZ}\) are blocked.
Tension \({F}_{0}\mathrm{=}{2.10}^{5}N\) is applied to node \(\mathit{NC001021}\). This tension value is consistent with the cross-section and elastic limit values, for a strand-type pretension cable.
The calculation of the equilibrium state of the girder and cable assembly is carried out in a single step, the behavior being elastic. Two additional calculations are then carried out to determine the stresses in the lower and upper skins (\(z\mathrm{=}\mathrm{\pm }p\mathrm{/}2\)) of the beam.
3.2. Calculation steps and functionalities tested#
The main calculation steps correspond to the functionalities that we want to validate:
operator DEFI_MATERIAU: definition of the behavioral relationships BPEL_BETON and BPEL_ACIER, in the particular case where the tension losses along the prestress cable are neglected (values by default parameters);
operator DEFI_CABLE_BP: determination of a constant tension profile along the prestress cable, the losses being neglected; calculation of the coefficients of the kinematic relationships between 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 an eccentric cable;
operator AFFE_CHAR_MECA: definition of a RELA_CINE_BP load;
operator STAT_NON_LINE, option COMPORTEMENT: calculation of the equilibrium state taking into account the load type RELA_CINE_BP.
Finally, we use the operator CALC_CHAMP option SIGM_ELNO in order to calculate the stresses in the lower skin and then in the upper skin of the beam.
3.3. Tested sizes and results#
The equilibrium value of the normal force in the cable is \(F\mathrm{=}\mathrm{1,95509}{10}^{5}N\). This value is used to calculate the numerical reference results using the analytic expressions explained in paragraph [§3].
3.3.1. Movement of the nodes of the concrete piece#
The values extracted from field DEPL from STAT_NON_LINE are compared to the theoretical reference values corresponding to plane \(Z\mathrm{=}0\).
The tolerance for relative deviation from the reference is equal to 0.1%.
Node |
Component |
Reference value |
Calculated value |
Relative variance |
NB001006 |
DX |
—2,036552.10—4 m |
—2,0365561834835.10—4 m |
2,05.10— 6% |
NB002006 |
DX |
—2,036552.10—4 m |
—2,0365561835042.10—4 m |
2,05.10— 6% |
NB001011 |
DX |
—4,073104.10—4 m |
—4,0731123669671.10—4 m |
2,05.10— 6% |
NB002011 |
DX |
—4,073104.10—4 m |
—4,0731123670073.10—4 m |
2,05.10— 6% |
NB001016 |
DX |
—6,109656.10—4 m |
—6,1096685504506.10—4 m |
2,05.10— 6% |
NB002016 |
DX |
—6,109656.10—4 m |
—6,1096685505104.10—4 m |
2,05.10— 6% |
NB001021 |
DX |
—8,146208.10—4 m |
—8,1462247339343.10—4 m |
2,05.10— 6% |
NB002021 |
DX |
—8,146208.10—4 m |
—8,1462247340137.10—4 m |
2,05.10— 6% |
NB001006 |
DZ |
3,818535.10—3m |
3,8185428440476.10—3m |
2,05.10— 6% |
NB002006 |
DZ |
3,818535.10—3m |
3,8185428440475.10—3m |
2,05.10— 6% |
NB001011 |
DZ |
1,527414.10—2m |
1,5274171376197.10—2m |
2,05.10— 6% |
NB002011 |
DZ |
1,527414.10—2m |
1,5274171376197.10—2m |
2,05.10— 6% |
NB001016 |
DZ |
3,436682.10—2m |
3,4366885596448.10—2m |
1,91.10— 6% |
NB002016 |
DZ |
3,436682.10—2m |
3,4366885596448.10—2m |
1,91.10— 6% |
NB001021 |
DZ |
6,109656.10—2m |
6,1096695504804.10—2m |
2,05.10— 6% |
NB002021 |
DZ |
6,109656.10—2m |
6,1096695504804.10—2m |
2,05.10— 6% |
Linear density of normal force on the mean plane of the concrete part (analysis with the plate model) ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~
The values extracted from field SIEF_ELNO from STAT_NON_LINE are compared to the theoretical reference values.
The component that the tests focus on is \({N}_{\mathit{XX}}\) (\({N}_{\mathit{XX}}\mathrm{=}{s}_{\mathit{xx}}p\)).
The tolerance for relative deviation from the reference is equal to 0.1%.
Node |
Mesh |
Reference value |
Calculated value |
Relative variance |
NB001001 |
|
—4.887725.105N/m |
—4.8877348399136.105N/m |
2.01.10— 6% |
NB002001 |
|
—4.887725.105N/m |
—4.8877348399728.105N/m |
2.01.10— 6% |
NB001011 |
|
—4.887725.105N/m |
—4.8877348402090.105N/m |
2.01.10— 6% |
NB002011 |
|
—4.887725.105N/m |
—4.8877348402511.105N/m |
2.01.10— 6% |
NB001021 |
|
—4.887725.105N/m |
—4.8877348403607.105N/m |
2.01.10— 6% |
NB002021 |
|
—4.887725.105N/m |
—4.88773484039.105N/m |
2.01.10— 6% |
3.3.2. Normal stress on the lower skin (z = ‑0.1 m) of the concrete piece#
The values extracted from field SIGM_ELNO from CALC_CHAMP are compared to the theoretical reference values.
The component that the tests focus on is SIXX.
The tolerance for relative deviation from the reference is equal to 0.1%.
Node |
Mesh |
Reference value |
Calculated value |
Relative variance |
|
NB001001 |
|
1,221931.106Pa |
1,221931.106Pa |
1,221931.106Pa |
2,22.10— 6% |
NB002001 |
|
1,221931.106Pa |
1,221931.106Pa |
1,221931.106Pa |
2,22.10— 6% |
NB001011 |
|
1,221931.106Pa |
1,221931.106Pa |
1,221931.106Pa |
2,22.10— 6% |
NB002011 |
|
1,221931.106Pa |
1,221931.106Pa |
1,221931.106Pa |
2,22.10— 6% |
NB001021 |
|
1,221931.106Pa |
1,221931.106Pa |
1,221931.106Pa |
2,22.10— 6% |
NB002021 |
|
1,221931.106Pa |
1,221931.106Pa |
1,221931.106Pa |
2,22.10— 6% |
3.3.3. Normal stress on the upper skin (z= 0.1 m) of the concrete piece#
The values extracted from field SIGM_ELNO from CALC_CHAMP are compared to the theoretical reference values.
The component that the tests focus on is SIXX.
The tolerance for relative deviation from the reference is equal to 0.1%.
Node |
Mesh |
Reference value |
Calculated value |
Relative variance |
NB001001 |
|
—6,109656.106Pa |
—6,1096685504454.106Pa |
2,05.10— 6% |
NB002001 |
|
—6,109656.106Pa |
—6,1096685505156.106Pa |
2,05.10— 6% |
NB001011 |
|
—6,109656.106Pa |
—6,1096685504816.106Pa |
2,05.10— 6% |
NB002011 |
|
—6,109656.106Pa |
—6,1096685504999.106Pa |
2,05.10— 6% |
NB001021 |
|
—6,109656.106Pa |
—6,1096685503914.106Pa |
2,05.10— 6% |
NB002021 |
|
—6,109656.106Pa |
—6,1096685505642.106Pa |
2,05.10— 6% |
3.4. notes#
The calculated values actually correspond to those theoretically expected. A state of flexion-compression is indeed obtained for the concrete beam.