Modeling A ============== Characteristics of modeling ----------------------------------- The figure below gives a simplified representation of the mesh of the beam. .. image:: images/10001BF4000069D500000D55AD49EC4615D37479.svg :width: 613 :height: 77 .. _RefImage_10001BF4000069D500000D55AD49EC4615D37479.svg: The concrete beam is represented by 20 elements of type DKT, supported by as many quadrangle meshes with 4 knots. A thickness :math:`p\mathrm{=}\mathrm{0,2}m` is assigned to them, as well as a concrete material for which the behaviors ELAS (Young's modulus :math:`{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 :math:`\mathit{DX}`, :math:`\mathit{DY}`, :math:`\mathit{DZ}`, and :math:`\mathit{DRY}` of node :math:`\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 :math:`\mathit{NC001001}` and :math:`\mathit{NC001021}` respectively. An area of right cross section :math:`{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 :math:`{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 :math:`{f}_{\mathit{prg}}\mathrm{=}\mathrm{1,77}{.10}^{9}\mathit{Pa}` is chosen. Node :math:`\mathit{NC001001}`'s degrees of freedom :math:`\mathit{DX}`, :math:`\mathit{DY}`, and :math:`\mathit{DZ}` are blocked. Tension :math:`{F}_{0}\mathrm{=}{2.10}^{5}N` is applied to node :math:`\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 (:math:`z\mathrm{=}\mathrm{\pm }p\mathrm{/}2`) of the beam. 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. Tested sizes and results ------------------------------ The equilibrium value of the normal force in the cable is :math:`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 [:ref:`§3 <§3>`]. 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 :math:`Z\mathrm{=}0`. The tolerance for relative deviation from the reference is equal to 0.1%. .. csv-table:: "**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 :math:`{N}_{\mathit{XX}}` (:math:`{N}_{\mathit{XX}}\mathrm{=}{s}_{\mathit{xx}}p`). The tolerance for relative deviation from the reference is equal to 0.1%. .. csv-table:: "**Node**", "**Mesh**", "**Reference value**", "**Calculated value**", "**Relative variance**" "NB001001 "," QD001001 ", "—4.887725.105N/m", "—4.8877348399136.105N/m", "2.01.10— 6%" "NB002001 "," QD001001 ", "—4.887725.105N/m", "—4.8877348399728.105N/m", "2.01.10— 6%" "NB001011 "," QD001011 ", "—4.887725.105N/m", "—4.8877348402090.105N/m", "2.01.10— 6%" "NB002011 "," QD001011 ", "—4.887725.105N/m", "—4.8877348402511.105N/m", "2.01.10— 6%" "NB001021 "," QD001020 ", "—4.887725.105N/m", "—4.8877348403607.105N/m", "2.01.10— 6%" "NB002021 "," QD001020 ", "—4.887725.105N/m", "—4.88773484039.105N/m", "2.01.10— 6%" 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%. .. csv-table:: "**Node**", "**Mesh**", "**Reference value**", "**Calculated value**", "**Relative variance**" "NB001001 "," QD001001 ", "1,221931.106Pa", "1,221931.106Pa", "1,221931.106Pa", "2,22.10— 6%" "NB002001 "," QD001001 ", "1,221931.106Pa", "1,221931.106Pa", "1,221931.106Pa", "2,22.10— 6%" "NB001011 "," QD001011 ", "1,221931.106Pa", "1,221931.106Pa", "1,221931.106Pa", "2,22.10— 6%" "NB002011 "," QD001011 ", "1,221931.106Pa", "1,221931.106Pa", "1,221931.106Pa", "2,22.10— 6%" "NB001021 "," QD001020 ", "1,221931.106Pa", "1,221931.106Pa", "1,221931.106Pa", "2,22.10— 6%" "NB002021 "," QD001020 ", "1,221931.106Pa", "1,221931.106Pa", "1,221931.106Pa", "2,22.10— 6%" 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%. .. csv-table:: "**Node**", "**Mesh**", "**Reference value**", "**Calculated value**", "**Relative variance**" "NB001001 "," QD001001 ", "—6,109656.106Pa", "—6,1096685504454.106Pa", "2,05.10— 6%" "NB002001 "," QD001001 ", "—6,109656.106Pa", "—6,1096685505156.106Pa", "2,05.10— 6%" "NB001011 "," QD001011 ", "—6,109656.106Pa", "—6,1096685504816.106Pa", "2,05.10— 6%" "NB002011 "," QD001011 ", "—6,109656.106Pa", "—6,1096685504999.106Pa", "2,05.10— 6%" "NB001021 "," QD001020 ", "—6,109656.106Pa", "—6,1096685503914.106Pa", "2,05.10— 6%" "NB002021 "," QD001020 ", "—6,109656.106Pa", "—6,1096685505642.106Pa", "2,05.10— 6%" notes --------- The calculated values actually correspond to those theoretically expected. A state of flexion-compression is indeed obtained for the concrete beam.