Modélisations ============= En fonction des modélisations, les objets sont maillés avec des éléments différents: **Modélisation A:** cube: 1 élément HEXA8 à 8 nœuds faces: 1 élément QUAD4 à 4 nœuds **Modélisation B:** cube: 6 éléments TETRA4 à 4 nœuds faces: 2 éléments TRIA3 à 3 nœuds **Modélisation C:** même modélisation qu’en A avec des éléments quadratiques **Modélisation D:** même modélisation qu’en B avec des éléments quadratiques **3.1** **Grandeurs testées et résultats de la modélisation A** La modélisation A est constituée d’un élément CUB8 pour le cube et d’un élément QUA4 pour chaque face. Le comportement est élastique (commandes MECA_STATIQUE puis STAT_NON_LINE (vérification)). On teste les valeurs données par EPSI_ELGA; EPSI_ELNO, SIEF_ELGA, SIEF_ELNO, SIEF_ELGA, SIGM_ELNO en différents points dans les directions principales des nappes d’armature, obtenues respectivement avec les commandes MECA_STATIQUE et STAT_NON_LINE. Pour **MECA_STATIQUE** : +----------------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Points d’intégration | EPSI_ELGA | SIEF_ELGA | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA1 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA2 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA3 – Point 1 | 1.8264 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA4 – Point 1 | 1.8264 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA5 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA6 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ +--------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Noeud | EPSI_ELNO | SIGM_ELNO | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | NO1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | NO1 | 1.8264 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | NO1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ Pour **STAT_NON_LINE** +----------------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Points d’intégration | EPSI_ELGA | SIEF_ELGA | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA1 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA2 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA3 – Point 1 | 1.8264 | 1.8263 | 0.003 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA4 – Point 1 | 1.8264 | 1.8263 | 0.003 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA5 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA6 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ +--------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Noeud | EPSI_ELNO | SIEF_ELNO | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA1 - NO1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA2 - NO1 | 1.8264 | 1.8263 | 0.003 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA3 - NO1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ La valeur de l’énergie potentielle totale est aussi testée à partir du calcul STAT_NON_LINE. La solution analytique est calculée à partir de toutes les déformations uniaxiales dans les grilles, à partir de l’équation: .. image:: images/Object_11.svg :width: 139 :height: 46 .. _RefImage_Object_11.svg: où :math:`A` désigne le tenseur d’élasticité. Les résultats obtenus sont les suivants: .. csv-table:: "", "*Code_Aster*", "Référence analytique", "Ecart" "Energie potentielle totale (:math:`J` )", "2.8173E10", "2.8173E10", "0" **3.2** **Grandeurs testées et résultats de la modélisation B** La modélisation B est constituée de six éléments TETRA4 pour le cube et de deux éléments TRIA3 pour chaque face. Le comportement est élastique en utilisant la commande MECA_STATIQUE puis la commande STAT_NON_LINE (vérification). On teste les valeurs données par EPSI_ELGA; EPSI_ELNO, SIEF_ELGA, SIEF_ELNO; SIEF_ELGA; SIGM_ELNO en différents points dans les directions principales des nappes d’armature, obtenues respectivement avec les commandes MECA_STATIQUE et STAT_NON_LINE. Pour **MECA_STATIQUE** : +----------------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Points d’intégration | EPSI_ELGA | SIEF_ELGA | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA1 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA2 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA3 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA4 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA5 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA6 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA11 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA21 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA31 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA41 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA51 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA61 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ +--------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Noeud | EPSI_ELNO | SIGM_ELNO | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA1 - NO1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA2 - NO1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA3 - NO1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA11 - NO1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA21 - NO1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA31 - NO1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ Pour **STAT_NON_LINE** +----------------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Points d’intégration | EPSI_ELGA | SIEF_ELGA | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA1 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA2 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA3 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA4 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA5 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA6 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA11 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA21 – Point 1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA31 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA41 – Point 1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA51 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA61 – Point 1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +----------------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ +--------------+------------------------------------------------------------------------------------------------+----------------------------------------------------------------------------+ | Noeud | EPSI_ELNO | SIEF_ELNO | | | (dans la direction principale de la nappe d’armature) | (dans la direction principale de la nappe d’armature) | | +-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | | *Code_Aster* | Référence | Ecart (:math:`\text{\%}` ) | *Code_Aster* | Référence | Ecart | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA1 - NO1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA2 - NO1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA3 - NO1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA11 - NO1 | 1.067 | 1.067 | 0.001 | 2.13397E11 | 2.13397E11 | 0 | | (nappe GEOZ) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA21 - NO1 | 1.8263 | 1.8263 | 0 | 3.6527E11 | 3.6527E11 | 0 | | (nappe GEOY) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+ | MA31 - NO1 | 2 | 2 | 0 | 4E11 | 4E11 | 0 | | (nappe GEOX) | | | | | | | +--------------+-------------------------------------------------------+-----------+----------------------------+-------------------------------------------------------+------------+-------+