Solution de référence ===================== Méthode de calcul utilisée pour la solution de référence -------------------------------------------------------- .. _1291536073: .. _1291535993: .. _1291535740: .. _1183277437: .. _1183275405: .. _1183275350: .. _1183275244: .. _1183275091: Les résultats de référence ont été obtenus à la fois de manière expérimentale [:ref:`1 <1>`] et avec le code HERCULE [:ref:`2 <2>`]. Le modèle éléments finis utilisé est identique a celui utilisé pour la modélisation A effectuée avec *Code_Aster* . +----------------------------------------------------------------+--------------------------------------------------------+ | |Maille: :math:`1019` : :math:`\mathrm{N6}-\mathrm{N17}` | | .. image:: images/1000000000000207000001594C43CE9088988969.png | :math:`1009` : :math:`\mathrm{N17}-\mathrm{N18}` | | :width: 3.3945in | :math:`1008` : :math:`\mathrm{N17}-\mathrm{N16}` | | :height: 2.1602in | | | | Masses des coins: :math:`4.444\mathrm{kg}` | | | Masses intermédiaires: :math:`0.783\mathrm{kg}` | | .. _RefImage_1000000000000207000001594C43CE9088988969.png: | | | | | | | | +----------------------------------------------------------------+--------------------------------------------------------+ .. _11832774011: .. _11832773321: Les dimensions géométriques utilisées dans les calculs avec HERCULE sont très légèrement différentes de celles présentées au §1.1: * :math:`L=\mathrm{692.50}\mathrm{mm}` ; :math:`l=\mathrm{438.50}\mathrm{mm}` ; :math:`H=\mathrm{473.25}\mathrm{mm}` Résultats de référence ---------------------- * Fréquences propres * Déplacements aux points constituant les coins de la table, * Réactions d'appuis aux ancrages, * Efforts internes aux "coins". +--------------------+----------------------------------+----------------------------------+------------------------------------------------+ | **Identification** | Référence [:ref:`bib-1 `] | Référence [:ref:`bib-2 `] | **Masses effectives** | | | **expérimentale** | **Hercule** | (% de la masse totale) | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | Mode | Fréquence(:math:`\mathrm{Hz}` ) | Fréquence(:math:`\mathrm{Hz}` ) | :math:`X` | :math:`Y` | :math:`Z` | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 1 | 110.0 | 110.857 | 94.2 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 2 | 117.0 | 115.471 | 0.0 | 0.0 | 94.4 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 3 | 134.0 | 135.936 | 0.0 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 4 | 214.0 | 213.541 | 0.0 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 5 | 416.0 | 417.332 | 0.0 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 6 | | 434.813 | 0.0 | 24.1 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 7 | | 464.097 | 0.0 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 8 | 553.0 | 557.262 | 0.0 | 0.0 | 0.3 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 9 | 821.0 | 821.746 | 0.0 | 18.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 10 | | 847.071 | 0.0 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 11 | 927.0 | 978.174 | 1.6 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 12 | | 991.842 | 0.0 | 2.2 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 13 | | 1021.669 | 1.8 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 14 | | 1040.240 | 0.0 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 15 | | 1056.948 | 0.0 | 0.0 | 0.2 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 16 | | 1088.861 | 0.0 | 18.8 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 17 | | 1093.157 | 1.2 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ | 18 | | 1107.870 | 0.0 | 0.0 | 0.0 | +--------------------+----------------------------------+----------------------------------+------------------------+-----------+-----------+ +----------------------------------------------------------+-----------------------------------------------------------------------------------------+ | **Identification** | **Valeurs obtenues avec combinaison** | | +---------------------------------------------+-------------------------------------------+ | | quadratiques des directions des excitations | de NEWMARK des directions des excitations | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | Déplacement : | | | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{N17}` :math:`\mathrm{DX}(m)` | 3.4246E-04 | 3.4265E-04 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{DY}(m)` | 4.3562E-06 | 4.8392E-06 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{DZ}(m)` | 3.0321E-04 | 3.0324E-04 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{DRX}(\mathrm{rad})` | 3.7031E-04 | 3.7612E-04 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{DRY}(\mathrm{rad})` | 4.7665E-05 | 5.2602E-05 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{DRZ}(\mathrm{rad})` | 5.1104E-04 | 5.2310E-04 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | Réactions : | | | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{N25}` :math:`\mathrm{FX}(N)` | 1.2536E+03 | 1.2790E+03 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FY}(N)` | 1.2473E+03 | 1.3868E+03 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FZ}(N)` | 1.2196E+03 | 1.2441E+03 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MX}(\mathrm{N.m})` | 3.2474E+02 | 3.2789E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MY}(\mathrm{N.m})` | 4.1310E+00 | 4.5579E+00 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MZ}(\mathrm{N.m})` | 3.4846E+02 | 3.5199E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | Efforts : | | | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`1019` :math:`\mathrm{N17}` :math:`\mathrm{FX}(N)` | 1.1312E+03 | 1.1486E+03 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FY}(N)` | 1.2431E+03 | 1.3793E+03 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FZ}(N)` | 1.0982E+03 | 1.1141E+03 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MX}(\mathrm{N.m})` | 2.2833E+02 | 2.2982E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MY}(\mathrm{N.m})` | 4.1301E+00 | 4.5580E+00 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MZ}(\mathrm{N.m})` | 2.2068E+02 | 2.2537E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`1009` :math:`\mathrm{N17}` :math:`\mathrm{FX}(N)` | 1.8813E+02 | 2.0079E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FY}(N)` | 1.0419E+03 | 1.0650E+03 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FZ}(N)` | 1.3175E+02 | 1.4833E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MX}(\mathrm{N.m})` | 2.2833E+02 | 2.2975E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MY}(\mathrm{N.m})` | 2.9165E+01 | 3.2490E+01 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MZ}(\mathrm{N.m})` | 1.6408E-01 | 1.6400E-01 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`1008` :math:`\mathrm{N17}` :math:`\mathrm{FX}(N)` | 2.9587E+02 | 3.3579E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FY}(N)` | 6.3879E+02 | 6.7526E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{FZ}(N)` | 2.6539E+02 | 2.7947E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MX}(\mathrm{N.m})` | 1.8400E-01 | 1.8500E-01 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MY}(\mathrm{N.m})` | 3.2361E+01 | 3.5570E+01 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ | :math:`\mathrm{MZ}(\mathrm{N.m})` | 2.2068E+02 | 2.2535E+02 | +----------------------------------------------------------+---------------------------------------------+-------------------------------------------+ Remarques * Les déplacements des coins :math:`(\mathrm{N9},\mathrm{N12},\mathrm{N17},\mathrm{N20})` sont identiques, * Les réactions aux appuis :math:`(\mathrm{N25},\mathrm{N26},\mathrm{N27},\mathrm{N28})` sont identiques, * Les efforts généralisés sont exprimés dans le repère global. Références bibliographiques --------------------------- 1. NEUBERT V. H. et EZELL W. H.: Dynamic behavior of a foundation like structure. ASME Colloquium on Mechanical Impedance Methods for Mechanical Vibrations, pp. 77-86, 1958. 2. HERCULE: code de calcul par éléments finis pour le génie civil développé par SOCOTEC.