Modeling A ============== Characteristics of modeling and meshing -------------------------------------------------- Line element: POU_D_TGM. .. image:: images/Object_8.svg :width: 492 :height: 86 .. _RefImage_Object_8.svg: Mechanical characteristics of the section (units homogeneous to meters): .. csv-table:: ":math:`A` "," :math:`\mathrm{IY}` "," :math:`\mathrm{IZ}` "," :math:`\mathrm{AY}` "," "," :math:`\mathrm{AZ}` "," :math:`\mathrm{JX}` "," :math:`\mathrm{JG}`" "8.0e-04", "2.666667e-08", "1.066667e-07", "1.191790e+00", "1.172840e+00", "7.093682e-08", "7.093682e-08", "1.438125e-12" Loading at point :math:`B`. .. csv-table:: "", ":math:`\mathrm{Fx}` "," :math:`\mathrm{Fy}` "," :math:`\mathrm{Fz}`" "Instant 1"," :math:`\mathrm{80 000}\text{N}` "," :math:`–\mathrm{150N}` "," :math:`–\mathrm{200N}`" "Instant 2"," :math:`\mathrm{80 000}\text{N}` "," :math:`–\mathrm{280N}` "," :math:`–\mathrm{400N}`" At time 1 the section remains elastic, at time 2 the section is partially plasticized. Fiber numbering ----------------------- The main inertia coordinate system :math:`(\mathrm{Y0},\mathrm{Z0})` of the beam must be rotated by 90° for the greatest inertia :math:`\mathrm{IZ}` to be along the :math:`Y` axis of the global coordinate system. The aim is to test the ORIENTATION keyword from the AFFE_CARA_ELEM command. +-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+ |**In the local frame of reference of the beam** |**In the global study benchmark** | +-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+ | | | + .. image:: images/Object_9.svg + .. image:: images/Object_10.svg + | :width: 186 | :width: 492 | + :height: 247 + :height: 86 + | | | + + + | | | +-----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+ Several "types" of fibers are used: • fibers whose numbers range from 1 to 32. They are assigned a non-zero area in the command file, • fibers whose numbers are 33, 34, 34, 35, 36. They are located at the 4 corners of the section. These fibers, in the command file, are assigned a cross section equal to zero. They are only used for post-processing, • fibers whose numbers range from 37 to 60. They are located on the outer edge of the section. These fibers, in the command file, are assigned a cross section equal to zero. They are only used for post-processing. Tested sizes and results ------------------------------ The quantity tested and analyzed is SIEF_ELGA, at the first Gauss point of element :math:`\text{E00200}`. It is the Gauss point closest to the embedment, :math:`\mathrm{Lo}=\mathrm{2.95491933m}`. Elastic behavior ~~~~~~~~~~~~~~~~~~~~~~~~ **Gauss point constraints:** **SIEF_ELGA** The constraints calculated by equation [:ref:`éq 2.1-1 <éq 2.1-1>`] and by*Code_Aster* are given in the two tables below (values in :math:`\text{MPa}`). The layout of the tables is based on the fiber layout diagram in the global coordinate system. The most used fiber is No. 36 with a constraint of :math:`\mathrm{390 }\text{MPa}`. .. csv-table:: "Stresses calculated in fibers by equation [:ref:`éq 2.1-1 <éq 2.1-1>`]", "", "Stresses calculated in fibers by*Code_Aster*." .. csv-table:: "35", "80", "168", "257", "257", "257", "346", "346", "390"", "80", "168", "257", "257", "257", "257", "346", "390" "21", "66", "154", "243", "243", "243", "331", "376", "", "21", "66", "154", "243", "243", "243", "243", "331", "376" "-7", "37", "126", "215", "215", "215", "303", "303", "348", "37", "126", "126", "215", "215", "215", "303", "348" "-35", "9", "98", "187", "187", "187", "275", "", "320", "", "-35", "9", "98", "187", "187", "187", "187", "275", "320" "-63", "-19", "70", "70", "158", "158", "158", "247", "247", "-63", "-19", "70", "70", "158", "158", "158", "158", "247", "291" "-91", "-47", "42", "130", "130", "130", "219", "", "", "-91", "-47", "42", "42", "130", "130", "130", "130", "219", "263" "-120", "-75", "13", "102", "102", "102", "191", "235", "", "-120", "-75", "13", "13", "102", "102", "102", "191", "235" "-148", "-103", "-15", "-15", "74", "74", "163", "", "", "-148", "-103", "-15", "-15", "" 74", "74", "163", "207" "-176", "-131", "-43", "-43", "46", "46", "134", "134", "", "-176", "-131", "-43", "-43", "46", "46", "46", "134", "179" "-190", "-146", "-57", "-57", "32", "32", "120", "", "", "-190", "-146", "-57", "-57", "-57", "32", "32", "32", "120", "165" The relative error between the two calculations is given in the table below. Whatever the fiber, it is still less than :math:`0.095\text{\%}`. .. csv-table:: "-0.022% "," -0.013% "," -0.009% "," -0.009% "," -0.008% "," -0.007% "," -0.007%" "-0.026% "," -0.012% "," -0.009% "," -0.009% "," -0.008% "," -0.007% "," -0.007%" "0.014% "," -0.010% "," -0.007% "," -0.007% "," -0.006% "," -0.006% "," -0.006%" "-0.010% "," 0.010% "," -0.004% "," -0.004% "," -0.005% "," -0.005% "," -0.005%" "-0.013% "," -0.029% "," 0.000% "," -0.003% "," -0.004% "," -0.004% "," -0.004%" "-0.014% "," -0.021% "," 0.012% "," 0.012% "," 0.000% "," -0.002% "," -0.003%" "-0.014% "," -0.019% "," 0.069% "," 0.069% "," 0.004% "," -0.001% "," -0.002%" "-0.015% "," -0.018% "," -0.094% "," 0.012% "," 0.002% "," 0.000%" "-0.015% "," -0.018% "," -0.043% "," -0.043% "," 0.029% "," 0.006% "," 0.003%" "-0.015% "," -0.018% "," -0.036% "," 0.036% "," 0.049% "," 0.009% "," 0.005%" Plastic behavior ~~~~~~~~~~~~~~~~~~~~~~~~ **Gauss point constraints:** **SIEF_ELGA** The table below gives the stress values, after partial plasticization of the section, obtained with*Code_Aster*. The behavior of the material is "almost perfect" elasto-plastic, the work-hardening slope is low. The maximum stress, which is beyond the elastic limit, therefore remains very close to the elastic threshold of :math:`\mathrm{400MPa}`. .. csv-table:: "8", "79", "223", "36", "366", "400", "400" "-16", "55", "199", "199", "342", "400", "400" "-64", "7", "151", "151", "294", "400", "400" "-112", "-41", "103", "103", "246", "390", "400" "-160", "-89", "55", "55", "198", "342", "400" "-208", "-137", "7", "7", "150", "294", "365" "-256", "-185", "-41", "41", "102", "246", "317" "-305", "-233", "-89", "-89", "54", "198", "269" "-353", "-281", "-137", "-137", "6", "150", "221" "-377", "-305", "-161", "-161", "-18", "126", "197" This calculation is carried out for the *Code_Aster* non-regression test.