Reference problem ===================== Geometry --------- The pipe is modelled by a hollow circular section 762mm in diameter and 22mm thick (S1) or 813mm in diameter and 23mm thick (S2). The modeling is done with SEG2 (element POU_D_T) only. .. image:: images/1000020100000284000001F447BF3B6482197C1E.png :width: 5.2516in :height: 4.0772in .. _RefImage_1000020100000284000001F447BF3B6482197C1E.png: +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------+ |**Symbol** |**Description** | +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------+ | |Point mass | + .. image:: images/1000020100000025000000191095571F33A0D4BC.png + + | :width: 0.3854in | | + :height: 0.2602in + + | | | + + + | | | +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------+ | |0D spring in translation and/or rotation| + .. image:: images/100002010000001F0000001DA492A6014E26BE4E.png + + | :width: 0.2063in | | + :height: 0.1929in + + | | | + + + | .. image:: images/10000201000000130000001116F2DD9430B470A4.png | | + :width: 0.198in + + | :height: 0.1772in | | + + + | | | +----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+----------------------------------------+ Figure 1: Benchmark 6 diagram The following table shows the coordinates of each of the points in the pipe: .. csv-table:: "**Knot**", "**X** **(m)**", "**Y** **(m)**", "**Z** **(m)**" "PT_1 ", "3.20", "12.29", "17.93" "PT_2 ", "3.20", "12.29", "17.90" "PT_3 ", "3.20", "13.44", "16.75" "PT_4 ", "3.20", "14.43", "16.75" "PT_5 ", "3.20", "16.56", "16.75" "PT_6 ", "3.20", "18.69", "16.75" "PT_7 ", "3.20", "20.39", "16.75" "PT_8 ", "3.20", "22.08", "16.75" "PT_9 ", "3.20", "23.30", "16.75" "PT_10 ", "3.20", "24.52", "16.75" "PT_11 ", "3.20", "24.59", "16.75" "PT_12 ", "4.31", "25.74", "16.50" "PT_13 ", "4.42", "25.74", "16.48" "PT_14 ", "5.84", "25.74", "16.03" "PT_15 ", "7.21", "25.74", "15.46" "PT_16 ", "8.53", "25.74", "14.77" "PT_17 ", "9.79", "25.74", "13.97" "PT_18 ", "10.97", "25.74", "13.07" "PT_19 ", "12.07", "25.74", "12.06" "PT_20 ", "13.08", "25.74", "10.96" "PT_21 ", "13.98", "25.74", "9.78" "PT_22 ", "14.78", "25.74", "8.52" "PT_23 ", "15.47", "25.74", "7.20" "PT_24 ", "16.04", "25.74", "5.83" "PT_25 ", "16.49", "25.74", "4.41" "PT_26 ", "16.81", "25.74", "2.95" "PT_27 ", "17.00", "25.74", "1.48" "PT_28 ", "17.07", "25.74", "-0.01" "PT_29 ", "17.00", "25.74", "-1.50" "PT_30 ", "16.81", "25.74", "-2.97" "PT_31 ", "16.49", "25.74", "-4.43" "PT_32 ", "16.04", "25.74", "-5.85" "PT_33 ", "15.47", "25.74", "-7.22" "PT_34 ", "14.78", "25.74", "-8.54" "PT_35 ", "13.99", "25.74", "-9.79" "PT_36 ", "9.53", "25.74", "-11.35" "PT_37 ", "8.78", "25.74", "-11.09" "PT_38 ", "6.66", "25.74", "-10.36" "PT_39 ", "5.63", "25.74", "-10.00" "PT_40 ", "4.55", "24.59", "-9.63" "PT_41 ", "4.55", "24.59", "-9.63" Table 1: Coordinates of pipe nodes The following table shows the types of elements that make up the pipe: .. csv-table:: "**Element**", "**Node 1**", "**Node 2**", "**Type**", "**Section**" "EL_1 "," PT_1 "," PT_2 ", "Tangent", "1" "EL_2 "," PT_2 "," PT_3 ", "Bend", "1" "EL_3 "," PT_3 "," PT_4 ", "Tangent", "1" "EL_4 "," PT_4 "," PT_5 ", "Tangent", "1" "EL_5 "," PT_5 "," PT_6 ", "Tangent", "1" "EL_6 "," PT_6 "," PT_7 ", "Tangent", "1" "EL_7 "," PT_7 "," PT_8 ", "Tangent", "1" "EL_8 "," PT_8 "," PT_9 ", "Tangent", "1" "EL_9 "," PT_9 "," PT_10 ", "Tangent", "1" "EL_10 "," PT_10 "," PT_11 ", "Tangent", "1" "EL_11 "," PT_11 "," PT_12 ", "Bend", "1" "EL_12 "," PT_12 "," PT_13 ", "Tangent", "1" "EL_13 "," PT_13 "," PT_14 ", "Tangent", "1" "EL_14 "," PT_14 "," PT_15 ", "Tangent", "1" "EL_15 "," PT_15 "," PT_16 ", "Tangent", "1" "EL_16 "," PT_16 "," PT_17 ", "Tangent", "1" "EL_17 "," PT_17 "," PT_18 ", "Tangent", "1" "EL_18 "," PT_18 "," PT_19 ", "Tangent", "1" "EL_19 "," PT_19 "," PT_20 ", "Tangent", "1" "EL_20 "," PT_20 "," PT_21 ", "Tangent", "1" "EL_21 "," PT_21 "," PT_22 ", "Tangent", "1" "EL_22 "," PT_22 "," PT_23 ", "Tangent", "1" "EL_23 "," PT_23 "," PT_24 ", "Tangent", "1" "EL_24 "," PT_24 "," PT_25 ", "Tangent", "1" "EL_25 "," PT_25 "," PT_26 ", "Tangent", "1" "EL_26 "," PT_26 "," PT_27 ", "Tangent", "1" "EL_27 "," PT_27 "," PT_28 ", "Tangent", "1" "EL_28 "," PT_28 "," PT_29 ", "Tangent", "1" "EL_29 "," PT_29 "," PT_30 ", "Tangent", "1" "EL_30 "," PT_30 "," PT_31 ", "Tangent", "1" "EL_31 "," PT_31 "," PT_32 ", "Tangent", "1" "EL_32 "," PT_32 "," PT_33 ", "Tangent", "1" "EL_33 "," PT_33 "," PT_34 ", "Tangent", "1" "EL_34 "," PT_34 "," PT_35 ", "Tangent", "1" "EL_35 "," PT_35 "," PT_36 ", "Bend", "1" "EL_36 "," PT_36 "," PT_37 ", "Tangent", "1" "EL_37 "," PT_37 "," PT_38 ", "Tangent", "1" "EL_38 "," PT_38 "," PT_39 ", "Tangent", "1" "EL_39 "," PT_39 "," PT_40 ", "Bend", "2" "EL_40 "," PT_40 "," PT_41 ", "Tangent", "2" Table 2: **Type of pipe elements** Material properties ---------------------- The characteristics are as follows: * :math:`\mathit{YOUNG}=206000\mathit{MPa}` * :math:`\mathit{NU}=0.30` * :math:`\mathit{RHO}=0\mathit{kg}/m3` The mass of the pipe is not considered in this test case. Boundary conditions and loads ------------------------------------- Boundary conditions ~~~~~~~~~~~~~~~~~~~~~~~~ In accordance with Figure, this study does not include fixed supports. On the other hand, some points are elastic supports whose characteristics are specified in the following tables: .. csv-table:: "**Knot**", "**X** **(m)**", "**Y** **(m)**", "**Z** **(m)**", "**Type**" "RESS_PT10_42_BASE ", "0.00", "0.00", "0.00", "-0.19", "K_T_D_L" "RESS_PT10_43_BASE ", "1.83", "3.69", "-0.03", "K_T_D_L" "RESS_PT30_50_BASE ", "2.78", "3.69", "3.69", "0.92", "K_T_D_L" "RESS_PT34_51_BASE ", "2.75", "3.71", "3.00", "K_T_D_L" "RESS_PT1_BASE ", "3.20", "12.29", "12.29", "17.93", "K_ TR_D_L" "RESS_PT20_BASE ", "13.08", "25.74", "25.74", "10.96", "K_ TR_D_L" "RESS_PT26_BASE ", "16.81", "25.74", "2.95", "K_ TR_D_L" "RESS_PT41_BASE ", "4.55", "24.59", "-9.63", "K_ TR_D_L" Table 3: Coordinates and type of elastic supports .. csv-table:: "**Node**", "**K** local **(N/m)**", "**K** X **(N/m)**", "**K** Y **(N/m)**", "**K** Z **(N/m)**", "**K** RX **(N/** rad **)**) **", "**K** RY **(N/** rad **)**) **", "**K** RZ **(N/** rad **)**" "RESS_PT10_42_BASE ", "1.75E+08", "-", "-", "-", "-", "-", "-", "-" "RESS_PT10_43_BASE ", "1.75E+08", "-", "-", "-", "-", "-", "-", "-" "RESS_PT30_50_BASE ", "3.50E+08", "-", "-", "-", "-", "-", "-", "-" "RESS_PT34_51_BASE ", "1.40E+08", "-", "-", "-", "-", "-", "-", "-" "RESS_PT1_BASE ", "-", "1.75E+21", "1.75E+21", "1.75E+21", "1.75E+21", "1.75E+21", "1.75E+21", "1.75E+21" "RESS_PT20_BASE ", "-", "4.37E+07", "4.37E+07", "3.50E+08", "4.37E+07", "-", "-", "-" "RESS_PT26_BASE ", "-", "7.87E+07", "3.50E+07", "3.50E+08", "7.87E+07", "-", "-", "-" "RESS_PT41_BASE ", "-", "1.75E+11", "1.75E+11", "1.75E+11", "1.13E+16", "1.13E+16", "1.13E+16", "1.13E+16", "1.13E+16" Table 4: **Stiffers** of elastic supports Loads ~~~~~~~~~~~ The loads applied to the pipe are as follows: * Point masses in different nodes (identical in all 3 directions); * Earthquake in X, Y, and Z directions. The spectrum is weighted by 1 in all three directions; .. csv-table:: "**Knot**", "**M** **(** kg **)**" "PT_4 ", "1 738.13" "PT_5 ", "954.97" "PT_6 ", "856.02" "PT_7 ", "1 031.15" "PT_9 ", "940.96" "PT_11 ", "691.75" "PT_12 ", "425.56" "PT_13 ", "690.17" "PT_15 ", "1 332.54" "PT_17 ", "1 332.54" "PT_19 ", "1 333.07" "PT_21 ", "1 332.89" "PT_23 ", "1 332.89" "PT_25 ", "1 332.89" "PT_27 ", "1 332.89" "PT_29 ", "1 332.89" "PT_31 ", "1 332.89" "PT_33 ", "1 331.14" "PT_35 ", "1 802.58" "PT_36 ", "1 316.60" "PT_37 ", "678.97" "PT_38 ", "1 843.74" Table 5: Values and locations of point masses The spectra used for modal analysis are shown in the following figure: .. image:: images/100002010000078C000004F58CA9D8C038C4CAA7.png :width: 5.272in :height: 3.4626in .. _RefImage_100002010000078C000004F58CA9D8C038C4CAA7.png: Figure 2: Computational spectrum — X and Z .. image:: images/100002010000078C000004F5119E9C2D83E20D8C.png :width: 5.3299in :height: 3.5008in .. _RefImage_100002010000078C000004F5119E9C2D83E20D8C.png: Figure 3: Computational spectrum - Y