1、STRESS INTENSITY K FACTORS FOR EXTERNAL SURFACE CRACKS IN THICK-WALLED CYLINDER VESSELSSTP-PT-082STP-PT-082 STRESS INTENSITY K FACTORS FOR EXTERNAL SURFACE CRACKS IN THICK-WALLED CYLINDER VESSELS Prepared by: Lucie Parietti Greg Thorwald, Ph.D. Quest Integrity USA, LLC Date of Issuance: February 7,
2、2017 This report was prepared by ASME Standards Technology, LLC (“ASME ST-LLC”) and sponsored by the American Society of Mechanical Engineers (“ASME”) Pressure Technology Codes the constraints and dimensions are shown in Figure 2-1. The left end of the cylinder is the cross-section symmetry plane an
3、d has an X-constraint. The top and bottom mesh surfaces are on the axial symmetry plane and have a Z-constraint outside the crack. The right end of the cylinder is unconstrained. A single node at the top of the cylinder has a Y-constraint. The green mesh zone is used to improve the mesh refinement n
4、ear the crack plane, and has the same elastic material properties as the red mesh zone in the cylinder model. The crack face pressure loading is applied to the crack face elements in the light blue mesh region. For the shallower crack depths, more elements are added through the thickness in the liga
5、ment outside the crack as shown in Figure 2-2. An example of the thickest cylinder, Y = 4 (t/Ri = 3) is shown in Figure 2-3. Figure 2-1: Quarter Symmetric Crack Mesh, case 149, t/Ri=2, a/c=0.5, a/t=0.6 STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels 7 Figure 2-2: Shallow Crack Mesh exam
6、ple, case 17, t/Ri=1, a/c=0.125, a/t=0.2 Figure 2-3: Thickest Cylinder Example, case 267, t/Ri=3, a/c=1.0, a/t=0.4 STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels8 3 AXIAL EXTERNAL FULL-WIDTH CRACKS The combination of geometry ratios and five load cases gives 100 axial external full-wid
7、th crack meshes. The full-width crack meshes are intended to model an infinitely long, partial-depth crack as a bounding solution for long axial surface cracks. The model “Run ID” numbers are used to uniquely identify each case, from 281 through 380. The non-dimensional G polynomial coefficient resu
8、lts are listed in Appendix B, Table 6 and in the corresponding Excel file delivered with this report. The axial full-width crack meshes are quarter symmetric models; the constraints and dimensions are shown in Figure 3-1. The left end of the cylinder is the cross-section symmetry plane and has an X-
9、constraint. The top and bottom mesh surfaces are on the axial symmetry plane and have a Z-constraint outside the crack. The right end of the cylinder is constrained in the X-direction to model the infinitely long partial-depth crack. A single node at the top of the cylinder has a Y-constraint. The g
10、reen mesh zone is used to improve the mesh refinement near the crack plane, and has the same elastic material properties as the red mesh zone in the cylinder model. The crack face pressure loading is applied to the crack face elements in the light blue mesh region. The full-width crack mesh does not
11、 need to be very long, since the geometry factor is constant along the crack front for the infinitely long crack being modeled. An example of the deepest full-width crack in the thickest cylinder is shown in Figure 3-2. Figure 3-1: External Full-Width Crack, case 281, t/Ri=1, a/t=0.2 STP-PT-082: Ext
12、ernal Cracks in Thick-Walled Cylinder Vessels 9 Figure 3-2: Thickest Cylinder, Full-Width Crack, case 377, t/Ri=3, a/t=0.8 STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels10 4 CIRCUMFERENTIAL EXTERNAL SURFACE CRACKS The combination of geometry ratios and four load cases gives 388 externa
13、l circumferential surface crack meshes. The model “Run ID” numbers are used to uniquely identify each case, from 381 through 940, with gaps for cases where the crack length is too long for the inside cylinder circumference (see Table 4). The non-dimensional G polynomial coefficient results are liste
14、d in Appendix C, Table 7 and in the corresponding Excel file delivered with this report. The circumferential surface crack meshes are quarter symmetric models for linear crack face pressure and in-plane bending load cases. Half symmetric models are needed for uniform crack face pressure and out-of-p
15、lane bending load cases. The quarter symmetric model constraints and dimensions are shown in Figure 4-1. The left end of the cylinder is the cross-section symmetry plane and has an X-constraint on the nodes outside the crack. The top and bottom mesh surfaces are on the axial symmetry plane and have
16、a Z-constraint. The right end of the cylinder is unconstrained for the crack face pressure load cases. The bending load cases are shown below. A single node at the top of the cylinder has a Y-constraint. The crack face pressure loading is applied to the crack face elements in the light blue mesh reg
17、ion. Figure 4-1: External Circumferential Surface Crack, case 414, t/Ri=1, a/c=0.125, a/t=0.2 A deeper and longer circumferential crack is shown in Figure 4-2. A circumferential crack in the thickest cylinder is shown in Figure 4-3, and the same size cylinder and same size crack for the half-symmetr
18、ic mesh for the uniform crack face pressure and out-of-plane bending load cases is shown in Figure 4-4. The combined loading for the in-plane bending about the z-axis plus the axial load in the x-direction is shown in Figure 4-5. The combined loading for the out-of-plane bending about the y-axis plu
19、s the axial load in the x-direction is shown in Figure 4-6 for the half symmetric mesh. STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels 11 Figure 4-2: Circumferential Surface Crack Case 666, t/Ri=2, a/c=0.25, a/t=0.8 Figure 4-3: Circumferential Surface Crack Case 866, t/Ri=3, a/c=0.125,
20、 a/t=0.4, Thickest Cylinder STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels12 Figure 4-4: Case 868, t/Ri=3, a/c=0.125, a/t=0.4, Half Symmetric Mesh to apply the Out-Of-Plane Bending Load about the y-Axis Figure 4-5: Combined in-Plane Bending plus Axial Load, case 423, t/Ri=1, a/c=0.125,
21、 a/t=0.6 STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels13 Figure 4-6: Combined Out-of-Plane Bending plus Axial load, Case 548, t/Ri=1.5, a/c=0.25, a/t=0.4 STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels14 5 CIRCUMFERENTIAL EXTERNAL 360 CRACKS The combination of geometry ra
22、tios and five load cases gives 100 circumferential external 360 crack meshes. The 360 crack meshes are intended to provide a bounding solution for the crack lengths that are longer than the external cylinder circumference. The model “Run ID” numbers are used to uniquely identify each case, from 1100
23、 through 1199. The non-dimensional G polynomial coefficient results are listed in Appendix D, Table 8 and in the corresponding Excel file delivered with this report. The circumferential external 360 crack meshes are quarter symmetric models; the constraints and dimensions are shown in Figure 5-1. Th
24、e left end of the cylinder is the cross-section symmetry plane and has an X-constraint on the nodes in the ligament region outside the crack. The top and bottom mesh surfaces are on the axial symmetry plane and have a Z-constraint. The right end of the cylinder is unconstrained. A single node at the
25、 top of the cylinder has a Y-constraint. The crack face pressure loading is applied to the crack face elements in the light blue mesh region on the left end of the cylinder. An example of the shallow crack mesh is shown in Figure 5-2. An example of the thickest cylinder with a deep 360 crack is show
26、n in Figure 5-3. Figure 5-1: External Circumferential 360 Crack, case 1130, t/Ri=1.5, a/t=0.6 STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels 15 Figure 5-2: 360 Crack, Case 1100, t/Ri=1, a/t=0.2, Shallow Crack example Figure 5-3: 360 Crack, Case 1195, t/Ri=3, a/t=0.8, Thickest Cylinder
27、STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels16 6 MESH REFINEMENT STUDY Several aspects of the crack mesh refinement were examined to confirm that sufficient mesh refinement was used for the crack models in this analysis. The number of elements along the crack front was varied from th
28、e default 1x crack front refinement to 3x refinement (three times as many crack front elements) and to 9x refinement (nine times as many crack front elements). The plot in Figure 6-1 shows that there is good agreement along most of the crack front, with some difference at the crack tip node. Free su
29、rface effects are expected at the crack tip node, so the observed difference in the K solutions results in this comparison is expected. Omitting the crack tip node from the non-dimensional geometry factor results curve-fit is discussed in the Results section below. The 3x crack front refinement leve
30、l was used for this analysis. Figure 6-1: Compare Crack Front Mesh Refinement for the Uniform Loading Case The number of elements in the contours around the crack front was varied from three to nine. The number of mesh contours sets the number of J-integral contours used to compute J and subsequentl
31、y K along the crack front. The plot in Figure 6-2 shows overall good agreement, except near the crack tip for the lower refinement with three contours. Five element contours were used in this analysis. STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels17 Figure 6-2: Compare the Number of C
32、ontours around the Crack Front for the Uniform Loading Case The number of elements through the remaining ligament in the thickness past the crack depth was varied from three to eight elements. The plot in Figure 6-3 shows no difference in the K results. The number of elements in the ligament for eac
33、h analysis varies depending on the crack depth. The additional length of the cylinder past the end of the crack tip was varied from 3 to 30 inches as a multiple of the cylinder OD, and again no difference in the K results was observed. The additional cylinder length past the axial crack length used
34、for this analysis was equal to twice the outside diameter. Figure 6-3: Compare the Number of Elements through the Thickness and the Cylinder Length for the Uniform Loading Case STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels18 The same study was conducted for the linear crack face press
35、ure load case. No difference in the K results were observed from the plots in Figure 6-4 through Figure 6-6, indicating that sufficient mesh refinement is available to compute accurate crack front J values. Figure 6-4: Compare Crack Front Mesh Refinement for the Linear Crack Face Pressure Load Case
36、Figure 6-5: Compare the number of Contours around the Crack Front for the Linear Crack Face Pressure Load Case STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels19 Figure 6-6: Compare the Number of Elements through the Thickness and the Cylinder Length for the Linear Crack Face Pressure Lo
37、ad Case STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels 20 7 RESULTS AND DISCUSSION The crack front stress intensity results are reported as a sixth order polynomial curve-fit to the non-dimensional G trend along the crack front for the external surface cracks. The axial full-width and
38、circumferential 360 cracks have a constant G value along the crack front, so a single value is reported. The results values are tabulated in appendices A through D. Since the crack tip node (at =0) J-integral value can be inconsistent due to stress triaxiality at the free surface as compared to the
39、overall crack front trend, the crack tip node is omitted from the polynomial curve-fit. This allows the overall curve-fit along the crack front of the non-dimensional geometry factor to extrapolate the solution to the crack tip location. An example of the external axial surface crack results plot of
40、 the non-dimensional G value versus the normalized crack front angle is shown in Figure 7-1; results from axial surface crack cases 1 to 8 are shown. The plot x-axis uses the normalized crack front angle 2/ along the crack front from 0 at the crack tip to 1 at the crack depth location. The solid cur
41、ves are for the uniform crack face pressure G0 load cases, and the dashed curves are for the linear crack face pressure G1 load cases. Examples of the external circumferential surface crack results plots are shown in Figure 7-2 (crack face pressure loads) and Figure 7-3 (bending loads) for cases 429
42、 to 444. Results plots for all the analysis cases are shown in the appendices. Figure 7-1: Results Plot Example for the Non-Dimensional G Results versus the Normalized Crack Front Angle for the External Axial Surface Crack STP-PT-082: External Cracks in Thick-Walled Cylinder Vessels 21 Figure 7-2: R
43、esults Plot Example for the External Circumferential Surface Crack, Uniform and Linear Crack Face Pressure Load Cases Figure 7-3: Results Plot example for the External Circumferential Surface Crack, In-Plane and Out-of-Plane Bending Load Cases STP-PT-082: External Cracks in Thick-Walled Cylinder Ves
44、sels22 7.1 Combined Load Post Processing The non-dimensional geometry G factor solutions for the bending load cases were obtained by subtracting the uniform load results from the combined bending plus axial load results. Subtracting the uniform load result curve from the combined result curve is don
45、e by subtracting a set of points along each curve at the same crack front angle locations. Then, a new polynomial curve is fit to the bending only result curve to report the bending only G5and G6coefficient values. Figure 7-4 shows an example for the in-plane bending G5result. The combined loading i
46、s the higher Gtotalvalues in the plot, and when the uniform G0result curve is subtracted, the bending only G5result curve is the lower value along the bottom of the plot. Examining the crack mesh deformed shapes showed the possibility of crack face closure for some of the larger crack cases, so a la
47、rge axial load is needed to avoid crack closure for the combined bending plus axial loading. To keep the crack front in tension the axial load is six times the bending load to prevent crack closure. For a consistent analysis approach, the same axial load was used for all the bending cases. We added
48、an automated post processing check to confirm that there was no crack closure for the combined bending plus axial load results. Figure 7-5 shows an example for the out-of-plane bending G6result. Just half of the crack front is used in the range from 2/ (left crack tip) to 2/ (crack depth) even thoug
49、h the half symmetric model crack front extends to 2/ at the right side crack tip so that the same sixth order polynomial can be used to report the crack front results. The combined loading G results are the higher values, and when the uniform G0results are subtracted the out-of-plane bending G6results are negative along the first half of the crack front (crack closure on half of the crack) going to zero at the crack depth location, which is on the bending neutral axis for the out-of-plane bending load case. The other half of the crack front trend in the 2/ to 2
