1、STP-PT-071STRESS INTENSITY FACTOR SOLUTIONS FOR INTERNAL CRACKS IN THICK-WALLED CYLINDER VESSELSSTP-PT-071 STRESS INTENSITY FACTOR SOLUTIONS FOR INTERNAL CRACKS IN THICK-WALLED CYLINDER VESSELS Prepared by: Greg Thorwald, Ph.D. Lucie Parietti Bruno Fletcher Joyce Wright Quest Integrity Group, LLC Da
2、te of Issuance: September 2, 2014 This report was prepared as an account of work sponsored by 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 and has an X-constraint. The top and bottom mesh surfa
3、ces 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 near the crack plane, and has the same elastic materi
4、al 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 ligament outside the crack as shown in Figure 2-2. An ex
5、ample 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 Sy m m et ry , fix XSy m m et ry , fix ZR ight end unc ons train edC rac k fac e tODRiF ix Y s ingle nodeacSTP-PT-071: Stress Intensity Factor Solutio
6、ns for Internal Cracks in Thick-Walled Cylinder Vessels 7 Figure 2-2: Shallow Crack Mesh Example, 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-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder V
7、essels 8 3 AXIAL INTERNAL FULL-WIDTH CRACKS The combination of geometry ratios and five load cases gives 100 axial internal full-width 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 mod
8、el “Run ID” numbers are used to uniquely identify each case, and are from 301 through 400. The non-dimensional G polynomial coefficient results are listed in Table 2 in Appendix B. The axial full-width crack meshes are quarter symmetric models; the constraints and dimensions are shown in Figure 3-1.
9、 The left end of the cylinder is the cross-section symmetry plane and 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 constrained in the X-direction to model the infinitely long partial-
10、depth crack. A single node at the top of the cylinder has a Y-constraint. The green 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
11、face elements in the light blue mesh region. The full-width crack mesh does not need to be very long, since the geometry factor is constant along the crack front for the infinitely long crack being modeled. Figure 3-1: Internal Full-Width Crack, Case 301, t/Ri=1, a/t=0.2 An example of the deepest fu
12、ll-width crack in the thickest cylinder is shown in Figure 3-2. Sy m m et ry , fix XSy m m et ry , fix ZR ight end, fix XC rac k fac etODRiF ix Y s ingle nodeaSTP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 9 Figure 3-2: Thickest Cylinder, Full-Widt
13、h Crack, Case 396, t/Ri=3, a/t=0.8 STP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 10 4 CIRCUMFERENTIAL INTERNAL SURFACE CRACKS The combination of geometry ratios and four load cases gives 264 internal circumferential surface crack meshes. The model
14、 “Run ID” numbers are used to uniquely identify each case and are from 501 through 1060, with gaps for cases where the crack length is too long for the inside cylinder circumference (see Figure 1-4). The non-dimensional G polynomial coefficient results are listed in Table 3 in Appendix C. The circum
15、ferential surface crack meshes are quarter symmetric models for the crack face pressure and in-plane bending load cases. Half symmetric models are needed for the out-of-plane bending load case. The quarter symmetric model constraints and dimensions are shown in Figure 4-1. The left end of the cylind
16、er 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 a Z-constraint. The right end of the cylinder is unconstrained for the crack face pressure load cases. The bending load cases are s
17、hown 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 region. Figure 4-1: Internal Circumferential Surface Crack, Case 533, t/Ri=1, a/c=0.125, a/t=0.2 A deeper and longer circumferential c
18、rack 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-symmetric mesh for the out-of-plane bending load case is shown in Figure 4-4. The combined loading for the in-plane bending about the z-ax
19、is 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 plus the axial load in the x-direction is shown in Figure 4-6 for the half symmetric mesh. Sy m m et ry , fix XSy m m et ry , fix ZR ight end unc ons train edC rac k f
20、ac etODRiF ix Y s ingl e nod eacSTP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 11 Figure 4-2: Circumferential Surface Crack Case 797, t/Ri=2, a/c=0.5, a/t=0.6 Figure 4-3: Circumferential Surface Crack Case 1033, t/Ri=3, a/c=1.0, a/t=0.4, Thickest C
21、ylinder STP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 12 Figure 4-4: Case 1036, t/Ri=3, a/c=1.0, 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 559,
22、t/Ri=1, a/c=0.25, a/t=0.6 Be nd in g ab ou t z -ax isAx ia l lo ad al on g x -ax is Co mb in e d lo a d in g ne ed ed to kee p the crack in t en sio n, a p p ly : in -p la n e be nd in g pl us ax ia l lo ad Cra ck le ng th ex ten ds pa st 90oan d b el ow the z -ax is be nd in g ne utra l ax isSTP-PT
23、-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 13 Figure 4-6: Combined Out-of-Plane Bending Plus Axial Load, Case 664, t/Ri=1.5, a/c=0.25, a/t=0.2 Be nd in g ab ou t y -ax isAx ia l lo ad al on g x -ax is Co mbi ne d lo ad in g ne ed ed to kee p ha lf th
24、e cra ck in t e n sio n , a p p ly : o u t-of -p la n e be nd in g pl us ax ia l lo adSTP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 14 5 CIRCUMFERENTIAL INTERNAL 360O CRACKS The combination of geometry ratios and five load cases gives 100 circumfe
25、rential internal 360o crack meshes. The 360o crack meshes are intended to provide a bounding solution for the crack lengths that are longer than the internal cylinder circumference. The model “Run ID” numbers are used to uniquely identify each case, and are from 1101 through 1200. The non-dimensiona
26、l G polynomial coefficient results are listed in Table 4 in Appendix D. The circumferential internal 360o crack meshes are quarter symmetric models; the constraints and dimensions are shown in Figure 5-1. The left end of the cylinder is the cross-section symmetry plane and has an X-constraint on the
27、 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 top of the cylinder has a Y-constraint. The crack face pressure loading is applied to the cr
28、ack face elements in the light blue mesh region on the left end of the cylinder. Figure 5-1: Internal Circumferential 360o Crack, Case 1131, t/Ri=1.5, a/t=0.6 An example of the shallow crack mesh is shown in Figure 5-2. An example of the thickest cylinder with a deep 360o crack is shown in Figure 5-
29、3. Sy m m etry , fix XSy m m et ry , fix ZR ight end unc ons train edC rack facetODRiF ix Y s ingl e nod eaSTP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 15 Figure 5-2: 360o Crack, Case 1101, t/Ri=1, a/t=0.2, Shallow Crack Example Figure 5-3: 360o
30、Crack, Case 1196, t/Ri=3, a/t=0.8, Thickest Cylinder STP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 16 6 MESH REFINEMENT STUDY Several aspects of the crack mesh refinement were examined to confirm that sufficient mesh refinement was used for the cr
31、ack models in this analysis. The number of elements along the crack front was varied from the 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
32、agreement along most of the crack front, with some difference at the crack tip node. Free surface effects are expected at the crack tip node, so the observed difference in the K results in this comparison is expected. Omitting the crack tip node from the non-dimensional geometry factor results curve
33、-fit is discussed in the Results section below. The 3x crack front refinement level was used for this analysis. Figure 6-1: Compare Crack Front Mesh Refinement 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
34、J-integral contours used to compute J and subsequently 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. 012345670 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 .
35、 4 1 . 6Stress intensity along crack front K, ksi inCr ack an gle , r adian sUn if orm load in g , v ar yin g cr ack fr ont mes h r e fine me n t1x cra c k front r ef i ne me nt3x cra c k front r ef i ne me nt9x cra c k front r ef i ne me ntSTP-PT-071: Stress Intensity Factor Solutions for Internal
36、Cracks in Thick-Walled Cylinder Vessels 17 Figure 6-2: Compare the Number of Contours around the Crack Front 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 result
37、s. The number of elements in the ligament for each analysis varies depending on the crack depth. The additional length of the cylinder past the end of the crack tip was varied from three to 30 inches as a multiple of the cylinder OD, and again no difference in the K results was observed. The additio
38、nal cylinder length past the axial crack length used for this analysis was 2*OD. 012345670 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Stress intensity along crack front K, ksi inC r ack angle, r adiansUn if orm load in g , v ar yin g n u mb e r of cr ack tu b e e le me n ts3 co nt ou r e l e men ts5 co nt ou r e
39、l e men ts7 co nt ou r e l e men ts9 co nt ou r e l e men tsSTP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 18 Figure 6-3: Compare the Number of Elements through the Thickness and the Cylinder Length A combination of mesh refinement settings were us
40、ed to compare the K results for the linear crack face pressure load case. No difference in the K results was observed from the plot in Figure 6-4, indicating that sufficient mesh refinement is available to compute accurate crack front J values. Figure 6-4: Compare Mesh Refinement for the Linear Crac
41、k Face Pressure Load Case The longest axial crack length was also used to compare the crack front refinement for case 175, t/Ri=2.5, a/c=0.03125, a/t=0.8. The analysis mesh has 285,865 nodes with 1,099 crack front nodes. The coarser comparison mesh has 135,146 nodes with 201 crack front nodes, and d
42、oes not include the 012345670 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Stress intensity along crack front K, ksi inCr ack an gle , r adian sUn if orm load , v ar y n u mb e r of e le me n ts: thi ckn e ss and le n g th3 elem en ts through t h i c kness, L=4. 47 elem en ts through t h i c kness, L=4. 45 elem en
43、ts through t h i c kness, L=4. 49 elem en ts through t h i c kness, L=4. 45 elem en ts through t h i c kness, L=3. 05 elem en ts through t h i c kness, L=10.05 elem en ts through t h i c kness, L=30.0012345670 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Stress intensity along crack front K, ksi inCr ack an gle , r
44、 adian sL in e ar cr ack f ace p r e ssu r e load in g , v ar yin g all p ar ame t e r s 3 elem T, 3 x c r ac k fro nt , 5 elem con t our , L= 4 . 43 elem T, 3 x c r ac k fro nt , 5 elem con t our , L= 4 . 47 elem T, 3 x c r ac k fro nt , 5 elem con t our , L= 4 . 49 elem T, 3 x c r ac k fro nt , 5
45、elem con t our , L= 4 . 45 e lem T, 1x c r ac k fro nt , 5 elem c ontour , L= 4 . 45 elem T, 9 x c r ac k fro nt , 5 elem con t our , L= 4 . 45 elem T, 3 x c r ac k fro nt , 3 elem con t our , L= 4 . 45 elem T, 3 x c r ac k fro nt , 7 elem con t our , L= 4 . 45 elem T, 3 x c r ac k fro nt , 9 elem c
46、on t our , L= 4 . 45 elem T, 3 x c r ac k fro nt , 5 elem con t our , L= 3 . 05 elem T, 3 x c r ac k fro nt , 5 elem con t our , L= 1 0 . 05 elem T, 3 x c r ac k fro nt , 5 elem con t our , L = 3 0 . 0STP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels
47、19 3-to-1 crack front mesh transition option. The plot in Figure 6-5 shows good agreement in the K results along the crack front, indicating that the more refined crack mesh is sufficient for the analysis. Figure 6-6 shows the coarser comparison axial crack mesh picture. Figure 6-5: Longest Axial Cr
48、ack Mesh Comparison 0510152025300 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Stress intensity K. ksi inCr ack an gle s (r ad) Str e ss i n t e n sity f act or f or lo n g e s t cr ack fr ontRun 175 - more refined me shRun 175 - coarser me shSTP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thic
49、k-Walled Cylinder Vessels 20 Figure 6-6: Comparison Coarser Mesh for the Longest Axial Internal Crack, Case 175 The mesh refinement results indicate that sufficient crack and cylinder mesh refinement was used in this analysis to obtain accurate crack front J-integral and stress intensity K results. STP-PT-071: Stress Intensity Factor Solutions for Internal Cracks in Thick-Walled Cylinder Vessels 21 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 internal
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