examination of non-intuitive stress intensity …...examination of non-intuitive stress intensity...

EXAMINATION OF NON-INTUITIVE STRESS INTENSITY SOLUTION TRENDS FOR THICK-WALL CYLINDER INTERNAL CRACKS FROM ASME STP-PT-072

G. Thorwald, Ph.D. (Quest Integrity Group, USA)

Abstract Accurate crack front stress intensity K solutions are needed for failure assessment and fatigue crack growth analysis, especially for a commonly used geometry like a cylinder for vessels and pipes. One way to compute K solutions is by using finite element analysis (FEA) with highly refined three-dimensional crack meshes for each geometry case. While computing K solutions for internal cracks in thick-wall cylinders, two types of non-intuitive result trends were observed and investigated to confirm that the results were correct. This project for ASME computed stress intensity K solutions for 744 crack cases across a range of cylinder radius, thickness and crack sizes for internal axial and internal circumferential cracks. Linear elastic finite element analyses of three-dimensional crack meshes were used to compute the crack front K solution values. The crack shapes used included semi-elliptical surface cracks, full-width partial-depth axial cracks, and 360o partial-depth circumferential cracks. The full-width and 360 o cracks provide a limiting K solution for very long surface cracks. A crack K solution is a function of a non-dimensional geometry factor, the applied loading, and the crack size. The crack results were tabulated by saving just the non-dimensional geometry factor value, G, for each case. For later use in a crack assessment, the K solution can be recovered using the K equation to combine the G factor with a load value and crack size. Plots of the G values versus crack size ratios were created to check that the computed results followed expected trends, as the cylinder and crack dimensions changed, as one of the ways to ensure accurate results. For example, as the crack size increases the K value is expected to also increase and the G factor would intuitively also be expected to increase. However, two instances of non-intuitive G result trends were observed. The first type of non-intuitive result trend was for the G results at the crack tip with a uniform loading that did not follow the expected increasing trend for some crack size cases. The second type of non-intuitive G results trend was for circumferential cracks with out-of-plane bending loading, where the G results trend did not consistently increase for all crack sizes. Detailed examination of the K and G results revealed the causes of the non-intuitive trends and that

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the K and G results were correct. One cause was mathematical and the other cause was geometric. This paper will review the project parameters and three-dimensional crack meshes used to compute the K and G solutions and will explain the non-intuitive result trends. These new G solutions will extend the G factor solutions available in the 2007 API 579-1/ASME FFS-1 Annex C tables to include the thick-wall cylinder cases. Extending the K solutions will enable fatigue crack growth and fracture assessment of a larger range of cracked cylinders. This project was funded by ASME PTCS through ASME ST-LLC. The G solution values are tabulated and reported in the ASME publication STP-PT-072. 1. Introduction This paper describes a part of the results from ASME report STP-PT-072 (Thorwald, 2014), which was sponsored by ASME Pressure Technology Codes & Standards and ASME Standards Technology, LLC (ASME ST-LLC). My colleagues contributing to this project include Lucie Parietti, Joyce Wright, and Bruno Fletcher. Accurate crack front stress intensity K values are needed to evaluate the stability of crack like flaws in a fracture analysis, and are needed to compute crack growth rates in a fatigue analysis. Crack front K values were computed for 744 crack cases using 3D crack meshes, generated by the FEACrack™ software (Quest, 2014). The finite element analysis (FEA) input files were run using the Abaqus™ solver (Dassault, 2014), which also computes the crack front J-integral values for several contours at each crack front node. The crack front results were post-processed using FEACrack to compute K from the J-integral values. The crack front K solutions are saved as non-dimensional geometry factors, G versus the crack front position. Trend plots of the G results versus geometry ratios were created as one of the ways to confirm accurate solutions. Some of the result plots gave unexpected trends. These non-intuitive G result trends are examined and explained in this paper. The crack analysis cases include internal axial and circumferential cracks. Surface crack, partial depth 360o circumferential crack, and partial depth full-length axial crack shapes were examined. The crack geometry cases are described by three sets of ratios: t/Ri, c/a, and a/t, where t is the cylinder thickness, Ri is the cylinder inside radius, a is the crack depth, and c is the half crack length (2c is the full surface crack length). The ratio values examined were: t/Ri = 1.0, 1.5, 2.0, 2.5, 3.0,



a/c = 0.03125, 0.0625, 0.125, 0.25, 0.5, 1.0, 2.0, and a/t = 0.2, 0.4, 0.6, 0.8. As the t/Ri ratio increases the cylinder becomes thicker compared to the radius. The a/c ratio varies from long cracks to short cracks. The a/t ratio varies from shallow to deep cracks. The t/Ri = 1.0 ratio matches existing crack results in the API 579/ASME-FFS Annex C tables (American, 2007) as a way to overlap this project with the previous K solutions and check the new K results. An example of a surface crack mesh is shown in Figure 1. The internal axial surface crack meshes use quarter symmetry to model one crack face and half of the total surface crack length, c. The green “mesh zone 1” labelled in the figure is used to adjust the crack mesh refinement near the crack plane, and the red “mesh zone 2” labelled in the figure is used to adjust the mesh refinement away from the crack. The geometry ratios for this crack mesh are: t/Ri = 2, a/c = 0.5, a/t = 0.6. Figure 2 shows a close up of the focused mesh region around the crack front. The focused mesh pattern with concentric rings of elements around the crack front and collapsed brick elements at the crack front is necessary to compute the J-integral. Several rings of elements are typically used so that several J contour values can be computed at each crack front node. Comparing the J contour values (the J path dependence) is a way to confirm sufficient mesh refinement and a valid result. Arranging the radial mesh lines perpendicular to the crack front also allows K to be computed from the crack face opening displacement as another way to check the results, which will be discussed later.



Figure 1: Internal axial surface crack mesh, quarter symmetric.

Figure 2: Close-up of the focused mesh region around the crack front.



Figure 3 shows another axial surface crack mesh case for a longer and shallower crack size: t/Ri = 1, a/c = 0.125, a/t = 0.2. Figure 4 shows the longest and deepest surface crack case for t/Ri = 2.5, a/c = 0.03125, a/t = 0.8.

Figure 3: Example of a longer and shallower surface crack mesh.

Figure 4: Example of the longest and deepest surface crack size.

To provide an upper bound solution for very long surface cracks, the partial-depth full-length axial surface crack shape is used, where the crack depth is kept constant along the length of the cylinder. An example of the partial-depth crack shape is shown in Figure 5 for t/Ri = 1, a/t = 0.2



Figure 5: Example of the partial-depth full-length surface crack mesh.

Figure 6 shows an example of the half symmetric internal circumferential surface crack used for the out-of-plane bending loading cases, where the bending is applied about the vertical y-axis. The crack mesh shown is for t/Ri = 3, a/c = 1, a/t = 0.4. To provide an upper bound solution for long circumferential cracks, the partial-depth 360o crack shape is used, where the crack depth is constant around the cylinder circumference. The example shown in Figure 7 is for t/Ri = 3, a/t = 0.8.



Figure 6: Example of an internal circumferential surface crack mesh, half symmetric mesh; out-of-plane bending applied about the vertical y-axis.

Figure 7: Example of a partial-depth 360o crack mesh.



2. Loading and Geometry Factors Crack face pressure distributions and global bending stresses were applied to the crack meshes as the loading cases to compute the crack front K values. Since the analysis was linear elastic, the K solutions could be combined by superposition to model various through-thickness and bending stress loading combinations. The global bending stresses were applied across the end of the cylinder for the circumferential crack cases, which could be combined with the uniform crack face pressure K solutions. The general form of the stress intensity K equation is shown in Equation 1 (Anderson, 2005), where σ is an applied load like crack face pressure or bending stress, G is the non-dimensional geometry factor, a is the crack depth, and the crack shape factor Q is a function of the crack depth, a, and half crack length, c. The a/c ratio in Equation 2 is changed to c/a when a > c for shorter deeper surface cracks.

�� = �� (1)

= 1 + 1.464 ��.��

� ≤ � (2)

After K is computed from the crack mesh analysis, the input values are used to obtain the non-dimensional G value to save as the result. The K value can be recovered later by providing specific crack sizes and a load value. For this project the load σ is one of the crack face pressure distributions or global bending stress. The two diagrams in Figure 8 show the uniform crack face pressure, p0, and the linear crack face pressure, p1, distribution that varies along the crack depth, a, direction. The stress intensity Equations 3 and 4 show the crack face pressure substituted for the general applied stress term. The non-dimensional results G0 and G1 are the values saved as the final crack front results for each crack face pressure distribution.



Uniform crack face pressure Linear crack face pressure

�� = �� (3) �� = ��

(4)

Figure 8: Crack face pressure loading distributions.

3. Results A typical set of non-dimensional G results is shown in Figure 9 for an axial internal surface crack (t/Ri = 1, a/c = 0.03125); each curve has a different a/t ratio. The upper four curves (solid lines) are for the uniform crack face pressure loading, and the lower four curves (dashed lines) are for the linear crack face pressure results. The plot x-axis gives the crack front position and starts at the crack tip location and ends at the crack depth location for the symmetric mesh. These typical results show the usual expected trend of increasing crack size resulting in increasing G result values. To ensure accurate crack front results were computed, several quality assurance approaches were used to examine the results. The t/Ri =1 ratio overlaps the existing results to allow comparison to the new results to ensure continuity with the new results. Mesh refinement convergence studies examined and confirmed that adequate mesh refinement was used in the crack meshes. The number of contours used to compute the crack front J-integral was examined, along with the contour path dependence. An independent calculation of the crack front stress intensity was done using the crack face opening displacements (Anderson) (Thorwald, 2014). This allowed comparison between the K results computed from the crack front J-integral and K from crack opening displacement, which showed excellent agreement.



Figure 9: Typical non-dimensional crack front G results for 8 cases.

Figure 10: Crack tip location G result trends showing unexpected G0 trend.

a/t=0.2 G0 trend (red curve) above the other a/t trends, examine these non-intuitive results

G0

at c

rack

tip

a/c ratio

G0 uniform crack face pressure, t/Ri = 3

Shorter cracksLonger cracks



Another way to examine sets of results is to plot the crack tip or crack depth position results versus the a/c ratio with a curve for each a/t ratio. Figure 10 shows the uniform crack face pressure G0 results at the crack tip location (t/Ri = 3). This type of plot compares the result trends for 28 cases: 4 a/t curves, 7 a/c ratios. The a/t = 0.2 curve (red line) has G values above the other curves at a/c = 1 and 2 (shorter cracks). Why does the a/t = 0.2 curve not follow the usual expected increasing trend as the crack size increases? Recall from Equation 1 that the non-dimensional G value is computed from the K solution and the input values of the loading and crack size. To examine the G trend, the chart in Figure 11 shows three curves versus the a/t ratio for the a/c = 1 values: the G values from Figure 10 at a/c = 1, the square-root term values, and computed K values. The solid blue curve shows the G values at a/c = 1 where the G value at a/t = 0.2 is slightly higher than the G value at a/t = 0.4 or 0.6, as seen in Figure 10, and the G value does not change much across the a/t ratio values. The square-root term, shown by the dashed red curve, is computed from the crack sizes, so it is not a constant. In this case the magnitude of the square-root term varies more than the G values. When the K values are computed using the G and square-root values its trend increases with the a/t ratio as expected. Similar curves comparing the a/c = 2 values from Figure 10 are shown in Figure 12. Again the square-root term varies in magnitude more than the G values. Computing K values from G and the square-root term confirms the increasing trend for increasing crack depth. The G and square-root values were also compared for cases where G did increase as expected and found that usually the G and square-root terms have similar magnitude. If the square-root term varies in magnitude more than the G values, then the G value trend plots can give non-intuitive trends that do not show the expected increase in value with increasing crack size, such as increasing crack depth. Since the square-root term in the K equation is not constant, the non-dimensional G value should be considered part of the K solution, but not necessarily relied on to give intuitively increasing trends in all cases. If there is a question about a G result trend, K should be computed using G and the square-root term for definitive comparison of the crack result trend.



Figure 11: Crack tip location G, square-root term, and K result trends.

Figure 12: Crack tip location G, square-root, and K result trends.

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8 1

G,

sqrt

te

rm,

K v

alu

es

crack front position, 2*phi/pi

Comparisons for G, sqrt, K terms, a/c=1, t/R=3

G=A0, a/c=1 sqrt(pi*a/Q) K

a/t ratio

G=A0, a/c=1 K, ksi √in

Val

ues:

G, s

quar

e-ro

ot te

rm, c

ompu

ted

K



Figure 13 shows an example of the usual increasing G trend as the crack depth a/t ratio increases (t/Ri = 1, at a/c = 0.5). How do the G and square-root magnitudes compare in this case? Figure 14 shows that the G and square-root values have similar magnitude and change a similar amount as the a/t ratio increases, which is in contrast to the previous non-intuitive cases where the magnitude of the square-root term changed more than G. As expected, the K computed from those values clearly increases as the crack depth increases. The other non-intuitive G result trend was observed for the out-of-plane bending load case, for the half symmetric circumferential surface crack. Figure 15 shows the G result trend plot versus the a/c ratio, and shows the half symmetric cylinder crack mesh cross-section (t/Ri = 1, a/c = 0.5, a/t = 0.6). The out-of-plane bending was applied about the vertical y-axis so that the left side of the cylinder axial stress is in compression and the right side of the cylinder axial stress is in tension. The crack tip G results are all negative in the chart for the crack tip on the left side of the cylinder in the compressive axial stress region. Two of the G result curves have a local dip in their trends (a/t = 0.2 red curve, and a/t = 0.6 green curve). Since applying only a bending load would put the left side of the cylinder into compression and cause crack face closure, a combined bending plus axial load was used to keep the entire crack face in tension to obtain the combined load K results. For the linear elastic analysis, superposition was used to subtract the axial load only K results from the combined load case to get the negative K results for the bending only load shown in the plot. Why does the G trend decrease and then increase again as the crack length a/c ratio changes? Examining where the crack tip is located around the cylinder circumference revealed that the local dip in the G values occurs when the crack tip is near the +90o position on the left side of the cylinder, which is where the axial stress magnitude is largest due to bending. Since all the G values on the left side of the cylinder are negative, the local decrease in the plot indicates an increase in absolute magnitude of the G value when the crack tip is near the largest axial stress. Shorter or longer crack lengths have the crack tip located above or below the +90o location and will have a smaller magnitude G value. The circumferential crack geometry and particular crack lengths are the cause of this non-intuitive trend.



Figure 13: Example of the usual increasing G trend as a/t increases.

Figure 14: The G and square-root terms have similar magnitude.

0

1

2

3

4

5

6

7

8

0 0.2 0.4 0.6 0.8 1

G,

sqrt

te

rm,

K v

alu

es

crack front position, 2*phi/pi

Comparisons for G, sqrt, K terms, a/c=0.5, t/R=1

G=A0, a/c=1 sqrt(pi*a/Q) K

a/t ratioG=A0, a/c=0.5 K, ksi √in

Val

ues:

G, s

quar

e-ro

ot te

rm, c

ompu

ted

K



Figure 15: Out-of-plane bending load G result trends at the crack tip.

4. Summary In this project crack front solutions were computed for 744 crack cases in thick-wall cylinders. The non-dimensional G results are available in the ASME publication STP-PT-072 and eventually will be in API 579/ASME FFS. The crack meshes, crack shape, and geometry cases were reviewed. The quality assurance methods used to ensure accurate crack results were briefly reviewed. One of the methods to check results relied on trend plots of the G results versus crack size ratios as a way to compare many cases in the same plot. Two types of non-intuitive result trends were observed in the results and examined and explained. The first cause of a non-intuitive G result trend was mathematical in the stress intensity K equation terms used to compute the G result values. The square-root term in the K equation is not constant and its magnitude may change more than the G values, which leads to the non-intuitive G result trend in the comparison plot. The second cause of a non-intuitive G result trend for the out-of-plane bending load case was geometric. When the circumferential surface crack tip is located near the side of the cylinder where the magnitude of axial stress is largest, the G value has a local change in the overall result trend. For longer or shorter crack lengths, the crack tip is located above or below the side of the cylinder in a lower stress region, which gives a lower magnitude G value.



The non-dimensional G result trends are useful for checking the crack results, but should not be relied on to always give an intuitively increasing trend as the crack size increases. When in doubt compute the K value to check the stress intensity results trend. 5. References

[1] Thorwald, G., Parietti, L., Fletcher, B., Wright, J. (June 20, 2014). STP-PT-072, Development Of Stress Intensity Factor Solutions For Internal Cracks In Thick-Walled Cylinder Vessels For ASME RFP-ASME ST-14-04: ASME Standards Technology, LLC, Two Park Ave., New York, NY 10016-5990, ISBN No. 978-0-7918-6957-4.

[2] Quest Integrity Group (2014), FEACrack software, version 3.2.025: http://www.questintegrity.com/products/feacrack-3D-crack-mesh-software

[3] Dassault Systemes SIMULIA Corp. (2014). Abaqus, version 6.13: SIMULIA, Rising Sun Mills, 166 Valley Street, Providence, RI 02909-2499, USA

[4] American Petroleum Institute and The American Society of Mechanical Engineers (June 5, 2007). API 579-1/ASME FFS-1, Annex C Table C.10 through Table C.14: API Publishing Services, 1220 L Street, N.W., Washington, D.C. 20005

[5] Anderson, T. L. (2005) Fracture Mechanics, Fundamentals and Applications, 3rd ed.: CRC Press, Taylor & Francis Group, p. 49. Anderson, pp. 44-

[6] Thorwald, G. (2014). Crack Front Stress Intensity Validation Using Two Methods for a Crack at a Material Boundary in a Nozzle Component: SIMULIA Community Conference, May 20-22, 2014, Providence, RI.



examination of non-intuitive stress intensity …...examination of non-intuitive stress intensity...

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