Review of the Stress Analysis of the MiniBooNE Horn MH1

1. Review of the Stress Analysis of the MiniBooNE Horn MH1 Larry Bartoszek, P.E. 1/20/00 BARTOSZEK ENGINEERING

2. Overview 1 • The MiniBooNE horn carries 170 kiloamps of current in a pulse 143 microseconds long. • The pulse repeats 10 times in a row, 1/15 sec between each pulse, then the horn is off until 2 seconds from the first pulse in train. • The horn is stressed by differential thermal expansion and magnetic forces. We need to design it to survive 200 million cycles with >95% confidence.

3. Overview 2 • Motivation for design lifetime: • The horn will eventually become very radioactive and require a complicated handling procedure in the event of a replacement. We don’t want to make many of these objects. • We can’t afford to make many horns. • The major design issue is fatigue. • Every component around the horn needs to survive 200 million pulses to get overall system reliability.

4. Analysis Outline 1: Fatigue theory • Discussion of fatigue in Al 6061-T6 • Presentation of data sources • Discussion of effects that modify maximum stress in fatigue • Discussion of scatter in the maximum stress data in fatigue • Discussion of multiaxial stress in fatigue

5. Analysis Outline 2: Allowable stress • Determination of allowable stress in fatigue • Perform a statistical analysis on the MIL-SPEC data to get confidence curves for a sample set of fatigue tests. • This yields a starting point for maximum stress that needs to be corrected for environmental factors. • The allowable is then compared with the calculated stress from the FEA results.

6. Analysis Outline 3: Calculated Stress • Description of the finite element model • FEA results on MH1 and the calculated stresses • Assumptions • Thermal analysis • Magnetic force analysis • Combined forces transient analysis • Results for stress ratio R and maximum calculated stress in horn

7. Sources of Fatigue Data for AL 6061-T6 used in analysis • MIL-SPEC Handbook #5, Metallic Materials and Elements for Aerospace Vehicles • ASM Metals Handbook Desk Edition • ASM Handbook Vol. 19, Fatigue and Fracture • “Aluminum and Aluminum Alloys”, pub. by ASM • “Atlas of Fatigue Curves”, pub. by ASM • “Fatigue Design of Aluminum Components and Structures”, Sharp, Nordmark and Menzemer

8. How well do sources agree? • For unwelded, smooth specimens, R=-1, room temperature, in air, N=5*107 • MIL-SPEC smax=13 ksi (89.6 MPa) • Atlas of Fatigue Curves smax=17 ksi (117.1 MPa) • Fatigue Design of Al… smax=16 ksi (110.2 MPa) • Metals Handbook (N=5*108) smax=14 ksi (96.5 MPa) • These numbers represent 50% probability of failure at 5*107 cycles.

9. Effects that lower fatigue strength, 1 • Geometry influences fatigue: • Tests are done on “smooth” specimens and “notched” specimens • Smooth specimens have no discontinuities in shape • Notched specimens have a standard shaped discontinuity to create a stress riser in the material • Notches reduce fatigue strength by ~1/2 • see graph on next slide

10. ASM data showing effect of notches on fatigue strength Graph from Atlas of Fatigue Curves

11. Effects that lower fatigue strength, 2 • Welding influences fatigue: • Welded and unwelded specimens are tested • Welding reduces fatigue strength by ~1/2 • see graph on next slide

12. ASM data showing effect of welding Graph from Atlas of Fatigue Curves

13. Effects that lower fatigue strength, 3 • The stress ratio influences fatigue strength: • Stress Ratio, R, is defined as the ratio of the minimum to maximum stress. • Tension is positive, compression is negative • R=Smin/Smax varies from -1£R£1 • R = -1 Þ (alternating stress) smax=16 ksi • R = 0 Þ (Smin=0) smax=24 ksi, (1.5X at R=-1) • R = .5 Þ smax=37 ksi, (2.3X at R=-1) • These values are for N=107 cycles, 50% confidence • Stress ratio is a variable modifier to maximum stress. Whole stress cycle must be known.

14. MIL-SPEC Data Showing Effect of R This is the page from the MIL-SPEC handbook that was used for the statistical analysis of the scatter in fatigue test data. The analytical model assumes that all test data regardless of R can be plotted as a straight line on a log-log plot after all the data points are corrected for R. The biggest problem with this data presentation style is that the trend lines represent 50% confidence at a given life and we need >95% confidence of ability to reach 200 x 106 cycles.

15. Effects that lower fatigue strength, 4 • Moisture reduces fatigue strength • For R = -1, smooth specimens, ambient temperature: • N=108 cycles in river water, smax= 6 ksi • N=107 cycles in sea water, smax~ 6 ksi • Hard to interpret this data point • N=5*107 cycles in air, smax= 17 ksi • See data source on next slide • Note curve of fatigue crack growth rate in humid air, second slide

16. ASM data on corrosion fatigue strength of many Al alloys Graph from “Atlas of Fatigue Curves” showing that the corrosion fatigue strength of aluminum alloys is almost constant across all commercially available alloys, independent of yield strength. Data from this graph was used to determine the moisture correction factor.

17. ASM data on effect of moisture on fatigue crack growth rate Graph from “Atlas of Fatigue Curves” This graph is for a different alloy than we are using, but the assumption is that moisture probably increases the fatigue crack growth rate for 6061 also. It was considered prudent to correct the maximum stress for moisture based on this curve and the preceding one.

18. Discussion of scatter in the maximum stress data in fatigue • The MIL-SPEC data is a population of 55 test specimens that shows the extent of scatter in the test results. • Trend lines in the original graph indicate 50% chance of part failure at the given stress and life. • The source gave a method of plotting all the points on the same curve when corrected for R. • We used statistical analysis to create confidence curves on this sample set.

19. Confidence Curves on Equivalent Stress data plot This graph plots all of the MIL-SPEC data points corrected for R by the equation at bottom. The y axis is number of cycles to failure, the x axis is equivalent stress in ksi. From this graph we concluded that the equivalent stress for >97.5% confidence at 2e8 cycles was 10 ksi.

20. Discussion of multiaxial stress in fatigue • Maximum stress in fatigue is always presented as result of uniaxial stress tests • Horn stresses are multiaxial. • We assumed that we could sensibly compare the uniaxial stress allowable with calculated multiaxial combined stresses • FEA provided stress intensities and principal normal stresses that were converted to combined stress • See next slide for combined stress expression

21. Expression for combined stress • Maximum Distortion Energy Theory provides an expression for comparing combined principal normal triaxial stresses to yield stress in uniaxial tension • We assumed this expression was valid comparing combined stress with uniaxial fatigue maximum stress limit • Sallow³ {[(S1-S2)2+(S2-S3)2+(S3-S1)2]/2}.5

22. Allowable Stress Determination 1 • Allowable stress starts as the equivalent stress for 97.5% confidence that material will not fail in 2e8 cycles • Seq = 10 ksi (68.9 MPa) • Allowable stress is then corrected by multiplicative factors, as described in Shigley’s “Mechanical Engineering Design” • Sallow = Seq*fR*fmoisture*fweld

23. Calculation of stress ratio correction factor: • First correction is for R,stress ratio • We determined that the minimum stress was thermal stress alone after the horn cooled between pulses just before the next pulse. • Maximum stress happened at time in cycle when magnetic forces and temperature were peaked simultaneously • R was calculated by taking the ratio in every horn element in the FEA of the maximum principal normal stresses at these two points in time • Results not significantly different for ratios of combined stress • Smax = Seq/(1-R).63 therefore: fR = 1 /(1-R).63

24. Finding the moisture correction factor: • Determining the fatigue strength moisture correction factor: • At R = -1, N = 108 in river water, smax = 6 ksi • At R = -1, N = 5*108 in air, smax = 14 ksi • 6 ksi/14 ksi = .43 • Moisture effect could be .43 smax in air • We used this number, and assumed that all of the aluminum was exposed to moisture

25. Other Correction Factors • From data above, • Welding correction factor, fweld = .5 • Welding correction factor only applied to welded areas • We assumed that there were no notches anywhere. • This is fair for the inner conductor • Stresses are so low on the outer conductor that it doesn’t matter • We did not include a size correction to go from sample size to horn size.

26. Description of the finite element model • We created a 2D axisymmetric model of the horn and first did a transient thermal analysis • We assumed 3000 W/m2-K convective heat transfer coefficient all along the inner conductor only • The only heat transfer from the outer conductor was by conduction to inner • The skin depth of the current was explicitly modeled (all heat was generated within 1.7mm of surface of conductors)

27. Plot of temperature of smallest radius of inner conductor vs time

28. High Temperature profile in cross-section The beam axis in the model is a vertical line (not shown) just to the left of the shape in the figure

29. Conclusions from thermal analysis • Temperature difference between hot end of pulse and cool end are not that different. • Heating of the inner conductor elongates it and pushes end cap along beam axis, putting itself in compression and the end cap in bending • There are only two areas of the horn that see significant stress • Middle of the end cap • Welded region immediately upstream of end cap

30. Magnetic Force FEA • Magnetic forces were modeled in the 2D axisymmetric model as element pressures using an analytical expression for the pressure as a function of radius in the horn. • This model was verified by a 3D 10° sector model of the horn. • We needed to model the magnetic forces in the 2D model to be able to combine thermal and magnetic stress effects.

31. Stress intensity caused by high temperature + magnetic force loads on horn end cap Stress units above are Pascals.

32. Conclusions from magnetic + thermal analysis • The magnetic field creates a pressure normal to the surface the current is flowing through • The magnetic field pressure is non-linear and maximum at small radii. • Stress ratio in the welded neck is ~ -.16 (low temperature thermal stress is small compression) • Stress ratio in the end cap varies from -.3 near beam axis to .5 at middle

33. Results of Finite Element Analysis • The following plot is a graph of the ratio of calculated principal normal stress to allowable stress for every element in the horn axisymmetric model • Stresses have not been combined in this graph • Values are maximum of S1 and S3 only • Allowable stress has been derated for moisture and welding everywhere

34. Summary graph for uncombined principal normal stresses

35. Combined Stress Results • The following graph presents the same results, but the principal normal stresses have been combined by the equation shown above • Allowable stress corrected for moisture everywhere, but welding only where appropriate in horn • The places where the ratio is >1 are welded areas that we have since thickened as a result of this analysis • Any stress value over 20% of allowable is in the inner conductor smallest radius tube section

36. Summary graph for combined stress data

37. Conclusion • After correcting the thickness of the welded region upstream of the end cap, the graphs indicate that the stress level everywhere in the horn during pulsing is below the maximum set by the 97.5% confidence level that the material will not fail in 2e8 cycles.