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 gt95
  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 dont want to
  make many of these objects.
• We cant 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.

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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

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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, N5107
• MIL-SPEC smax13 ksi (89.6 MPa)
• Atlas of Fatigue Curves smax17 ksi (117.1 MPa)
• Fatigue Design of Al smax16 ksi (110.2 MPa)
• Metals Handbook (N5108) smax14 ksi (96.5 MPa)
• These numbers represent 50 probability of
  failure at 5107 cycles.

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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
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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

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ASM data showing effect of welding
Graph from Atlas of Fatigue Curves
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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
• RSmin/Smax varies from -1R1
• R -1 Þ (alternating stress) smax16 ksi
• R 0 Þ (Smin0) smax24 ksi, (1.5X at R-1)
• R .5 Þ smax37 ksi, (2.3X at R-1)
• These values are for N107 cycles, 50 confidence
• Stress ratio is a variable modifier to maximum
  stress. Whole stress cycle must be known.

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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 gt95 confidence of ability to reach 200 x
106 cycles.
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Effects that lower fatigue strength, 4

• Moisture reduces fatigue strength
• For R -1, smooth specimens, ambient
  temperature
• N108 cycles in river water, smax 6 ksi
• N107 cycles in sea water, smax 6 ksi
• Hard to interpret this data point
• N5107 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.
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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.

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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 gt97.5
confidence at 2e8 cycles was 10 ksi.
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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

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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
• Sallow ³ (S1-S2)2(S2-S3)2(S3-S1)2/2.5

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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 Shigleys
  Mechanical Engineering Design
• Sallow SeqfRfmoisturefweld

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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

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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 5108 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

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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 doesnt matter
• We did not include a size correction to go from
  sample size to horn size.

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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)

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Plot of temperature of smallest radius of inner
conductor vs time
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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
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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

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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.

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

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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
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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 gt1 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

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Summary graph for combined stress data
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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.