Mean-Stress Fatigue Corrections (structures/fatigue/goodman-diagram)
Use when a fluctuating stress cycle is superimposed on a steady mean
stress and the infinite-life margin must be quantified: compute the
allowable stress amplitude per modified Goodman, Gerber, and Soderberg,
place the design point on the Haigh diagram, and decide whether the
applied amplitude is acceptable.
Domain quick reference
- Mean stress Sm is the steady component of the cycle,
Sm = (Smax + Smin) / 2; stress amplitude Sa is the oscillating
component, Sa = (Smax - Smin) / 2; stress ratio R = Smin / Smax.
- The endurance limit Se is the fully reversed (R = -1) stress
amplitude the material survives for infinite life.
- Modified Goodman line: allowable amplitude
Sa = Se * (1 - Sm / Sut), a straight line from (0, Se) to (Sut, 0).
- Gerber parabola: Sa = Se * (1 - (Sm / Sut)^2), above the Goodman
line, less conservative.
- Soderberg line: Sa = Se * (1 - Sm / Sy), below the Goodman line,
most conservative, safe against yield at the mean stress.
- The Haigh diagram plots allowable amplitude against mean stress for
each criterion; a design point (Sm, Sa) above the line fails the
infinite-life check for that criterion.
- Infinite-life check: the applied amplitude must not exceed the
allowable amplitude, Sa <= Sa_allowable.
- FAR-25 and CS-25 set the certification context for fatigue
substantiation of transport airplane structure; the mean-stress
correction itself is standard mechanical engineering methodology.
Workflow
- Gather material allowables: endurance limit Se, ultimate strength
Sut, yield strength Sy, all in the same stress unit.
- Resolve the load case into mean stress Sm and applied amplitude Sa
(from Smax and Smin if the cycle extrema are given; R = Smin / Smax
confirms the cycle type).
- Compute the allowable amplitude for each criterion:
Goodman Se * (1 - Sm / Sut), Gerber Se * (1 - (Sm / Sut)^2),
Soderberg Se * (1 - Sm / Sy).
- Compare the applied amplitude against each allowable; any
exceedance fails the infinite-life check under that criterion.
- Plot the design point and the three lines on the Haigh diagram to
show which criterion governs and the margin.
- Report the verdict and the governing criterion.
Pitfalls
- Mixing stress units (MPa vs ksi) between Se, Sut, Sy, Sm, Sa.
- Using a mean stress above the ultimate strength: the Goodman
allowable goes non-positive and no positive amplitude is acceptable.
- Forgetting that the endurance limit assumes fully reversed loading;
any nonzero mean stress reduces the allowable amplitude.
- Treating Soderberg as an alternative to yield checks: it bounds
amplitude, it does not replace a static yield check at the peak
stress Sm + Sa.
- Applying these corrections to compressive mean stress regimes
without noting the conservative assumption; the formulas here target
tensile mean stress.
- Reporting the Gerber or Goodman verdict without stating which
criterion it came from; the three lines disagree by design.
Behavior contract (gate 3)
The mean-stress correction logic is exercised by the gate 3 contract
test: scripts/test_goodman.py against scripts/goodman_logic.py (stdlib
unittest, offline). Run:
python3 scripts/test_goodman.py
Compliance
- FAR-25 and CS-25 are cited as reference-only certification context
(compliance: STANDARDS-REF, gated: false); no text is quoted from
either regulation.
1---2name: goodman-diagram3description: Use when a fluctuating load case must be checked against a mean-stress fatigue limit or a Haigh diagram / fatigue diagram comparison is needed for a structure. Determine the allowable stress amplitude for infinite life under mean stress: compute the modified Goodman, Gerber, and Soderberg allowable amplitudes from the endurance limit, ultimate strength, and yield strength, plot the Haigh diagram with the design point, and give the infinite-life verdict when the applied amplitude exceeds the allowable. Trigger: goodman, gerber, soderberg, haigh diagram, fatigue diagram, mean stress, endurance limit, allowable amplitude, infinite life, stress ratio.4license: Apache-2.05---67# Mean-Stress Fatigue Corrections (structures/fatigue/goodman-diagram)89Use when a fluctuating stress cycle is superimposed on a steady mean10stress and the infinite-life margin must be quantified: compute the11allowable stress amplitude per modified Goodman, Gerber, and Soderberg,12place the design point on the Haigh diagram, and decide whether the13applied amplitude is acceptable.1415## Domain quick reference1617- Mean stress Sm is the steady component of the cycle,18 Sm = (Smax + Smin) / 2; stress amplitude Sa is the oscillating19 component, Sa = (Smax - Smin) / 2; stress ratio R = Smin / Smax.20- The endurance limit Se is the fully reversed (R = -1) stress21 amplitude the material survives for infinite life.22- Modified Goodman line: allowable amplitude23 Sa = Se * (1 - Sm / Sut), a straight line from (0, Se) to (Sut, 0).24- Gerber parabola: Sa = Se * (1 - (Sm / Sut)^2), above the Goodman25 line, less conservative.26- Soderberg line: Sa = Se * (1 - Sm / Sy), below the Goodman line,27 most conservative, safe against yield at the mean stress.28- The Haigh diagram plots allowable amplitude against mean stress for29 each criterion; a design point (Sm, Sa) above the line fails the30 infinite-life check for that criterion.31- Infinite-life check: the applied amplitude must not exceed the32 allowable amplitude, Sa <= Sa_allowable.33- FAR-25 and CS-25 set the certification context for fatigue34 substantiation of transport airplane structure; the mean-stress35 correction itself is standard mechanical engineering methodology.3637## Workflow38391. Gather material allowables: endurance limit Se, ultimate strength40 Sut, yield strength Sy, all in the same stress unit.412. Resolve the load case into mean stress Sm and applied amplitude Sa42 (from Smax and Smin if the cycle extrema are given; R = Smin / Smax43 confirms the cycle type).443. Compute the allowable amplitude for each criterion:45 Goodman Se * (1 - Sm / Sut), Gerber Se * (1 - (Sm / Sut)^2),46 Soderberg Se * (1 - Sm / Sy).474. Compare the applied amplitude against each allowable; any48 exceedance fails the infinite-life check under that criterion.495. Plot the design point and the three lines on the Haigh diagram to50 show which criterion governs and the margin.516. Report the verdict and the governing criterion.5253## Pitfalls5455- Mixing stress units (MPa vs ksi) between Se, Sut, Sy, Sm, Sa.56- Using a mean stress above the ultimate strength: the Goodman57 allowable goes non-positive and no positive amplitude is acceptable.58- Forgetting that the endurance limit assumes fully reversed loading;59 any nonzero mean stress reduces the allowable amplitude.60- Treating Soderberg as an alternative to yield checks: it bounds61 amplitude, it does not replace a static yield check at the peak62 stress Sm + Sa.63- Applying these corrections to compressive mean stress regimes64 without noting the conservative assumption; the formulas here target65 tensile mean stress.66- Reporting the Gerber or Goodman verdict without stating which67 criterion it came from; the three lines disagree by design.6869## Behavior contract (gate 3)7071The mean-stress correction logic is exercised by the gate 3 contract72test: scripts/test_goodman.py against scripts/goodman_logic.py (stdlib73unittest, offline). Run:74python3 scripts/test_goodman.py7576## Compliance7778- FAR-25 and CS-25 are cited as reference-only certification context79 (compliance: STANDARDS-REF, gated: false); no text is quoted from80 either regulation.
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