Fatigue Life Calculator – S-N Curve, Goodman Diagram & Marin Factor Analysis

Q: What is the difference between Goodman, Gerber, and Soderberg?

Goodman uses a straight line from $ Se $ on the alternating axis to $ Su $ on the mean axis — moderately conservative. Gerber uses a parabola to the same intercept — less conservative and closer to experimental data for ductile steels. Soderberg uses yield strength instead of ultimate — the most conservative. Most machine design textbooks recommend modified Goodman.

Q: Does the endurance limit exist for all materials?

Steels and titanium alloys exhibit a distinct knee in the S-N curve near $ 10^6 $ cycles (endurance limit). Aluminum, copper, and most non-ferrous alloys do not — their S-N curves continue to decline, and a fatigue strength at a specified life (e.g., $ 5 \times 10^8 $ cycles) is used instead.

Q: How do I handle variable-amplitude loading?

For varying stress amplitudes, Miner's linear damage rule sums cycle ratios: $ \sum ni/Ni \leq 1.0 $. This is a first-order approximation; load sequence effects and small-cycle thresholds are not captured.

Q: When should I use strain-life instead of stress-life?

Strain-life (Coffin-Manson) is appropriate for low-cycle fatigue (below $ 10^3 $ cycles) where significant plastic deformation occurs. The stress-life approach in this module applies to high-cycle fatigue ($ 10^3 $ to $ 10^7+ $ cycles) where stresses remain nominally elastic.

Q: Can I use this module for weld fatigue?

Weld fatigue follows different S-N curves classified by joint detail category (BS 7608, EN 1993-1-9). The Marin factor approach does not apply to welds. Use code-specific fatigue detail categories for welded joints.

Check	IND	US	EU	ISO
Modified Goodman utilization	implemented	—	—	—
Estimated fatigue life	implemented	—	—	—

How engineers analyze fatigue life

Fatigue is the most common cause of mechanical failure — responsible for an estimated 80–90 % of all structural and machine component failures. Unlike static overload, fatigue failure occurs at stress levels well below the material's yield strength through the gradual accumulation of micro-damage over millions of load cycles.

The design process centers on the S-N curve, which relates the applied stress amplitude to the number of cycles to failure. For steels, a distinct endurance limit exists near $1 0^{6}$ cycles: stress amplitudes below this level can theoretically be sustained indefinitely. But the raw endurance limit from a polished laboratory specimen must be corrected for real-world conditions — surface finish, component size, loading type, temperature, and reliability — using Marin modification factors.

When the component also sees a steady (mean) stress in addition to the alternating component, the allowable alternating stress decreases. The Goodman, Gerber, and Morrow diagrams provide different mean-stress correction models.

Types and configurations

Loading type	Stress pattern	Typical component
Rotating bending	Fully reversed ( $R = - 1$ )	Shafts, axles
Axial (push-pull)	Various $R$ -ratios	Connecting rods, bolts
Torsion	Reversed or pulsating shear	Drive shafts, springs
Combined	Multiaxial alternating + mean	Crankshafts, gear teeth

The module handles uniaxial fatigue with user-specified alternating and mean stress components. Multiaxial fatigue requires equivalent stress approaches (von Mises for proportional loading) before entry.

Engineering workflow

Determine loading — Identify the alternating stress amplitude $S_{a}$ and mean stress $S_{m}$ at the critical location. For rotating bending, $S_{a}$ equals the bending stress and $S_{m} = 0$ .
Get material data — Ultimate tensile strength $S_{u}$ , and either the measured endurance limit or the estimate $S_{e}^{'} \approx 0.5 S_{u}$ (for $S_{u} \leq 1400$ MPa steels).
Apply Marin factors — Surface finish $k_{a}$ , size $k_{b}$ , load type $k_{c}$ , temperature $k_{d}$ , and reliability $k_{e}$ to get the modified endurance limit $S_{e}$ .
Select mean-stress method — Goodman (linear, moderately conservative), Gerber (parabolic, less conservative), or Morrow (uses true fracture strength).
Check infinite life — If $S_{a} \leq S_{e}$ after mean-stress correction, the component has infinite life at the specified reliability.
Estimate finite life — If $S_{a} > S_{e}$ , use the Basquin equation to predict cycles to failure between $1 0^{3}$ and $1 0^{6}$ .

Key quantities and formulas

Modified Goodman criterion

\frac{S _{a}}{S _{e}} + \frac{S _{m}}{S _{u}} = \frac{1}{S F}

This is the most widely used mean-stress correction for steel machine components.

Marin endurance limit

S_{e} = k_{a} k_{b} k_{c} S_{e}^{'}

where $k_{a} = a S_{u}^{b}$ (surface finish, from Shigley Table 6-2), $k_{b}$ depends on the characteristic dimension $d$ , and $k_{c}$ depends on load type (1.0 bending, 0.85 axial, 0.59 torsion).

Basquin finite-life equation

S_{f} = a N^{b}, a = \frac{( f S _{u} ) ^{2}}{S _{e}}, b = - \frac{1}{3} lo g_{10} (\frac{f S _{u}}{S _{e}})

where $f \approx 0.8 - 0.9$ is the fatigue strength fraction at $1 0^{3}$ cycles.

Gerber parabola (alternative)

\frac{S _{a}}{S _{e}} + (\frac{S _{m}}{S _{u}})^{2} = 1

Worked example

Problem: A machined AISI 1040 steel shaft ( $S_{u} = 590$ MPa) of 30 mm diameter experiences rotating bending with $S_{a} = 180$ MPa and steady torsion giving $S_{m} = 50$ MPa (von Mises equivalent).

Uncorrected endurance limit: $S_{e}^{'} = 0.5 \times 590 = 295$ MPa.
Surface factor (machined): $k_{a} = 4.51 \times 59 0^{- 0.265} = 0.832$ .
Size factor (30 mm): $k_{b} = 1.24 \times 3 0^{- 0.107} = 0.862$ .
Load factor (bending): $k_{c} = 1.0$ .
Modified endurance limit: $S_{e} = 0.832 \times 0.862 \times 1.0 \times 295 = 211$ MPa.
Goodman check: $180/211 + 50/590 = 0.853 + 0.085 = 0.938$ . Safety factor: $1/0.938 = 1.07$ — marginal, may need diameter increase.
Finite life estimate: $a = (0.9 \times 590)^{2} /211 = 1494$ , $b = - \frac{1}{3} lo g_{10} (531/211) = - 0.133$ . $N = (180/1494)^{1/ - 0.133} = 3.4 \times 1 0^{5}$ cycles — finite but adequate for many applications.

Common mistakes and checks

Using uncorrected endurance limit — The textbook $S_{e}^{'} = 0.5 S_{u}$ applies only to a polished 7.5 mm rotating-bending specimen. Real components require all Marin corrections; omitting surface finish alone can overpredict life by an order of magnitude.
Ignoring mean stress — Preloaded bolts, pressurized components, and rotating shafts under gravity all have nonzero mean stress. Even a modest mean stress significantly reduces the allowable alternating stress.
Wrong load factor $k_{c}$ — Using bending factor for an axial loading case overstates the endurance limit by 18 %. Identify the actual loading type at the critical location.
Extrapolating beyond $1 0^{6}$ cycles — The Basquin equation is valid between $1 0^{3}$ and $1 0^{6}$ cycles. Beyond $1 0^{6}$ , the S-N curve flattens at the endurance limit for steels (but not for aluminum or other non-ferrous alloys).
Neglecting notch sensitivity — Applying the full theoretical $K_{t}$ to fatigue calculations is conservative. The fatigue concentration factor is $K_{f} = 1 + q (K_{t} - 1)$ , where notch sensitivity $q < 1$ for ductile materials at mild notches.

FAQ

What is the difference between Goodman, Gerber, and Soderberg?

Goodman uses a straight line from $S_{e}$ on the alternating axis to $S_{u}$ on the mean axis — moderately conservative. Gerber uses a parabola to the same intercept — less conservative and closer to experimental data for ductile steels. Soderberg uses yield strength instead of ultimate — the most conservative. Most machine design textbooks recommend modified Goodman.

Does the endurance limit exist for all materials?

Steels and titanium alloys exhibit a distinct knee in the S-N curve near $1 0^{6}$ cycles (endurance limit). Aluminum, copper, and most non-ferrous alloys do not — their S-N curves continue to decline, and a fatigue strength at a specified life (e.g., $5 \times 1 0^{8}$ cycles) is used instead.

How do I handle variable-amplitude loading?

For varying stress amplitudes, Miner's linear damage rule sums cycle ratios: $\sum n_{i} / N_{i} \leq 1.0$ . This is a first-order approximation; load sequence effects and small-cycle thresholds are not captured.

When should I use strain-life instead of stress-life?

Strain-life (Coffin-Manson) is appropriate for low-cycle fatigue (below $1 0^{3}$ cycles) where significant plastic deformation occurs. The stress-life approach in this module applies to high-cycle fatigue ( $1 0^{3}$ to $1 0^{7} +$ cycles) where stresses remain nominally elastic.

Can I use this module for weld fatigue?

Weld fatigue follows different S-N curves classified by joint detail category (BS 7608, EN 1993-1-9). The Marin factor approach does not apply to welds. Use code-specific fatigue detail categories for welded joints.

Use the PhyCalcPro calculator

Estimate fatigue life and mean-stress-adjusted endurance in the Fatigue Assessment Calculator.

Purpose

Estimate fatigue life and mean-stress-adjusted allowable alternating stress using S-N curves, Marin modification factors, and Goodman, Gerber, or Morrow mean-stress corrections. Supports rotating bending, axial, and torsion load types.

Physics & theory

Fatigue failure occurs below yield after many stress cycles. The S-N curve relates alternating stress amplitude $S_{a}$ to life $N$ . Endurance limit $S_{e}^{'}$ at $1 0^{6}$ cycles is modified by Marin factors: surface finish $k_{a}$ , size $k_{b}$ , load type $k_{c}$ , giving $S_{e} = k_{a} k_{b} k_{c} S_{e}^{'}$ .

Mean stress $S_{m}$ reduces allowable alternating stress. Modified Goodman: $S_{a} / S_{e} + S_{m} / S_{u} = 1$ . Gerber uses parabolic mean-stress locus; Morrow uses true fracture strength. Basquin log-linear relation between $1 0^{3}$ and $1 0^{6}$ cycles predicts finite life: $S_{f} = a N^{b}$ .

Governing equations

\frac{S _{a}}{S _{e}} + \frac{S _{m}}{S _{u}} = 1 (Modified Goodman)

S_{e} = k_{a} k_{b} k_{c} S_{e}^{'}, k_{a} = a S_{u}^{b} (surface finish)

S_{f} = a N^{b}, a = \frac{( f S _{u} ) ^{2}}{S _{e}}, b = - \frac{1}{3} lo g_{10} (\frac{f S _{u}}{S _{e}})

Numerical method

Closed-form Marin factors (Shigley Table 6-2), mean-stress correction, and Basquin life prediction (engine). Surface finish, size, load type, and method selectable. Infinite life flagged when $S_{a} \leq S_{e}$ after mean-stress correction.

Inputs

Parameter	Description
`alternatingStress`, `meanStress`	$S_{a}$ , $S_{m}$
`ultimateStrength`, `enduranceLimit`	Material fatigue data
`surfaceFinish`, `loadType`	Marin factors
`characteristicDiameter`	Size factor (rotating bending)
`meanStressMethod`	goodman, gerber, or morrow

Outputs

Modified endurance limit, allowable alternating stress, predicted cycles to failure, infinite-life flag
Marin factor breakdown

Design codes & checks

Indicative: Modified Goodman utilization, estimated fatigue life
ISO: ISO 12107 fatigue of metallic materials
US: ASME VIII-2 fatigue screening (reference)

Assumptions & limitations

Uniaxial stress state; multiaxial fatigue needs equivalent stress approaches.
No notch sensitivity $K_{f}$ unless user adjusts endurance limit.
Constant amplitude loading; variable amplitude needs Miner's rule extension.
No environmental corrosion-fatigue interaction.

Verification

CI: fatigue-indicative-01.json
Engineer sign-off: validation-master-checklist.md

References

Shigley, J. E., & Budynas, R. G. Mechanical Engineering Design, 11th ed., Ch. 6.
Dowling, N. E. Mechanical Behavior of Materials, 5th ed.
ISO 12107:2012. Metallic materials — Fatigue testing — Statistical planning.
Peterson, R. E. Stress Concentration Factors, 4th ed.
Bannantine, J. A., Comer, J. J., & Handrock, J. L. Fundamentals of Metal Fatigue Analysis.

Fatigue Life Assessment — S-N Curves, Goodman & Marin Factors Guide

Assumptions

Limitations

Engineering checks