Goodman Fatigue Safety Factor Calculator — Fatigue Calculation & Factor of Safety Formula
Calculate the Modified Goodman fatigue safety factor from maximum/minimum or mean/alternating stress, with endurance-strength, target safety-factor and optional first-cycle yield checks.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Stress cycle definition
Maximum / minimum stress
Mean / alternating stress
Material properties
Fatigue strength basis
Design target
Results
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Method, application and limitations
Review the calculation method, intended application and engineering assumptions before using the result in a design decision.
Formula and calculation method
The calculator evaluates fatigue factor of safety using the Modified Goodman criterion. It combines alternating stress, mean stress, corrected endurance strength and ultimate tensile strength into a dimensionless Goodman safety factor.
Modified Goodman fatigue factor of safety formula
nG = 1 / [σa / Se + σm,G / Sut]
where:
- nG = Modified Goodman fatigue safety factor [-],
- σa = alternating stress amplitude,
- σm,G = mean stress used by the Goodman calculation,
- Se = corrected component endurance limit or fatigue strength at the reference life,
- Sut = ultimate tensile strength.
All stress and strength terms must use consistent units. The safety factor is dimensionless.
Mean and alternating stress
When maximum and minimum stresses are entered, the calculator converts them to mean and alternating stress:
σm = (σmax + σmin) / 2
σa = (σmax − σmin) / 2
The reverse relationships are:
σmax = σm + σa
σmin = σm − σa
The stress ratio reported by the calculator is:
R = σmin / σmax
when σmax is non-zero.
Treatment of compressive mean stress
This implementation deliberately does not take beneficial Goodman credit for compressive mean stress. The value used in the Goodman equation is:
σm,G = max(σm, 0)
Therefore, for σm ≤ 0, the fatigue calculation reduces to the alternating-stress contribution:
nG = Se / σa
for a non-zero alternating stress.
Endurance-strength basis
The preferred input is a corrected component endurance limit or fatigue strength appropriate to the actual design condition.
If the preliminary steel-estimate mode is selected, the calculator uses:
Se = min(0.5 Sut, 700 MPa)
This is only a preliminary specimen-level estimate. The calculator does not apply separate surface, size, loading, temperature, reliability or notch correction factors to this estimate.
Required fatigue safety factor and utilization
The entered required safety factor nreq is compared directly with the calculated Goodman factor:
nG ≥ nreq
The corresponding Goodman design utilization reported by the calculator is:
UG = nreq [σa / Se + σm,G / Sut]
Therefore, UG ≤ 1 corresponds to meeting the requested Goodman fatigue target.
Allowable alternating stress
For the entered mean stress and required safety factor, the Goodman-based allowable alternating stress is:
σa,allow,G = max{0, Se[1 / nreq − σm,G / Sut]}
Optional first-cycle yield check
If yield strength is supplied, the calculator also checks the largest absolute stress reached during the cycle:
σpeak = max(|σmax|, |σmin|)
ny = Sy / σpeak
where Sy is yield strength.
The yield-based allowable alternating stress at the current mean stress is:
σa,allow,y = max{0, Sy / nreq − |σm|}
The calculator reports the smaller allowable amplitude:
σa,allow = min(σa,allow,G, σa,allow,y)
If yield strength is not supplied, only the Goodman fatigue criterion is used to determine this allowable alternating stress.
The alternating-stress margin is:
Margin = σa,allow − σa
When to use this calculator
Use this calculator for a fatigue calculation when a component is subjected to repeated or fluctuating uniaxial stress and the mean-stress effect is to be evaluated with the Modified Goodman criterion.
It is particularly useful when the stress cycle is known either as maximum and minimum stress or directly as mean and alternating stress.
Typical applications
- preliminary fatigue screening of shafts, rods, brackets and other machine components under repeated normal stress,
- checking the effect of tensile mean stress on allowable alternating stress,
- calculating a Goodman fatigue factor of safety,
- comparing the calculated fatigue safety factor with a required design factor,
- determining the allowable alternating stress for the current mean stress,
- checking first-cycle yielding when yield strength is available,
- classifying a stress cycle as fully reversed, tension-tension, compression-compression or tension-compression loading.
Required information
For a normal fatigue calculation, provide:
- either σmax and σmin, or σm and σa,
- ultimate tensile strength Sut,
- a corrected endurance limit or fatigue strength Se, unless the preliminary steel estimate is selected.
Yield strength Sy is optional. When it is provided, the calculator also performs a first-cycle yield check. The required safety factor is optional and defaults to 1.0.
When not to use this calculator as the only fatigue analysis
This calculator does not predict the number of cycles to failure and is not an S-N fatigue-life solver. It also does not calculate cumulative damage from variable-amplitude load histories.
A more detailed method may be required for multiaxial loading, complex stress histories, low-cycle fatigue, plastic strain, weld fatigue, contact fatigue, crack-growth analysis, strongly temperature-dependent behavior or components governed by a specific design standard.
Stress concentrations, residual stresses, surface condition, size, reliability, temperature and other fatigue modifiers must already be represented appropriately in the stress or corrected fatigue-strength data supplied to the calculator.
How to interpret the result
The primary result nG is the Modified Goodman fatigue safety factor. It measures the proportional margin between the evaluated stress state and the Goodman fatigue boundary used by this calculator.
Goodman fatigue safety factor
- nG > nreq — the evaluated Goodman fatigue criterion satisfies the requested safety-factor target.
- nG = nreq — the stress state is at the requested Goodman design boundary.
- nG < nreq — the requested Goodman fatigue target is not satisfied.
If no required factor is entered, the calculator uses nreq = 1.0.
Design utilization
The reported design utilization includes the required safety factor:
- utilization < 1 — the Goodman target is satisfied,
- utilization = 1 — the Goodman target is exactly reached,
- utilization > 1 — the requested Goodman target is exceeded.
Increasing alternating stress increases utilization directly. Increasing tensile mean stress also increases utilization because it reduces the available fatigue margin.
Allowable alternating stress and fatigue margin
The allowable alternating stress is the maximum stress amplitude permitted by the active Goodman and, when available, first-cycle yield checks for the entered mean stress and required factor.
- positive margin — the current alternating stress remains below the calculated allowable value,
- zero margin — the current alternating stress lies on the controlling target boundary,
- negative margin — the current alternating stress exceeds the calculated allowable value.
First-cycle yield factor
If yield strength is entered, the calculator evaluates the largest absolute stress at either end of the cycle. The yield criterion is satisfied when:
ny ≥ nreq
A fatigue result can therefore meet the Goodman target while the first-cycle yield target does not. In that case, the overall result is reported as not meeting the target because yielding governs.
If yield strength is omitted, first-cycle yielding is not verified and the calculator reports this limitation.
Mean stress effect
A positive tensile mean stress reduces the allowable alternating stress under the Modified Goodman relation. For negative mean stress, this implementation takes no beneficial fatigue credit and uses zero mean stress in the Goodman term.
Fully reversed and static cases
For fully reversed loading, σm ≈ 0 and the Goodman calculation reduces to an endurance-strength comparison.
If σa ≈ 0, the stress state is classified as static or non-cyclic. The calculator does not interpret this as an infinite-fatigue-life result because no fatigue cycle is present. If yield strength is supplied, the static/first-cycle strength check remains relevant.
What a passing result means
A passing status means that the evaluated Modified Goodman criterion and, when available, the first-cycle yield criterion satisfy the entered target. It does not establish complete component safety.
Additional checks may still be required for stress concentration, multiaxial stress, finite-life fatigue, variable-amplitude damage, buckling, fracture, wear, contact loading, thermal effects or other applicable failure modes.
Calculation example
Example: Goodman fatigue safety factor from maximum and minimum stress
A component experiences a repeated tensile stress varying between 20 MPa and 220 MPa. The corrected component endurance limit is 280 MPa, the ultimate tensile strength is 700 MPa, the yield strength is 450 MPa and the required safety factor is 1.5.
Input data
- σmax = 220 MPa
- σmin = 20 MPa
- Se = 280 MPa
- Sut = 700 MPa
- Sy = 450 MPa
- nreq = 1.5
1. Calculate mean stress
σm = (σmax + σmin) / 2
σm = (220 + 20) / 2 = 120 MPa
2. Calculate alternating stress
σa = (σmax − σmin) / 2
σa = (220 − 20) / 2 = 100 MPa
The stress ratio is:
R = 20 / 220 = 0.091
Because both stress extrema are positive, this is tension-tension fluctuating loading.
3. Calculate the Goodman factor of safety
The mean stress is tensile, so σm,G = 120 MPa.
nG = 1 / [σa / Se + σm / Sut]
nG = 1 / [100 / 280 + 120 / 700]
nG = 1 / (0.3571 + 0.1714)
nG = 1.892
4. Compare with the required safety factor
1.892 > 1.5
The Modified Goodman fatigue criterion satisfies the requested safety-factor target.
The corresponding design utilization is:
UG = 1.5 × (0.3571 + 0.1714) = 0.793
5. Check first-cycle yielding
The maximum absolute stress in the cycle is 220 MPa:
ny = Sy / σpeak
ny = 450 / 220 = 2.045
Because 2.045 > 1.5, the first-cycle yield target is also satisfied.
6. Calculate allowable alternating stress
The Goodman-based allowable amplitude is:
σa,allow,G = 280 × [1 / 1.5 − 120 / 700]
σa,allow,G = 138.67 MPa
The yield-based allowable amplitude is:
σa,allow,y = 450 / 1.5 − 120
σa,allow,y = 180 MPa
The controlling allowable alternating stress is therefore:
σa,allow = min(138.67, 180) = 138.67 MPa
The alternating-stress margin is:
138.67 − 100 = 38.67 MPa
Interpretation
For these inputs, the requested factor of safety of 1.5 is satisfied for both the Modified Goodman fatigue check and the first-cycle yield check. The Goodman fatigue criterion governs because its calculated safety factor is lower.
This result is a constant-amplitude uniaxial fatigue screening result. It does not predict cycles to failure and does not account automatically for variable-amplitude damage, stress concentrations or other component-specific fatigue effects unless they are already reflected in the entered stresses and corrected endurance strength.
Assumptions and limitations
- The fatigue calculation uses the Modified Goodman linear mean-stress criterion.
- The evaluated loading is treated as uniaxial and constant-amplitude.
- The method is intended primarily for predominantly elastic high-cycle fatigue screening; it is not a strain-life or low-cycle fatigue model.
- The alternating stress amplitude is non-negative and is defined as half of the maximum-to-minimum stress range.
- Maximum stress must be greater than or equal to minimum stress when the maximum/minimum input mode is used.
- Positive mean stress is treated as tensile and reduces the fatigue margin.
- Negative mean stress receives no beneficial Goodman credit in this implementation. The Goodman mean-stress term is limited to max(σm, 0).
- The preferred Se input is a corrected component endurance limit or fatigue strength appropriate to the actual material, geometry, surface condition, loading mode, temperature, reliability and design environment.
- The optional steel estimate Se = min(0.5Sut, 700 MPa) is a preliminary specimen-level estimate only. Separate Marin-type correction factors are not calculated by this tool.
- If yield strength is supplied, first-cycle yielding is checked using the largest absolute stress at the two cycle extrema.
- If yield strength is not supplied, the calculator does not verify first-cycle yielding.
- The calculator does not automatically calculate notch sensitivity, theoretical or fatigue stress-concentration factors. Their effects must be incorporated in the stress or fatigue-strength inputs when applicable.
- The calculator does not perform multiaxial fatigue reduction or calculate equivalent alternating and mean stresses from combined normal and shear loading.
- The calculator does not calculate S-N curve life, cycles to failure, Miner cumulative damage or variable-amplitude spectrum damage.
- Residual stress, corrosion, fretting, surface treatments, manufacturing effects and environmental degradation are not modeled separately.
- A zero alternating stress is treated as a static or non-cyclic stress state and is not interpreted as proof of infinite fatigue life.
- The calculated safety factor applies only to the evaluated Goodman and optional first-cycle yield criteria. Other failure mechanisms may require separate verification.
- Final design limits, material fatigue data and required safety factors should be selected according to the applicable design requirements, validated material data and governing engineering standards.
Frequently asked questions
The calculator uses the Modified Goodman fatigue factor of safety formula:
nG = 1 / [σa/Se + σm,G/Sut].
Here σa is alternating stress, Se is corrected endurance strength, Sut is ultimate tensile strength and σm,G is the tensile mean-stress term used by the calculator.
First determine mean stress and alternating stress. For maximum and minimum stresses, use σm = (σmax + σmin)/2 and σa = (σmax − σmin)/2. Then divide σa by the corrected endurance strength, add the tensile mean-stress term divided by ultimate tensile strength, and take the reciprocal.
The Goodman factor of safety is a dimensionless margin obtained from a linear interaction between alternating stress and tensile mean stress. A larger value represents a larger proportional margin to the Modified Goodman fatigue boundary. In this calculator, the result is compared with the required safety factor entered by the user.
There is no universal fatigue safety factor that is appropriate for every component. The required value depends on uncertainty, material data, loading variability, consequence of failure, manufacturing quality, inspection strategy and the governing design requirements. This calculator therefore lets you enter nreq; if it is omitted, the computational default is 1.0.
Mean stress is σm = (σmax + σmin)/2. Alternating stress amplitude is σa = (σmax − σmin)/2. The calculator can also accept σm and σa directly and reconstruct σmax and σmin.
A positive tensile mean stress increases the Goodman interaction term and therefore reduces the calculated fatigue safety factor and allowable alternating stress. This is why two cycles with the same stress amplitude can produce different Goodman results when their mean stresses differ.
The calculator does not take beneficial fatigue credit for compressive mean stress. When σm is negative, the Goodman calculation uses σm,G = 0. The original signed mean stress is still retained for stress-cycle reporting and the optional yield check.
Use a corrected component endurance limit or fatigue strength that represents the actual material and design condition whenever reliable data are available. Surface finish, component size, loading mode, temperature, reliability, notch effects and environment can materially change fatigue strength and should be reflected in the selected value when applicable.
In the steel-estimate mode, the calculator uses Se = min(0.5Sut, 700 MPa). This is a preliminary uncorrected specimen estimate and should not be treated as a fully corrected endurance limit for a finished component.
No. It calculates a Modified Goodman fatigue safety factor and related allowable-stress quantities. It does not solve an S-N curve for cycles to failure and does not perform cumulative-damage calculations for variable-amplitude loading.
A stress cycle can satisfy the Goodman fatigue relation while still reaching a stress high enough to cause first-cycle yielding. When Sy is supplied, the calculator therefore computes ny = Sy/max(|σmax|, |σmin|) and requires that check to meet the selected target as well.
For fully reversed loading, σm ≈ 0 and the Goodman relation reduces to an endurance-strength check based on σa/Se. For static loading, σa ≈ 0; the calculator identifies the case as non-cyclic and does not make an infinite-fatigue-life claim.