Fatigue Calculator (Steel • Copper • Bolt • Shaft • Rope) — S‑N Curve Life Estimator
This fatigue life calculator estimates component life using the Basquin (S-N) model with Goodman, Gerber, or Soderberg mean stress corrections. Enter your material properties, stress state, and Marin modification factors — surface finish, size, reliability, temperature, and load type — to get corrected endurance limit (Sₑ), fatigue life (Nf), and safety factor. Miner's rule support enables variable-amplitude damage accumulation across multiple load blocks.
Steel & Metal Fatigue Life Calculator (S‑N Curve Predictor)
Steel · Copper · Bolt · Shaft · Rope wire · S‑N curve plot — complete predictor with Miner’s rule & visual S‑N (Excel/CSV + PDF/print)
Accuracy note: this is an engineering estimator. Always validate with your organization’s standard, test data, and applicable codes/standards before design approval.
Inputs (Material + Stress + Factors)
Mode: SteelSteel fatigue calculator (stress-life / S‑N)
Preset materials + endurance limitSteel defaults: \( S'_e \approx 0.5 S_u \) (rule-of-thumb), then corrected using Marin factors. Use your test data when available.
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Fatigue Calculator
S‑N / Basquin Method — Complete User Guide
Step-by-step instructions, all formulas with LaTeX notation, input validation tips, and accuracy notes for mechanical engineers, students, and designers.
📋 Table of Contents
- Overview & Purpose
- Quick-Start (5-Step Workflow)
- Units System & Conversion
- Material Inputs & Basquin Parameters
- Stress State Definitions
- Marin Modification Factors
- Notch & Stress Concentration
- Mean Stress Correction Methods
- Life Estimation — Basquin Equation
- Miner's Rule — Cumulative Damage
- Safety Factor Computation
- Bolt Fatigue Helper
- Shift / HSE Fatigue Score
- Common Input Mistakes
- Accuracy & Disclaimer
Overview & Purpose of the Calculator
This tool implements the classical Shigley S‑N (stress-life) fatigue analysis methodology, combining the Basquin power-law curve with Marin correction factors, mean-stress interaction diagrams (Goodman, Gerber, Soderberg), and Miner's linear cumulative damage rule.
It is intended for:
- Mechanical engineering students completing fatigue coursework or projects
- Designers performing preliminary life estimates on rotating shafts, fasteners, springs, and wire rope
- Engineers comparing failure theories and mean-stress corrections on the same data
- HSE / Occupational Health professionals scoring human fatigue risk during shift work
Quick-Start: 5-Step Workflow
Choose Units & Material
Select Metric (MPa / mm) or Imperial (ksi / in) from the units dropdown. Then pick a material preset — the tool pre-fills UTS, yield strength, and Basquin coefficients. Override any value with your own test data.
Enter Stress State
Choose one of three input modes: σ_max & σ_min, amplitude & mean, or R‑ratio & σ_max. The calculator automatically derives the alternating stress σ_a and mean stress σ_m from whichever pair you provide.
Set Marin Modification Factors
Select surface finish, enter diameter (for size factor), choose reliability level, temperature, and load type. Optionally enter K_t (stress concentration) and notch sensitivity q.
Choose Mean-Stress Criterion
Select Goodman (conservative, most common), Gerber (less conservative), or Soderberg (most conservative). This affects the equivalent alternating stress used for life prediction.
Click Calculate & Review
Hit Calculate. Review σ_a, σ_m, K_f, S_e (corrected endurance limit), N_f (fatigue life in cycles), time estimate, and safety factor. Use the S‑N Plot tab to visualise the operating point on the Basquin curve.
Units System & Conversion Factors
All internal computation runs in MPa and mm. Inputs are converted before calculation; outputs are converted back for display.
| Quantity | Metric Unit | Imperial Unit | Conversion (Metric → Imperial) |
|---|---|---|---|
| Stress | MPa | ksi | 1 ksi = 6.8948 MPa |
| Length / Diameter | mm | in | 1 in = 25.4 mm |
| Temperature | °C | °F | °F = (°C × 9/5) + 32 |
| Force (Bolt) | N | lbf | 1 lbf = 4.4482 N |
| Area (Bolt) | mm² | in² | 1 in² = 645.16 mm² |
Material Inputs & Basquin Parameters
Required Material Properties
| Symbol | Name | Typical Range (MPa) | Notes |
|---|---|---|---|
| S_u | Ultimate Tensile Strength (UTS) | 200 – 2 000 | Obtained from tensile test or material datasheet. Required for mean-stress corrections. |
| S_y | Yield Strength | 170 – 1 800 | Used by Soderberg criterion and static yield check. |
| σ′f | Fatigue Strength Coefficient (Basquin) | ≈ 1.1–1.5 × S_u | Y-intercept of the Basquin log-log line at 2N = 1. |
| b | Basquin Exponent | −0.05 to −0.12 | Must be negative. Common default: −0.085 to −0.095 for steels. |
Built-in Material Presets
| Preset Name | S_u (MPa) | S_y (MPa) | σ′_f (MPa) | b |
|---|---|---|---|---|
| Steel 1018 | 440 | 370 | 660 | −0.095 |
| Steel A36 | 550 | 250 | 825 | −0.090 |
| Steel 4140 Q&T | 950 | 800 | 1 400 | −0.085 |
| Stainless 316 | 620 | 290 | 900 | −0.095 |
| Copper C110 | 220 | 70 | 320 | −0.110 |
| Copper C122 | 240 | 80 | 350 | −0.110 |
| Brass C360 | 350 | 200 | 520 | −0.105 |
Stress State Definitions & Input Modes
Fundamental Stress Quantities
All three input modes ultimately compute the same two fundamental quantities:
| Mode | Inputs Required | Derivation | Best Used When… |
|---|---|---|---|
| σ_max / σ_min | Max & min stress | Direct formula above | Load cycle limits are known from FEA or measurement |
| σ_a / σ_m | Amplitude & mean | σ_max = σ_m + σ_a; σ_min = σ_m − σ_a | Haigh diagram analysis or test data in amplitude form |
| R / σ_max | Stress ratio & peak stress | σ_min = R × σ_max | Test machine settings (e.g., R = 0.1 for tensile-dominated cycles) |
Common R‑Ratio Reference Points
| R Value | Cycle Type | Physical Meaning |
|---|---|---|
| R = −1 | Fully reversed | Equal tension and compression; σ_m = 0 |
| R = 0 | Zero-to-max | One-sided tensile pulsing; σ_m = σ_a |
| R = 0.1 | Near zero-to-max | Standard ASTM fatigue test ratio |
| 0 < R < 1 | Fluctuating tension | Mean tensile stress present |
| R > 1 or R < −1 | Compressive cycling | Negative mean stress — increases life |
Marin Modification Factors — Corrected Endurance Limit
The corrected endurance limit \(S_e\) accounts for real-world conditions that reduce the laboratory-measured endurance limit \(S_e'\):
Unmodified Endurance Limit \( S_e' \)
For steels, the laboratory specimen endurance limit is estimated from UTS:
Marin Modification Factors — Visual Summary
Polished (best) → As-forged (worst)
Larger diameter → lower kb
50 % (1.0) → 99.9 % (0.753)
Linear drop above 25 °C, floor at 300 °C
Enter manually; default = 1.0
Torsion (0.59) · Axial (0.85) · Bending (1.0)
Surface Factor \( k_a \) — Marin Power-Law
| Surface Finish | a | b | Typical k_a (S_u = 800 MPa) |
|---|---|---|---|
| Polished | 1.58 | −0.085 | ≈ 0.96 |
| Ground | 1.34 | −0.085 | ≈ 0.82 |
| Machined / Cold-drawn | 4.51 | −0.265 | ≈ 0.72 |
| Hot-rolled | 57.7 | −0.718 | ≈ 0.45 |
| As-forged | 272 | −0.995 | ≈ 0.34 |
Size Factor \( k_b \)
Leave the diameter field blank (or 0) if the size factor is not applicable — the tool sets \(k_b = 1.0\) in that case.
Reliability Factor \( k_c \)
| Reliability | k_c | Application context |
|---|---|---|
| 50 % | 1.000 | Mean value; no reliability margin |
| 90 % | 0.897 | General engineering design (default) |
| 95 % | 0.868 | Structural / critical components |
| 99 % | 0.814 | Aerospace / medical / safety-critical |
| 99.9 % | 0.753 | Extreme reliability requirements |
Temperature Factor \( k_d \)
Below 25 °C: \(k_d = 1.0\). Above 300 °C: \(k_d = 0.7\) (clamped). For cryogenic or very high-temperature applications, consult material-specific data.
Notch & Stress Concentration Factor \( K_f \)
Geometric discontinuities (holes, fillets, keyways, grooves) elevate local stress. The fatigue stress concentration factor \(K_f\) accounts for this through the notch sensitivity \(q\):
Where:
- \(K_t\) = theoretical stress concentration factor (from charts, FEA, or Peterson's tables)
- \(q\) = notch sensitivity (0 = notch insensitive; 1 = fully sensitive to the notch)
| Scenario | K_t | q | K_f Result |
|---|---|---|---|
| No notch / smooth shaft | 1.0 | any | 1.0 (no amplification) |
| Small fillet, ductile steel | 2.0 | 0.8 | 1.8 |
| Sharp notch, brittle material | 3.5 | 1.0 | 3.5 |
| Moderate hole, cast iron | 2.5 | 0.2 | 1.3 |
The notch-corrected alternating stress fed into the life calculation is:
Mean Stress Correction Methods — Haigh Diagram
A tensile mean stress reduces fatigue life. The three available criteria plot differently on a Haigh (modified Goodman) diagram:
Haigh Diagram (Modified Goodman)
✓ Safe zone
Design point below and left of the criterion line. Component expected to survive fatigue loading.
✗ Unsafe zone
Design point above or right of the criterion line. Fatigue failure is predicted within the design life.
Note: Se = corrected endurance limit; Sy = yield strength; Su = ultimate tensile strength. Compressive mean stress (σm < 0) is generally beneficial and not plotted here.
Modified Goodman Criterion
Gerber Criterion
Soderberg Criterion
These expressions rearrange to give an equivalent fully-reversed alternating stress for life prediction:
Life Estimation — The Basquin Equation (S‑N Model)
The Basquin (power-law) S‑N relationship describes the connection between applied alternating stress amplitude and the number of cycles to failure:
Solving for \(N_f\) (cycles to failure):
where \(\sigma_{a,\text{eq}}\) is the mean-stress-corrected equivalent alternating stress (from Section 08), and \(\sigma_f'\), \(b\) are the Basquin material constants.
Time-to-Failure Conversion
If you enter an operating frequency, the tool converts \(N_f\) to a time estimate:
Life Region Classification
| Condition | Region Label | Design Guidance |
|---|---|---|
| \(\sigma_{a,\text{eq}} \leq S_e\) (Steel) | Below S_e — possible infinite-life zone | Life may be theoretically unlimited; verify with real scatter and factor of safety |
| \(\sigma_{a,\text{eq}} > S_e\) (Steel) | Above S_e — finite-life | Finite life predicted by Basquin; inspect or replace at computed interval |
| Any stress (Non-ferrous) | Long-life region (estimate) or Finite-life | No true endurance limit; use life estimate conservatively |
Miner's Rule — Linear Cumulative Damage
When a component experiences multiple blocks of loading at different stress levels, Miner's rule sums the damage fractions:
σ_max = 280 MPa
n₁ = 50 000 cycles
Damage: n₁/N₁
σ_max = 200 MPa
n₂ = 300 000 cycles
Damage: n₂/N₂
D < 1 → Safe
D ≥ 1 → Failure
predicted
Each block uses the same material properties, Marin factors, and mean-stress correction as the main calculation. \(N_{f,i}\) is the Basquin life for the equivalent alternating stress of block \(i\).
How to Add Miner Blocks
- Click "Add Load Block" in the main tab.
- Enter σ_max, σ_min (in your selected unit), and the number of applied cycles \(n_i\) for that block.
- Repeat for each loading level. Damage D updates after each block entry.
- Click Calculate to refresh the main result and damage simultaneously.
Safety Factor Computation
The factor of safety \(n_f\) measures how far the operating stress state is from the failure boundary under the chosen criterion:
| n_f Value | Interpretation | Typical Application |
|---|---|---|
| < 1.0 | Failure predicted — component is unsafe | Requires immediate redesign |
| 1.0 – 1.5 | Marginal — very limited margin | Not recommended for general use |
| 1.5 – 2.0 | Acceptable margin for well-controlled conditions | Automotive, consumer machinery |
| 2.0 – 3.0 | Good design margin | Structural, industrial equipment |
| > 3.0 | High margin — may be over-designed or conservative inputs | Safety-critical, initial sizing |
Bolt Fatigue Helper
For threaded fasteners under fluctuating axial loads, the bolt tab converts applied forces to stresses using the tensile stress area \(A_s\):
Click "Apply to Main Calculator" to transfer these stresses. The tool sets the stress input mode to amplitude & mean automatically.
Shift / HSE Fatigue Score
The Shift tab provides a simple indicative human fatigue score for occupational health and safety purposes. It is not an official FAID, FRMS, or HSE-certified tool.
Fatigue Score Model
Where the five sub-scores are:
| Variable | Definition | Formula |
|---|---|---|
| S | Sleep deficit | (8 − sleep hours) / 8 |
| W | Excessive wakefulness | (hours awake − 16) / 8 |
| L | Long shift penalty | (shift length − 10) / 6 |
| N | Night shift binary | 1 if night shift, 0 otherwise |
| Q | Workload stress | (workload_rating − 1) / 2 |
| B | Breaks bonus | min(break_minutes / 120, 0.2) |
| Score Range | Band | Recommended Action |
|---|---|---|
| 0 – 24 | Low | Proceed normally with standard monitoring |
| 25 – 49 | Moderate | Increase supervisor check-ins; schedule breaks |
| 50 – 74 | High | Consider task reallocation; avoid safety-critical solo work |
| 75 – 100 | Severe | Do not assign safety-critical tasks; require adequate rest before next shift |
Common Input Mistakes & How to Fix Them
Positive Basquin exponent b
Entering b = +0.09 instead of b = −0.09. The tool rejects positive values with an error.
✅ Fix
Always enter b as a negative number, e.g. −0.085. Values between −0.05 and −0.12 are physically realistic.
Switching units mid-session
Entering stresses in MPa, then switching to imperial — displayed values are not auto-converted.
✅ Fix
Set your unit system before entering any values. If you switch, re-select the material preset to restore correct values.
Zero or blank diameter with size factor applied
Leaving the diameter field blank assumes k_b = 1.0 (small specimen). For large shafts this is non-conservative.
✅ Fix
Always enter the actual shaft or section diameter. Even a rough diameter (e.g. 50 mm) gives a more realistic k_b.
Forgetting the notch factor
Leaving K_t = 1.0 for a shaft with keyways or holes greatly overestimates fatigue life.
✅ Fix
Look up K_t from Peterson's chart for your geometry. For a typical shaft keyway: K_t ≈ 2.0–2.5. Enter q ≈ 0.7–0.9 for medium-strength steels.
Using Miner's D = 1.0 as the hard failure limit
Real components often fail at D = 0.3–0.7 due to sequence effects and scatter.
✅ Fix
Apply a damage threshold of D ≤ 0.3–0.5 for safety-critical components, not the theoretical D = 1.0 limit.
Applying polished surface factor to as-machined parts
Using k_a for polished condition when the actual part is machined overstates S_e by 20–30%.
✅ Fix
Match the surface finish preset to your actual manufacturing process — or enter a custom k_e factor in the miscellaneous field.
⚡ Accuracy, Assumptions & Disclaimer
This calculator is built on widely-used engineering approximations compiled from textbook sources (primarily Shigley's Mechanical Engineering Design and similar references). It is designed to give useful first-pass estimates, not code-certified results.
- All Marin factor equations are simplified approximations; real parts may differ due to microstructure, surface treatments, and residual stresses.
- Material preset values are typical mid-grade estimates — certified properties from your supplier should always be used for final design.
- Non-ferrous metals (copper, brass) do not have a true endurance limit; long-life results should be interpreted with extra caution.
- Basquin extrapolation beyond the data-fitted range (particularly below 10³ cycles) is unreliable — use Coffin-Manson for low-cycle fatigue.
- Miner's rule ignores sequence effects and statistical scatter in fatigue data.
- The Shift/HSE score is an indicative awareness tool — it does not replace a validated fatigue risk management system (FRMS).
- Results must be validated with physical test data, applicable standards (ASTM E466, EN 3987, ISO 1143, etc.), and review by a qualified professional engineer before use in any safety-critical application.
All Formulas at a Glance
| Formula Name | Expression | Section |
|---|---|---|
| Alternating stress | \(\sigma_a = (\sigma_{\max} - \sigma_{\min})/2\) | 05 |
| Mean stress | \(\sigma_m = (\sigma_{\max} + \sigma_{\min})/2\) | 05 |
| Stress ratio | \(R = \sigma_{\min}/\sigma_{\max}\) | 05 |
| Unmodified endurance limit (steel) | \(S_e' = \min(0.5 S_u,\; 700\,\text{MPa})\) | 06 |
| Corrected endurance limit | \(S_e = k_a k_b k_c k_d k_e k_\text{load} S_e'\) | 06 |
| Surface factor | \(k_a = a \cdot S_u^b\) | 06 |
| Size factor | \(k_b = (d/7.62)^{-0.107}\) | 06 |
| Temperature factor | \(k_d = 1 - 0.3(T-25)/275\) | 06 |
| Fatigue stress concentration | \(K_f = 1 + q(K_t - 1)\) | 07 |
| Goodman equivalent stress | \(\sigma_{a,\text{eq}} = \sigma_a / (1 - \sigma_m/S_u)\) | 08 |
| Basquin S‑N equation | \(\sigma_a = \sigma_f'(2N_f)^b\) | 09 |
| Life to failure | \(N_f = \frac{1}{2}(\sigma_{a,\text{eq}}/\sigma_f')^{1/b}\) | 09 |
| Miner's cumulative damage | \(D = \sum n_i / N_{f,i}\) | 10 |
| Safety factor (Goodman) | \(n_f = 1/(\sigma_a/S_e + \sigma_m/S_u)\) | 11 |
| Bolt stresses | \(\sigma_m = F_m/A_s,\; \sigma_a = F_a/A_s\) | 12 |
