Fatigue Analysis Calculator — Endurance Limit, S-N Life & Safety Factor (Shigley Marin / Goodman)

Governing standard: Shigley §6 — Marin / Goodman-Gerber-Soderberg· Shigley DE-Goodman / Gerber / Soderberg · Marin factors · Palmgren–Miner (cycle counting per ASTM E1049) · Coffin-Manson LCF (constants from ASTM E606 tests) · Peterson notch sensitivity

Page last updated 17 October 2026

Video walkthrough (3:30)

Read the transcript

Here's a steel shaft, 40 mm across, bent as it turns. The tool says it's SAFE, with a fatigue safety factor of 1.58.

Give it a sharper shoulder and a hot-rolled surface. Same load, and it fails, in about half a million cycles.

This is the MechanixCalc fatigue calculator. It corrects the endurance limit for the real part, applies the notch, and checks the shaft against a mean-stress line and against yield. Let's go through it, step by step.

A shaft that turns under a bending load sees the stress at its surface swing up and down, every turn. The swing has a mean, σm, and an alternating part, σa.

Each cycle does a little damage. Millions of them can start a crack at a notch, far below the yield stress. So the tool checks the swing against a mean-stress line, and against yield.

Step one is the material, structural steel S355. The tool reads its guaranteed strength at the bar's size, so this shaft gets 470 MPa, not the 510 of the thinnest sections.

Step two is the endurance limit. Half the strength, 235 MPa, is corrected for a machined surface, the 40 mm size and 95% reliability. That leaves 150.5 MPa.

Step three is the notch. A shoulder fillet has a Kt of 2. With a notch sensitivity of 0.8, the fatigue factor Kf is 1.8.

Step four is the stress, in rotating bending: 40 MPa mean, and 40 MPa alternating. On the Haigh diagram, the point sits inside the Goodman line.

Step five shows the checks. The fatigue factor is 1.58, and first-cycle yield is 2.33. Both clear the design factor of 1.5, so the verdict is SAFE.

The stress stays below the endurance limit, so the life is infinite.

Now make the fillet sharper, a Kt of 2.5. The factor falls to 1.30.

That's above 1, so the shaft won't fail. But it's below 1.5, so the verdict is CHECK.

At a Kt of 3, it's 1.10. Still CHECK, and the life still reads infinite, but the margin is thin.

Now leave the surface hot-rolled. The endurance limit drops to 118.6 MPa.

The factor is 0.91, below 1, and the shaft fails, at about 525,000 cycles.

The fix is the surface and the notch. Grind the shaft, and go back to a Kt of 2.

The factor is 1.65, and the verdict turns green.

The mean-stress line is your choice. Gerber, the best fit for ductile steel, gives 1.91.

Soderberg, the most conservative, gives 1.44, a CHECK. Goodman, in between, is the default.

What the tool does not check is listed too: rainflow counting, welded joints, crack growth, corrosion, surface treatments and high temperature. So silence never reads as a pass.

Every input, every formula and every check goes into a PDF report. A report your reviewer can follow.

That's a shaft in fatigue, checked. Try it on your own design, on MechanixCalc.com.

The MechanixCalc fatigue analysis calculator covers the full Shigley §6 design workflow in one tool: enter the material (Sut, Sy), the alternating and mean stresses, and the surface, size, load and reliability correction factors, and the tool returns the corrected endurance limit Se, the Goodman (or Gerber or Soderberg) fatigue safety factor, the estimated S-N life, and the Haigh diagram — all in a single pass. A dedicated stress-concentration panel interpolates Peterson Kt and estimates Kf (Neuber / Peterson notch sensitivity) for shoulder fillets, circular grooves, transverse holes and keyways; Miner's linear damage rule handles variable-amplitude spectra; and Coffin-Manson covers low-cycle strain-life regimes below about 10 000 cycles.

It is built for machine-design, structural and rotating-machinery engineers who need a defensible fatigue assessment — from a quick Goodman safety-factor check to a full variable-amplitude damage sum or a multiaxial Dang Van / Crossland critical-plane analysis — and who need to hand a reviewer a textbook-cited, fully worked calculation rather than a spreadsheet.

What this calculator does

Goodman, Gerber and Soderberg fatigue safety factor with von Mises combined bending + torsion (σ_a_eq, σ_m_eq)
Marin endurance-limit correction factors ka (surface), kb (size), kc (load type), ke (reliability) — 8 material presets plus Custom. Room-temperature only: the temperature factor kd is fixed at 1.0 and is not an input
S-N (Wöhler) curve with Basquin slope b and operating-point overlay on the log-log plot
Palmgren-Miner linear damage rule for variable-amplitude loading — up to six stress-range levels with per-level di = n/Nf and cumulative damage D. Enter pre-counted blocks (count them per ASTM E1049); the tool does not perform rainflow counting itself
Coffin-Manson strain-life (LCF) for low-cycle fatigue below ~10 000 cycles, with elastic/plastic separation and transition-life Nt (ASTM E606)
Stress concentration factor Kt/Kf interactive interpolator (Peterson/Neuber) for shoulder fillets, grooves, transverse holes and keyways
Multiaxial fatigue via von Mises equivalent stresses and Dang Van / Crossland critical-plane criteria; Haigh diagram; branded PDF engineering report

Method & formulas

Marin corrected endurance limit and Goodman safety factor

The rotating-beam endurance limit Se_prime is estimated as 0.5·Sut for steels (up to the 1 400 MPa cap, above which Se_prime = 700 MPa). This rule and the flat 10⁶-cycle endurance knee are steel-only: for cast irons the tool uses Se_prime = 0.4·Sut, and non-ferrous alloys (aluminium, titanium, copper) have no endurance limit at all, so the tool flags them, makes no infinite-life claim and caps the verdict at CHECK until you enter your material's fatigue strength. Marin correction factors then reduce it for the real part: ka (surface finish), kb (cross-section size), kc (loading mode — bending, axial or torsion) and ke (reliability percentile). The corrected endurance limit is Se = ka·kb·kc·kd·ke·Se_prime. The temperature factor kd is fixed at 1.0 — the tool assumes room-temperature service and does not take a temperature input; above about 250 °C you must apply kd yourself (Shigley Table 6-4).

The fatigue stress-concentration factor Kf (= 1 + q·(Kt − 1), where q is the notch sensitivity you enter — the Neuber / Peterson panels estimate it) is applied to both the alternating and the mean stresses. The von Mises equivalent alternating and mean stresses combine the normal and shear contributions before the mean-stress criterion is applied, so torsion loading is handled without double-counting the √3 shear conversion.

Corrected endurance limit

Se = ka · kb · kc · kd · ke · Se_prime

where Se_prime = 0.5·Sut (≤ 700 MPa; 0.4·Sut for cast iron); ka = surface factor; kb = size factor; kc = load factor; kd = temperature factor; ke = reliability factor

Goodman fatigue safety factor

SF = 1 / (σ_a_eq / Se + σ_m_eq / Sut)

where σ_a_eq = von Mises alternating stress including Kf; σ_m_eq = von Mises mean stress; Se = corrected endurance limit; Sut = ultimate tensile strength

S-N curve and estimated fatigue life

The S-N curve is anchored at Sf(10³) = f·Sut and Se at 10⁶ cycles, where f is Shigley’s fatigue-strength fraction (Fig. 6-18). f = 0.9 only below about 490 MPa; above that it falls — 0.83 at 800 MPa, 0.80 at 1000 MPa — because a stronger steel is proportionally weaker at 10³ cycles. Using a flat 0.9 would flatten the Basquin line and overstate the finite life. The slope is b = −log(f·Sut / Se) / log(10⁶ / 10³). To account for a non-zero mean stress, the fully-reversed equivalent amplitude σ_ar is computed from the chosen criterion (Goodman: σ_ar = σ_a / (1 − σ_m/Sut); Gerber: σ_ar = σ_a / (1 − (σ_m/Sut)²); Soderberg: σ_ar = σ_a / (1 − σ_m/Sy)) before reading off the S-N curve — ignoring the mean stress when looking up N_life would be unconservative.

Basquin S-N relation

Sf(N) = a · N^b for 10³ ≤ N ≤ 10⁶

where a = f·Sut / (10³)^b; b = −log(f·Sut / Se) / 3 (log base 10); f = Shigley fatigue-strength fraction (0.9 below ~490 MPa, falling to ~0.78 at 1400 MPa); Sf = fatigue strength at N cycles

Miner's damage rule and Coffin-Manson low-cycle fatigue

For variable-amplitude loading, each stress-range block contributes a partial damage fraction di = ni / Nf,i, where Nf,i is the S-N life at the amplitude σa,i = Δσ_i / 2. Total damage D = Σdi; failure is predicted when D ≥ 1 (Palmgren 1924 / Miner 1945; ASTM E1049 governs the CYCLE COUNTING that produces the blocks, not the damage sum). Cycles below Se accumulate zero damage — for a spectrum whose higher blocks exceed Se that is optimistic, because those cycles remove the endurance limit; a Miner-Haibach extension below the knee is not implemented.

For low-cycle fatigue (plastic strain is significant, typically N < 10 000 cycles) the Coffin-Manson relation separates elastic and plastic strain contributions. The transition life Nt marks the cross-over from predominantly elastic to predominantly plastic failure; the total strain amplitude at 2Nf reversals is fitted by bisection.

Coffin-Manson strain-life

ε_a = (σ'_f / E) · (2Nf)^b + ε'_f · (2Nf)^c

where ε_a = total strain amplitude; σ'_f = fatigue strength coefficient; E = elastic modulus; b = fatigue strength exponent; ε'_f = fatigue ductility coefficient; c = fatigue ductility exponent; 2Nf = reversals to failure

Worked example

A polished 1040 steel specimen (Sut = 600 MPa, Sy = 450 MPa) is loaded in bending — a hand case: use fully idealized Marin factors (ka = kb = kc = kd = ke = 1.0) and no notch (Kf = 1). Alternating bending stress σ_a = 100 MPa, mean bending stress σ_m = 50 MPa. Find the GOVERNING safety factor — which means checking both the Goodman fatigue criterion and the Langer first-cycle yield line, and taking the worse of the two.

Given

Ultimate tensile strength Sut600 MPa
Alternating stress σ_a100 MPa
Mean stress σ_m50 MPa
Marin factors ka·kb·kc·kd·ke1.0 (all idealized)
Fatigue stress-concentration factor Kf1.0 (no notch)

Result

Corrected endurance limit Se300 MPa
Goodman fatigue safety factor n_f2.40
Langer first-cycle yield factor n_Y3.00
Governing safety factor2.40 (fatigue governs)

Estimate the rotating-beam endurance limit: Se_prime = 0.5 × 600 = 300 MPa.
Apply Marin factors: Se = 1.0 × 1.0 × 1.0 × 1.0 × 1.0 × 300 = 300 MPa.
Apply Goodman criterion: SF = 1 / (σ_a/Se + σ_m/Sut) = 1 / (100/300 + 50/600).
Evaluate each term: 100/300 = 1/3; 50/600 = 1/12.
Sum: 1/3 + 1/12 = 4/12 + 1/12 = 5/12.
Fatigue safety factor: n_f = 12/5 = 2.40.
Now the first-cycle yield check (Langer): n_Y = Sy / (σ_a + σ_m) = 450 / (100 + 50) = 3.00.
Governing factor = min(n_f, n_Y) = min(2.40, 3.00) = 2.40 — fatigue governs here. On a material with a low Sy/Sut ratio (annealed stainless, for instance) n_Y would be the smaller of the two and yielding would govern instead.

Illustrative example with fully idealized Marin factors (all = 1.0) and no notch. Real components have surface, size and loading corrections that reduce Se and therefore reduce n_f. Note that n_f alone is not the verdict: a part that yields on its first cycle never reaches its fatigue life, so the tool reports the WORSE of n_f and n_Y. Verify against your actual geometry, surface finish, material certificate and load history.

Frequently asked questions

Which standard does this fatigue calculator use?

The headline Goodman / Gerber / Soderberg safety factor follows Shigley's Mechanical Engineering Design (Budynas & Nisbett) §6, as do the Marin endurance-limit correction factors. These are textbook methods, not the product of a standards body — Goodman is J. Goodman (1899) and Gerber is Gerber (1874). The cumulative-damage sum is the Palmgren-Miner rule (1924/1945); ASTM E1049 is the cycle-counting practice that produces the input blocks. Coffin-Manson coefficients come from strain-controlled testing per ASTM E606, itself a test method rather than a design standard. Stress concentration is from Peterson's Stress Concentration Factors (3rd ed.) with the notch sensitivity q entered by you (the Neuber and Peterson panels estimate it). If you need an ASME-governed fatigue assessment (BPVC VIII Div. 2 Part 5 design fatigue curves, or B31 piping), this tool is not that — use it for machine-element design to Shigley. The governing method and all factor values are shown in the generated PDF report.

How does the calculator handle stress concentration (notches)?

A dedicated Stress Concentration panel interpolates the geometric concentration factor Kt from Peterson chart fits for shoulder fillets, circular grooves, transverse holes and keyways. The notch sensitivity q (a function of Sut and notch radius r — the Neuber and Peterson panels estimate it) reduces Kt to the fatigue factor Kf = 1 + q(Kt − 1). Kf is then applied to the alternating and mean stresses before computing the safety factor, so a notched component is never reported as passing on its nominal stress alone.

What is Miner's rule and when should I use it?

The Palmgren-Miner linear damage rule predicts cumulative fatigue life when a component sees more than one stress amplitude — for example, a vehicle axle experiencing different load levels over its service life. Each amplitude σa,i contributes a partial damage fraction di = ni/Nf,i (where ni = applied cycles and Nf,i = S-N life at that amplitude); failure is predicted when the sum D = Σdi reaches 1.0. It is simple and widely used for preliminary design, but it assumes no load-sequence effects and no damage below the endurance limit — both are conservative approximations.

When should I use Coffin-Manson instead of the S-N approach?

The Basquin S-N curve is calibrated for high-cycle fatigue (typically N > 10 000 cycles), where strains are predominantly elastic. Below about 10 000 cycles — for example, in pressure-vessel thermal cycling, forging tools or earthquake loading — plastic strain is significant and the S-N approach over-predicts life. The Coffin-Manson strain-life relation separates elastic and plastic contributions and gives much more accurate life estimates in the low-cycle regime. In the calculator, enter the fatigue strength and ductility coefficients (σ'f, ε'f, b, c) from your material test data or look-up tables.

Why does the calculator sometimes report a FAIL when the fatigue safety factor is above 1.5?

Because the fatigue factor is not the whole check. Shigley §6-14 requires a first-cycle yield check alongside the mean-stress criterion, and the governing factor is the worse of the two: n_f (Goodman/Gerber/Soderberg) and n_Y = Sy/(σ_a + σ_m), the Langer line. A design can sit comfortably inside the Goodman line and still yield on its very first cycle — this happens whenever the mean stress is high and the material has a low yield-to-ultimate ratio, which is typical of annealed austenitic stainless steels and many normalised steels. The hero verdict, the PDF report and the drawing all report the governing factor and name which of the two is driving it.

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