Fatigue Analysis — Engineering Reference

1. What it is

Fatigue is the progressive, localised, and permanent structural damage that occurs in a material subjected to cyclic loading at stresses well below the static yield. After enough cycles a crack initiates at a stress riser (notch, inclusion, surface defect, weld toe, fretting site), propagates with each subsequent cycle, and finally precipitates a brittle-looking fast fracture — even in a ductile, well-characterised material that would never fail under the same load applied monotonically. The fracture surface carries the signature: a smooth, beach-marked initiation-and-propagation zone, terminating in a rough, dimpled fast-fracture overload zone.

The discipline divides cyclic-load problems into three regimes by cycle count to failure:

Regime	Cycles N_f	Strain state	Governing approach	Examples
VLCF (very low)	< 100	Macro plastic, large Δε	Strain-life + ductility	Seismic events, single-impact tests
LCF (low cycle)	10² – 10⁴	Plastic strain dominates	Strain-life (Coffin-Manson)	Pressure-vessel start-stop, turbine disks, nuclear
HCF (high cycle)	10⁵ – 10⁹	Nominally elastic, Δσ < σ_y	Stress-life (S-N, Basquin)	Rotating shafts, bearings, springs, fasteners
VHCF / gigacycle	> 10⁹	Sub-surface initiation	Ultrasonic test S-N, no clear limit	Rail axles, wind-turbine main shafts, aero gas-turbine

Within those regimes the analysis cascade is:

1. Cycle counting (constant-amplitude or rainflow per ASTM E1049 for random history)
2. Cycle-by-cycle damage (S-N or strain-life curve)
3. Damage accumulation (Miner’s linear rule, or one of its many correction schemes)
4. Crack-growth integration if a crack is detected or assumed (Paris-Erdogan; covered in fracture-mechanics)

Fatigue analysis is the first failure mode every cyclic-load designer checks — and the one most often missed by engineers trained only in static stress.

1.1 The three-stage life of a fatigue crack

A useful mental model partitions the total life N_f into three sequential stages:

1. Stage I — crystallographic initiation. Persistent slip bands (PSBs) form in surface grains favourably oriented for shear; intrusions and extrusions develop after ~10³–10⁵ cycles. Crack initiates along {111} (FCC) or {110} (BCC) planes at angles near 45° to the maximum normal stress. Length ≈ 1–3 grain diameters (~10–100 μm). This stage consumes 50–90 % of total HCF life in clean, polished specimens; almost none in welded joints, cast components, or parts with pre-existing flaws.
2. Stage II — long-crack growth. Crack reorients perpendicular to the maximum normal stress and grows by a striation-forming mechanism (one striation per cycle in many alloys, though not always one-to-one). Growth rate da/dN follows the Paris-Erdogan law da/dN = C · (ΔK)^m with m ≈ 2–4 for metals. Stage II is the regime addressed by fracture-mechanics damage-tolerance analysis — see fracture-mechanics.
3. Stage III — fast fracture. Once K_max approaches K_Ic (or J_Ic in ductile case), the remaining cross-section overloads catastrophically in a single cycle.

The strain-life and S-N curves both quote N_f as total life — they do not separate initiation and propagation. Damage-tolerance methods (FAR 25.571, aerospace) deliberately count only Stage II from an assumed initial-flaw size; safe-life methods (rail, automotive) use total-life curves with a large factor of safety to cover initiation scatter.

2. Why it matters

Most mechanical failures in service are fatigue, not overload. Industry-wide failure-survey papers (Forsyth 1969, Findlay & Harrison 2002 in Eng Fail Anal surveying ~200 service failures) routinely attribute 50–90 % of in-service mechanical failures to fatigue, with corrosion-fatigue and stress-corrosion cracking contributing further to the cyclic-failure tally. The Versailles and Sankey-Junction railway disasters (1842, 1846) drove Wöhler’s original work on Prussian railway axles; the Comet airliner pressurisation cracks (1954) recast aerospace structural design around fatigue; the I-35W Mississippi River Bridge collapse (2007) traced to gusset-plate fatigue.

Every modern cyclic-load industry now has a dedicated fatigue-design code:

• Aerospace — FAR 25.571, DEF STAN 00-970, AC 25.571-1D damage-tolerance philosophy
• Pressure vessels & nuclear — ASME BPVC III NH (high-T), VIII Div 2 (fatigue), B31.3 piping
• Offshore / marine — DNV-RP-C203, API RP 2A WSD
• Rail — AAR M-1003, EN 13261/13262 (axles, wheels)
• Bridges & welded steel — Eurocode 3 part 1-9, BS 7608, AASHTO LRFD
• Automotive — internal OEM specs based on SAE J1099, FKM-Richtlinie (Germany)
• Wind turbines — IEC 61400, GL-Wind, DNV-OS-J101

Single-event overload analysis is necessary but not sufficient. A part can pass every static stress check, every yield check, every buckling check — and still fail in service after a million benign cycles. The engineering judgement is to identify which members see cyclic load, what spectrum, and which S-N or ε-N rule applies.

3. First principles

3.1 Cyclic-stress descriptors

For a periodic stress history with min and max σ_min, σ_max:

σ_a   = (σ_max − σ_min) / 2        (alternating stress amplitude)
σ_m   = (σ_max + σ_min) / 2        (mean stress)
Δσ    = σ_max − σ_min   = 2·σ_a    (stress range)
R     = σ_min / σ_max              (R-ratio, dimensionless)
A     = σ_a / σ_m                  (amplitude ratio)

R-ratio shorthand defines the load case:

R	Name	σ_m	Typical use
−1	Fully reversed	0	Rotating bending, lab S-N tests
0	Repeated (zero-to-max)	σ_max/2	Pressure cycling, gear-tooth bending
+0.1	Tension-tension	> 0	Aerospace skin pressurisation (used by ASTM E466)
> 1	All-compression	< 0	Springs in compression, rolling contact

Industry shorthand: “fully reversed” usually means R = −1, “pulsating tension” R = 0, “tension-tension” R > 0. ASTM E466 actually mandates R = 0.1 as the default lab-test condition — close to fully tensile, slightly off zero to keep the load cell in tension and avoid grip-slip on stress reversal. Beware: published S-N data is not always at R = −1. Always check the test report’s R value before applying mean-stress corrections.

3.2 The S-N curve — Wöhler (1858)

August Wöhler’s investigation of broken Prussian railway axles (published 1858–1870) established the canonical experimental curve: σ_a versus N (cycles to failure) on log-log axes. For ferrous and many titanium alloys the curve flattens near 10⁶–10⁷ cycles at the fatigue (endurance) limit σ_e — below which infinite life is achieved. Aluminum, copper, magnesium, and most non-ferrous alloys show no clear limit; the curve continues to droop, so a fatigue strength at a chosen design life (typ 5×10⁸) is quoted instead.

Why does ferrous steel have a knee? The classical explanation (Cottrell-Bilby 1949) is that interstitial carbon and nitrogen atoms diffuse to dislocation cores and pin them, raising the local CRSS until macroscopic slip can no longer initiate at the imposed stress. Modern very-high-cycle (VHCF) fatigue tests using ultrasonic-rig piezo loading at 20 kHz, going out to 10⁹–10¹⁰ cycles, show that the “endurance limit” is in fact a slowing-down, not a hard limit — sub-surface initiation at non-metallic inclusions (so-called “fish-eye” failures) continues beyond the conventional knee. For rail axles and wind-turbine main shafts that see > 10⁹ cycles in service, the classical σ_e is not an infinite-life guarantee.

3.3 HCF — Stress-Life (Basquin 1910)

In the HCF regime the σ–N curve is approximated by a power law:

σ_a = σ'_f · (2 N_f)^b              (Basquin equation; 2 N_f = reversals to failure)

with σ’_f the fatigue-strength coefficient (≈ σ_u for many steels) and b the fatigue-strength exponent (typically −0.05 to −0.12 for steel, more negative for higher-strength alloys).

The Basquin slope corresponds to a S-N log slope of |1/b|. For b = −0.085 a factor-of-2 change in σ_a corresponds to a (2)^(−1/−0.085) = 2^11.8 ≈ 3500× change in N_f. Fatigue life is extraordinarily sensitive to stress amplitude — getting the σ_a estimate right within 10 % matters; getting K_t right within 10 % matters; getting σ_u right within 10 % barely matters at all. The lever is the local stress, not the bulk strength.

3.4 LCF — Strain-Life (Coffin 1954, Manson 1954)

Below ~10⁴ cycles, plastic strain per cycle becomes significant and stress-controlled descriptions break down. Coffin and Manson (independently, both 1954) related plastic strain amplitude to life:

Δε_p / 2 = ε'_f · (2 N_f)^c         (Coffin-Manson)

with ε’_f the fatigue ductility coefficient (≈ true fracture strain for many metals) and c the fatigue ductility exponent (≈ −0.5 to −0.7).

Physically: in the LCF regime each cycle imposes detectable bulk plastic strain that drives dislocation multiplication, sub-structure formation, and ultimately void-nucleation-and-coalescence on the same mechanism as monotonic ductile fracture — just repeated. A material with high monotonic ductility (high true fracture strain ε_f) generally has high cyclic ductility ε’_f.

3.5 Combined Morrow / Mason-Coffin strain-life

In the strain-life framework the elastic and plastic contributions are added (Smith 1958, Morrow 1965):

Δε / 2 = (σ'_f / E) · (2 N_f)^b  +  ε'_f · (2 N_f)^c
       ↑ elastic part            ↑ plastic part

The first term dominates for large N (HCF, asymptote → Basquin). The second dominates for small N (LCF). The crossover at Δε_e = Δε_p is the transition life 2 N_t ≈ ((ε’_f · E / σ’_f))^(1/(b−c)) — typically 10³–10⁴ cycles. Above 2N_t, run S-N; below, run strain-life.

For pure-elastic loading (HCF), Δε/2 ≈ Δσ/(2E) = σ_a/E and the strain-life equation reduces algebraically to Basquin. In that sense strain-life subsumes stress-life — but stress-life remains the workhorse for HCF design because it is easier to apply with elastic FE results (no notch-root plasticity correction needed).

3.6 Mean-stress correction models

Lab S-N data is almost always fully reversed (R = −1, σ_m = 0); real loads are not. Mean stress shifts the allowable amplitude:

Model	Equation	Notes
Goodman (1899)	σ_a / σ_e + σ_m / σ_u = 1 / n	Linear, slightly conservative, the industry default
Soderberg	σ_a / σ_e + σ_m / σ_y = 1 / n	Replaces σ_u with σ_y — over-conservative; rarely used today
Gerber (1874)	σ_a / σ_e + (σ_m / σ_u)² = 1 / n	Parabolic; better fit to data; non-conservative when σ_m near σ_u
Morrow (1968)	σ_a / (σ’_f − σ_m) = (2 N_f)^b	Strain-life equivalent; preferred for fatigue analysis under tensile σ_m
Smith-Watson-Topper (1970)	σ_max · ε_a · E = (σ’_f)² · (2 N_f)^(2b) + σ’_f · ε’_f · E · (2 N_f)^(b+c)	Strain-based, handles mean stress + multiaxial well; non-negative σ_max only

Compressive mean stress is beneficial (shot-peen, cold-roll, induction-hardened residual compression). Tensile mean stress is detrimental — exactly what welds, sharp section changes, and assembly preload tend to produce.

Choosing a mean-stress model: Goodman for HCF on steel where σ_m is moderate (σ_m/σ_u < 0.3); Morrow for strain-life on ductile metals; SWT for strain-life on cast iron, high-strength steel and welds. Gerber tends to be non-conservative near σ_m → σ_u. Never apply mean-stress correction to a compressive σ_m unless you have direct test data showing benefit — most “compression is beneficial” claims rely on residual-stress superposition, not on improved S-N at compressive applied σ_m.

3.7 Miner’s linear damage rule (Palmgren 1924, Miner 1945)

D = Σ (n_i / N_i)            (Palmgren-Miner)
failure when D = 1

with n_i cycles applied at stress amplitude σ_a,i for which the constant-amplitude S-N curve gives N_i cycles to failure. Conceptually clean — damage accumulates linearly and is sequence-independent. In practice Miner predicts within a factor of 2 in only ~5 % of published comparison cases (Schütz 1996); industry routinely uses D_fail = 0.3–0.5 as a conservative correction for variable-amplitude spectra, especially when overloads precede small cycles.

Why Miner under-predicts so often. Three reasons: (i) sequence effects — a tensile overload retards subsequent crack growth (Elber 1970 plasticity-induced closure), a compressive overload accelerates it; Miner has no memory of sequence. (ii) Cycles below the CA-endurance-limit knee are not zero-damage when the spectrum also contains cycles above the knee — pre-existing damage extends the slope below the knee (Haibach’s second-slope correction). (iii) Mean-stress relaxation during early cyclic plasticity changes the effective σ_m on subsequent cycles, but Miner uses initial σ_m throughout.

Alternative cumulative-damage rules (Corten-Dolan 1956 power-law, Marco-Starkey 1954 σ-dependent exponent, double-linear damage rule) all reduce to Miner in the constant-amplitude limit and improve it modestly under specific load sequences; none is universally better than modified Miner with Haibach extension below the knee.

3.8 Rainflow counting (ASTM E1049-17)

For random load histories, rainflow counting (Endo 1968, Matsuishi & Endo 1968) extracts closed hysteresis loops from a sequence of reversals. The “rainflow” name comes from the rooftop analogy: turn the time history sideways; rain drops fall, run along the surface, drip off at each reversal — each drip-path defines one cycle. The algorithm produces a histogram of (σ_a, σ_m) pairs that can be fed directly into Miner’s rule. It is the standard preprocessing step in any modern fatigue post-processor (nCode DesignLife, FE-SAFE, MSC Fatigue).

Pseudo-algorithm (4-point method, ASTM E1049):

1. Extract local extrema (peaks and valleys) from the time history.
2. Maintain a stack S of unprocessed extrema.
3. Read next extremum; push onto S. If |S| < 4, continue.
4. Let A, B, C, D be the last four points on S.
   If |B − C| ≤ |A − B| and |B − C| ≤ |C − D|:
       Output a cycle (σ_a = |B − C|/2, σ_m = (B + C)/2).
       Remove B and C from S.
       Go to step 3 (re-test with new last four).
5. End of stream: output residual half-cycles from S.

A variant (Downing-Socie 1982, one-pass) is the implementation in most commercial codes. Both yield the same cycle histogram for any input sequence.

Why does rainflow work? Each extracted (σ_a, σ_m) pair corresponds to a closed hysteresis loop in σ-ε space — exactly the unit of damage the strain-life equation predicts. Half-cycles (open at the end of the record) are typically counted as one full cycle with half the damage, or kept aside as “residual” to be merged with the next stationary block in continuous-monitoring applications.

3.9 Notch sensitivity

The static stress-concentration factor K_t (geometric, from charts; see mechanics-of-materials) over-predicts the fatigue penalty. The empirical fatigue notch factor K_f is smaller:

K_f = 1 + q · (K_t − 1)              (Peterson 1959)
q   = 1 / (1 + a / r)                (Peterson)        or
q   = 1 / (1 + √(ρ / r))              (Neuber 1958)

where r is the notch radius and a (Peterson) or ρ (Neuber) is a material-and-strength-dependent length. Typical values for steel: a ≈ 0.025 mm at σ_u = 1400 MPa, rising to a ≈ 0.5 mm at σ_u = 350 MPa. Higher-strength steels are more notch-sensitive — one of the practical reasons high-σ_u steel doesn’t always buy fatigue performance.

For sharp notches (r → 0) Neuber gives q → 1 and K_f → K_t; for blunt notches K_f approaches 1 (the part behaves as if un-notched). Cast iron’s graphite flakes act as internal pre-cracks, giving very low q (~0.2) and unusually small fatigue penalty for added stress concentrators — one reason gray cast iron is dimensionally insensitive in cylinder-block design.

4. HCF + S-N — design procedure (Shigley / Marin / Juvinall framework)

The lab-test endurance limit σ_e’ is derived on a polished 7.5 mm rotating-bending specimen, R = −1. A real part does not see that environment. Marin (1962) introduced multiplicative modifying factors; modern Shigley distils them as:

σ_e  =  C_S · C_d · C_L · C_R · C_T · C_E · σ_e'

Factor	Description	Typical range
C_S (surface finish)	Polished 1.0; ground 0.9; machined 0.7; hot-rolled 0.5; as-forged 0.4; corroded 0.2 (steel, σ_u = 500 MPa) — drops further with higher σ_u
C_d (size)	1.0 for d ≤ 8 mm; (d/7.62)^(−0.107) for 8 < d ≤ 51 mm; ~0.7 for d > 250 mm
C_L (loading)	Bending 1.0; axial 0.85; torsion 0.59 (per von Mises distortion-energy)
C_R (reliability)	1.0 for 50 %; 0.897 for 90 %; 0.814 for 99 %; 0.753 for 99.9 %
C_T (temperature)	1.0 at room; ~0.85 at 300 °C; ~0.5 at 500 °C (steel)
C_E (environment, fretting, residual stress)	Variable; 0.5–0.8 in mild corrosion, 0.2–0.4 in salt water

For ferrous steels with σ_u < 1400 MPa, the un-modified rotating-bending fatigue limit is approximately:

σ_e' ≈ 0.5 · σ_u

Above σ_u = 1400 MPa the relation flattens to σ_e’ ≈ 700 MPa regardless of further strength gain. Aluminum has no clear limit; design fatigue strength at 5×10⁸ cycles is roughly σ_e ≈ 0.4 · σ_u for wrought aluminum (Juvinall).

Worked Example A — Shaft sizing for fatigue, Goodman line

Problem. A 25 mm-diameter shaft in AISI 1045 hot-rolled (σ_u = 565 MPa, σ_y = 310 MPa, σ_e’ ≈ 0.5 · 565 = 282.5 MPa), machined finish, sees rotating bending σ_a = 80 MPa superposed on a constant-tension mean σ_m = 100 MPa. Design for 99.9 % reliability. Find n (safety factor).

Step 1 — Modify endurance limit.

• C_S (machined, σ_u = 565 MPa) ≈ 0.70
• C_d (25 mm) = (25 / 7.62)^(−0.107) = 3.28^(−0.107) ≈ 0.875
• C_L (bending) = 1.0
• C_R (99.9 %) = 0.753
• σ_e = 0.70 · 0.875 · 1.0 · 0.753 · 282.5 = 130.4 MPa

Step 2 — Goodman criterion.

σ_a / σ_e + σ_m / σ_u = 1 / n
80 / 130.4 + 100 / 565 = 0.6135 + 0.1770 = 0.7905
n = 1 / 0.7905 = 1.265

Step 3 — Yield check (first cycle). σ_max = σ_a + σ_m = 80 + 100 = 180 MPa < σ_y = 310 MPa, so n_y = 310 / 180 = 1.72. First-cycle yielding is not the failure mode.

Step 4 — Engineering judgement. n = 1.27 against fatigue is marginal — industrial practice for a rotating shaft is n ≥ 1.5 (some critical applications ≥ 2.0). Options: shot-peen the fillet regions (compressive residual stress lifts σ_e by 15–30 % — see joining-welding for the same principle on weld toes); polish to ground finish (raise C_S from 0.7 to 0.9); upsize to 28 mm (drops σ_a by (25/28)³ ≈ 0.71). Upsize is usually cheapest.

5. LCF + strain-life — Coffin-Manson framework

For pressure-vessel shakedown, gas-turbine disk LCF, seismic structural cycling, and notch-root analysis even on HCF components, the strain-life approach is required. Procedure:

1. Acquire the cyclic stress-strain curve (ASTM E606): σ_a = K’ · ε_p,a^n’, not the monotonic one — cyclic-softened steels behave very differently from virgin material.
2. Compute notch-root strain via Neuber (1961): K_t² · σ_nom · ε_nom = σ_local · ε_local. With the cyclic σ-ε curve this gives σ_local, ε_local at the notch root.
3. Enter the Δε/2-vs-2N_f curve at the computed local strain amplitude; read off 2 N_f.
4. Correct for mean stress: SWT or Morrow.

Worked Example B — Coffin-Manson LCF on AISI 4340

Problem. AISI 4340 Q&T to σ_y = 1000 MPa; E = 200 GPa. Tabulated strain-life parameters (Bannantine 1990): σ’_f = 1655 MPa, b = −0.076, ε’_f = 0.73, c = −0.62. Imposed total strain amplitude Δε / 2 = 0.005 (0.5 %), fully reversed. Find N_f.

Step 1 — Set up combined Morrow equation.

Δε / 2 = (σ'_f / E) · (2 N_f)^b  +  ε'_f · (2 N_f)^c
0.005  = (1655 / 200 000) · (2 N_f)^(−0.076) + 0.73 · (2 N_f)^(−0.62)
0.005  = 0.008275 · (2 N_f)^(−0.076) + 0.73 · (2 N_f)^(−0.62)

Step 2 — Solve numerically (Newton or bisection).

Trial 2 N_f	Elastic term	Plastic term	Total
500	0.008275 · 0.621 = 0.00514	0.73 · 0.0286 = 0.0209	0.0260
5 000	0.008275 · 0.522 = 0.00432	0.73 · 0.00682 = 0.00498	0.00930
50 000	0.008275 · 0.438 = 0.00362	0.73 · 0.00163 = 0.00119	0.00481
40 000	0.008275 · 0.449 = 0.00371	0.73 · 0.00184 = 0.00135	0.00506

So 2 N_f ≈ 40 000, hence N_f ≈ 20 000 cycles. At this life the plastic and elastic contributions are roughly comparable — we are right at the LCF/HCF transition for 4340.

Step 3 — Engineering judgement. At Δε/2 = 0.5 % a designer would not run a stress-life analysis: the plastic term carries roughly a third of the damage. A pure Basquin (HCF) calculation would over-predict life by an order of magnitude.

6. Damage accumulation under variable amplitude

Worked Example C — Rainflow + Miner

Problem. A shaft in 1045 (σ_e = 130 MPa, σ_u = 565 MPa from Example A, log-log slope b = −0.085) sees a measured 24-hour load history. Rainflow counting (per ASTM E1049-17) yields the following per-day cycle bins, fully reversed equivalent (mean correction already applied via Goodman):

Bin	σ_a (MPa)	n_i (cycles/day)	N_i from Basquin (cycles)	n_i / N_i (per day)
1	200	10	(200/σ’_f)^(1/b) ≈ 5.0 × 10⁴	2.0 × 10⁻⁴
2	150	100	≈ 5.0 × 10⁵	2.0 × 10⁻⁴
3	100	1 000	≈ 5.0 × 10⁶	2.0 × 10⁻⁴
4	80	10 000	(< σ_e: infinite per CA model)	~0 (or 2.0 × 10⁻⁴ with modified Miner extending S-N below knee)

Per-day damage (un-modified Miner, knee at σ_e): D = 0.0006. Life = 1 / D = 1670 days ≈ 4.6 years.

With modified Miner extending the S-N below the knee at slope b’ = −0.13 (Haibach 1970, the standard variable-amplitude correction): bin 4 contributes 2 × 10⁻⁴ also, D = 0.0008, life = 1250 days ≈ 3.4 years. The modified-Miner result is the design number for variable-amplitude road, rail, and wind-load spectra — un-modified Miner is non-conservative because it ignores damage from cycles below the constant-amplitude endurance limit.

Observation. Despite contributing only 10/11 110 = 0.09 % of the cycle count, the σ_a = 200 MPa events drive 25 % of total damage. Spectrum-truncation studies should always start from a damage-weighted histogram, not a cycle-count histogram.

Worked Example D — Welded steel detail, Eurocode 3 part 1-9

Problem. A transverse load-carrying fillet weld in S355 structural steel (σ_y = 355 MPa) carries a stress range Δσ = 95 MPa for n = 5 × 10⁶ cycles per year. The detail is “transverse load-carrying fillet weld, weld-toe failure”, Eurocode 3 detail category FAT 56 (σ_C = 56 MPa at N_C = 2 × 10⁶ cycles, slope m = 3 below N_D = 5 × 10⁶, m = 5 between N_D and N_L = 10⁸ cutoff). Find the design life in years and compare to a 25-year service requirement.

Step 1 — Determine which slope applies. At Δσ = 95 MPa we are above σ_C = 56 MPa → in the m = 3 region. The Wöhler equation:

N · (Δσ)^m = N_C · (σ_C)^m
N = (2 × 10⁶) · (56 / 95)^3 = 2 × 10⁶ · 0.205 = 4.10 × 10⁵ cycles to failure

Step 2 — Life in years. Years = N / n = 4.10 × 10⁵ / 5 × 10⁶ = 0.082 years ≈ 30 days.

Step 3 — Apply partial safety factor. Eurocode 3 uses γ_Mf = 1.35 on damage-tolerant safe-life details. With γ_Mf the allowable Δσ at 2 × 10⁶ is 56 / 1.35 = 41.5 MPa; the applied 95 MPa exceeds it by 2.3× — design fails fatigue by a wide margin.

Step 4 — Remedies. Three real-world fixes, ranked by cost-effectiveness:

1. TIG-dressing or burr-grinding the weld toe raises the detail category by typ. one bin (FAT 56 → FAT 71); recomputed life N = 2 × 10⁶ · (71/95)³ = 8.4 × 10⁵ cycles, still inadequate but a 2× improvement.
2. Full-penetration butt weld replacing the fillet — FAT 80 as-welded → 1.2 × 10⁶ cycles; FAT 90 ground flush + NDT → 1.7 × 10⁶ cycles.
3. Reduce nominal stress range — usually achieved by adding cross-section or modifying load path. Δσ from 95 to 56 MPa restores life to the FAT-56 endurance plateau (~10⁷ cycles, 2 years per the m = 5 region).

Comment. A common practitioner’s mistake is to upgrade base-metal grade (S355 → S690) hoping for longer fatigue life. Eurocode 3 detail categories are independent of grade — the upgrade buys nothing for welded fatigue at the same geometry. The only fatigue-life levers on a welded detail are geometry, residual-stress state (post-weld treatment), and applied stress range.

Worked Example E — Paris-Erdogan crack-growth integration (bridge to fracture-mechanics)

Problem. A wide steel plate (σ_u = 565 MPa, K_Ic = 60 MPa·√m) carries a remote tensile stress range Δσ = 200 MPa, R = 0.1. An NDT-detected edge crack of length a₀ = 1 mm exists. Paris constants (long crack, R ≈ 0.1): C = 6.9 × 10⁻¹² (units: m/cycle, MPa·√m), m = 3.0. Find remaining life N_p.

Step 1 — Stress intensity range. For an edge crack in a wide plate, K = 1.12 · σ · √(π a). The range:

ΔK = 1.12 · Δσ · √(π a)

Step 2 — Critical crack length. Crack becomes unstable when K_max = K_Ic:

σ_max = Δσ / (1 − R) = 200 / 0.9 = 222.2 MPa
a_c = (K_Ic / (1.12 · σ_max))² / π
    = (60 / (1.12 · 222.2))² / π
    = (60 / 248.9)² / π = (0.241)² / π = 0.0185 m = 18.5 mm

Step 3 — Integrate da/dN. Paris-Erdogan:

da/dN = C · (ΔK)^m = C · (1.12 · Δσ · √(π a))^m

For m = 3.0, the integration from a₀ to a_c is analytic:

N_p = (a_c^(1−m/2) − a₀^(1−m/2)) / [C · (1.12 · Δσ · √π)^m · (1 − m/2)]
    = (a_c^(−0.5) − a₀^(−0.5)) / [6.9 × 10⁻¹² · (1.12 · 200 · √π)³ · (−0.5)]

Compute the bracket: 1.12 · 200 · √π = 1.12 · 200 · 1.7725 = 397.0. Cubed: 6.26 × 10⁷. C · 397.0³ = 6.9 × 10⁻¹² · 6.26 × 10⁷ = 4.32 × 10⁻⁴. (1 − m/2) = −0.5. Denominator: 4.32 × 10⁻⁴ · (−0.5) = −2.16 × 10⁻⁴.

Numerator: a_c^(−0.5) − a₀^(−0.5) = (0.0185)^(−0.5) − (0.001)^(−0.5) = 7.35 − 31.62 = −24.27.

N_p = −24.27 / −2.16 × 10⁻⁴ = 1.12 × 10⁵ cycles

Step 4 — Inspection-interval recommendation. Damage-tolerance practice sets the in-service inspection interval at N_p / 2 = 5.6 × 10⁴ cycles, so at most one cycle’s worth of inspection-missed growth can occur before the next inspection. Combined with a probabilistic POD curve this becomes the calendar-time inspection schedule for the bridge or airframe.

Comment. The total fatigue life is N_initiation + N_propagation. For this plate, an S-N analysis at Δσ = 200 MPa, σ_a = 100 MPa, σ_m = 111 MPa would predict total N ≈ 5–10 × 10⁵ cycles to failure of an un-cracked specimen. The 1.1 × 10⁵ propagation life is one-fifth to one-tenth of that — initiation dominates clean structure, propagation dominates pre-cracked or NDT-monitored structure. Damage-tolerance methods integrate only the propagation part and treat initiation as having already occurred.

7. Edge cases & gotchas

7.1 Welded joints

Weld-toe and weld-root fatigue is the dominant failure mode in welded steel structures. The base-metal S-N curve does not apply. Eurocode 3 part 1-9, BS 7608, and IIW recommendations (XIII-2151-07) instead define detail categories — discrete S-N curves indexed by joint geometry, characterised by the fatigue strength at 2 × 10⁶ cycles. Selected Eurocode 3 categories (FAT, MPa, normal stress, R-ratio − 1, base metal grade-independent):

FAT (MPa @ 2×10⁶)	Detail	Notes
160	Rolled / extruded section, machined, polished	Best ferritic-steel base detail
125	Plain plate, oxy-cut edges, rolled	Practical “base metal” reference
100	Longitudinal continuous butt weld, ground flush	Best welded detail
90	Transverse butt weld, ground flush	High-quality nuclear / aerospace
80	Transverse butt weld, as-welded	Routine code minimum
71	Cruciform K-butt, as-welded	Typical structural connection
56	Transverse fillet weld, load-carrying	The “weak link” in many fabrications
36	Cruciform fillet weld with weld-root fatigue	Avoid by detailing — full-penetration if possible

The choice of detail dominates the fatigue life; base-metal grade upgrade buys nothing in a welded fatigue calculation.

7.2 Corrosion fatigue

Salt-water service eliminates the ferritic fatigue limit and drops the fatigue strength at 10⁷ cycles by 30–70 %. DNV-RP-C203 specifies separate S-N curves “in air”, “in seawater with cathodic protection”, and “in free corrosion”. No CA-endurance-limit knee.

7.3 Multiaxial fatigue

Real components see proportional and non-proportional combinations of bending, torsion, and pressure. Equivalent-stress criteria:

• Sines (1955) — combines σ_a (alternating) and σ_m (hydrostatic) — HCF, proportional only.
• Findley (1959) — critical-plane shear + normal stress; widely used in automotive shaft design.
• Fatemi-Socie (1988) — critical-plane shear strain + normal stress; strain-life version, used in nCode and FE-SAFE.
• Brown-Miller — critical-plane shear + normal strain; classic.

Use von Mises equivalent stress only for in-phase proportional HCF on ductile materials — it falls apart non-proportionally.

7.4 Variable temperature & creep-fatigue interaction

Above ~0.4 T_melt (homologous temperature), time-dependent deformation (creep) interacts with cyclic damage. Hold-time at peak stress accumulates creep damage that adds to cycle-by-cycle fatigue damage. ASME BPVC III Subsection NH provides a linear-damage-summation framework: D_total = D_fatigue + D_creep, failure at D_total = 1. Used for nuclear pressure vessel components, gas-turbine blades, and the high-temperature primary loop of next-generation reactors.

7.5 Environmental effects beyond corrosion

• Hydrogen embrittlement — H₂ ingress (cathodic protection, pickling, plating, HER on cathode) into high-strength steel (σ_u > 1000 MPa) cuts fatigue life by 5–100×. ASTM F519 baking specification; design rule: avoid σ_u > 1100 MPa in cathodically-protected service.
• Liquid metal embrittlement — Zn on stainless or brass-on-aluminum contact at elevated T; relevant to galvanised fasteners on austenitic frames.
• Vacuum vs air — vacuum extends fatigue life by 2–10× because oxide-rupture-and-reform damage at the crack tip is suppressed.

7.6 Other gotchas

• Sequence effects. A single tensile overload retards subsequent crack growth (Elber 1970 — plasticity-induced crack closure); a compressive overload accelerates it. Miner ignores both.
• Mean-stress relaxation. Under cyclic plasticity, applied σ_m relaxes towards zero over the first 10–100 cycles. Strain-life analyses must use the stabilised mean stress, not the initial one.
• Casting + AM defects. Pores, lack-of-fusion, and inclusions dominate fatigue life in cast and additively-manufactured parts. HIP (hot isostatic pressing) closes internal pores and can recover most of the wrought-equivalent life; surface defects need machining or peening.
• Fretting fatigue. Small relative motion (< 100 μm) at a clamped interface (bolted joint, shrink-fit hub) initiates cracks at the contact edge. Reduces effective σ_e by up to 70 %.
• Foreign object damage (FOD). Turbine blades hit by ingested debris carry a notch with K_t = 3–8; fatigue initiates there. The reason aero engines have FOD inspection intervals shorter than calendar overhauls.
• Rolling-contact fatigue (RCF). Bearings, gears, rails — sub-surface initiation under Hertzian contact (depth ≈ 0.78 a). Statistical Weibull L_10 life; see bearings §3.
• Threaded fasteners. Fatigue concentrates at the first engaged thread (typ K_f ≈ 3.0 for rolled, > 5 for cut threads). Preload reduces Δσ on the bolt provided the joint stays clamped — see fasteners-bolts.
• Strain-life vs stress-life boundary. Run both at the suspected transition; pick the more conservative. Pure stress-life over-predicts life in any plastic regime.
• Ratcheting under combined steady + cyclic loads (e.g. constant internal pressure + cyclic mechanical bending on a pressure pipe). Each cycle accumulates a small unidirectional plastic strain; even small per-cycle ratchet strains × 10⁶ cycles → gross deformation. ASME BPVC III NB-3222.5 shakedown check.
• Through-life inspection. Damage-tolerance design requires inspection intervals such that the largest crack escaping detection at one inspection cannot grow to critical size before the next. Driven by NDT detection probability curves (POD, MIL-HDBK-1823A) and the Paris da/dN integration in fracture-mechanics.

8. Standards & software

Codes (cite the dated revision in a real design package)

• ASME BPVC Section VIII Div 2 (2023) — design-by-analysis fatigue for pressure vessels; uses smooth-bar S-N with stress-intensity reduction.
• ASME BPVC III NH (2021) — elevated-temperature fatigue + creep, nuclear.
• EN 1993-1-9:2005 — Eurocode 3 part 1-9, fatigue of welded steel structures (the detail-category system shown in §7.1).
• BS 7608:2014+A1:2015 — British equivalent of EN 1993-1-9.
• IIW Recommendations XIII-2151r4-07/XV-1254r4-07 — international source for welded-joint S-N curves.
• DNV-RP-C203 (2021) — offshore + marine fatigue; air, CP, and free-corrosion S-N curves.
• AAR M-1003 (latest) — rail; rolling-stock fatigue qualification.
• AASHTO LRFD Bridge Design (9th, 2020) Section 6.6 — fatigue of welded steel bridges.
• AISI S100-16 — cold-formed steel fatigue.
• FAR 25.571 — aircraft damage tolerance and fatigue evaluation of structure.
• API 579-1 / ASME FFS-1 Annex 14 — fitness-for-service fatigue assessment of in-service equipment.
• FKM-Richtlinie (6th ed., 2012) — German analytical strength assessment, the de-facto European auto-industry guideline.

Test standards

• ASTM E466-21 — Force-controlled constant-amplitude axial fatigue testing.
• ASTM E606/E606M-21 — Strain-controlled fatigue testing.
• ASTM E739-10(R2015) — Statistical analysis of S-N data (fitting + confidence bands).
• ASTM E1049-85(R2017) — Cycle counting in fatigue analysis (the rainflow standard).
• ISO 12106:2017 — Strain-controlled axial fatigue, international equivalent of E606.
• ISO 12107:2012 — Statistical planning + analysis of S-N data.

Software

• nCode DesignLife (HBK) — industry-leading fatigue post-processor; reads FE results (ANSYS, Abaqus, NASTRAN), runs S-N, ε-N, Dang Van, weld, vibration fatigue.
• FE-SAFE (Dassault Simulia) — bundled with Abaqus; critical-plane multiaxial, weld, thermo-mechanical fatigue.
• MSC Fatigue (Hexagon) — paired with MSC NASTRAN/Patran.
• ANSYS Mechanical Fatigue Tool — built-in S-N with mean-stress correction; entry-level.
• Femfat (Magna) — Austrian automotive-favoured; strong on welded structures and gears.
• DesignLife Welds module / Verity (Battelle) — mesh-insensitive structural-stress method for welds.
• Pylife (BMW Group, open source) — Python library, rainflow + Miner + ε-N, used in automotive R&D.
• fatpack (Python, MIT) — lightweight rainflow + Miner, popular in offshore-wind analysis.
• eFatigue.com (Univ. of Illinois, ASTM-sponsored) — free online calculators with vetted strain-life database.
• MMPDS-19 (2024, formerly MIL-HDBK-5) — the certified aerospace-metallic-material database; ε-N parameters, S-N curves, allowables.

9. Residual-stress engineering for fatigue life

Compressive residual stress at the surface (or notch root) directly subtracts from the applied tensile σ_max — equivalent to a beneficial mean-stress shift. Tensile residual stress (welding shrinkage, machining tear, grinding burn) adds to applied tensile stress and shortens fatigue life. The principal techniques engineers deploy:

Technique	Mechanism	Typical σ_residual at surface	σ_e improvement	Depth
Shot peening (steel ball, ceramic, glass bead)	Surface yielding under repeated impacts; sub-surface elastic constraint locks in compression	−400 to −800 MPa	15–50 %	0.1–0.5 mm
Laser shock peening (LSP)	Plasma-driven shockwave plastically yields the surface	−500 to −1000 MPa	30–80 %	1–2 mm
Cavitation peening	Imploding cavitation bubbles drive surface plasticity	−300 to −600 MPa	10–30 %	0.05–0.2 mm
Roller burnishing / deep rolling	Hard roller imposes plastic deformation	−600 to −1200 MPa	30–100 %	0.5–2 mm
Cold expansion (Split Sleeve, FTI)	Oversize mandrel pulled through fastener hole → tangential compression	−500 to −900 MPa	3–5× life on fastener-hole fatigue	0.5–1 mm
Autofrettage (high-pressure shock + release)	Wall yields, residual hoop compression at bore	−400 to −600 MPa	2–5× fatigue life	through-thickness
Induction hardening (selective surface heat-treat)	Martensitic surface layer with volume expansion → core constraint	−300 to −500 MPa	30–60 %	0.5–5 mm
Case carburising / nitriding	Hard, compressively-stressed surface case from diffused C or N	−400 to −1000 MPa (nitriding deeper)	50–100 %	0.1–2 mm

Compressive residuals can relax under service load — particularly under cyclic plasticity (large Δε), elevated temperature (creep), and overloads above local yield. Aerospace LSP and FTI cold-expansion are specified to MIL-STD requirements precisely because their durability under service is documented.

Welding introduces tensile residuals of magnitude approaching σ_y at the weld toe, in the worst orientation for fatigue. Post-weld heat treatment (PWHT) reduces residual stress to 10–30 % of σ_y but does not restore the parent-metal S-N curve — the geometric stress riser remains. PWHT + weld-toe grinding gives best results.

10. Fractography — reading the fracture surface

A failed component tells you exactly how it failed if you know how to read the fracture surface. Fatigue fractures are diagnosable by eye and confirmable by SEM. Three regions, always present in a classical fatigue failure:

1. Initiation site — usually at the surface (sharpest stress riser); ratchet marks (sharp radial steps) indicate multiple simultaneous initiation sites, a sign of high stress amplitude relative to σ_e. Sub-surface “fish-eye” initiation at an inclusion is the signature of VHCF in clean steel.
2. Stable-growth zone — smooth, often with beach marks (macroscopic concentric arcs centred on the initiation site). Beach marks correspond to changes in load amplitude, temperature, or environment — a beach mark per shift, per flight, per startup cycle. Striations are the microscopic equivalent visible by SEM (one striation per cycle in many alloys; spacing 0.01–1 μm tracks the local da/dN).
3. Fast-fracture zone — rough, dimpled (ductile) or cleavage-faceted (brittle), at the geometrically-thin remaining ligament. The ratio of stable-growth area to fast-fracture area gives a qualitative measure of stress level: ~95 % stable + 5 % fast = low-stress HCF; ~50/50 = moderate stress; > 90 % fast fracture = single overload (not fatigue).

Feature	Meaning	Typical scale
Beach marks (macro)	Load-history changes	mm
Striations (SEM)	One per cycle (approx)	0.01–1 μm
Ratchet marks (radial)	Multiple initiation sites	mm
Fish-eye + inclusion	Sub-surface VHCF initiation	0.1–1 mm fish-eye, 10–50 μm inclusion
River pattern	Cleavage, brittle fracture (not fatigue)	μm
Dimples	Ductile fracture (overload)	μm
Final-fracture lip	Shear lip at free surface	mm

Fractographic verification is required before accepting a fatigue diagnosis on a failed part. Misdiagnosis of overload as fatigue (or vice versa) leads to the wrong corrective action — adding cycles of inspection to a part that actually failed once by overload buys nothing.

11. Engineering judgement & design philosophies

Three formal design philosophies dominate cyclic-load engineering. They are not interchangeable.

Safe-life. Calculate total life N_f from S-N or ε-N curves with material-property safety factors and scatter bands (typ. scatter factor of 4 on cycles, 1.5 on stress). Retire the component at a fraction of N_f. Used for: rail axles, automotive crank shafts, helicopter rotor heads. Strength: no inspection burden in service. Weakness: retires sound parts; assumes the S-N curve covers actual service.

Fail-safe. Design redundancy so that any single primary-structure failure can be detected and repaired before catastrophe. Used for: older transport aircraft, certain bridge designs. Strength: simple inspection and repair philosophy. Weakness: doubles or triples structural mass; not always achievable in slender structures.

Damage-tolerance. Assume an initial flaw of detectable size (per MIL-A-83444 / FAR 25.571 — typ. a₀ = 1.27 mm for fastener holes), integrate Paris-Erdogan da/dN crack growth, define inspection intervals so the largest undetected flaw cannot grow to critical before the next inspection. Used for: modern commercial aircraft, gas-turbine disks, offshore structures. Strength: rational basis for inspection; no false retirements. Weakness: requires reliable POD curves and accurate K-factor solutions; covered in fracture-mechanics.

In practice a complete safety case combines all three: safe-life sizing for the bulk of the structure, fail-safe redundancy at critical load paths, damage-tolerance inspection of high-K_t details.

12. Reference data

S-N coefficients (steel, axial R = −1, ASTM E466)

Material	σ_u (MPa)	σ_e’ (MPa, 10⁶)	b (Basquin)	Source
AISI 1018 normalised	440	220	−0.110	Bannantine 1990
AISI 1045 hot-rolled	565	280	−0.085	Stephens 2000
AISI 1045 Q&T 500 °C	725	360	−0.090	Stephens 2000
AISI 4140 Q&T 540 °C	1100	550	−0.080	Bannantine 1990
AISI 4340 Q&T 425 °C	1240	620	−0.076	Stephens 2000
AISI 304 stainless	590	250	−0.080	NIST
ASTM A572 Gr 50	450	220	−0.107	AISC

Strain-life parameters (Bannantine 1990; eFatigue.com)

Material	σ’_f (MPa)	b	ε’_f	c
AISI 1045 HR	948	−0.092	0.26	−0.445
AISI 4340 Q&T 1240 MPa	1655	−0.076	0.73	−0.620
AISI 4340 Q&T 1470 MPa	2000	−0.091	0.48	−0.600
AISI 304	1000	−0.110	0.17	−0.510
Al 7075-T6	1466	−0.143	0.41	−0.619
Al 2024-T351	1100	−0.124	0.22	−0.590
Ti-6Al-4V (annealed)	2030	−0.104	0.84	−0.690

Fatigue endurance limit / fatigue strength — selected alloys at 21 °C

Alloy / condition	σ_u (MPa)	σ_e (MPa)	σ_e / σ_u	Cycle base	Source
AISI 1018 normalised	440	220	0.50	10⁶	ASM Vol 19
AISI 1045 normalised	620	310	0.50	10⁶	ASM Vol 19
AISI 1095 spheroidised	660	280	0.42	10⁶	Stephens 2000
AISI 4140 Q&T 400 °C	1400	590	0.42	10⁶	Stephens 2000
AISI 4340 Q&T 540 °C	1100	540	0.49	10⁶	MMPDS-19
ASTM A36 / S235 plate	450	200	0.44	10⁶	AISC
ASTM A992 / Gr 50 plate	450	220	0.49	10⁶	AISC
Cast iron (gray, class 30)	210	100	0.48	10⁶	ASM Vol 1
Ductile iron (60-40-18)	415	165	0.40	10⁶	ASM Vol 1
6061-T6 aluminum	310	95	0.31	5×10⁸	Boller-Seeger
7075-T6 aluminum	570	160	0.28	5×10⁸	MMPDS-19
2024-T351 aluminum	470	140	0.30	5×10⁸	MMPDS-19
Ti-6Al-4V annealed	950	510	0.54	10⁷	MMPDS-19
304 SS annealed	590	240	0.41	10⁶	NIST cryo
316L SS annealed	560	230	0.41	10⁶	NIST
Inconel 718 (aged)	1370	620	0.45	10⁷	Special Metals
Copper C11000	220	80	0.36	5×10⁸	ASM
CFRP UD (R = 0.1, fibre-direction)	2200	~1200	0.55	10⁷	Talreja 2012

Aluminum, copper and most non-ferrous metals do not exhibit a true endurance limit — the values above are the fatigue strength at the cycle base shown. Use them with explicit life requirements; do not assume infinite life below them.

Surface-finish factor C_S for steel (Juvinall / Shigley)

Surface	C_S at σ_u = 400 MPa	C_S at σ_u = 1000 MPa	C_S at σ_u = 1600 MPa
Polished / ground / mirror	1.00	1.00	0.95
Machined / cold-drawn	0.90	0.78	0.68
Hot-rolled	0.78	0.55	0.40
As-forged	0.68	0.40	0.25
Corroded (water)	0.55	0.36	0.22
Corroded (salt water)	0.30	0.20	0.13

Notch fatigue factor K_f vs theoretical K_t (steel, Neuber a = 0.1 mm at σ_u = 700 MPa)

Notch root r (mm)	K_t = 1.5	K_t = 2.0	K_t = 2.5	K_t = 3.0
0.1	1.25	1.50	1.75	2.00
0.5	1.42	1.83	2.25	2.67
1.0	1.45	1.91	2.36	2.82
2.0	1.48	1.95	2.43	2.90
5.0	1.49	1.98	2.47	2.96
∞ (blunt notch)	1.50	2.00	2.50	3.00

Sharp notches (r → 0) give K_f → 1; blunt notches give K_f → K_t. The Neuber length a scales inversely with σ_u — high-strength alloys are more notch-sensitive.

Bolt fatigue — first-engaged-thread K_f (cut vs rolled)

Bolt grade	σ_u (MPa)	K_f cut threads	K_f rolled threads (after heat treat)	Endurance limit σ_e (MPa, R ≥ 0)
SAE 5 / Grade 5 / 8.8	830	3.0	2.2	80–100
SAE 7 / Grade 7	1030	3.0	2.2	90–110
SAE 8 / Grade 8 / 10.9	1030	3.0	2.2	100–130
ASTM A490 / Grade 12.9	1230	3.8	3.0	110–140

Rolled threads (formed by cold deformation, residual compression at the root) consistently outperform cut threads (machined or ground, residual tensile or zero). Per Shigley, rolled threads give ~30 % longer life than cut at equivalent K_f. In high-cycle fatigue applications always specify rolled threads, heat-treated after rolling preserved.

Weld fatigue detail categories — Eurocode 3 part 1-9 selected FAT values

FAT (MPa @ 2×10⁶, m = 3 below 5×10⁶, m = 5 above)	Detail
160	Rolled / extruded plates and sections, no welds
140	Sheared / oxy-cut edges, no surface imperfections
125	Plain plate, machine-cut edges, no welds
112	Longitudinal continuous automatic butt weld, ground flush, NDT
100	As 112 but without NDT
90	Transverse butt weld, ground flush, NDT
80	Transverse butt weld, as-welded
71	Cruciform full-penetration K-butt, as-welded
56	Transverse load-carrying fillet weld, weld-toe failure
50	As 56 but with longer attachment
45	Cover plate end on flange
36	Cruciform fillet, weld-root failure

Typical fatigue-design service lives (industrial benchmark)

Application	Cycles per service life	Equivalent calendar time	Governing regime
Aircraft fuselage pressurisation	10⁵	30 years	LCF/HCF, damage-tolerance
Aircraft wing-bending GAG	10⁵–10⁶	30 years	Variable amplitude, DT
Helicopter rotor blade	10⁸	10⁰⁰⁰ flight hours	HCF, safe-life
Wind-turbine main shaft	> 10⁹	20 years	VHCF
Wind-turbine blade root bolt	~10⁹	20 years	HCF, preloaded fatigue
Automotive crankshaft	10⁸	240 000 km	HCF
Automotive engine valve spring	10⁹	240 000 km	HCF
Rail axle	10⁹–10¹⁰	30 years	VHCF safe-life
Bridge girder (Class A road)	10⁸	100 years (50 trucks/hr)	HCF welded
Pressure-vessel start-stop	10³–10⁴	30 years	LCF
Gas-turbine HP disk	10⁴	25 000 hours	LCF + creep
Industrial pump shaft	10⁹	25 000 hours @ 3000 rpm	HCF
Robotic-arm joint reducer	10⁸	10 years (industrial duty)	HCF
Office chair gas spring	10⁵	10 years (50 cycles/day)	HCF
Smartphone hinge (foldable)	2 × 10⁵	5 years	LCF/HCF

Fatigue safety factors (typical, against σ_e and σ_y)

Industry	n_fatigue (σ_e)	n_yield (σ_y)	Source
AISC structural steel	1.5	1.67 LRFD / 1.5 ASD	AISC 360-22
ASME pressure-vessel	1.5–2.5 on Sa-curve	1.5 on σ_y	ASME BPVC VIII Div 2
Eurocode 3 welded fatigue	γ_Mf = 1.0 / 1.15 / 1.35	γ_M0 = 1.0	EN 1993-1-9 §3
Auto crankshaft (typ. OEM)	1.5	1.5	FKM Richtlinie
Aerospace damage-tolerance	2× life or 2× crack-growth	1.5 on σ_y	FAR 25.571
Helicopter (safe-life)	4–8 on cycles, 1.5 on stress	1.5	DEF STAN 00-970
Wind-turbine (DNV-GL)	1.25–1.35	1.1	DNVGL-ST-0376
Marine/offshore (DNV-RP-C203)	3 on life (DFF = 3)	—	DNV-RP-C203
Bicycle frame (ISO 4210)	typically 2 on life	1.5	ISO 4210-6
Consumer-product appliance	1.5–2.0	1.5	UL standard mfr practice

NDT methods for fatigue-crack detection — POD ranges

Method	Minimum detectable crack (typ.)	POD-90/95 size	Surface only?
Visual (10× magnifier)	1 mm length	3–5 mm	yes
Liquid penetrant (LPI)	0.5 mm length surface-breaking	1–2 mm	yes
Magnetic particle (MPI)	0.3 mm length, near-surface ferrous	1 mm	near surface
Eddy current (EC)	0.25 mm length surface	0.75 mm	near surface
Ultrasonic (UT, conventional)	1.5 mm half-depth	3 mm	volumetric
Phased-array UT (PAUT)	0.5 mm	1.5 mm	volumetric
Radiography (RT)	2 % wall, or ~1 mm length	depends on orientation	volumetric
Acoustic emission (AE)	active-crack only	growth-rate-dependent	active monitoring

POD curves are statistical and vary by inspector training, surface preparation, geometry, and material. MIL-HDBK-1823A is the reference for POD characterisation and aerospace damage-tolerance inspection scheduling.

13. Cross-references

• mechanics-of-materials — stress, K_t, principal stresses, Mohr’s circle (prerequisite).
• materials-steel — σ_u, σ_y, microstructure that drives σ_e ≈ 0.5 · σ_u and the 1400 MPa knee.
• materials-aluminum — no clear fatigue limit; design at 5×10⁸ cycles.
• materials-composites — CFRP fatigue is matrix-dominated; very different curves (flat slope, scatter dominates).
• beam-theory — provides σ from M·c/I that feeds σ_a, σ_m here.
• fasteners-bolts — preloaded-bolt fatigue; first-engaged-thread K_f.
• bearings — rolling-contact fatigue; L_10 Weibull treatment; sub-surface initiation.
• gears-power-transmission — gear-tooth bending fatigue (AGMA 2001) and pitting (Hertzian RCF).
• joining-welding — weld geometry → detail category; residual stress effects.
• fracture-mechanics — companion note; Paris-Erdogan crack-growth integration takes over after initiation.
• pipe-fittings — ASME BPVC Section VIII Div 2 fatigue rules.
• fem-fea — planned; fatigue post-processing (nCode, FE-SAFE) consumes FE stress results.
• vibration-dynamics — vibration-induced HCF (resonance + small amplitude → huge cycle count).
• manipulator-design — robot joint actuators see 10⁷-cycle service; gear-tooth + bearing fatigue dominate end-of-life.
• scientific — fatigue post-processors read .odb (Abaqus), .rst (ANSYS), .op2 (NASTRAN).

14. Citations

1. Stephens, R. I.; Fatemi, A.; Stephens, R. R.; Fuchs, H. O. Metal Fatigue in Engineering, 2nd ed. Wiley, 2000. ISBN 978-0-471-51059-8. The canonical modern reference.
2. Suresh, S. Fatigue of Materials, 2nd ed. Cambridge University Press, 1998. ISBN 978-0-521-57847-9. Materials-science treatment; crack mechanisms.
3. Dowling, N. E. Mechanical Behavior of Materials, 5th ed. Pearson, 2020. ISBN 978-0-13-485819-1. Strong on combining yield, fatigue, and fracture into one framework.
4. Bannantine, J. A.; Comer, J. J.; Handrock, J. L. Fundamentals of Metal Fatigue Analysis. Prentice Hall, 1990. ISBN 978-0-13-340191-1. The strain-life tabular database used in this note.
5. Juvinall, R. C.; Marshek, K. M. Fundamentals of Machine Component Design, 6th ed. Wiley, 2017. ISBN 978-1-119-32352-0. Marin factor framework.
6. Shigley, J. E.; Budynas, R. G.; Nisbett, J. K. Shigley’s Mechanical Engineering Design, 11th ed. McGraw-Hill, 2020. ISBN 978-0-07-339821-1. Shaft-fatigue design procedure.
7. Schijve, J. Fatigue of Structures and Materials, 2nd ed. Springer, 2009. ISBN 978-1-4020-6807-2. Aerospace damage-tolerance perspective.
8. Wöhler, A. “Versuche über die Festigkeit der Eisenbahnwagen-Achsen.” Zeitschrift für Bauwesen, 1858–1870. Original S-N curves on Prussian railway axles.
9. Basquin, O. H. “The exponential law of endurance tests.” ASTM Proceedings 10 (1910): 625–630. The power-law σ-N fit.
10. Coffin, L. F. “A study of the effects of cyclic thermal stresses on a ductile metal.” Trans. ASME 76 (1954): 931–950. Plastic-strain life law.
11. Manson, S. S. “Behavior of materials under conditions of thermal stress.” NACA Report 1170 (1954). Independent strain-life formulation.
12. Goodman, J. Mechanics Applied to Engineering. Longmans, Green, 1899. Mean-stress straight-line.
13. Gerber, W. “Bestimmung der zulässigen Spannungen in Eisen-Constructionen.” Z. Bayer Arch. Ing. Ver. 6 (1874): 101–110. Parabolic mean-stress.
14. Smith, K. N.; Watson, P.; Topper, T. H. “A stress-strain function for the fatigue of metals.” J. Materials 5(4) (1970): 767–778. The SWT parameter.
15. Morrow, J. “Cyclic plastic strain energy and fatigue of metals.” Internal Friction, Damping, and Cyclic Plasticity, ASTM STP 378 (1965): 45–87.
16. Miner, M. A. “Cumulative damage in fatigue.” J. Applied Mechanics 12 (1945): A159–A164. Linear damage rule.
17. Palmgren, A. “Die Lebensdauer von Kugellagern.” Z. Verein Deutscher Ingenieure 68 (1924): 339–341. Pre-Miner linear-summation hypothesis for bearings.
18. Endo, T.; Matsuishi, M. “Fatigue of metals subjected to varying stress.” Japan Society of Mech. Eng. (1968). Rainflow algorithm.
19. Neuber, H. “Theory of stress concentration for shear-strained prismatical bodies.” J. Appl. Mech. 28 (1961): 544–550. Neuber’s notch rule.
20. Peterson, R. E. Stress Concentration Factors. Wiley, 1974. Empirical fatigue notch factor.
21. Elber, W. “Fatigue crack closure under cyclic tension.” Eng. Fracture Mech. 2 (1970): 37–45. Plasticity-induced closure → overload retardation.
22. Haibach, E. Betriebsfestigkeit. VDI-Verlag, 1989. Modified Miner with second-slope below the CA-limit knee.
23. ASTM E466-21 — Standard Practice for Conducting Force Controlled Constant Amplitude Axial Fatigue Tests of Metallic Materials.
24. ASTM E606/E606M-21 — Standard Test Method for Strain-Controlled Fatigue Testing.
25. ASTM E1049-85 (R2017) — Standard Practices for Cycle Counting in Fatigue Analysis.
26. EN 1993-1-9:2005 — Eurocode 3: Design of steel structures — Part 1-9: Fatigue.
27. BS 7608:2014+A1:2015 — Guide to fatigue design and assessment of steel products.
28. DNV-RP-C203 (2021) — Fatigue design of offshore steel structures.
29. IIW XIII-2151r4-07 / XV-1254r4-07 — Recommendations for Fatigue Design of Welded Joints and Components.
30. MMPDS-19 (2024) — Metallic Materials Properties Development and Standardization Handbook.