Fatigue Strength Calculator (Basquin, Wöhler, Goodman)

Calcola la vita a fatica di un componente metallico (curva di Wöhler S-N, Basquin). Correzione Goodman per tensione media σ_m. Limite di fatica di acciaio e alluminio, coefficiente di sicurezza e regime LCF/HCF. Gratis, in 5 lingue.

Calculation parameters
Was this calculator useful?

What is fatigue in materials

Fatigue is the phenomenon where a metallic component fails under cyclic stresses far below its static strength σ_UTS. It is the most insidious failure mode because: (1) it occurs without macroscopic plastic deformation as warning; (2) can appear after years of apparently normal service; (3) is responsible for 50-90% of in-service failures of shafts, connecting rods, gears, turbine blades, welded joints, railway wheels, pressurised pipes. The first systematic study was by August Wöhler (1858) on railway axle rollers: he showed that a rod under millions of alternating stress cycles could fail at stresses equal to half its static limit, and introduced the famous σ-N curve (Wöhler or S-N curve). Modern fatigue design is regulated in Eurocode 3 part 1-9 for steel, ASME BPVC Section VIII Div 2 for pressure vessels, DNV-GL RP-C203 for offshore, FKM Richtlinie for German mechanical components.

The Wöhler S-N curve and Basquin's law

The S-N curve is experimentally obtained by subjecting specimens to alternating stresses σ_a of various amplitudes and recording the cycles N to failure. In log-log scale the curve shows a linear region from 10³ to 10⁶ cycles described by Basquin's law (1910):

σ_a = σ'_f · (2N)^b  or  σ_a = C · N^b

where σ'_f is the fatigue strength coefficient (typically 1.3-1.7 times σ_UTS), b is the Basquin exponent (negative, -0.05 to -0.15 for common metals). Steels have a knee around 10⁶ cycles beyond which the curve becomes horizontal (fatigue limit σ_lim); aluminium has no true fatigue limit but the curve continues to fall until 10⁸-10⁹ cycles. Practical simplification: taking two points (10³ cycles, 0.9·σ_UTS) and (10⁶ cycles, 0.5·σ_UTS for steels; 0.35·σ_UTS for Al at 5·10⁸): σ_a = 0.9·σ_UTS · (N/10³)^(-m) with m ≈ 0.085 steels, 0.072 aluminium.

Fatigue limit: steels vs aluminium

The fatigue limit (endurance limit) σ_lim is the stress below which the material can withstand infinite life. Only some metals have it:

  • Ferritic steels and some carbon steels: true fatigue limit at σ_lim ≈ 0.5·σ_UTS (widely verified rule).
  • Aluminium, copper, titanium, austenitic steels (AISI 304, 316): NO true fatigue limit. Design uses "conventional limit" at 5·10⁸ cycles (Al) or 10⁷ (SS) with f_end ≈ 0.35-0.40.
  • Composite materials: much better than metals, flatter S-N curve. σ_lim ≈ 0.7-0.8·σ_UTS.
  • Real-service reduction factors: lab-specimen σ_lim must be reduced for real conditions: surface roughness (Ka = 0.7-0.9 rolled, 0.3-0.5 forged), size (Kb = 0.8-0.9), reliability (Kc = 0.7-0.9 for 99%), temperature (Kd reduced above 400 °C steels), corrosion (Kf = 0.3-0.7 seawater vs clean air). Marin formula: σ_lim_real = Ka·Kb·Kc·Kd·Ke·Kf · σ_lim_specimen.

Haigh diagram and Goodman correction

In reality applied stress is not purely alternating: the cycle has a mean stress σ_m superimposed on the alternating amplitude σ_a. The Haigh diagram shows the (σ_m, σ_a) pair at infinite-life fatigue limit. Reference theoretical curves:

  • Goodman (1899): line σ_a/σ_lim + σ_m/σ_UTS = 1. Simple, conservative for ductile and brittle materials.
  • Soderberg (1930): line σ_a/σ_lim + σ_m/σ_y = 1 (uses yield stress). Very conservative.
  • Gerber (1874): parabola σ_a/σ_lim + (σ_m/σ_UTS)² = 1. Less conservative, accurate for ductile.
  • ASME elliptic: ellipse (σ_a/σ_lim)² + (σ_m/σ_UTS)² = 1.

Goodman correction gives equivalent alternating stress at σ_m = 0:

σ_a_eq = σ_a / (1 − σ_m/σ_UTS)

Compare σ_a_eq with σ_lim for safety factor and fatigue life. If σ_m < 0 (mean compression), Goodman doesn't apply: mean compression increases fatigue life (often ignored for safety).

HCF vs LCF: two fatigue regimes

Metal fatigue has two regimes:

  • HCF (High Cycle Fatigue, N > 10⁴-10⁵ cycles): cyclic stress in elastic range (σ_a < σ_y). Basquin's law σ_a = C·N^b applies. Typical of continuous-service components (rotating shafts at 1500-3000 rpm, springs, welded joints under variable loads). Life 10⁶-10⁹ cycles. Failure "beach mark" (crack growth from microstructural defect).
  • LCF (Low Cycle Fatigue, N < 10⁴ cycles): cyclic stresses locally exceeding yield (cyclic plastic strain ε_a > 0.2%). Basquin doesn't apply, uses Coffin-Manson law (1954): ε_a_pl · N^c = C_2 with c ≈ -0.5. Typical of thermal shock components (turbine blades, combustion chambers), extreme manoeuvres (aircraft takeoff/landing), pulsating pressures. Life 10²-10⁴ cycles.

This calculator covers only HCF. For LCF specific ε-N characterization tests and dedicated software are needed.

Factors reducing real fatigue life

An ideal lab specimen has higher fatigue life than a real component. Reduction factors (Marin formula):

  • Surface roughness (Ka): hot-rolled Ka = 0.70; cold-rolled 0.85; ground 0.90; polished 1.00.
  • Size (Kb): small specimens (φ 8 mm) have higher σ_lim than large components. Kb = 1.24·d^(-0.107) up to d = 250 mm.
  • Reliability (Kc): average σ_lim = 50% reliability; for 99% Kc = 0.814; for 99.9% Kc = 0.753.
  • Temperature (Kd): above 350 °C carbon steels degrade; above 500 °C creep-fatigue interaction.
  • Stress concentration (Kf): notches, holes, section changes, welds amplify stress locally. K_t (static) > K_f (fatigue) for ductile materials; Peterson: K_f = 1 + q·(K_t − 1).
  • Corrosive environment (Ke): seawater Ke = 0.3-0.5; industrial air 0.7-0.8; clean 1.0. Corrosion removes the fatigue limit (curve keeps falling).
  • Machining residual stresses: shot-peening introduces compressive surface stresses improving life 30-100%. MIG/MAG welding leaves tensile residual stresses reducing σ_lim 20-50%; requires stress relief heat treatment.

Typical fatigue applications

  • Rotating shafts: turbines, engines (rods), compressors. 25-100 Hz frequency, > 10⁹ cycles in service life.
  • Suspension springs: automotive (500,000 km ≈ 10⁷ cycles), railway (10⁹ cycles).
  • Welded joints: offshore structures, bridges. Eurocode 3 part 1-9 provides detail categories (36-160 MPa) at 2·10⁶ cycles.
  • Gears: bending fatigue at tooth root (Lewis-AGMA) and Hertz contact fatigue (pitting).
  • Railway wheels and crane runners: rolling contact fatigue (RCF).
  • Pressure vessels: pressurisation/depressurisation cycles. ASME BPVC VIII Div 2 provides specific σ-N design curves.
  • Gas turbine blades: thermomechanical fatigue (TMF), creep interaction, often LCF regime.
  • Bolted joints: fretting fatigue.
  • Aeronautical components: damage tolerance analysis (DTA) mandatory for FAR/EASA 25.571 certification.
  • Wind turbines: onshore 20 years ≈ 10⁷-10⁸ blade bending cycles.

How to use the calculator

Enter ultimate tensile strength σ_UTS in MPa (S235 = 340-470; S355 = 470-630; quenched-tempered 42CrMo4 = 900-1100; 6061-T6 aluminium = 310; 7075-T6 = 570). Enter alternating stress amplitude σ_a in MPa (half of σ_max − σ_min). Enter mean stress σ_m in MPa (0 for pure alternating R = -1). Choose material: 1 = steel (endurance limit at 10⁶ cycles, f_end = 0.50); 2 = aluminium (no true endurance, reference 0.35·σ_UTS at 5·10⁸). Enter cycle frequency in Hz (1500 rpm shaft = 25 Hz; auto suspension = 5-15 Hz). Calculator returns σ_lim, Goodman equivalent σ_a_eq, safety factor SF, fatigue life N in cycles (10⁹ if endurance), life in hours, Basquin exponent and assessment with regime classification.

Frequently Asked Questions

What is the R ratio and how does it affect fatigue life?

R ratio is defined as R = σ_min / σ_max. Typical cases: R = -1 fully reversed (σ_m = 0, e.g. rotating shaft), most severe; R = 0 zero-tension (0 → σ_max, e.g. repeated spring), σ_m = σ_a; R = 0.1 preloaded (bolts), high σ_m; R = 0.5 small swing on high preload. Higher R (toward +1) means less severe. Calculator uses σ_m and σ_a; if you have σ_min and σ_max: σ_a = (σ_max − σ_min)/2, σ_m = (σ_max + σ_min)/2. Standard tests (ASTM E466) typically at R = -1 (rotating bending) or R = 0 (pulsating).

Is the rule σ_lim = 0.5·σ_UTS always valid?

Only for ferritic steels (low/medium carbon, quenched-tempered) with σ_UTS < 1400 MPa. For very hard steels (σ_UTS > 1400 MPa) the relation saturates: σ_lim ≈ 700 MPa regardless (dominated by non-metallic inclusions as crack initiators). For austenitic stainless f_end ≈ 0.40. For grey cast iron 0.4-0.45. For aluminium no true endurance, σ_lim at 5·10⁸ cycles reference (f_end ≈ 0.35). For titanium Ti-6Al-4V: f_end ≈ 0.55-0.60 (exceptionally fatigue-resistant).

How does the calculator handle negative σ_m (compression)?

In mean compression (σ_m < 0) fatigue life increases vs pure alternating, because compression closes microcracks slowing propagation. Formulas exist (modified Goodman with |σ_m|, or Morrow with σ'_f) but are minor improvements. Practical design ignores compression benefit and treats as σ_m = 0 (conservative). Calculator accepts only σ_m ≥ 0; for predominantly compressive cycles use σ_m = 0 for conservative result.

What is Eurocode 3 double-slope S-N curve?

EC3 part 1-9 (Steel fatigue) uses double-slope S-N curves: slope m₁ = -3 between 10⁴ and 5·10⁶ (dominant regime); m₂ = -5 between 5·10⁶ and 10⁸ (variable amplitude); then cut-off at 10⁸. Reference is detail category Δσ_C at 2·10⁶ cycles (36, 40, 45, 50, 56, 63, 71, 80, 90, 100, 112, 125, 140, 160 MPa) for 90 structural detail types. Different from classical Wöhler "infinite life above endurance" approach. For welded structure design use EC3 directly, not this simplified calculator.

Why does welding reduce fatigue limit so much?

Because welding introduces: (1) tensile residual stresses typically equal to material σ_y, acting as preload; (2) geometric discontinuities (bead concavity, undercut) with stress concentration K_t = 2-5; (3) HAZ (heat-affected zone) with brittle microstructure; (4) internal defects (porosity, slag inclusions, lack of fusion). Welded joint fatigue limit typically 50-100 MPa regardless of base material σ_UTS. Eurocode 3 treats welded joint fatigue independently of σ_UTS: applies tabulated Δσ_C for joint type. Post-weld heat treatment (PWHT) and grinding improve σ_lim 50-100%.

How to apply Miner Rule for time-variable load?

Palmgren-Miner rule (1945) handles variable-amplitude loads: if component sees n_1 cycles at σ_a1, n_2 at σ_a2, etc., accumulated damage D = Σ (n_i / N_i), where N_i is failure cycles at σ_ai. Failure at D = 1 (practically D_crit = 0.5-0.8 for safety). Procedure: (1) analyse measured or simulated dynamic load → σ(t) time series; (2) Rainflow counting (ASTM E1049) → extract half-cycles as σ_max/σ_min pairs with n_i; (3) for each pair compute σ_a, σ_m and get N_i from S-N with Goodman; (4) sum n_i/N_i. If D_tot < 0.5 wide safety. If D_tot > 1 failure predicted. This calculator does step (3) for single cycle; full Miner needs software (fe-safe, ncode, ANSYS Fatigue).

How to compare with professional software (fe-safe, ncode DesignLife, ANSYS)?

Professional software like fe-safe (SIMULIA), ncode DesignLife (HBM), ANSYS Fatigue Module, MSC.Fatigue integrate fatigue with FEA: from stress-strain results at each 3D component node under realistic cycle load, apply multiaxial criteria (Sines, Findley, McDiarmid, Dang Van), material-specific S-N curves (FKM, BS 7608, DNV databases), mean correction (Goodman, Gerber, Morrow, SWT), Rainflow + Miner, crack propagation (Paris law). This calculator is analytical uniaxial estimate, ±30-50% accuracy. For critical components (aerospace, offshore, medical) always use FEA + validated fatigue software.

Comments (0)

No comments yet. Be the first!