Machine design: failure theories and fatigue

In short

Failure theories answer one question: a tensile test gives one number (yield or ultimate strength), but a real machine part carries a combination of normal and shear stresses — so when does it fail? Fatigue answers the harder question: why a shaft stressed to only 40% of its yield strength snaps after ten million reversals. Together they decide the diameter of every shaft, the fillet radius of every step and the surface finish specified on every drawing. Static theories (Rankine, Tresca, von Mises) and the fatigue block (S-N curve, endurance limit, modifying factors, Soderberg/Goodman/Gerber, Miner's rule) are a guaranteed GATE question and the standard screening question in design interviews at L&T, Tata Motors, Ashok Leyland, Mahindra, BHEL, Godrej & Boyce, Cummins, Kirloskar, ISRO/DRDO and every captive R&D centre.

Explained

Design philosophy, factor of safety and the stress state

• Failure in design means loss of function: yielding (permanent set), fracture, fatigue cracking, buckling, excessive deflection, wear or creep. A theory of failure predicts only the first two.
• Factor of safety (FOS) = strength / working stress. For ductile materials strength = yield strength *Syt* (yielding is failure); for brittle materials (cast iron, ceramics) strength = ultimate strength *Sut*, since they fracture with almost no yield.
• Typical FOS: 1.25–2 for known loads and ductile material, 2–3 for average service, 3–5 for shock loads/uncertain data, 10–12 historically for cast iron in older practice. Codes fix it: IS 2825 for unfired pressure vessels bases allowable stress on a fraction of tensile and yield strength (of the order of *Sut*/4 and *Syt*/1.5 — use the current edition's table); IS 800:2007 uses the limit state approach with partial safety factors (γm = 1.10 on yielding, 1.25 on ultimate) instead of a single FOS.
• Any 2-D stress state (σx, σy, τxy) must first be reduced to principal stresses; all theories are written in terms of σ1 ≥ σ2 ≥ σ3. In plane stress the out-of-plane principal stress is zero — and that zero must still be counted in the shear-stress theory (it is σ3 when both in-plane principals are tensile, σ2 when they are of opposite sign).

Static theories of failure

1. Maximum principal (normal) stress theory — Rankine. Failure when the largest principal stress reaches the uniaxial strength. Simple, and the correct theory for brittle materials in tension. Unsafe for ductile materials in pure shear (it predicts Ssy = Syt, whereas tests give about 0.5–0.6 Syt). Yield locus: a square.
2. Maximum shear stress theory — Tresca/Guest. Failure when τmax reaches Syt/2. Predicts Ssy = 0.5 Syt. Most conservative of the ductile theories and therefore the usual code choice (ASME shaft code, many pressure-vessel rules). Locus: a hexagon inscribed in the von Mises ellipse.
3. Maximum distortion energy theory — von Mises–Hencky (shear strain energy). Failure when the energy of distortion equals that at uniaxial yield; hydrostatic stress is assumed not to cause yielding. Predicts Ssy = 0.577 Syt. Best agreement with experiment for ductile metals; the default in FEA post-processing ("von Mises stress"). Locus: an ellipse.
4. Maximum principal strain theory — St. Venant. Failure when the largest principal strain reaches the yield strain: σ1 − μσ2 ≤ Syt/N. Rarely used; poor agreement in biaxial tension.
5. Total strain energy theory — Haigh. Failure when total stored strain energy reaches the uniaxial value. Superseded by von Mises, because hydrostatic pressure clearly does not yield metals.
6. Mohr–Coulomb / modified Mohr theory. For brittle materials whose compressive strength *Suc* far exceeds *Sut* (grey cast iron: Suc ≈ 3–4 × Sut): σ1/Sut − σ3/Suc ≤ 1/N when σ1 > 0 > σ3.
7. Ranking in the first quadrant (both principal stresses tensile) the three ductile theories nearly coincide; in the second and fourth quadrants (opposite signs, e.g. torsion) they diverge most — Tresca is up to 15% more conservative than von Mises. Pure shear is the worst case and the favourite exam case.

Fatigue: the phenomenon

• Fatigue is progressive, localised, permanent damage under fluctuating stress, leading to fracture at a stress far below the static strength. Roughly 80–90% of in-service mechanical failures are fatigue failures.
• Three stages: (i) crack initiation at a surface discontinuity — a fillet, keyway, tool mark, inclusion or corrosion pit; (ii) crack propagation, one small increment per cycle, leaving beach marks / striations; (iii) sudden final fracture when the remaining section can no longer carry the load.
• Fracture surface signature: a smooth, shiny, often rust-stained propagation zone with concentric beach marks, plus a rough crystalline final-fracture zone — no necking and no warning, which is what makes fatigue dangerous.
• Stress cycle terms: σmax, σmin, mean σm = (σmax + σmin)/2, amplitude σa = (σmax − σmin)/2, range = 2σa, stress ratio R = σmin/σmax. Completely reversed loading: σm = 0, R = −1. Repeated loading: σmin = 0, R = 0.
• S-N (Wöhler) curve: fatigue strength against number of cycles, plotted log-log from a rotating-beam test on a polished 7.5 mm specimen.
  • Low-cycle fatigue (LCF): N < 10³ cycles, plastic strain dominates, designed by strain-life (Coffin–Manson) — pressure-vessel start-ups, press tools, aircraft landing gear.
  • High-cycle fatigue (HCF): N > 10³, essentially elastic, designed by stress-life.
  • Endurance limit (Se′): for ferrous metals the S-N curve becomes horizontal near 10⁶ cycles — below Se′ life is "infinite". Non-ferrous metals (aluminium, copper) and austenitic stainless steels have no true endurance limit; a fatigue strength at 5 × 10⁸ cycles is quoted instead.
• Approximate estimates (no test data): steel Se′ ≈ 0.5 Sut (take 700 MPa if Sut > 1400 MPa); cast iron ≈ 0.4 Sut; aluminium alloys ≈ 0.3–0.4 Sut. For the loading mode: bending ≈ 0.5 Sut, axial ≈ 0.45 Sut, torsion ≈ 0.29 Sut. Fatigue strength at 10³ cycles ≈ 0.9 Sut in bending, 0.75 Sut in axial loading.

From specimen to component: endurance limit modifying factors

The laboratory Se′ is multiplied by factors to get the component endurance limit Se (Shigley's lettering; Indian texts such as Bhandari use ka surface, kb size, kc reliability — the physics is identical):

• ka — surface finish: ground ≈ 0.9–1.0, machined/cold-drawn ≈ 0.75–0.85, hot-rolled ≈ 0.6–0.7, as-forged ≈ 0.4–0.5 (lower as Sut rises). The single biggest controllable factor, because cracks start at the surface.
• kb — size: 1.0 up to 7.5 mm, ≈ 0.85 for 7.5–50 mm, ≈ 0.75 above 50 mm (or kb = 1.24 d⁻⁰·¹⁰⁷ for 2.8–51 mm). Larger sections have more volume of highly stressed surface and more defects.
• kc — loading: 1.0 bending, 0.85 axial, 0.59 torsion (used when Se′ is taken as the bending value).
• kd — temperature: ≈ 1.0 up to about 350 °C for steel; falls above it (creep-fatigue interaction).
• ke — reliability: 1.0 for 50%, 0.897 for 90%, 0.868 for 95%, 0.814 for 99%, 0.753 for 99.9% (based on an 8% standard deviation).
• kf — miscellaneous: corrosion, plating, fretting, residual stress, decarburisation.
• Stress concentration is handled separately by the fatigue stress concentration factor Kf, applied to the *alternating* stress (local yielding at the notch relieves the mean stress in ductile materials; conservative practice applies it to both components). Kf = 1 + q(Kt − 1), where Kt is the theoretical (elastic) factor from charts and q is the notch sensitivity (0 = fully insensitive, as in grey cast iron, which is already full of internal notches; 1 = fully sensitive, as in high-strength steel).
• Remedies for stress concentration: generous fillet radii, relief grooves/notches, drilled holes at crack-prone corners, gradual section change, removing material to spread the flow of stress, and inducing compressive residual stress by shot peening, cold rolling, nitriding, carburising or autofrettage.

Fluctuating stress: mean-stress criteria

• A fluctuating stress is split into σm and σa. A tensile mean stress reduces fatigue strength; a compressive mean stress is harmless or beneficial (hence shot peening).
• On a diagram with σm on the x-axis and σa on the y-axis:
  • Soderberg line: (0, Se) to (Syt, 0) — most conservative, guards against yielding automatically.
  • Goodman line: (0, Se) to (Sut, 0) — straight, slightly conservative, widely used.
  • Gerber parabola: through the same end points but bulging outward — fits test data best, so it is the least conservative.
  • ASME elliptic: ellipse through (0, Se) and (Syt, 0).
  • Langer yield line: σa + σm = Syt, guarding against first-cycle yielding.
• The modified Goodman diagram is the *lower envelope* of the Goodman line and the Langer line; the safe region is the area bounded by them. For σm < 0 the boundary is taken as the horizontal line σa = Se.
• Combined fatigue loading (bending + torsion, the shaft case): compute the von Mises equivalent of the alternating components (σ′a) and of the mean components (σ′m) separately, then enter Goodman/Soderberg with these. Check first-cycle yielding separately with σ′max.
• Cumulative damage — Palmgren–Miner rule: Σ(ni/Ni) = 1 at failure, where ni cycles are applied at a level whose life is Ni. Linear, ignores load sequence and the fact that overloads can retard or accelerate cracking; still the standard design tool.
• Fracture mechanics view (damage-tolerant design): the crack-tip stress intensity K = Yσ√(πa); the crack grows per Paris law da/dN = C(ΔK)^m (m ≈ 3 for steels) and fails when K reaches the fracture toughness KIC. This underlies inspection intervals on pressure vessels, boiler drums and aircraft structures rather than general machine design.
• Special fatigue modes: corrosion fatigue (no endurance limit survives in a corrosive medium), fretting fatigue at press-fits and splines, thermal fatigue from repeated temperature gradients, contact/surface fatigue (pitting) in gears and bearings, and creep-fatigue at high temperature.

Indian standards and codes

• IS 800:2007, Section 13 — fatigue design of steel structures and welded joints: detail categories, S-N curves with the fatigue strength quoted at 5 million cycles, a constant-amplitude fatigue limit, and Miner's rule for variable amplitude. Fatigue check is required only where the number of load cycles exceeds the stated threshold.
• IS 2825 (unfired pressure vessels) — allowable stresses and the design margin on tensile/yield strength; fatigue (cyclic) analysis is required for vessels with many pressure cycles.
• IS 2062 (hot-rolled structural steel: E250, Syt 250 MPa, Sut 410 MPa min) and IS 1570 (schedules of wrought steels, including alloy steels such as 40Cr4Mo2, 20MnCr5, 42CrMo4 used for shafts and gears) supply the strength data.
• IS 1608 (tensile test) and Charpy impact testing give the static properties; fatigue test methods follow IS/ASTM practice (rotating bending, ASTM E466 for axial fatigue).
• IS 1367 governs fastener property classes (8.8, 10.9, 12.9), relevant because bolt pre-load is what keeps the alternating stress in a bolt small.
• IS 3073 / IS 10719 surface roughness specification on drawings is a *fatigue* requirement, not cosmetic.
• Welded fatigue detail: toe grinding, full-penetration butt welds in place of fillet welds, and avoiding weld terminations in high-stress zones.

Key points

• Factor of safety — strength ÷ working stress; use Syt for ductile, Sut for brittle materials.
• Rankine (MPST) — σ1 ≤ Syt/N; correct for brittle, unsafe for ductile in shear.
• Tresca (MSST) — σ1 − σ3 ≤ Syt/N; predicts Ssy = 0.5 Syt; most conservative; used by codes.
• von Mises (DET) — best fit for ductile metals; Ssy = 0.577 Syt; the stress FEA plots.
• Pure shear is the discriminating case — the theories agree in biaxial tension and differ most in torsion.
• Mohr–Coulomb — for brittle materials with Suc ≫ Sut (grey cast iron).
• Fatigue — failure below yield stress under repeated loading; crack initiation, propagation, sudden fracture.
• Beach marks and no necking — the visual signature of a fatigue fracture.
• Endurance limit — the stress below which steel has infinite life; absent in aluminium and copper alloys.
• Se′ ≈ 0.5 Sut for steel in reversed bending, as a first estimate.
• Modifying factors — surface finish, size, loading type, temperature, reliability; surface finish matters most.
• Kf = 1 + q(Kt − 1) — notch sensitivity converts the elastic Kt into the fatigue factor.
• Soderberg > Goodman > Gerber in conservatism (Soderberg most conservative); modified Goodman adds the Langer yield line.
• Compressive mean stress helps — the whole basis of shot peening, cold rolling and nitriding.
• Miner's rule — Σni/Ni = 1; linear damage, ignores load order.
• LCF is strain-controlled (Coffin–Manson), HCF is stress-controlled (Basquin/S-N).
• Combined fatigue — take von Mises of alternating and mean parts separately, then apply Goodman.
• IS 800:2007 Section 13 is the Indian fatigue code for steelwork and welds; IS 2825 covers vessels.

Formulas and examples

Example 1 — the three theories compared. At a point: σx = 80 MPa, σy = −40 MPa, τxy = 30 MPa; Syt = 300 MPa (ductile). Find the FOS by each theory.

• Mean = (80 − 40)/2 = 20 MPa; radius = √(60² + 30²) = √4500 = 67.08 MPa.
• In-plane principals 20 ± 67.08, so σ1 = 87.08 MPa, σ2 = 0 (out-of-plane), σ3 = −47.08 MPa.
• Rankine: N = 300/87.08 = 3.45.
• Tresca: σ1 − σ3 = 87.08 − (−47.08) = 134.16 → N = 300/134.16 = 2.24.
• von Mises: σ′ = √(80² − 80(−40) + (−40)² + 3 × 30²) = √(6400 + 3200 + 1600 + 2700) = √13900 = 117.9 → N = 300/117.9 = 2.54.
• Tresca is the most conservative; design to von Mises if data are reliable, to Tresca if a code demands it.

Example 2 — endurance limit of a real shaft. A machined steel shaft, d = 40 mm, Sut = 600 MPa, carries a completely reversed bending moment of 250 N·m. A step fillet gives Kf = 1.5. Reliability 90%. Find the FOS for infinite life.

• Se′ = 0.5 × 600 = 300 MPa. ka (machined, 600 MPa) = 0.78; kb (40 mm) = 0.85; ke (90%) = 0.897.
• Se = 300 × 0.78 × 0.85 × 0.897 = 178.4 MPa; with Kf: Se = 178.4/1.5 = 118.9 MPa.
• σa = 32 × 250 000/(π × 40³) = 8 × 10⁶/201 062 = 39.79 MPa.
• FOS = 118.9/39.79 = 2.99. Note Se fell from 300 to 119 MPa — a 60% loss purely from finish, size, reliability and the fillet.

Example 3 — Soderberg versus Goodman sizing. A round bar carries an axial force fluctuating between 20 kN and 60 kN. Sut = 600 MPa, Syt = 450 MPa, corrected Se = 220 MPa, required FOS = 2.

• Pm = 40 kN, Pa = 20 kN → σm = 40 000/A, σa = 20 000/A (A in mm²).
• Soderberg: (20 000/A)/220 + (40 000/A)/450 = 0.5 → (90.91 + 88.89)/A = 0.5 → A = 359.6 mm² → d = √(4 × 359.6/π) = 21.4 mm, use 22 mm.
• Goodman: (90.91 + 66.67)/A = 0.5 → A = 315.2 mm² → d = 20.0 mm.
• Soderberg demands about 7% more diameter (14% more area) because it measures the mean stress against yield, not ultimate.

Example 4 — cumulative damage (Miner's rule). Sut = 700 MPa, Se = 350 MPa at 10⁶ cycles, S = 0.9 Sut = 630 MPa at 10³ cycles (log-log S-N line). One duty block: 2 000 cycles at 500 MPa, 5 000 at 450 MPa, 20 000 at 400 MPa. How many blocks to failure?

• Slope: b = (log 350 − log 630)/3 = (2.5441 − 2.7993)/3 = −0.08509; log N = 3 + (log S − 2.7993)/(−0.08509).
• S = 500: log N = 3 + (2.6990 − 2.7993)/(−0.08509) = 4.180 → N₁ ≈ 15 120 cycles.
• S = 450: log N = 3 + (2.6532 − 2.7993)/(−0.08509) = 4.717 → N₂ ≈ 52 170 cycles.
• S = 400: log N = 3 + (2.6021 − 2.7993)/(−0.08509) = 5.318 → N₃ ≈ 208 200 cycles.
• Damage per block = 2000/15 120 + 5000/52 170 + 20 000/208 200 = 0.1323 + 0.0958 + 0.0961 = 0.3242.
• Blocks to failure = 1/0.3242 = 3.09 blocks, i.e. 3.09 × 27 000 ≈ 83 300 cycles.

Example 5 — combined fatigue on a shaft. d = 30 mm; completely reversed bending M = 150 N·m; steady torque T = 200 N·m. Sut = 700 MPa, Syt = 500 MPa, corrected Se = 200 MPa (Kf included). Find the FOS by Goodman and check yielding.

• πd³ = π × 27 000 = 84 823 mm³.
• σa = 32 × 150 000/84 823 = 56.59 MPa (alternating, bending). τm = 16 × 200 000/84 823 = 37.73 MPa (mean, torsion).
• σ′a = √(56.59² + 3 × 0²) = 56.59 MPa; σ′m = √(0² + 3 × 37.73²) = 1.732 × 37.73 = 65.35 MPa.
• Goodman: 1/N = 56.59/200 + 65.35/700 = 0.2830 + 0.0934 = 0.3763 → N = 2.66.
• Yield check: σ′max = √(56.59² + 3 × 37.73²) = √(3202 + 4271) = 86.4 MPa → N = 500/86.4 = 5.8. Fatigue governs, as it almost always does on a rotating shaft.

Where it is used

• Automotive and tractor design (Tata Motors, Mahindra, Ashok Leyland, Bajaj, TVS): crankshafts, connecting rods, leaf and coil springs, steering knuckles, axle shafts — all sized on modified Goodman diagrams and validated on rig fatigue tests to a target number of cycles.
• Power and process plant (BHEL, L&T, Thermax, Godrej & Boyce): boiler drums, turbine blades and rotors, pressure-vessel nozzles under start-stop thermal-pressure cycling, designed per IS 2825 / ASME with cyclic-life checks.
• Railways and infrastructure (Indian Railways, RDSO, bridge fabricators): rail-wheel contact fatigue, axle and bogie design, welded bridge girders checked under IS 800:2007 Section 13 detail categories.
• Aerospace and defence (HAL, ISRO, DRDO): damage-tolerant design — crack growth by Paris law, fracture toughness, inspection intervals, and LCF of turbine discs.
• Design and CAE job roles: a design engineer picks the theory and the FOS; a CAE/FEA engineer reports von Mises stress and feeds alternating/mean stresses into nCode, fe-safe or Ansys Fatigue; an NDT/reliability engineer finds the crack before it finds the part.
• Failure investigation: reading a fracture surface — beach marks, ratchet marks, origin at a keyway or tool mark — is routine work in plant maintenance and insurance/forensic engineering.

Interview questions

• Q: Which theory of failure would you use for grey cast iron and why? A: Maximum principal stress (Rankine) or Mohr–Coulomb against ultimate strength, because cast iron is brittle, fractures without yielding and is much stronger in compression than tension.
• Q: Why is von Mises preferred over Tresca for ductile metals? A: It matches biaxial test data better, especially in shear, where Tresca under-predicts strength by about 15%; Tresca is retained in codes only because it is safe and simple.
• Q: What is the shear yield strength of a steel of Syt = 400 MPa? A: 200 MPa by maximum shear stress theory and 231 MPa (0.577 × 400) by distortion energy theory.
• Q: A shaft fails after years of trouble-free running at a stress well below yield. Explain. A: Fatigue — the fluctuating stress exceeded the component endurance limit at a notch, and a crack initiated and grew to the critical size.
• Q: How would you tell a fatigue fracture from a ductile overload fracture? A: A fatigue fracture shows no necking, a smooth zone with beach marks radiating from a surface origin, and a small rough final-fracture zone; overload failure shows gross plastic deformation and a fibrous/cup-and-cone surface.
• Q: Does aluminium have an endurance limit? A: No — its S-N curve keeps falling, so a fatigue strength at a specified life (commonly 5 × 10⁸ cycles) is used instead of an endurance limit.
• Q: Why does shot peening improve fatigue life? A: It introduces compressive residual stress in the surface layer, which lowers the effective tensile mean stress where cracks start, and work-hardens the surface.
• Q: Difference between Kt and Kf, and what is notch sensitivity? A: Kt is the elastic theoretical stress concentration factor from geometry; Kf is the measured fatigue reduction factor; notch sensitivity q = (Kf − 1)/(Kt − 1) tells how much of the theoretical concentration the material actually feels — near zero for grey cast iron, near one for hard steel.
• Q: Soderberg or Goodman — which one would you use in practice? A: Goodman for most ductile components as it fits data reasonably and is simple, with a Langer yield line added; Soderberg where I want yielding automatically guarded and can afford extra material.
• Q: Why is a compressive mean stress not harmful in fatigue? A: Cracks grow only when the crack faces are pulled apart; compression keeps them closed, so the Goodman boundary is taken as flat at σa = Se for negative mean stress.
• Q: How do you design for bending plus steady torque on a shaft? A: Reduce the alternating and the mean components separately to von Mises equivalents, apply Goodman or Soderberg to the pair, and check first-cycle yielding with the peak von Mises stress.
• Q: State Miner's rule and one limitation. A: Failure when Σ(ni/Ni) = 1; it assumes damage adds linearly and ignores load sequence effects, so an overload followed by low cycles can give non-conservative predictions.

Common mistakes

• Forgetting that in plane stress the third principal stress is zero and must be used in Tresca — using only σ1 − σ2 when both are tensile gives a wrong, unsafe answer.
• Using Sut for a ductile material or Syt for a brittle one when computing the FOS.
• Writing the distortion-energy formula as √(σ1² + σ2²) — the cross term −σ1σ2 (and the 3τ² in the σ-τ form) is the whole point.
• Mixing up √(σ² + 4τ²) (Tresca) with √(σ² + 3τ²) (von Mises) for combined bending and torsion.
• Applying the raw rotating-beam Se′ = 0.5 Sut to a real part — the surface, size, loading, reliability factors and Kf typically halve it.
• Applying Kf to the mean stress as well without saying so, or applying it to both and then also using Soderberg — the result can be grossly oversized.
• Reading the Goodman diagram with σa on the x-axis, or quoting the FOS as the ratio of Se to σa alone, ignoring the mean stress entirely.
• Mixing units: N·m with mm in 32M/πd³, or MPa with N/m² — always convert the moment to N·mm and the diameter to mm to get MPa.

Related topics

Design of shafts, keys, bearings and springs · Bolted, riveted and welded joints · Strength of materials: stress, strain and Mohr's circle · Gear design · Material science: heat treatment and surface hardening · Fracture mechanics and non-destructive testing.