Fatigue is the gradual cracking of a material under repeated loading at stresses well below its static strength. Fatigue analysis predicts how many load cycles a part will survive before a crack forms or grows to failure. It combines three things: the stress range at each point in the part, the number of cycles at each range, and material data, usually in the form of an S-N curve, which plots stress amplitude S against the number of cycles to failure N.
Fatigue is one of the most common causes of mechanical failure in service. A shaft, bracket or weld that passes a static strength assessment with a comfortable margin can still crack after a few million cycles. A fatigue analysis shows where cracks will start, gives an estimate of the life, and shows which change, whether a larger radius, a better surface finish or a relocated weld, would extend it.
How fatigue cracks form
A fatigue crack goes through three phases. It initiates at a local stress raiser on the surface: a fillet, a hole, a machining mark, an inclusion or a weld toe. It then grows a small distance with each cycle, often leaving visible beach marks on the fracture surface. Finally, the remaining section can no longer carry the peak load and breaks suddenly. Crack growth is driven mainly by the stress range, the difference between the maximum and minimum stress in a cycle. The mean stress has a secondary but real effect.
A stress cycle is described by its range Δσ = σmax − σmin, its amplitude σa = Δσ / 2, its mean σm = (σmax + σmin) / 2 and its stress ratio R = σmin / σmax. A rotating shaft in bending sees fully reversed loading, R = −1, with zero mean. A part loaded from zero to a peak and back sees R = 0.
High-cycle and low-cycle fatigue
| High-cycle fatigue | Low-cycle fatigue | |
|---|---|---|
| Typical life | More than about 10⁴ to 10⁵ cycles | Fewer than about 10⁴ cycles |
| Stress level | Elastic, below yield | Local plastic strain in every cycle |
| Approach | Stress-life (S-N curve) | Strain-life (ε-N curve) |
| Material data | S-N curve, endurance limit or fatigue strength | Cyclic stress-strain curve and strain-life constants |
| Examples | Shafts, springs, vibrating brackets, fan blades | Pressure cycling, thermal cycling, repeated overloads |
Most machinery operates in high-cycle fatigue. Low-cycle fatigue applies where each cycle causes local yielding, as in pressure equipment cycled at every start-up and shutdown or components under severe thermal cycles. There, strain amplitude is related to life by the strain-life equation:
Δε / 2 = (σ′f / E)(2N)b + ε′f(2N)c
where σ′f and b describe the elastic part of the curve, ε′f and c the plastic part, E is Young’s modulus and 2N the number of load reversals to failure.
The S-N curve
An S-N curve is produced by testing specimens at constant stress amplitude until they fail, and plotting amplitude against life on logarithmic axes. Reference data is often for small polished specimens under fully reversed loading. Over much of its length the curve follows the Basquin relation:
σa = σ′f (2N)b
where σ′f is the fatigue strength coefficient and b the fatigue strength exponent, typically between −0.05 and −0.12 for metals. The shallow slope makes life very sensitive to stress. With b = −0.1, a 10% cut in stress amplitude almost triples the life.
The endurance limit
Steels and titanium alloys show a knee in the curve, typically between 10⁶ and 10⁷ cycles, below which constant-amplitude loading in a benign environment does not cause failure. That stress is the endurance limit. For polished steel specimens it is roughly half the ultimate tensile strength, up to a strength of about 1,400 MPa, beyond which it levels off at around 700 MPa. Aluminium and copper alloys have no true endurance limit, so a fatigue strength is quoted at a stated life, commonly 5 × 10⁸ cycles. Corrosion removes the endurance limit even in steel. Under variable loading, large cycles can start a crack that smaller cycles then continue to grow, so standards usually extend the curve below the knee rather than stopping it.
Mean stress effects
A tensile mean stress reduces the stress amplitude a part can tolerate. A compressive mean stress helps, which is why shot peening and rolled threads improve fatigue life. Three classic relations adjust the allowable amplitude for mean stress, where Se is the endurance limit or the fatigue strength at the required life, Su the ultimate tensile strength and Sy the yield strength:
Goodman: σa / Se + σm / Su = 1
Gerber: σa / Se + (σm / Su)² = 1
Soderberg: σa / Se + σm / Sy = 1
Take a steel with Su = 600 MPa and Sy = 400 MPa, an endurance limit after allowing for surface and size of Se = 200 MPa, and a mean stress of 150 MPa. The allowable stress amplitude by each relation is:
| Relation | Allowable amplitude | Character |
|---|---|---|
| Gerber | 187.5 MPa | A parabola that fits test data for ductile metals well; the least conservative of the three |
| Goodman | 150 MPa | A straight line to the ultimate strength; widely used and moderately conservative |
| Soderberg | 125 MPa | A straight line to the yield strength; the most conservative |
For design, the right-hand side becomes 1/n, where n is the factor of safety. The relations are applied to tensile mean stress; for a compressive mean, a cautious and common practice is to take no credit for it.
Stress concentrations
Most fatigue cracks start at a local stress raiser: a shoulder fillet, a cross-hole, a keyway, a thread root or a groove. The theoretical stress concentration factor Kt is the ratio of peak to nominal stress in an elastic material. Under cyclic load, the effective factor is the fatigue notch factor Kf, which is usually smaller, especially for small radii and lower-strength materials:
Kf = 1 + q (Kt − 1)
where q is the notch sensitivity, between 0 (no effect) and 1 (full effect). High-strength steels are more notch-sensitive than mild steel, so a sharp corner costs them more of their fatigue advantage. Increasing a fillet radius is often the cheapest life improvement available. FEA gives the peak elastic stress at the notch directly, so Kt is already in the result. Typical values are covered in our stress concentration factors guide.
Surface finish, size and residual stress
Laboratory S-N data comes from small, polished specimens. Real parts are rougher, larger and loaded differently, so the endurance limit is reduced by modifying factors for surface, size, load type, temperature and reliability. Surface finish often has the largest effect, and the effect grows with strength. Typical surface factors for a steel of 600 MPa ultimate strength are:
| Surface | Approximate surface factor |
|---|---|
| Ground | 0.92 |
| Machined or cold drawn | 0.83 |
| Hot rolled | 0.58 |
| As forged | 0.47 |
Residual stress matters as much as roughness. Shot peening, roller burnishing of fillets and thread rolling leave the surface in compression and extend life. Grinding damage, hard chromium plating and welding leave tensile residual stress and shorten it.
Welded joints
Welds behave differently from machined parts, for three reasons. The weld toe is a sharp notch with small flaws already present, so the initiation phase is short. Residual stresses near the weld can approach yield, so the full applied stress range is damaging whatever the applied mean stress. And because life is dominated by crack growth from those flaws, the fatigue strength of an as-welded joint depends little on the strength of the parent steel: a higher-strength grade does not improve it.
Weld fatigue standards therefore work in stress range, place each joint geometry in a detail category with its own S-N curve, and assess it on nominal, hot-spot or notch stress. A common slope for these curves is 3, so doubling the stress range cuts the life by a factor of eight. Toe grinding, peening and similar treatments can extend the life of critical welds, and the most effective change is often to move the weld away from the highest stress.
Variable loading and Miner’s rule
Few parts see constant amplitude. A load history is broken into individual cycles, usually by rainflow counting, and the damage from each block of cycles is added using the Palmgren-Miner rule:
D = Σ ni / Ni
where ni is the number of cycles applied at stress range i and Ni the life at that range from the S-N curve. Failure is predicted when D reaches 1. Suppose a bracket sees, each year, 10,000 cycles at a range with a life of 100,000 cycles, 200,000 cycles at a range with a life of 2 million, and 5 million cycles at a range with a life of 50 million. Each block contributes 0.1, so D = 0.3 per year and the predicted life is a little over three years. Miner’s rule ignores the order in which cycles occur, which can matter, and some standards set a design limit on D below 1.
What a fatigue study needs
- The load history or spectrum: which loads occur, how often and in what combination, from duty cycles, measured data or the code or standard you specify.
- Fatigue data for the actual material and condition, or at least its tensile and yield strength.
- Surface finish and treatments: machined, ground, shot peened, plated or as-welded.
- Weld locations, types and any improvement treatment.
- The required life, in cycles, hours or years, and the environment, including temperature and corrosion.
The results show life or damage across the part, the locations that govern, and the factor of safety on fatigue strength, so that design effort goes where it extends life most.
When to bring in a specialist
A fatigue prediction is only as useful as the loads and details behind it, and its value lies in the design change it points to. SixGate™’s fatigue analysis combines stress results with your load history and material data, reported against the code or standard you specify, to show where a design is likely to crack and an estimate of how long it will last. The outcome includes the changes that extend life: radii, weld positions, section sizes, surface treatment or material. Where the cycles come from vibration, the study can start from a vibration analysis of the same model. The same engineers can then put those radii and weld positions into the detailed design and onto the drawings, so the part that goes into production is the one that was assessed.