Fatigue analysis predicts failure under repeated or fluctuating loading, often at stresses below the static yield strength. This cheat sheet covers S-N curve interpretation, stress amplitude, mean stress effects, endurance limits, and cumulative damage. College engineering students need these tools to estimate safe life, compare materials, and design parts that survive cyclic service.
Key Facts
- Stress range is Δσ = σmax - σmin, and stress amplitude is σa = Δσ/2.
- Mean stress is σm = (σmax + σmin)/2, and the stress ratio is R = σmin/σmax.
- For many metals in the finite-life region, Basquin's law is σa = A N^b, where A and b come from the S-N curve fit.
- A common log form of the S-N relation is log(σa) = log(A) + b log(N).
- Goodman mean stress correction is σa/Se + σm/Sut = 1 for the failure boundary under tensile mean stress.
- Gerber mean stress correction is σa/Se + (σm/Sut)^2 = 1 and is less conservative than Goodman for ductile metals.
- Soderberg mean stress correction is σa/Se + σm/Sy = 1 and is more conservative because it uses yield strength.
- Miner's rule for cumulative fatigue damage is D = Σ(ni/Ni), and failure is predicted when D is approximately 1.
Vocabulary
- S-N curve
- A graph showing the cyclic stress level S that causes failure after a number of cycles N.
- Fatigue life
- The number of loading cycles a material or component can withstand before fatigue failure.
- Endurance limit
- The stress amplitude below which some materials can survive a very large number of cycles without fatigue failure.
- Mean stress
- The average stress during a loading cycle, calculated as σm = (σmax + σmin)/2.
- Stress concentration factor
- A factor Kt that measures how much a notch, hole, fillet, or geometry change increases local elastic stress.
- Miner's rule
- A linear damage rule that estimates fatigue damage from variable loading using D = Σ(ni/Ni).
Common Mistakes to Avoid
- Using maximum stress instead of stress amplitude is wrong because most S-N curves are based on alternating stress or stress amplitude, not σmax.
- Ignoring mean stress is wrong because tensile mean stress usually reduces fatigue life, while compressive mean stress can improve it.
- Reading the S-N curve on linear axes is wrong when the curve uses log stress, log cycles, or both, because equal spacing does not represent equal changes.
- Applying an endurance limit to every material is wrong because many aluminum alloys and nonferrous metals do not have a true horizontal fatigue limit.
- Using Kt directly for all fatigue calculations can be wrong because fatigue notch sensitivity may require Kf instead of the theoretical stress concentration factor.
Practice Questions
- 1 A shaft cycles between σmax = 180 MPa and σmin = 20 MPa. Find Δσ, σa, σm, and R.
- 2 A component experiences 10,000 cycles at a stress level with Ni = 100,000 cycles and 50,000 cycles at a second stress level with Ni = 500,000 cycles. Use Miner’s rule to find D.
- 3 For a steel part with Se = 240 MPa, Sut = 600 MPa, and σm = 120 MPa, use the Goodman relation to estimate the allowable stress amplitude σa.
- 4 Explain why a polished specimen S-N curve may overpredict fatigue life for a real machined part with notches, surface roughness, and variable loading.
Understanding Fatigue & S-N Curve Reference
Fatigue failure usually develops in stages. A tiny crack begins at a local weak point, often on the surface. It may start at a machining mark, a scratch, a corrosion pit, a thread root, or the edge of a hole.
Repeated loading makes the crack grow a little during each cycle. For much of the part's life, this crack can be too small to see. Near the end, the remaining uncracked section becomes too small to carry the load.
Final fracture can then happen suddenly. This is why a part can look normal before it breaks.
An S-N curve comes from tests on many similar specimens. Each specimen is loaded repeatedly at one stress level until it fails or reaches a chosen cycle count. The results naturally scatter because real materials are not perfectly uniform.
Surface finish, heat treatment, specimen size, loading direction, and the surrounding environment all affect the result. A polished laboratory sample often lasts longer than a rough production part.
Water, salt, high temperature, or corrosive chemicals can shorten life further. Students should treat curve data as evidence for a specific material condition, not as a universal property that applies without adjustment.
Local geometry deserves close attention. A sharp change in shape forces stress lines to crowd together. This produces a stress concentration near shoulders, keyways, holes, grooves, and threads.
The nominal stress from a simple beam or shaft equation may therefore be much lower than the stress at the critical point. The theoretical concentration factor describes the effect of shape. The fatigue concentration factor describes how strongly the real material responds under cycling.
It can be lower because a material is not equally sensitive to every notch size. Larger radii, smoother transitions, and better surface finishing reduce the chance of crack initiation.
Mean stress corrections matter because a cycle centered in tension is usually more damaging than a cycle centered near zero. Tensile loading helps a crack stay open, so it can advance more easily. Compressive mean stress can partly close a crack and may improve fatigue resistance.
Goodman, Gerber, and Soderberg lines turn this effect into a design check. Goodman is commonly used for a cautious practical estimate. Gerber can fit some ductile metal data better.
Soderberg is especially conservative because it guards against yielding as well as fatigue. In coursework, identify the material strengths supplied, state the chosen model, and keep tensile and compressive signs consistent.
Miner's rule is useful when a machine sees several load levels, such as a vehicle axle on smooth roads, rough roads, and potholes. It adds the fraction of life used at each level. The method is simple, but real damage does not always add perfectly.
A severe overload early in service can create a crack that changes later behavior. Load order, residual stress, corrosion, and vibration frequency can all matter. Use Miner's rule as an engineering estimate, then check whether the loading history or safety requirement calls for testing, inspection intervals, or a more detailed crack growth analysis.