Machine Design: Fatigue Analysis

Fatigue analysis sits at the center of reliable machine design because many real components do not fail from a single overload. They fail after thousands or millions of load cycles that are individually well below the material’s static strength. A rotating shaft, a bolted joint under vibration, a gear tooth, or a spring can look perfectly safe in a static stress check and still crack over time. The purpose of fatigue analysis is to predict that risk under cyclic loading and to estimate life using tools such as S-N curves, endurance limits, the modified Goodman diagram, and cumulative damage methods like Miner’s rule.

What fatigue is and why designers care

Fatigue is progressive damage caused by fluctuating stress. A typical fatigue failure proceeds in three stages:

1. Crack initiation, often at stress concentrators such as fillets, keyways, threads, inclusions, or surface defects.
2. Crack growth under repeated loading.
3. Final fast fracture when the remaining cross-section can no longer carry the load.

A defining feature of fatigue is that it is sensitive to details that static strength calculations may ignore: surface finish, size effects, residual stresses, mean stress, and stress concentrations. That is why fatigue analysis is not just “stress below yield.” It is a dedicated design check that connects the loading spectrum to material fatigue behavior.

Cyclic stress basics: mean and alternating components

Many machine elements experience a stress history that can be idealized as a repeated cycle between a maximum and minimum stress. Two quantities are used throughout fatigue design:

• Mean stress: ${\sigma }_m=\frac{{\sigma }_{\max }+{\sigma }_{\min }}{2}$
• Alternating (amplitude) stress: ${\sigma }_a=\frac{{\sigma }_{\max }-{\sigma }_{\min }}{2}$

A fully reversed cycle has ${\sigma }_m=0$ and ${\sigma }_{\min }=-{\sigma }_{\max }$. A pulsating load from 0 to a maximum has a positive mean stress. Mean stress matters because tensile mean stress generally reduces fatigue strength, while compressive mean stress can improve it.

In rotating bending, a shaft surface point often sees a nearly fully reversed stress even if the external bending moment is constant in magnitude, because the material point alternates between tension and compression as it rotates.

S-N curves: the backbone of fatigue life estimation

An S-N curve (Wöhler curve) relates stress amplitude to the number of cycles to failure, typically plotted as stress on the vertical axis and cycles to failure $N$ on a logarithmic horizontal axis. It is generated from fatigue tests under controlled conditions, commonly at a specified stress ratio.

Two broad behaviors are commonly used in design:

• For many steels, an endurance limit (or fatigue limit) is observed: below a certain stress amplitude, specimens can survive “infinite life” in a practical sense, often defined around $10^6$ to $10^7$ cycles in testing.
• For many nonferrous alloys (such as aluminum), there is no clear endurance limit; the S-N curve continues to drop, so design targets are set for a finite life.

In practice, designers rarely use a raw lab curve without adjustment. Real components differ from polished test specimens, and the fatigue strength must be corrected for surface condition, size, reliability, and other factors before using it as an allowable.

Endurance limit in design

When an endurance limit is applicable, it is often treated as a threshold stress amplitude for high-cycle fatigue. But “endurance limit” in a design context is not a single number; it is a corrected value that reflects the actual part and service environment.

Key influences on the usable endurance limit include:

• Surface finish: rougher surfaces reduce fatigue strength because they act like micro-notches.
• Size effect: larger parts tend to have lower endurance strength due to a higher probability of flaws and a greater stressed volume.
• Stress concentration and notch sensitivity: geometric features increase local stress; the fatigue penalty depends on how sensitive the material is to notches.
• Residual stress and manufacturing: shot peening can introduce beneficial compressive residual stress; machining marks can be detrimental.
• Reliability requirement: higher reliability typically means a lower allowable fatigue strength.

The practical outcome is that the endurance limit used in calculations is usually lower than the value suggested by idealized test data.

Mean stress effects and the modified Goodman diagram

Because mean stress shifts fatigue behavior, design often combines ${\sigma }_a$ and ${\sigma }_m$ in a failure criterion. The modified Goodman diagram is one of the most widely used tools for this purpose. It plots alternating stress versus mean stress and provides a boundary between “safe” and “unsafe” combinations.

Conceptually:

• At zero mean stress, the allowable alternating stress is near the corrected endurance limit.
• As tensile mean stress increases, the allowable alternating stress decreases.
• The boundary is typically drawn as a straight line between the endurance limit on the ${\sigma }_a$ axis and the ultimate tensile strength on the ${\sigma }_m$ axis.

This approach is popular because it is straightforward, conservative for many steels, and easy to apply in early-stage design. It is especially useful for components that must survive a large number of cycles with a combination of steady load (mean) plus fluctuating load (alternating), such as a shaft carrying torque plus bending, or a fastener under preload plus vibration.

In design workflows, the modified Goodman diagram is often used to compute a factor of safety against fatigue by checking where the operating point $({\sigma }_m,{\sigma }_a)$ sits relative to the failure line.

Life estimation when “infinite life” is not the goal

Not every component can be designed below the endurance limit, and not every material provides one. When stresses exceed the high-cycle threshold, a finite-life approach is used. The S-N curve then becomes a direct life estimator: given a stress amplitude (with mean stress accounted for using a diagram like modified Goodman), read or compute the expected cycles to failure $N$.

This is common in:

• Lightweight aluminum structures where endurance limits are not clear.
• Components where size or packaging constraints prevent keeping stress amplitudes low.
• Applications with occasional high stress cycles that dominate damage.

Variable amplitude loading and Miner’s rule

Real machines rarely experience perfectly constant amplitude cycles. Loads vary with speed, duty cycle, operator behavior, and environmental conditions. Miner’s rule is a practical method to estimate cumulative fatigue damage under variable amplitude loading.

The method treats fatigue damage as linear in cycle fraction:

• For each stress level $i$, determine the number of cycles applied $n_i$.
• From the S-N curve (with appropriate corrections), determine cycles to failure at that stress level $N_i$.
• The cumulative damage is $D=\sum \frac{n_i}{N_i}$.

Failure is predicted when $D$ reaches approximately 1. In equation form: $$\sum _i\frac{n_i}{N_i}\approx 1$$

Miner’s rule is widely used because it is simple and often reasonable for engineering estimates, but it has limitations. It does not inherently account for load sequence effects, interactions between high and low amplitude cycles, or crack growth mechanics. Even so, it remains a standard tool for duty-cycle based fatigue life estimation in machine design.

Practical design insight: how fatigue failures are prevented

Fatigue design is most effective when it shapes geometry, manufacturing, and loading assumptions early. Practical steps include:

• Reduce stress concentrations: generous fillet radii, smooth transitions, and avoiding sharp corners.
• Improve surface condition: controlled machining, polishing in critical areas, and preventing corrosion pitting.
• Manage mean stress: compressive residual stresses from shot peening or rolling can improve fatigue performance.
• Control assembly and preload: in bolted joints, proper preload can reduce alternating stress in the fastener and shift cyclic load into the joint members.
• Validate the duty cycle: fatigue predictions are only as good as the load spectrum used. Field measurements and realistic assumptions matter.

Bringing it together in a typical workflow

A disciplined fatigue analysis in machine design often follows this sequence:

1. Define cyclic loading and compute ${\sigma }_{\max }$ and ${\sigma }_{\min }$ at critical locations.
2. Convert to ${\sigma }_m$ and ${\sigma }_a$, accounting for stress concentrations where appropriate.
3. Select fatigue properties from S-N data and determine the corrected endurance limit if an infinite-life approach is plausible.
4. Apply a mean-stress correction using the modified Goodman diagram to evaluate safety or determine an equivalent allowable alternating stress.
5. If loads vary, break the spectrum into blocks and apply Miner’s rule to estimate life.
6. Iterate the design using geometry changes, material selection, surface treatments, or altered operating limits.

Fatigue analysis is not an optional refinement; it is a primary design requirement for components under cyclic loading. When S-N curves, endurance limits, modified Goodman mean-stress correction, and Miner’s rule are used with clear assumptions and sound engineering judgment, designers can predict failure risk, set realistic life targets, and build machines that last in the real world.