Fatigue Strength Fundamentals: S-N Curves, Endurance Limit, Stress Concentration Factors, and Safety Factors for Cyclic Loads

Introduction

Most mechanical failures in service occur not from a single overload but from cyclic loading — fatigue. A shaft that easily survives a static load of 10 kN may fracture after millions of cycles at a load of only 3 kN. Fatigue failures are insidious because they often occur with little visible deformation, the crack initiates at a stress concentration far smaller than the nominal stress, and the final fracture appears suddenly. Understanding fatigue behavior is not peripheral knowledge for mechanical designers — it is central to designing any component subject to repeated loading.

This article explains the S-N curve, endurance limit, stress concentration factors, and the framework for calculating fatigue safety factors with sufficient rigor to be genuinely useful in daily design work.

The S-N Curve

The S-N (stress vs. number of cycles) curve describes the relationship between applied cyclic stress amplitude and the number of cycles to failure. It is determined experimentally by testing many specimens at various stress levels and recording cycles to fracture. Key features of the S-N curve:

• High-cycle regime (N > 10⁴): Failure is controlled by fatigue crack initiation and growth. Applied stress is below the yield strength.
• Low-cycle regime (N < 10⁴): Applied stress approaches or exceeds yield strength; significant plastic deformation occurs per cycle. Analyzed differently (strain-based approach).
• Endurance limit (for steels): At a sufficiently low stress amplitude, steel specimens survive 10⁷ cycles without fracture. This stress is the endurance limit S_e. Below this level, infinite life is theoretically predicted.

Important note: not all materials have a true endurance limit. Aluminum alloys, titanium, and most non-ferrous metals show continuously declining S-N curves with no plateau. For these materials, a fatigue strength at a specified life (typically 10⁷ or 10⁸ cycles) is used instead of an endurance limit.

Estimating the Endurance Limit

For steel, the endurance limit of a polished test specimen S’_e can be estimated from ultimate tensile strength S_ut:

• For S_ut ≤ 1400 MPa: S’_e ≈ 0.5 × S_ut
• For S_ut > 1400 MPa: S’_e ≈ 700 MPa (does not continue increasing)

This is the endurance limit of the ideal test specimen, not the actual component. A series of modifying factors must be applied to obtain the component endurance limit S_e:

S_e = k_a × k_b × k_c × k_d × k_e × S’_e

Where: k_a is the surface finish factor (machined, ground, as-cast, etc.); k_b is the size factor (larger cross-sections have lower endurance); k_c is the loading type factor (bending, axial, torsion); k_d is the temperature factor; k_e is the reliability factor (accounting for statistical scatter).

The product of these factors is typically 0.3–0.7, meaning the real component endurance limit may be only 30–70% of the ideal specimen value. This is why the empirical estimation of S’_e = 0.5 S_ut can be dangerously misleading if applied directly to component design without modification factors.

Stress Concentration Factors

Fatigue cracks initiate at locations of locally elevated stress: notches, holes, grooves, fillets, keyways, and surface defects. The stress concentration factor K_t is defined as the ratio of maximum local stress to nominal stress at a cross-section. Values of K_t for standard geometric features are tabulated in Shigley’s, Peterson’s, and other references.

However, K_t is a theoretical elastic value. In fatigue, the effective stress concentration factor K_f is used instead:

K_f = 1 + q × (K_t – 1)

Where q is the notch sensitivity factor (0 to 1), which depends on material and notch radius. High-strength steels (q approaching 1.0) are more notch-sensitive than low-strength steels. This means high-strength steels provide less benefit than expected when used in components with stress concentrations—a critical insight for designers who assume that upgrading to higher-strength steel always improves fatigue life.

Practical Implication: Fillets Matter

Increasing fillet radius from 0.5 mm to 2 mm can reduce K_t from 2.5 to 1.5 at a shoulder—a dramatic improvement in fatigue life with zero material cost change. Conversely, a sharp corner at the base of a keyway (K_t = 2.5–3.0) often initiates fatigue cracks that cause shaft failures. Design rule: at every stress concentration, specify the largest fillet radius that function will permit.

Fatigue Safety Factor Calculation

The modified Goodman line is the most widely used approach for determining fatigue safety factor under combined mean and alternating stress:

σ_a/S_e + σ_m/S_ut = 1/n_f

Where σ_a is the stress amplitude, σ_m is the mean stress, S_e is the component endurance limit (with all k factors applied), S_ut is ultimate tensile strength, and n_f is the fatigue safety factor.

Required safety factors: for critical rotating machinery components with well-characterized loads, n_f = 1.5–2.0. For components with uncertain loading, difficult maintenance access, or failure consequences affecting personnel safety, n_f = 2.0–3.0. Document the assumed load spectrum and safety factor rationale on the design calculation sheet, not just the result.

Design Strategies to Improve Fatigue Life

• Maximize fillet radii at all section changes and stress raisers.
• Specify surface treatments that introduce compressive residual stress: shot peening, roller burnishing, surface hardening.
• Minimize surface roughness at fatigue-critical locations (finer surface finish → higher k_a).
• Avoid surface defects, handling damage, and corrosion pits — these are fatigue crack initiation sites.
• Relocate stress concentrations away from regions of high nominal stress where possible.
• In welded structures, post-weld treatments (toe grinding, shot peening) improve fatigue life significantly.

Summary Table

FAQ

Q: We upgraded our shaft material from 1045 to 4340 (much higher S_ut). Our fatigue failures continue. Why?

A: High-strength steel has higher notch sensitivity (q closer to 1.0), which offsets much of the endurance limit gain when stress concentrations are present. If fatigue cracks are initiating at a keyway, thread root, or sharp fillet, the effective K_f is higher in 4340 than in 1045, partially or completely canceling the benefit of higher S_ut. The correct fix is to eliminate or reduce the stress concentration (larger fillet, better keyway geometry, or roll the thread root) rather than simply upgrading material. Also verify that the k_a factor (surface finish) is consistent between materials — 4340’s theoretical improvement disappears if the surface is rough or corroded.

Q: How do I determine the load spectrum for fatigue calculation on a custom machine?

A: For existing machines or similar applications, strain gauge measurements during representative operating cycles provide actual load data. For new designs without measurement data, use worst-case loading with conservative assumptions and document them explicitly. When in doubt, conservative safety factors (n_f ≥ 2.0) compensate for load uncertainty. For safety-critical components, dynamic simulation or FEA with measured input loads is strongly recommended. Never use nominal rated loads as the basis for fatigue calculations without verifying they represent actual peak cyclic loads — rated loads are often steady-state values that underestimate dynamic peaks.

Q: Our component has a finite design life (5 million cycles), not infinite. How does the S-N approach change?

A: For finite life design, read the allowable stress amplitude directly from the S-N curve at the required cycle count rather than using the endurance limit. If you have the S-N curve for the material, this is straightforward. If you do not have full S-N data, use the two-point approximation between 10³ cycles (S = 0.9 S_ut for bending) and 10⁶ cycles (S = S_e) to construct a slope and interpolate. Apply all the same k modification factors to the S-N curve ordinate. The finite life allowable stress is always higher than the endurance limit, which is why finite-life components can be lighter if the service life is truly limited and well-defined.