Calculating fatigue strength means determining the maximum stress a material can withstand for a specific number of loading cycles before it fails. The process combines material properties, real-world correction factors, and the type of loading your part will experience. For steel, a common starting point is estimating the endurance limit at roughly half the ultimate tensile strength, then adjusting downward for surface finish, part size, temperature, and other conditions.
Fatigue Strength vs. Endurance Limit
These two terms are related but not identical. The endurance limit (sometimes called the fatigue limit) is the stress level below which a material can theoretically survive an infinite number of cycles. Steel and titanium exhibit a clear endurance limit, typically visible as a flat horizontal line on a stress-life curve beyond about 10 million cycles. If the stress stays below that threshold, the part never fails from fatigue.
Fatigue strength, on the other hand, refers to the stress that causes failure at a specific number of cycles. You might need a part to last 100,000 cycles or 1 million cycles rather than forever. In that case, you read the allowable stress from the material’s stress-life curve at that cycle count. Aluminum, copper, and most other nonferrous metals don’t have a true endurance limit. Their stress-life curves keep sloping downward, so for these materials you always work with fatigue strength at a defined life, typically 100 million or 500 million cycles.
The S-N Curve: Your Primary Tool
The stress-life curve (called an S-N curve) is a plot of applied stress on the vertical axis against number of cycles to failure on the horizontal axis, with the cycle axis almost always on a log scale. Building one requires testing multiple specimens: you start at a high stress that causes failure quickly, then reduce the stress for each successive test until specimens survive the target number of cycles without breaking. The highest stress where a specimen doesn’t fail is the fatigue threshold for that material.
For steels, testing typically runs to at least 10 million cycles. For nonferrous alloys without a clear threshold, tests often extend to 100 million or 500 million cycles. If you don’t have test data for your exact material, published S-N curves are available in engineering handbooks and material databases, but they represent ideal laboratory conditions. Real parts need correction factors.
Estimating the Base Endurance Limit
For steel, the uncorrected endurance limit (S’e) is commonly estimated as half the ultimate tensile strength (Sut) for steels up to about 200 ksi (1400 MPa). Above that, the estimate caps at 100 ksi (700 MPa) because very high-strength steels become increasingly sensitive to surface defects and inclusions. Steel cleanliness matters significantly: vacuum-melted bearing steels, for example, can achieve roughly three times the fatigue life at a given stress compared to conventionally processed versions, or tolerate about 50% higher stress for the same life.
For other materials, there’s no universal shortcut. You need either published fatigue data or actual test results.
Marin Correction Factors
The base endurance limit assumes a polished, small-diameter specimen tested at room temperature under pure bending. Real parts differ on every count. The Marin equation adjusts the estimate:
Se = ka × kb × kc × kd × ke × S’e
Each factor accounts for one real-world condition:
- ka (surface finish) reflects how rough the part surface is. A machined surface on a 65 ksi steel gives ka ≈ 0.89, while an as-forged surface on the same steel drops it to 0.63. For a stronger 125 ksi steel, those values fall to 0.75 (machined) and 0.33 (as-forged). Higher-strength steels are more sensitive to surface condition because tiny surface irregularities act as stress concentrators. The factor is calculated as ka = a × Sut^b, where a and b depend on the finish: ground surfaces use a = 1.58 MPa and b = −0.085, machined surfaces use a = 4.51 MPa and b = −0.265, and as-forged surfaces use a = 272 MPa and b = −0.995.
- kb (size factor) accounts for the fact that larger parts have more volume where a flaw could initiate a crack. For axial loading, kb = 1 (size doesn’t matter). For bending or torsion with diameters between about 3 mm and 51 mm, kb = (d / 7.62)^−0.1133 using millimeters. Larger parts in bending or torsion typically see kb between 0.60 and 0.75.
- kc (load type) adjusts for whether the part sees bending, axial, or torsional loading, since fatigue behavior differs with load type.
- kd (temperature) is 1.0 at room temperature and stays close to that up to about 250°C. Above 300°C it drops noticeably: at 400°C it’s around 0.92, at 500°C about 0.77, and at 550°C roughly 0.67.
- ke (miscellaneous effects) captures anything else, including reliability level, residual stresses, or corrosion.
Multiply all these together and you often end up with a corrected endurance limit that’s 40% to 70% lower than the textbook base value. This is why skipping correction factors leads to dangerously optimistic designs.
Accounting for Mean Stress
The calculations above assume fully reversed loading, where stress swings equally between tension and compression with zero average. Most real parts carry some steady load on top of the cyclic load, creating a nonzero mean stress. Three classic relations handle this:
The Goodman equation draws a straight line between the endurance limit (at zero mean stress) and the ultimate tensile strength (at zero alternating stress): σa/Se + σm/Su = 1. This is the most commonly used approach and tends to be slightly conservative for ductile metals.
The Soderberg equation replaces ultimate strength with yield strength: σa/Se + σm/Sy = 1. This is more conservative and ensures the part never yields, making it useful when permanent deformation is unacceptable.
The Gerber equation uses a parabolic curve: σa/Se + (σm/Su)² = 1. This fits experimental data for ductile metals more closely than the linear models, but it’s less conservative, so it’s typically used when you have good material data and want a tighter design.
In each equation, σa is the alternating stress amplitude, σm is the mean stress, Se is the corrected endurance limit, Su is the ultimate tensile strength, and Sy is the yield strength. You solve for whichever unknown you need. To find the allowable alternating stress for a known mean stress, rearrange the equation for σa.
Finite Life Calculations
When your part doesn’t need infinite life, you use the S-N curve directly or apply Basquin’s equation: S = A × N^B, where S is the reversing stress, N is the number of cycles to failure, and A and B are material constants derived from test data. The exponent B is negative, reflecting the fact that allowable stress decreases as required life increases. You find A and B by fitting to two known points on the S-N curve, often the stress at 1,000 cycles and the endurance limit at 10 million cycles.
To use this in practice: if you know the applied stress, solve for N to find the expected life. If you know the required life, solve for S to find the maximum allowable stress.
Variable Loading and Cumulative Damage
Real parts rarely see one constant stress level. They experience a spectrum of loads. The Palmgren-Miner rule handles this by treating fatigue damage as additive. For each stress level, you calculate the fraction of life consumed: divide the number of cycles at that stress (n) by the total cycles to failure at that stress from the S-N curve (Nf). Then sum those fractions across all stress levels:
n1/Nf1 + n2/Nf2 + n3/Nf3 + … = D
Failure is predicted when D reaches 1.0. In practice, the rule is imperfect. Block loading tests have shown failure at damage sums as low as 0.1 in some cases, meaning the rule can be unconservative. Many engineers use a damage sum of 0.5 or lower as the failure criterion to add a safety margin. For complex, real-world load histories, you also need a cycle-counting method (rainflow counting is the most common) to break irregular load signals into individual cycles before applying the rule.
Combined Loading: Equivalent Stress
When a part experiences bending and torsion simultaneously, you need an equivalent stress to compare against the S-N curve. The von Mises equivalent stress is the standard approach. For a two-dimensional stress state with normal stresses in two directions and shear stress, the equivalent stress is:
σVM² = σxx² + σyy² − σxx × σyy + 3τxy²
This collapses a complex stress state into a single number you can plug into any of the fatigue equations above. When the different stress components are not in phase (they peak at different times), the calculation becomes more involved and may require frequency-domain methods, but for most mechanical design problems where loads are synchronized, the basic von Mises formula works well.
Putting It All Together
A typical fatigue strength calculation follows this sequence. First, get the ultimate tensile strength of your material and estimate the base endurance limit (half of Sut for steel). Second, apply each Marin correction factor for your part’s actual surface finish, size, loading type, temperature, and reliability requirement. Third, check whether you have a mean stress; if so, apply Goodman, Soderberg, or Gerber to find the adjusted allowable alternating stress. Fourth, if the part has a finite life requirement, use the S-N curve or Basquin’s equation instead of the endurance limit. Fifth, if loads vary, use the Palmgren-Miner rule to accumulate damage across the load spectrum.
Each step reduces the allowable stress from the idealized textbook value toward what the part can actually handle. The gap between those two numbers is often surprisingly large, which is why fatigue failures account for the majority of mechanical failures in service.