1 Background and context

The Soderberg relation is a fatigue design criterion used to estimate whether a part under repeated loading will remain safe over time. It relates alternating stress and mean stress through a conservative linear boundary. Engineers use it when a component must withstand many cycles without developing fatigue cracks.

1.1 Fatigue in cyclic loading

Fatigue is the progressive weakening of a material caused by repeated stress. Even when the applied loads are below the level needed for immediate failure, microscopic damage can accumulate with each cycle. Over many repetitions, small cracks may form and eventually lead to fracture.

In practical design, fatigue is especially important for parts that rotate, vibrate, or undergo fluctuating service loads. Shafts, springs, fasteners, and machine arms are common examples. The key issue is not only how large the stress is, but also how it varies with time.

1.2 Historical development of fatigue criteria

Early fatigue studies showed that materials could fail under stresses well below their static strength. This led to the development of engineering criteria that combined information about repeated loading with basic material properties. The Soderberg relation emerged as one of several simplified rules intended for design calculations.

The method reflects a safety-oriented approach. Instead of attempting to predict the exact life of a component, it defines an allowable region of stress states considered acceptable for long-term service. This made it useful in eras when detailed fatigue testing was limited and design methods needed to remain practical.

1.3 Relationship to other design theories

The Soderberg relation belongs to a family of stress-based fatigue criteria. It is closely related to the Goodman and Gerber relations, which also describe how mean stress affects fatigue strength. Among these, the Soderberg line is usually the most conservative because it uses yield strength as a limiting factor for the mean stress component.

It differs from static failure theories, which focus on single overload events rather than repeated cycles. In fatigue design, a part may be safe under one large load but fail after many smaller fluctuations. The Soderberg criterion is intended to address this cumulative effect.

2 Mathematical formulation

The relation is usually expressed in terms of alternating stress and mean stress. These two quantities describe the amplitude of stress variation and the average stress level around which the fluctuation occurs. The criterion compares their combined effect against material limits.

2.1 Alternating stress and mean stress

Alternating stress is half the stress range in a loading cycle. It represents how far the stress swings above and below its average value. Mean stress is the average of the maximum and minimum stresses in the cycle.

For a fully reversed load, the mean stress is zero and the alternating stress equals the stress amplitude. For a load that never changes sign, the mean stress may be significant and can reduce fatigue resistance. These two parameters are central to the Soderberg formulation.

2.2 Standard Soderberg equation

The standard form of the criterion is

\[ \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} \le \frac{1}{n} \]

where \(\sigma_a\) is the alternating stress, \(\sigma_m\) is the mean stress, \(S_e\) is the endurance limit, \(S_y\) is the yield strength, and \(n\) is the factor of safety.

The equation defines an allowable combination of alternating and mean stress. If the left-hand side is less than or equal to the reciprocal of the safety factor, the design is considered acceptable under the simplified criterion. In many engineering texts, the inequality is rearranged to solve directly for one unknown stress component.

2.3 Graphical representation

The Soderberg relation is often shown as a straight line on a plot of mean stress versus alternating stress. The line connects the endurance-limit intercept on the alternating-stress axis to the yield-strength intercept on the mean-stress axis. Any point below the line is treated as acceptable for the chosen safety factor.

This diagram gives a quick visual check during design. It helps engineers judge how changes in static preload or fluctuating load affect fatigue margin. Because the boundary is linear, the plot is easy to construct and interpret.

2.4 Assumptions of linearity

The criterion assumes that the combined influence of mean stress and alternating stress can be represented by a linear relation. This is a simplification, since actual fatigue behavior may be curved or depend on material history. The linear form is chosen for ease of use and for its conservative character.

It also assumes that the material properties used in the equation are applicable to the component and loading condition. In practice, local stress concentrations, residual stresses, and surface condition may alter the true fatigue response. The Soderberg line does not explicitly model those effects.

3 Material properties used in the relation

The relation depends on several basic strength properties. These properties serve as reference values in the design equation. Their accuracy and relevance strongly influence the usefulness of the result.

3.1 Yield strength

Yield strength is the stress at which a material begins to deform plastically. In the Soderberg criterion, it limits the allowable mean stress. This choice reflects the idea that sustained average stress should remain below the onset of permanent deformation.

Using yield strength makes the method cautious. It discourages designs in which the static component of load approaches the material’s plastic range. That is one reason the relation is often preferred when safety margin is more important than maximum material utilization.

3.2 Endurance limit

The endurance limit is the stress level below which a material can endure very large numbers of cycles without fatigue failure, at least in idealized laboratory conditions. It is used as the reference for alternating stress in the Soderberg equation. For many steels, this value can be estimated from standardized data.

Not all materials have a clear endurance limit. Some alloys, especially nonferrous ones, may continue to weaken slowly as the number of cycles increases. In such cases, engineers may use an equivalent fatigue strength at a specified life instead of a true endurance limit.

3.3 Ultimate tensile strength

Ultimate tensile strength is the maximum stress a material can withstand in a monotonic tensile test. Although it is not part of the original Soderberg equation, it is often included in comparisons with other fatigue criteria. Relations such as Goodman and Gerber use it as the mean-stress limit rather than yield strength.

This difference highlights the conservative nature of the Soderberg approach. Replacing ultimate strength with yield strength produces a lower allowable mean stress and a smaller design region. The result is a more cautious but often less efficient design.

3.4 Safety factors

A safety factor is included to account for uncertainty in loads, material data, geometry, and service conditions. It reduces the usable stress range and adds margin against unexpected variations. In the Soderberg relation, the safety factor appears directly in the allowable stress equation.

Higher safety factors are often selected when loading is poorly known or when failure would have serious consequences. Lower factors may be used only when the analysis is well supported and the environment is controlled. The chosen value reflects engineering judgment rather than a fixed rule.

4 Engineering application

The Soderberg relation is used mainly in preliminary and routine fatigue design. It provides a simple way to screen component dimensions and loading conditions. Because of its conservative basis, it is often favored in applications where reliability is prioritized.

4.1 Design of machine components

Machine elements that experience repeated stress are common candidates for Soderberg analysis. Rotating shafts, threaded fasteners, crank members, and spring components are typical examples. Designers use the relation to estimate whether a chosen size and material combination will resist fatigue under expected service loads.

The method is especially useful when a part has both a steady preload and a fluctuating operating load. It helps determine whether the static stress and cyclic stress together remain within acceptable limits. This is a common concern in bolted joints and rotating machinery.

4.2 Evaluation of fluctuating loads

Real loading often changes with time in irregular ways. Engineers may simplify such histories into an equivalent cycle characterized by a mean value and an alternating component. The Soderberg relation then provides a practical check on the resulting stress state.

This procedure is not a detailed life prediction model. Instead, it offers a design screening tool. If the calculated stress point lies safely below the criterion line, the part is considered more likely to survive prolonged cycling.

4.3 Conservative design practice

The method is valued for caution. By using yield strength as the mean-stress limit, it restricts allowable loading more than several competing criteria do. That makes it suitable for designs where excess deformation or fatigue cracking would be unacceptable.

This conservatism may lead to heavier or larger components, but it can also improve robustness. In many industrial settings, a design that is somewhat overbuilt is preferable to one that is close to the edge of failure. The Soderberg relation fits that philosophy well.

4.4 Comparison with experimental fatigue data

In experimental fatigue testing, actual failure points do not always fall exactly on the Soderberg line. Some materials show greater endurance than the criterion predicts, while others are more sensitive to mean stress than expected. The relation is therefore best treated as a design approximation.

Its usefulness lies in providing a repeatable and easy-to-apply rule rather than an exact prediction. Engineers often compare the calculated result with test data, service experience, or more refined analyses before finalizing a design. This combination of theory and evidence improves confidence in the outcome.

5 Interpretation and limitations

The Soderberg relation should be interpreted as a simplified safety check. It is not a complete description of fatigue behavior. Its assumptions make it reliable for certain cases and less suitable for others.

5.1 Conservative nature of the criterion

The criterion is intentionally conservative because it limits the mean stress by yield strength. This can substantially reduce the allowable stress region compared with other methods. As a result, it tends to err on the side of safety.

That conservatism is useful when uncertainty is high. However, it may also lead to designs that are larger or more costly than necessary. Engineers must balance simplicity and caution against efficiency and material usage.

5.2 Validity for ductile materials

The relation is most commonly applied to ductile materials that exhibit a measurable yield strength and a recognizable fatigue response. It is less directly suited to brittle materials, which may fail with little plastic deformation. For ductile metals, the assumptions are often adequate for engineering design.

Even for ductile materials, local conditions matter. If a component contains sharp notches or severe stress raisers, the nominal stress used in the equation may not represent the true local stress state. In such cases, additional correction methods are needed.

5.3 Effects of surface finish and size

Surface roughness can shorten fatigue life by acting as a site for crack initiation. Larger parts may also behave differently from small laboratory specimens because of statistical and geometric effects. These influences are not built into the basic Soderberg relation.

Design practice often introduces correction factors for surface condition, size, reliability, and loading type. Such adjustments improve the realism of the stress estimate. Without them, the nominal criterion may overstate the true endurance of the component.

5.4 Limitations under complex loading

The relation is most suitable for simple uniaxial stress states and proportional cyclic loading. Multiaxial stress, nonproportional variation, impact effects, and variable-amplitude histories complicate the analysis. In those situations, more advanced fatigue methods may be required.

The criterion also does not explicitly capture crack growth behavior or cumulative damage. It is a threshold-based design rule rather than a full fracture mechanics model. For complex service conditions, engineers often supplement it with additional analysis.

Several other fatigue criteria are used in design practice. They differ mainly in how they treat mean stress and how conservative they are. Comparing them helps clarify the place of the Soderberg relation within fatigue theory.

6.1 Goodman relation

The Goodman relation is a linear fatigue criterion that uses ultimate tensile strength as the mean-stress limit. Like the Soderberg line, it is simple and easy to apply. It is usually less conservative because ultimate strength is higher than yield strength.

Goodman is often used when a moderate level of conservatism is acceptable. It provides a compromise between safety and efficient material use. Many design handbooks present it alongside Soderberg for comparison.

6.2 Gerber relation

The Gerber relation uses a parabolic curve rather than a straight line. It is based on the idea that the effect of mean stress on fatigue strength follows a nonlinear trend. For some ductile metals, it can fit experimental data better than linear methods.

Because it allows more alternating stress at intermediate mean stresses, the Gerber relation is generally less conservative. Engineers may prefer it when test data support the use of a less cautious criterion. Still, it is not usually as simple to apply as the Soderberg rule.

6.3 Modified Goodman relation

The modified Goodman relation is a refined linear criterion that also uses ultimate tensile strength for the mean-stress limit. It is widely used in engineering because it balances simplicity with reasonable correlation to many test results. The formula is easy to calculate and convenient for design charts.

Compared with Soderberg, the modified Goodman method typically permits a larger operating envelope. Its reduced conservatism can be attractive in optimization work. However, in safety-critical applications, designers may still prefer Soderberg.

6.4 ASME elliptic criterion

The ASME elliptic criterion combines alternating and mean stresses through an elliptical boundary. It is intended to provide a different compromise between conservative linear rules and more permissive nonlinear ones. The result is often used for shaft design and similar applications.

This criterion can offer a balanced estimate in some cases. Its shape reflects the idea that both stress components contribute jointly to fatigue risk. Like the other relations, it remains a simplified engineering approximation rather than a complete fatigue theory.

7 Practical use in analysis

In practice, the Soderberg relation is often embedded in standard design workflows. It appears in hand calculations, spreadsheets, and engineering software. Its value lies in providing a clear decision rule for cyclic loading problems.

7.1 Stress-life approach

The relation is commonly used within the stress-life, or S-N, framework. In this approach, material response is described by the number of cycles to failure at different stress levels. The Soderberg line helps determine whether the combined mean and alternating stress should be regarded as acceptable for long life.

This makes the method suitable for high-cycle fatigue problems. It is less appropriate for low-cycle conditions dominated by significant plastic strain. In such cases, strain-based methods are usually more informative.

7.2 Factor of safety calculation

To compute a factor of safety, engineers rearrange the Soderberg equation using the known stresses and material strengths. The resulting value indicates how far the design is from the allowable boundary. A larger factor means a greater margin against fatigue failure.

This calculation is often done during iterative sizing. If the factor is too low, the component may be enlarged, the load reduced, or the material changed. The process continues until the design meets the chosen reliability target.

7.3 Design charts and diagrams

Design charts are widely used to present the criterion visually. They allow rapid comparison between stress states and allowable limits without repeated algebra. Such diagrams are useful in teaching, preliminary sizing, and hand analysis.

In many charts, the Soderberg line is shown together with Goodman, Gerber, or other criteria. This side-by-side view helps engineers see how conservative each method is. It also makes the effect of changing mean stress easy to interpret.

7.4 Worked engineering examples

A typical example begins with a shaft or bolt subjected to a steady load plus a cyclic component. The engineer determines the mean and alternating stresses from the loading history, then compares them with the endurance limit and yield strength. If the calculated point lies within the Soderberg boundary, the design is considered acceptable under the chosen safety factor.

Such examples illustrate the method’s practical appeal. The steps are straightforward, the required data are familiar, and the result is easy to explain. For this reason, the Soderberg relation remains a standard reference in introductory and applied fatigue analysis.