The Goodman Diagram: A Thorough Guide to Fatigue Design and Mean-Stress Effects

Adminn Misc 2. November 2025 | 0

In the realm of mechanical design and materials science, the Goodman Diagram stands as a foundational tool for evaluating fatigue life under combined mean and alternating stresses. It helps engineers transform complex loading histories into a clear visual assessment of safety margins and failure risk. This article delves into the theory, construction, interpretation, and practical application of the Goodman Diagram, with practical examples and comparisons to alternative fatigue criteria. The aim is to equip you with both the intuition and the practical steps to apply the Goodman Diagram confidently in real-world design tasks.

What is a Goodman Diagram?

A Goodman Diagram is a graphical representation used to assess the safe operating envelope for a component or material under uniaxial fatigue loading with a non-zero mean stress. On the diagram, the y-axis typically denotes the alternating stress (often denoted Sa or Δσ/2), while the x-axis represents the mean stress (Sm or σm). The line that connects the endurance limit of the material in the absence of mean stress to the material’s strength at maximum mean stress forms the boundary of safe operation. This boundary is known as the Goodman line.

In essence, the Goodman Diagram consolidates two critical fatigue constraints into a single linear relationship: the endurance limit at zero mean stress and a strength measure (either the yield strength, Sy, or the ultimate tensile strength, Su) at high mean stress. The interior of the diagram corresponds to safe loading states, while points outside the boundary indicate a probability of fatigue failure under the given cycle. The Goodman Diagram is widely used because it offers a straightforward visual check, a simple calculation, and can be adapted to many materials and service conditions.

Origins, theory and how the Goodman Diagram is used

Historical context and fundamental ideas

The Goodman approach emerged as engineers sought a practical way to account for mean stress effects in fatigue, acknowledging that a non-zero mean stress reduces fatigue life. Early fatigue criteria sought simple relationships; the Goodman criterion introduced a linear interpolation between a material’s endurance limit and its ultimate tensile strength. This linearization made it feasible to estimate safe operating stress states without resorting to complex life prediction models for every design, while still capturing the essential physics: higher mean stresses generally worsen fatigue performance.

Although the original formulation was developed for uniaxial loading, the underlying concept has been extended and adapted to multiaxial fatigue problems, either by adopting equivalent uniaxial representations or by applying generalized versions of the criterion. The core idea remains: combine mean and alternating stresses through a relationship that defines a safe zone in the stress space.

Endurance limit, ultimate strength, and the linear boundary

In the standard Goodman diagram, two material properties determine the boundary:

Endurance limit Se (or S’e): the stress amplitude below which a specimen can endure an infinite number of cycles when the mean stress is zero. For many ferrous metals, Se is well defined; for non-ferrous metals or alloys, a fatigue limit may be absent or difficult to pin down, requiring alternative references
Ultimate tensile strength Su (or Sy for yield-based variants): a measure of maximum tensile strength; the intercept on the mean-stress axis corresponds to Su (or Sy) depending on which variant is used

The Goodman line can be expressed by a simple relation, often written as:

Sa / Se + Sm / Su = 1

where Sa is the alternating stress (the stress range divided by two for a uniaxial load), Sm is the mean stress, Se is the endurance limit, and Su is the ultimate tensile strength (or Sy if the design uses yield as the strength limit). If this relationship yields a value less than or equal to 1, the loading state is typically considered safe; greater than 1, it is unsafe under the assumed loading regime.

In practice, many designers prefer to use Se and either Su or Sy depending on data availability and the material’s behaviour. Some engineers adopt a more conservative approach, using the yield strength Sy in place of Su to reflect yielding as a primary driver of failure, while others use Su to capture ultimate material limits. The choice affects the slope and intercept of the Goodman line and, consequently, the safe region drawn on the diagram.

How to read and construct a Goodman Diagram

Axes, intercepts and the safe region

The conventional Goodman Diagram layout places mean stress Sm on the horizontal axis and alternating stress Sa on the vertical axis. The safe region lies below and to the left of the straight boundary line that connects (Sm = Su, Sa = 0) to (Sm = 0, Sa = Se). The exact orientation may vary depending on whether a design uses yield or ultimate strength for the intercept, but the basic concept remains constant: a linear relationship defines the boundary beyond which fatigue failure is probable.

When drawing by hand or by CAD, you can annotate three key points for a quick reference:

At Sm = 0, Sa = Se (the endurance limit for zero mean stress)
At Sa = 0, Sm = Su (the maximum mean stress allowing zero alternating stress, depending on the strength reference)
Between these points, the line forms the boundary of safe operation

Reading a Goodman Diagram for a given load

To assess a given loading state, determine the mean stress Sm and the alternating stress Sa from the service load. Locate Sa on the vertical axis and Sm on the horizontal axis, then check whether the point (Sm, Sa) lies inside the safe region (below the Goodman line). If it does, the state is considered acceptable under the Goodman criterion; if it lies above the line, the state is considered unsafe and would require design changes such as reducing Sm, Sa, or both, increasing Se, or using a stronger material.

For example, suppose you’re evaluating a shaft subject to a fluctuating torque that produces a mean stress Sm of 150 MPa and an alternating stress Sa of 180 MPa. If Se is 250 MPa and Su is 600 MPa, the Goodman relation yields 180/250 + 150/600 = 0.72 + 0.25 = 0.97, which is slightly under 1, indicating a marginally safe condition under this simplified criterion. A small reduction in either stress component could ensure a robust margin.

What if Se is not readily available?

Some materials do not exhibit a well-defined endurance limit. In such cases, engineers often use a “modified Goodman” or substitute Su for the endurance limit Se, effectively restoring a similar linear boundary but with a different intercept. An alternative approach is to use the S-N curve data to define a different safe boundary specific to the material, or to adopt a different fatigue criterion such as the Soderberg or Gerber lines. The choice depends on material data, safety requirements, and industry standards.

Constructing a Goodman Diagram: step by step

What data you need

To construct a Goodman Diagram for a given material, you typically need:

Endurance limit Se (or fatigue limit) or an equivalent fatigue limit at the target life
Ultimate tensile strength Su, and optionally yield strength Sy
Material-specific considerations such as surface finish, size effect, temperature, and reliability targets if available

Steps to plot a basic Goodman Diagram

Draw the coordinate axes: Sm (horizontal) and Sa (vertical).
Mark the intercepts: (0, Se) on the vertical axis and (Su, 0) on the horizontal axis (or (Sy, 0) if using yield-based criteria).
Connect these two intercepts with a straight line. That line is the Goodman boundary for the chosen material data.
Shade or annotate the region below/left of the line as the safe region. The region above/to the right is unsafe.
Optionally plot a few representative loading points to illustrate safe vs unsafe states.

In practice, you may also want to overlay additional criteria, such as a Soderberg line or a Gerber parabola, to compare how sensitive the design is to different fatigue theories. The Goodman Diagram provides a clear, linear baseline for such comparisons, facilitating a straightforward engineering decision.

Practical applications of the Goodman Diagram

Where and when is the Goodman Diagram most useful?

The Goodman Diagram is particularly valuable in mechanical design contexts where components experience fluctuating loads that include both tension and compression or where load cycles are not symmetric. Common applications include:

Automotive powertrains: shafts, gears, and crank mechanisms subject to torque fluctuations
Aerospace components: shafts, landing gear pins, and control linkages subjected to variable flight loads
Industrial machinery: pulleys, belts, and rotating equipment with fluctuating loads
Structural components under vibratory service: where mean stresses arise from sustained loads or asymmetrical cycles

How the Goodman Diagram informs design decisions

Using the Goodman Diagram, engineers can:

Identify safe operating envelopes for new components
Quantify the margin of safety for a given load state
Evaluate the impact of design changes (e.g., reducing mean stress by redesigning geometry, redistributing loads, or altering support conditions)
Compare materials and coatings by considering how changes in Se, Su, or Sy affect the safe region
Integrate with finite element analysis by translating local stress states into Sm and Sa to assess fatigue risk regionally

Goodman diagram vs alternative fatigue criteria

Soderberg, Gerber and other alternatives

The Goodman criterion is one of several linear or nonlinear relationships used to account for mean stress effects in fatigue. Other well-known criteria include:

Soderberg criterion: This approach uses the yield strength Sy as the intercept on the mean-stress axis, giving a more conservative boundary for many materials. The relation is Sa/Se + Sm/Sy ≤ 1. It tends to be more conservative than Goodman for ductile metals with significant yield behaviour under load.
Gerber criterion: A nonlinear, parabolic relationship using the yield or ultimate strength and endurance data. It is more accurate for some materials where fatigue strength reduces with mean stress in a curved fashion, particularly for ductile metals with high safety margins.
Other approaches: Various empirical or physics-based criteria exist, including the S-N curve-based approaches for multiaxial loading, Miner’s rule with mean-stress corrections, or energy-based methods. In practice, many engineers use a combination of criteria to ensure robust design choices, especially for critical components.

When the design requires a conservative approach or when material data are sparse, the Soderberg or Gerber criteria may be preferred. The Goodman Diagram remains a useful first-pass tool for quickly assessing the impact of mean stress and comparing materials and load cases before applying more complex life-prediction methods.

Materials and practical considerations

Common materials and typical data ranges

Material data used in the Goodman Diagram reflect the fatigue behaviour under uniaxial loading. Some typical ranges (illustrative, not universal) are as follows:

Structural steels: Se in the range of a few hundred MPa to over 600 MPa depending on alloy and surface finish; Su often around 600–900 MPa; Sy typically around 400–700 MPa
Aluminium alloys: Se often lower than steels; Su commonly in the range 300–500 MPa for many alloys; Sy somewhat lower for various grades
Titanium alloys and composites: Varied fatigue properties with Se and Su tightly linked to microstructure, heat treatment, and notched factors

For accurate design, use material property data that matches the component’s service conditions, including temperature, surface finish, size effects, and whether the material will operate near its endurance limit. Fatigue properties can be highly sensitive to these factors, sometimes more than to the mean-stress correction itself.

Factoring in real-world service conditions

In practice, several factors influence the applicability and accuracy of the Goodman Diagram:

Surface finish: rougher surfaces often reduce Se due to increased stress concentration and microstructural imperfections
Size effects: larger components may exhibit lower fatigue strength because of larger critical flaws and higher probability of defects
Notches and stress concentrations: local mean stresses at notches can be significantly higher than nominal values, requiring representation as local Sm and Sa on the diagram
Temperature: elevated temperature can alter material strength and endurance limits, shifting the boundary
Load history: random or irregular loading can complicate the translation to Sm and Sa; spectral methods or rainflow counting may be required for precise life predictions

Practical examples and worked problem

Example 1: Simple uniaxial loading with a non-zero mean

Given a steel component with Se = 250 MPa and Su = 600 MPa. Suppose the service load shows a mean stress Sm = 120 MPa and an alternating stress Sa = 180 MPa. Use the Goodman diagram to assess safety.

Compute the Goodman value: Sa/Se + Sm/Su = 180/250 + 120/600 = 0.72 + 0.20 = 0.92.

Since 0.92 is less than 1, the loading state lies within the safe region of the Goodman Diagram under these assumptions. If you want a comfortable design margin, you could aim for Sa/Se + Sm/Su ≤ 0.8 or 0.9, depending on reliability requirements.

Example 2: Pushing mean stress toward the material strength

With the same material data (Se = 250 MPa, Su = 600 MPa), consider Sm = 240 MPa and Sa = 100 MPa. The Goodman value is 100/250 + 240/600 = 0.40 + 0.40 = 0.80. The state is still safe, but shows how mean-stress loading reduces the allowable alternating stress margin. If Sm rises to 300 MPa, the boundary becomes Sa/Se + Sm/Su = 1 threshold; at Sa = 0 MPa, Sm would reach Su, i.e., 600 MPa, as the zero-fatigue stress limit in this simplified view.

Interpreting and applying the Goodman Diagram in design workflows

Step-by-step integration into design processes

Gather material data: Se, Su (or Sy), and any other relevant properties; ensure data match service temperatures and finishes
Define the loading scenario: estimate Sm and Sa from the expected load history or design load cases
Plot the Goodman boundary using Se and Su (or Sy) and the linear relation Sa/Se + Sm/Su = 1
Map the service loads onto the diagram to determine safety; adjust design as needed to place all service states within the safe region
Consider conservatism: apply a design margin and compare with Soderberg or Gerber curves if applicable
Iterate with geometry changes, material selection, or heat treatment to achieve robust fatigue performance

Why the Goodman Diagram remains relevant in modern practice

Despite advances in probabilistic life prediction, multiaxial loading, and sophisticated finite element analyses, the Goodman Diagram remains attractive for its transparency, ease of communication, and the ability to quickly check multiple load cases. It is especially valuable in early-stage design, where fast decision support is essential, and in education to illustrate the interplay between mean stress and fatigue strength.

Common pitfalls and best practices

Pitfalls to avoid

Assuming Se applies at all sizes and finishes: real components may have modified endurance limits due to size effects and surface conditions
Using the wrong strength intercept: choose Su or Sy consistently with the intended criterion and material data
Ignoring multiaxial effects: the Goodman Diagram is uniaxial in its simplest form; multiaxial loading requires careful interpretation or equivalent stress formulations
Neglecting variable amplitude and random loading: for complex histories, simple Sm and Sa pairs may misrepresent fatigue life; consider spectrum methods
Overlooking notch effects: local stress concentration can invalidate the global mean stress estimate

Best practices for robust application

Use conservative material data and verify with alternative fatigue criteria when possible
Quantify uncertainties in Se and Su and reflect them in design margins
Validate critical designs with finite element analysis to estimate local Sm and Sa values around stress concentrators
Document the loading scenario and the chosen criterion clearly for project traceability

Multiaxial extensions and further reading

While the linear Goodman Diagram is a powerful tool for uniaxial fatigue assessment, real-world components often experience multiaxial loading. In such cases, the concept extends via equivalent stress formulations (e.g., using von Mises or maximum-shear-stress criteria) and through more advanced diagrams or criteria that account for the interaction of multiple stress components. For multiaxial fatigue, you may encounter:

Equivalent Good man analysis using an equivalent uniaxial stress state
Generalized Fatigue Criteria (e.g., critical plane approaches)
Use of S-N curves with mean-stress corrections in conjunction with finite-element stress analysis

If you are interested in digging deeper, consult advanced texts on fatigue design and materials science, where you will find detailed derivations, case studies, and comparisons of Goodman, Soderberg, Gerber, and other criteria in multiaxial contexts.

Practical tips for communicating about the Goodman Diagram

Provide clear axis labels and units on diagrams to avoid misinterpretation
Include a quick worked example within a report to illustrate how to interpret the boundary for a given load
Use consistent terminology: Sa for alternating stress, Sm for mean stress, Se for endurance limit, Su for ultimate strength or Sy for yield strength
Offer a short summary: “If Sa/Se + Sm/Su ≤ 1, the state is safe; otherwise, redesign” for non-technical stakeholders

Summary: why engineers rely on the Goodman Diagram

The Goodman Diagram is a pragmatic, intuitive, and widely accepted tool that helps engineers quantify the interaction between mean stress and alternating stress. Its linear boundary is easy to interpret and communicate, enabling rapid decision-making in design, inspection planning, and risk assessment. By using Se and Su (or Sy) as anchors, the diagram encapsulates essential fatigue physics—how the material’s fatigue resistance deteriorates with increasing mean stress—and translates it into a simple, actionable criterion. Whether you are selecting a material, sizing a shaft, or evaluating a structural component under fluctuating loads, the Goodman Diagram offers a reliable first-pass check and a solid foundation for more advanced life-prediction analyses.

A final note on best practice

In professional practice, you should treat the Goodman Diagram as part of a broader design strategy. It’s a powerful screening tool, but fatigue life is sensitive to many variables beyond the uniaxial, zero-defect idealised conditions assumed in the basic diagram. Always corroborate with material testing, numerical simulations, and, where feasible, life prediction models that reflect the service environment. When used thoughtfully, the Goodman Diagram remains a cornerstone of fatigue design that blends engineering intuition with quantitative insight, helping you deliver safer, more reliable mechanical systems.