Draw the modified Goodman line, Gerber parabola, and Soderberg line in real time. Instantly calculate the safety factor and predicted fatigue life for your operating stress point.
Material /Load Parameters
Ultimate Tensile Strength S_u
MPa
Material ultimate tensile strength
Endurance Limit S_e
MPa
Corrected endurance limit after stress concentration and surface factors
Mean Stress σ_mean
MPa
Alternating Stress Amplitude σ_alt
MPa
Results
Safety Factor n
—
Goodman criterion
Goodman Utilization
—
σa/Se + σm/Su（< 1: Safe）
Predicted Life N_f
—
cycles
FailureMode
—
prediction
Goodman Diagram
S-N Curve (life estimate from current stress amplitude)
What is the Modified Goodman Diagram?
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What exactly is a "Goodman Diagram"? I see it's a graph with lines, but what does it tell us about fatigue?
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Basically, it's a fatigue failure map. It tells you whether a component will fail under a combination of steady and fluctuating loads. The vertical axis is the alternating stress ($\sigma_a$), which is the "wiggle" part. The horizontal axis is the mean stress ($\sigma_m$), the constant background load. The lines on the diagram, like the Modified Goodman line, define the safe zone. Try moving the $\sigma_m$ and $\sigma_a$ sliders in the simulator above—you'll see your "operating point" move. If it stays under the lines, the part is safe from fatigue failure.
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Wait, really? So it's not just about the maximum stress? Why does the constant "mean" stress matter for fatigue?
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Exactly! That's the key insight. A high constant tension (mean stress) makes it easier for tiny fatigue cracks to grow. Think of stretching a paperclip back and forth—it's much easier to snap if you're already pulling it taut. The Modified Goodman criterion accounts for this by drawing a straight, conservative line from the endurance limit ($S_e$) on the alternating axis to the ultimate strength ($S_u$) on the mean axis. In the simulator, you can see how changing the material's $S_u$ or $S_e$ pivots this line, changing the safe zone.
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I see the Gerber and Soderberg lines too. Why are there different lines, and what's the "safety factor" number calculating?
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Great question. Different theories handle the interaction between mean and alternating stress differently. Modified Goodman is linear and conservative. Soderberg is even more conservative, using the yield strength. Gerber is a parabola that fits some experimental data better but is less safe. The safety factor is the ratio of the distance from your operating point to the failure line versus the distance from the origin to your point. It answers: "How much could I scale up both stress components before hitting the failure line?" Watch the safety factor update live as you drag your operating point closer to the Goodman line.
Physical Model & Key Equations
The core of the Modified Goodman criterion is a linear interaction equation between the alternating stress amplitude and the mean stress. Failure is predicted when the following relationship is reached:
Where:
$\sigma_a$ = Alternating stress amplitude (from cyclic loading)
$\sigma_m$ = Mean stress (constant component)
$S_e$ = Endurance limit of the material (fatigue strength for completely reversed loading)
$S_u$ = Ultimate tensile strength of the material
If the left-hand side is less than 1, the component is predicted to have infinite life (no fatigue failure).
The safety factor ($n$) for fatigue under the Modified Goodman criterion is calculated by finding how much the stress state can be increased proportionally before it reaches the failure line. This is derived from the equation above:
$$
n = \frac{1}{\frac{\sigma_a}{S_e}+ \frac{\sigma_m}{S_u}}
$$
This is the number displayed by the simulator. A safety factor $n > 1$ means the operating point is inside the safe region. If $n \leq 1$, failure is expected. This equation shows that the safety factor depends on a weighted sum of the two stress components, normalized by the material's fundamental limits.
Frequently Asked Questions
Yes, a stress point inside the modified Goodman line is theoretically judged to have infinite life (10^7 or more cycles). However, this is a criterion based on the assumption of a complete fatigue limit, and in actual materials, factors such as surface finish and stress concentration must be considered. It is recommended to check the safety factor and design with sufficient margin.
The modified Goodman is the most common and widely used for ductile materials. The Gerber parabola is slightly less conservative than the modified Goodman but may provide better agreement with experimental data in some cases. The Soderberg criterion, based on the yield point, is the most conservative and is suitable for brittle materials or when plastic deformation must be absolutely avoided. Please switch between them and compare according to the application.
The safety factor is calculated by extending a straight line (load line) from the origin through the operating point (σm, σa) to the failure limit line, and dividing the distance to the intersection point by the distance from the origin to the operating point. This tool automatically calculates it and displays it numerically and with color coding. The load line is based on the assumption that the ratio of mean stress to stress amplitude remains constant as the load increases.
The Se and Su values for common metallic materials are listed in material databases and mechanical design handbooks. Se is obtained from fully reversed fatigue tests, and Su is the value from a static tensile test. If unknown, an approximate value of Se ≈ 0.5Su for steel can be used, but for accurate design, please input measured values or reliable standard values.
Real-World Applications
Automotive Engine Crankshafts: These experience massive torsional and bending loads that are a combination of constant and wildly fluctuating forces from combustion cycles. Engineers use Goodman diagrams to select the right steel alloy and define safe operating RPM limits, ensuring the crankshaft lasts for the life of the engine without a fatigue fracture.
Aircraft Wing Spars: The main structural beam in a wing bears the steady stress from the aircraft's weight plus alternating stresses from turbulence, gusts, and maneuvers. Fatigue analysis via the Goodman diagram is critical for certifying the wing's life in flight cycles, directly impacting maintenance schedules and safety.
Wind Turbine Blade Roots: The connection between the blade and the hub is subjected to a constant mean stress from gravity and centrifugal force, plus a large alternating stress from wind gusts and tower shadow effects. Using a conservative Goodman criterion ensures these massive composite or metallic structures can survive decades of cyclic loading in harsh environments.
Reciprocating Pump and Compressor Rods: These connecting rods are under high tension from the pressure load, plus a fully reversed stress from the reciprocating motion. The Modified Goodman line helps determine the maximum allowable pressure and speed to prevent a fatigue failure that could cause catastrophic system rupture.
Common Misconceptions and Points of Caution
When starting to use this tool, there are several pitfalls that engineers, especially those with less field experience, often fall into. First and foremost is the point that "the endurance limit $S_e$ cannot be used as-is from the material catalog value". The value listed in the catalog is an ideal value for small, mirror-polished test specimens. The fatigue strength of an actual component can be significantly reduced by the size effect, surface roughness, manufacturing method (machining marks or heat treatment), and service environment (corrosion). For example, even for a steel with a tensile strength $S_u=600\text{MPa}$, for a large component with a rough surface, it is not uncommon for the endurance limit $S_e$ to drop from $300\text{MPa}$ to below $150\text{MPa}$. When setting parameters in the tool, it is essential to use an effective $S_e$ that considers this "fatigue strength reduction factor".
Secondly, "do not feel reassured by looking only at the value of the safety factor $n$". You cannot definitively say it's absolutely safe just because $n=2.0$. This calculation is based on the assumption of "constant amplitude stress". However, in actual machinery, "variable amplitude stress", where large and small stresses are mixed randomly, is almost always the case. There are phenomena not captured by a simple Goodman diagram, such as the "overload effect", where a single large overload can accelerate fatigue damage from subsequent smaller stresses. Use the tool's results as a first approximation, and always combine them with durability tests on actual machinery or more advanced cumulative damage calculations (like Miner's rule) for your judgment.