Fatigue: Modified Goodman Diagram

In engineering design, components often fail not from a single overload but from repeated cyclic loading, a phenomenon known as fatigue. Predicting this failure is critical for safety and longevity, especially when stresses combine steady and fluctuating components. The modified Goodman diagram provides a powerful graphical method to assess fatigue life under such combined stresses, incorporating real-world factors that affect material performance.

Understanding Fatigue Failure and Stress Components

Fatigue failure occurs when a material cracks and fractures after many cycles of stress below its ultimate tensile strength. To analyze this, you must break down the operational stress into two key components: mean stress ($S_m$), which is the average stress over a cycle, and alternating stress ($S_a$), which is the amplitude of the stress variation. For example, in a connecting rod, the mean stress might come from a constant compressive load, while the alternating stress arises from cyclic tension during engine operation. These components interact significantly; a high mean stress reduces the amount of alternating stress a material can withstand without failing, which is why combined stress analysis is essential. Fatigue life is typically characterized by an endurance limit ($S_e$), the stress level below which a material can theoretically endure an infinite number of cycles under completely reversed loading ($S_m=0$).

The Goodman Diagram: A Graphical Tool for Fatigue Analysis

The classic Goodman diagram is a plot with alternating stress ($S_a$) on the vertical axis and mean stress ($S_m$) on the horizontal axis. It features a failure line that connects two critical material properties: the endurance limit ($S_e$) on the $S_a$ axis (where $S_m=0$) and the ultimate tensile strength ($S_u$) on the $S_m$ axis (where $S_a=0$). This straight line represents all combinations of $S_a$ and $S_m$ that are expected to cause fatigue failure after a very high number of cycles, typically considered infinite life. Any stress state represented by a point ($S_m$, $S_a$) that falls below this line indicates safe operation, while a point above the line predicts failure. The linear relationship is a conservative simplification, but it forms the basis for more refined models.

The Modified Goodman Diagram: Incorporating Real-World Factors

The standard Goodman diagram uses idealized, laboratory-derived material properties, but actual component performance is worse. The modified Goodman diagram adjusts for this by using a corrected endurance limit. The failure line still connects a point on the alternating stress axis to the ultimate strength ($S_u$) on the mean stress axis, but the intercept is no longer the baseline $S_e$. Instead, it is a modified endurance limit ($S_e^{\prime }$) that accounts for various degradation factors. This modification makes the diagram a practical and essential tool for design engineers, ensuring predictions align with real-world behavior where components have surface imperfections, vary in size, and operate under different reliability requirements and temperatures.

Applying Modification Factors to the Endurance Limit

The baseline endurance limit, typically obtained from standardized tests on polished, small-diameter specimens, must be reduced for your specific application. This is done by applying four primary modification factors, each a multiplier less than or equal to one. Their product defines the overall reduction factor applied to $S_e$.

• Surface Finish Factor ($k_{surface}$): Machining marks, corrosion, or roughness act as stress concentrators. A ground surface has a factor near 1.0, while a forged or corroded surface can have a factor as low as 0.3.
• Size Factor ($k_{size}$): Larger components have a greater volume of material, increasing the probability of containing a defect that can initiate a crack. For bending and torsion, the factor decreases for diameters above approximately 10 mm.
• Reliability Factor ($k_{reliability}$): Standard endurance limit data represents a 50% survival probability. For higher reliability (e.g., 99.9%), you must use a factor below 1.0 to account for statistical scatter in fatigue data.
• Temperature Factor ($k_{temperature}$): Material strength properties change with temperature. At elevated temperatures, the endurance limit generally decreases, requiring a factor less than 1.0.

The modified endurance limit is calculated as: $$S_e^{\prime }=k_{surface}\times k_{size}\times k_{reliability}\times k_{temperature}\times S_e$$ This $S_e^{\prime }$ is then used as the alternating-stress intercept for the failure line on the modified Goodman diagram.

Practical Application and a Worked Example

Using the modified Goodman diagram involves a clear step-by-step process. First, calculate the mean ($S_m$) and alternating ($S_a$) stresses from your load history, often requiring peak and valley stress values. Second, determine all applicable modification factors and compute $S_e^{\prime }$. Third, plot the failure line from ($S_m=0$, $S_a=S_e^{\prime }$) to ($S_m=S_u$, $S_a=0$). Finally, plot your operating stress point ($S_m$, $S_a$). If it lies below the line, the design is safe for infinite fatigue life.

Consider a machined steel component with an ultimate strength $S_u=800$ MPa and a laboratory endurance limit $S_e=400$ MPa. The applied stresses are $S_m=300$ MPa and $S_a=150$ MPa. The modification factors are: $k_{surface}=0.9$ (machined), $k_{size}=0.85$, $k_{reliability}=0.814$ (for 99% reliability), and $k_{temperature}=1.0$ (room temperature).

1. Calculate the modified endurance limit:

$$S_e^{\prime }=0.9\times 0.85\times 0.814\times 1.0\times 400\text{ MPa}\approx 249\text{ MPa}$$

1. Construct the failure line between (0, 249) and (800, 0).

1. Evaluate the safety of point (300, 150). The equation for the failure line is $S_a=S_e^{\prime }\times (1-S_m/S_u)$. Substituting the coordinates:

$$S_a=249\times (1-300/800)=249\times 0.625=155.6\text{ MPa}$$

The allowable alternating stress at $S_m=300$ MPa is approximately 155.6 MPa. Since your applied $S_a$ (150 MPa) is less than this, the point falls below the line, indicating the component is safe from infinite-life fatigue failure.

Common Pitfalls

1. Using the Baseline Endurance Limit Uncorrected: The most significant error is directly plotting with the textbook $S_e$ value. This grossly overestimates fatigue life and leads to unsafe designs. Always remember that the "modified" in the diagram's name mandates the use of $S_e^{\prime }$ calculated from all relevant factors.

1. Incorrect Stress Component Separation: Mean and alternating stresses must be derived correctly from the load cycle. A frequent mistake is miscalculating the mean as a simple midpoint without considering stress ratios or misidentifying the cyclic amplitude. Double-check your formulas: $S_m=(S_{max}+S_{min})/2$ and $S_a=|S_{max}-S_{min}|/2$.

1. Omitting Stress Concentration Effects: The Goodman diagram analyzes nominal stresses. For components with notches, holes, or sharp changes in geometry, you must apply a fatigue stress concentration factor ($K_f$) to the alternating stress component before plotting the point. Failing to do so ignores a major source of fatigue initiation.

Summary

• The modified Goodman diagram is a graphical tool for assessing infinite-life fatigue failure under combined mean and alternating stresses.
• Its failure line connects the modified endurance limit ($S_e^{\prime }$) on the alternating stress axis to the ultimate tensile strength ($S_u$) on the mean stress axis.
• The modified endurance limit is found by applying reduction factors for surface finish, size, reliability, and temperature to the laboratory endurance limit ($S_e$).
• An operating point ($S_m$, $S_a$) plotted below this line indicates a safe design.
• Common errors include using the uncorrected $S_e$, miscalculating stress components, and neglecting stress concentrations.