Haigh Diagram

Mechanical components, from a car's axle to an airplane's wing, are rarely subjected to simple, constant forces. Instead, they endure complex, repetitive stress cycles that can lead to fatigue failure, often without any visible warning. Predicting when a part will fail under these variable loads is one of the most critical challenges in engineering design. The core problem lies in understanding how not just the magnitude of stress oscillations (alternating stress) but also the constant, underlying stress (mean stress) combine to determine a component's lifespan. This is the knowledge gap that the Haigh diagram expertly fills, providing a visual map to navigate the complex world of fatigue.

This article delves into the Haigh diagram, exploring its fundamental concepts and practical applications across two comprehensive chapters. In the first chapter, ​​Principles and Mechanisms​​, we will dissect the concepts of mean and alternating stress and explore how the diagram is constructed. You will learn about the competing philosophies—Goodman, Gerber, and Soderberg—used to define the boundaries of safe operation and uncover the underlying physics of crack closure that explains why mean stress is so crucial. The second chapter, ​​Applications and Interdisciplinary Connections​​, will demonstrate how engineers use the diagram as a dynamic tool. We will see how it translates complex loads into simple equivalents, helps define factors of safety, and even connects the fields of materials science, manufacturing, and probabilistic design to create safer, more reliable products.

Imagine you are bouncing on a trampoline. Two things determine how risky your bouncing is. First, how high are your bounces? A series of small hops is much safer than soaring leaps that brush the treetops. Second, where is the trampoline? Is it on the solid ground of the first floor, or is it perched precariously on a tenth-story balcony? A misstep from the tenth floor is far more catastrophic.

The life of a mechanical part under cyclic loading is much like this. It’s not just the magnitude of the stress oscillations that matters, but also the "level" around which they oscillate. In engineering, we dissect this loading into two fundamental characters that tell the whole story: the ​​alternating stress​​, $\sigma_a$, which is like the height of your bounce, and the ​​mean stress​​, $\sigma_m$, which is like the height of the balcony your trampoline is on.

For a stress cycle that swings between a minimum value $\sigma_{\min}$ and a maximum value $\sigma_{\max}$, we define them quite simply:

• The ​​mean stress​​, $\sigma_m$, is the average or midpoint of the stress cycle: $\sigma_m = \frac{\sigma_{\max} + \sigma_{\min}}{2}$.
• The ​​alternating stress​​, $\sigma_a$, is the amplitude of the stress swing: $\sigma_a = \frac{\sigma_{\max} - \sigma_{\min}}{2}$.

A positive (tensile) mean stress, like a high balcony, is generally bad news for fatigue life. A negative (compressive) mean stress, like having your trampoline in a pit, is often beneficial. Our journey now is to understand precisely how these two stresses conspire to determine the fate of a material, and how we can create a map to navigate the safe regions.

To visualize the combined threat of mean and alternating stress, engineers use a wonderful tool called the ​​Haigh diagram​​. It’s a simple map where we plot the mean stress, $\sigma_m$, on the horizontal axis and the alternating stress, $\sigma_a$, on the vertical axis. Every possible cyclic loading condition—a car axle spinning, a bridge vibrating, an airplane wing flexing—can be represented as a single point $(\sigma_m, \sigma_a)$ on this map.

The crucial question is: which parts of this map are safe, and which are a "danger zone" leading to fatigue failure? To draw this boundary, we first need to anchor our map with a few key landmarks based on physical limits.

1. ​​The Vertical Axis ($\sigma_m = 0$)​​: This line represents loading with zero mean stress—our trampoline is on the ground floor. This is called ​​fully reversed loading​​. For any material, there is a maximum alternating stress it can endure for a very large number of cycles (say, a million or a billion) without failing. This limit is called the ​​endurance limit​​, denoted as $S_e$. This gives us our first definitive point on the map of infinite life: the coordinates are $(0, S_e)$.

2. ​​The Horizontal Axis ($\sigma_a = 0$)​​: This line represents a situation with zero alternating stress—no bouncing at all, just a constant, steady load. In this case, failure isn't fatigue, but simple static fracture. A material will break when this steady stress reaches its ​​ultimate tensile strength​​, $S_u$. This gives us our second landmark on the map: the coordinates are $(S_u, 0)$.

Our "safe" operating zone for infinite life must be a region in the first quadrant of this diagram that is bounded by these two points. Any loading condition $(\sigma_m, \sigma_a)$ that falls inside this safe zone is predicted to last forever. Any point outside is predicted to fail. The challenge lies in drawing the line that connects our landmarks.

Nature doesn't always provide simple straight lines, and so engineers have developed several models, or philosophies, to draw the boundary on the Haigh diagram. Each has its own story and purpose.

The Goodman Line: A Simple, Straight-Line Guess

The most straightforward approach is to assume the boundary is simply a straight line connecting our two landmarks, $(0, S_e)$ and $(S_u, 0)$. This is the essence of the ​​modified Goodman criterion​​. The equation for this line is elegant in its simplicity:

$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = 1$

There is a beautiful intuition behind this formula. Think of the term $\sigma_a/S_e$ as the "fraction of fatigue life used up" by the alternating stress. Similarly, $\sigma_m/S_u$ is the "fraction of static strength used up" by the mean stress. The Goodman criterion states that failure occurs when the sum of these two fractions equals one. It's a linear trade-off: the more static load you have, the less alternating load you can tolerate, and vice-versa.

The Gerber Parabola: A Better Fit for Ductile Materials

While the Goodman line is simple and useful, experiments show it's often too conservative, especially for ductile materials like many steels and aluminum alloys. These materials can often withstand a bit more punishment than the straight line suggests. A better fit to much of the experimental data is provided by a parabolic curve, also connecting $(0, S_e)$ and $(S_u, 0)$. This is the ​​Gerber criterion​​:

$\frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_u}\right)^2 = 1$

Why a parabola? It suggests a more complex, nonlinear interaction between mean and alternating stress. Think about how ductile materials fail: they stretch, deform, and absorb energy. Failure criteria based on energy often involve terms squared (like kinetic energy, $\frac{1}{2}mv^2$). The quadratic term $(\sigma_m/S_u)^2$ hints at this more physical, energy-related interaction.

In a hypothetical test on a nickel alloy, for example, with $S_e = 450 \, \mathrm{MPa}$ and $S_u = 900 \, \mathrm{MPa}$, suppose we apply a high mean stress of $\sigma_m = 540 \, \mathrm{MPa}$ and find that the material fails when the alternating stress reaches $\sigma_a \approx 285 \, \mathrm{MPa}$. The Goodman model would have predicted failure at a much lower $\sigma_a = 180 \, \mathrm{MPa}$, being overly cautious. The Gerber model, however, predicts failure at $\sigma_a = 288 \, \mathrm{MPa}$, a near-perfect match! This shows how the Gerber parabola can better capture the true behavior of ductile materials.

The Soderberg Line: The Cautious Engineer's Choice

Both Goodman and Gerber are concerned with preventing ultimate fracture (breaking in two), which is why they use $S_u$ as their static anchor point. But what if you are designing a precision instrument where even the slightest permanent bending would be a catastrophic failure? In this case, the critical static limit isn't fracture, but the onset of permanent deformation, which occurs at the ​​yield strength​​, $S_y$.

For such applications, the ultra-cautious ​​Soderberg criterion​​ is the tool of choice. It also draws a straight line, but it connects the endurance limit $(0, S_e)$ to the yield strength $(S_y, 0)$:

$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = 1$

Since a material's yield strength is always less than its ultimate strength ($S_y S_u$), the Soderberg line always lies below the Goodman and Gerber curves. It is the most conservative of the three. It's the philosophy you choose when you can't afford any plastic deformation, perhaps in a system that might experience rare but significant overloads that could otherwise cause yielding.

Let's see how these differing philosophies lead to different conclusions. For a material with $S_e = 400 \, \mathrm{MPa}$, $S_y = 600 \, \mathrm{MPa}$, and $S_u = 900 \, \mathrm{MPa}$, subjected to a load of $(\sigma_m, \sigma_a) = (220 \, \mathrm{MPa}, 300 \, \mathrm{MPa})$:

• The ​​Soderberg​​ criterion predicts a safety factor of $0.895$. Failure is expected.
• The ​​Goodman​​ criterion predicts a safety factor of $1.006$. Marginally safe.
• The ​​Gerber​​ criterion predicts a safety factor of $1.22$. Comfortably safe. Your design decision—and the fate of your component—depends entirely on which philosophy you adopt!

So far, we have looked at these criteria as empirical "rules." But physics is about asking why. Why is a tensile mean stress so detrimental? The answer lies in the microscopic world of cracks.

Virtually all materials contain unimaginably small flaws or cracks. Fatigue is the story of these tiny cracks growing, cycle by cycle, until they reach a critical size. A crack can only grow when it is pulled open. If its faces are pressed together, it's effectively asleep. This simple idea is the basis of ​​crack closure​​.

Now, picture a microscopic crack under our two-part loading:

• ​​With a Tensile Mean Stress ($\sigma_m > 0$):​​ The mean stress acts as a constant background pulling force, propping the crack open. This makes it easier for the alternating stress to pull the crack faces further apart. The crack is open for a larger portion of the cycle, and the effective stress range driving its growth is larger. The crack grows faster, and the component's life is shorter.

• ​​With a Compressive Mean Stress ($\sigma_m 0$):​​ The mean stress acts as a background "clamping" force, squeezing the crack faces shut. The alternating stress first has to work to overcome this clamping force just to get the crack open. Only the very peak of the tensile part of the cycle contributes to crack growth. The effective stress range is drastically reduced, crack growth slows down, and fatigue life is extended.

This physical picture of crack opening and closing provides a beautiful, fundamental justification for the trends we see in our Haigh diagram models. It's the "why" behind the lines on our map.

Understanding this principle allows us to not only predict fatigue but to actively combat it.

The Gift of Compression

If compressive stress is so good for fatigue life, can we build it into our parts intentionally? Absolutely! A common technique is called ​​shot peening​​, where the surface of a part is bombarded with millions of tiny spherical projectiles. Each impact acts like a tiny hammer blow, creating a dimple. The surrounding material tries to push back, creating a thin surface layer with a high degree of compressive residual stress.

This built-in compressive layer acts as a permanent, beneficial negative mean stress. For a crack trying to start at the surface, it's like trying to grow out of that trampoline pit—it's incredibly difficult. This is why critical components like engine crankshafts, gears, and springs are almost always shot-peened. The presence of this compressive stress and the resulting crack closure effects "mutes" the material's sensitivity to applied mean stress, often making the less-sensitive Gerber parabola a better predictive model even for materials that might otherwise follow a steeper line.

The Dangers of the Compressive Zone

If a little compression is good, is a lot better? Can we just extend our Goodman or Gerber lines into the negative $\sigma_m$ quadrant and claim ever-increasing fatigue strength? A naive look at the equations might suggest so, but a wise engineer knows that a model is only a model. We cannot ignore other ways to fail!.

If the mean stress becomes too compressive, the lowest point of the stress cycle, $\sigma_{\min} = \sigma_m - \sigma_a$, could become so negative that it exceeds the material's ​​compressive yield strength​​, causing it to permanently deform. Or, if the part is long and slender like a rod, it might simply ​​buckle​​.

Therefore, sound engineering practice dictates that we treat the benefits of compressive mean stress with caution. Often, the allowable alternating stress is simply capped at the baseline endurance limit, $S_e$, for any compressive mean stress. The physical world always has the last word over a simple equation.

There is one last, crucial piece of wisdom to impart. The neat lines we've drawn on our Haigh diagram are a beautiful and powerful simplification. In reality, they are not sharp lines, but ​​fuzzy bands​​.

Why? Because material properties are not deterministic numbers. If you test one hundred "identical" steel bars, you will find that their ultimate strengths and endurance limits vary. There is an inherent statistical scatter in these properties. The value of $S_e$ or $S_u$ we use is just the average of a distribution.

This means that our failure boundary is also a distribution. A loading point near the line isn't simply "safe" or "unsafe"; it has a certain probability of failure. A design engineer never designs to the average-property line. Instead, they calculate where a "99.9% survival" line might lie—a line shifted downwards from the average by some number of standard deviations—and ensure their design point is safely below that.

This final concept transforms our view from a simple deterministic picture to a more profound probabilistic one. It acknowledges the inherent uncertainty in the real world and highlights the true role of engineering design: not just to apply formulas, but to manage risk and build reliability in the face of the unknown. The Haigh diagram, in its full, fuzzy glory, is not just a map of failure, but a guide to creating things that last.

Now that we have acquainted ourselves with the principles behind the Haigh diagram, let us embark on a journey to see how this elegant map of stress is used in the real world. It is one thing to understand the lines and points on a chart; it is another entirely to use that chart to navigate treacherous waters, to build stronger ships, and even to account for the unpredictable storms of uncertainty. The Haigh diagram is not merely a piece of academic theory; it is a dynamic tool at the heart of modern engineering, a conceptual space where materials science, manufacturing, and even statistical theory converge.

Imagine you are trying to describe a journey. You could list every turn, every landmark, but a far better way would be to point to your location on a map. The Haigh diagram serves a similar purpose for stress cycles. A machine part might be pulled and pushed in a complex cycle, with a certain minimum and maximum stress. This cycle has a specific "character," defined by its stress ratio, $R = \sigma_{\min}/\sigma_{\max}$. On the Haigh diagram, every possible stress cycle with the same character, a constant $R$, falls neatly onto a single straight line passing through the origin. A fully reversed load ($R=-1$) corresponds to the vertical axis ($\sigma_m=0$), while a pulsating, zero-to-tension load ($R=0$) follows a line with slope 1, where $\sigma_a = \sigma_m$. The diagram provides a universal language to describe any kind of cyclic loading.

But its true power comes from translation. Our most reliable fatigue data, the classic S-N curves, are typically measured under the simplest condition: fully reversed loading ($R=-1$). What do we do when our real-world component is subjected to a much more complex cycle, with both a mean stress $\sigma_m$ and an alternating stress $\sigma_a$? We need a way to ask: "What fully reversed stress would be just as damaging as our complex stress state?"

The Haigh diagram provides the answer through the concept of an ​​equivalent stress​​. Using a failure criterion like the linear Goodman relation, we can project our operating point $(\sigma_m, \sigma_a)$ back to the vertical axis to find its equivalent fully reversed stress, $\sigma_{a,eq}$. The geometry of the diagram provides the translation formula. Similarly, if we believe a parabolic Gerber criterion better represents our material, we can perform the same translation using its curved failure line. This ability to convert any complex loading into a simple, universally understood equivalent is the first great application of the diagram. It allows engineers to apply standard reference data to virtually any service condition.

Once we can place our operating stress on the map, the next crucial question is: where are the cliffs? Where are the boundaries of failure? Here, the Haigh diagram reveals that "failure" itself is not a single, absolute concept. The choice of failure line is a profound statement about our design philosophy.

Do we define failure as the point of ultimate fracture? If so, we might choose the ​​Goodman criterion​​, a straight line connecting the material's endurance limit on the alternating stress axis to its ultimate tensile strength on the mean stress axis. Or are we designing a part for a high-precision machine where even the slightest permanent deformation is unacceptable? In that case, we must be more conservative. We would define failure as the onset of yielding. This leads to the ​​Soderberg criterion​​, which anchors its failure line at the material's yield strength instead of its ultimate strength. Because the yield strength is always lower than the ultimate strength, the Soderberg line encloses a smaller, safer operating region.

The choice between Goodman and Soderberg is not merely mathematical; it is an engineering judgment call, balancing performance against safety. But the story doesn't end there. A component can fail in more than one way. It might survive a million cycles of a small load, only to yield and deform permanently on the very first cycle if the load is large enough. A complete design must guard against both. The Haigh diagram allows us to superimpose different failure criteria on the same plot. For instance, we can draw the fatigue failure line (like Goodman) and, on top of it, draw the line for first-cycle yielding (often called the Langer line). The true safe operating region is the area bounded by the innermost of all relevant failure criteria. The diagram forces us to think holistically about all the ways a part can fail.

With a map of our stress state and the boundaries of failure clearly marked, we can now practice the art of engineering design.

First, how safe are we? We define a ​​Factor of Safety​​, $n$, not as some arbitrary multiplier, but as a beautiful geometric concept. Imagine your operating point $(\sigma_m, \sigma_a)$ on the diagram. Now, draw a straight line from the origin, through your point, and extend it until it hits the failure boundary. The safety factor $n$ is simply the ratio of the length of this line (from the origin to the failure boundary) to the distance of your operating point from the origin. If $n=2$, it means we could double both our mean and alternating stresses before we would expect failure. This visual interpretation is wonderfully intuitive; it tells us exactly how much "room" we have before we hit the cliff.

This brings us to a remarkable application where we don't just avoid the cliff, but actively push it further away: ​​surface engineering​​. Many components, like axles and shafts, fail from cracks that start at the surface. What if we could make the surface inherently resistant to fatigue? This is precisely what processes like shot peening and case hardening do. By bombarding the surface with small beads or inducing chemical changes, we create a layer of material that is permanently in compression. This built-in ​​compressive residual stress​​ acts as a protective shield.

On the Haigh diagram, the effect is dramatic. If a component is under a remote load with zero mean stress, the material at a critical notch would normally operate on the vertical axis. But with a compressive residual stress, say $\sigma_r = -120\,\mathrm{MPa}$, the effective mean stress at that critical point is no longer zero, but $-120\,\mathrm{MPa}$. Since compressive mean stresses are beneficial for fatigue life (they help hold cracks closed), this shift dramatically increases the amount of alternating stress the part can withstand before failing. By quantifying this shift, we can calculate the exact improvement factor. A well-designed surface treatment can easily increase the allowable load on a component by 25% or more, a huge gain achieved by cleverly manipulating the internal stress state. This is a beautiful interplay between manufacturing (shot peening), materials science (residual stress), and design (the Haigh diagram).

The Haigh diagram's utility extends into even more advanced and interdisciplinary realms, forcing us to refine our models based on deeper physical insights.

Consider the distinction between High-Cycle Fatigue (HCF) and Low-Cycle Fatigue (LCF). HCF involves many cycles at low stresses, where the material behaves elastically. LCF involves few cycles at high loads that cause significant plastic deformation. In an LCF scenario, if we impose a large mean strain, the material doesn't sustain a high mean stress indefinitely. It relaxes! With each plastic cycle, the mean stress ebbs away. Therefore, a failure model like Goodman, which is anchored to the ultimate tensile strength—a stress state that is physically unattainable in a stabilized plastic cycle—becomes unrealistic. A yield-based model like Soderberg, which respects the physical limits of plastic flow, becomes a much more defensible choice for the elastic part of the damage calculation. Here, an understanding of cyclic plasticity, a topic deep in materials science, directly informs our choice of model on the Haigh diagram.

Perhaps the most profound extension of the Haigh diagram is into the world of ​​reliability and probabilistic design​​. So far, we have treated material properties like endurance limit ($S_e$) and ultimate strength ($S_u$) as fixed, deterministic numbers. But in reality, they are not. If you test one hundred "identical" steel samples, you will get a statistical distribution of strengths. A responsible design cannot be based on the average strength; it must account for this variability to guarantee a certain probability of survival.

This is where reliability engineering meets fatigue analysis. Using methods like the First-Order Reliability Method (FORM), we can no longer draw a single, sharp failure line. Instead, we must define our design strengths ($S_{e,d}$ and $S_{u,d}$) not as fixed values, but as functions of their statistical distributions and our desired reliability. For a target survival probability of, say, 99.9%, we calculate conservative design strengths that account for the uncertainty in both $S_e$ and $S_u$. The more uncertain a property is, the more we penalize it. The resulting design criterion, $\frac{\sigma_a}{S_{e,d}} + \frac{\sigma_m}{S_{u,d}} \le 1$, looks like the old Goodman line, but it is imbued with statistical rigor. It represents a boundary on the Haigh diagram that we are 99.9% certain the true failure line lies beyond. This transforms the diagram from a deterministic tool into a sophisticated map for managing risk.

In the end, the Haigh diagram is far more than a simple plot. It is a canvas for reason, a bridge connecting the abstract world of stress analysis to the tangible reality of manufactured parts, their hidden internal states, and even the statistical uncertainties of the materials from which they are made. It is a testament to the power of a good visualization to unify disparate ideas and guide us toward building a safer, more reliable world.