Soderberg Relation

Most engineered components are not subjected to simple, static loads but to millions of cycles of vibrating, twisting, and bending stress that can lead to failure far below the material's static strength. This phenomenon, known as fatigue, is a primary cause of mechanical failure. The critical challenge for engineers is to design components that can safely withstand these complex loading conditions. This requires a predictive framework that accounts for the damaging interplay between the steady (mean) and fluctuating (alternating) components of stress.

This article provides a comprehensive exploration of one of the foundational tools for this purpose: the Soderberg relation. Across two chapters, you will gain a clear understanding of this vital fatigue design criterion. The first chapter, ​​"Principles and Mechanisms"​​, deconstructs the concepts of mean and alternating stress, introduces the Haigh diagram as a design map, and derives the Soderberg relation, comparing its conservative, yield-based philosophy with the alternative Goodman and Gerber criteria. The second chapter, ​​"Applications and Interdisciplinary Connections"​​, demonstrates how this theory is put into practice, addressing real-world complexities like stress concentrations, residual stresses, and multiaxial loading, while also clarifying the model's fundamental limitations.

Imagine you are designing a bridge. You calculate the maximum weight it will ever have to support and build it strong enough. But what if that weight isn't a single, static load? What if it's a relentless procession of trucks, bouncing and vibrating, day in and day out? Or think of an airplane wing, flexing up and down in turbulence, or a spinning crankshaft in an engine, enduring billions of stress cycles. Most things in our engineered world don't just sit still; they are shaken, twisted, and bent, over and over again. This cyclic loading can cause a material to fail at stress levels far below what it could handle in a single pull. This insidious, slow-developing failure is called ​​fatigue​​.

Our task, as scientists and engineers, is to predict when fatigue will occur, to design against it, and to ensure our creations are safe. To do this, we need a map—a way to visualize the boundary between a safe, long life and a premature failure.

Let's first simplify the problem. Any complex, repeating stress pattern, no matter how jagged, can be broken down into two essential components. Think of yourself jumping up and down. Now, imagine doing that while wearing a heavy backpack. The jumping is much harder with the backpack on. The stress on your legs has two parts: the steady, constant load from the backpack, and the fluctuating load from your jumping.

In materials science, we do the same thing. We describe any cyclic stress by its average value, the ​​mean stress​​ ($\sigma_m$), and its fluctuating part, the ​​alternating stress​​ ($\sigma_a$). For a simple sine wave of stress varying between a maximum ($\sigma_{\max}$) and a minimum ($\sigma_{\min}$), these are easily defined:

• ​​Mean Stress:​​ $\sigma_m = \frac{\sigma_{\max} + \sigma_{\min}}{2}$ (the steady load, your backpack)
• ​​Alternating Stress:​​ $\sigma_a = \frac{\sigma_{\max} - \sigma_{\min}}{2}$ (the size of the fluctuation, your jump)

The central question of fatigue design is: How much alternating stress ($\sigma_a$) can a material withstand for a given mean stress ($\sigma_m$)? A positive, tensile mean stress—like a heavy backpack—is generally bad news, making the material weaker against the fluctuating load. Our job is to quantify this relationship.

To visualize this interplay, engineers use a simple but powerful tool called a ​​Haigh diagram​​. It's a graph where we plot the mean stress ($\sigma_m$) on the horizontal axis and the alternating stress ($\sigma_a$) on the vertical axis. Each possible loading condition on a component is a single point ($\sigma_m, \sigma_a$) on this map. Our goal is to draw a "border of safety" on this map. Any point inside this border represents a load that the material can endure for its target lifetime. Any point outside signifies eventual failure.

But how do we draw this border? We can't test every single combination of mean and alternating stress. That would take forever! Instead, we use a bit of logic and ingenuity. We start by looking at the simplest cases—the two axes of our map.

Anchoring Our Map: The Two Extremes

What happens at the absolute extremes of our Haigh diagram?

1. ​​Purely Alternating Stress ($\sigma_m = 0$):​​ This is the case of zero mean stress—no backpack, just jumping. Decades of experiments have shown that many materials, particularly steels, have a remarkable property. Below a certain level of alternating stress, called the ​​endurance limit​​ ($S_e$), they can withstand an effectively infinite number of cycles without failing. This gives us our first secure point, our first anchor for the border of safety: the point ($0, S_e$) on the vertical axis.

2. ​​Purely Static Stress ($\sigma_a = 0$):​​ This is the case of a purely static load—wearing the backpack but standing perfectly still. The component will fail when the static stress exceeds its strength. But here, we face a crucial philosophical choice that defines everything that follows. What do we mean by "failure"?

   • Do we mean the very first moment the material begins to permanently deform or stretch, an event known as ​​yielding​​? The stress at which this occurs is the ​​yield strength​​ ($S_y$).
   • Or do we mean the moment the material actually breaks apart? This happens at a higher stress, the ​​ultimate tensile strength​​ ($S_u$).

This choice splits our path. The one we take depends on how cautious we need to be. The decision between using $S_y$ or $S_u$ as our second anchor point on the horizontal axis is the fundamental difference between the various fatigue criteria.

The Soderberg Philosophy: Absolute Prudence

Let's be supremely cautious. Suppose we are designing a component where any permanent change in shape, no matter how small, is unacceptable. Think of a precision bearing or a critical aircraft part. In this case, we must guard against yielding at all costs. Our static failure point must be the yield strength, $S_y$. This gives us our second anchor point at ($S_y, 0$).

Now we have two anchors: ($0, S_e$) and ($S_y, 0$). What's the simplest way to connect them to form our safety boundary? A straight line! This beautifully simple, linear connection is the essence of the ​​Soderberg relation​​. The equation for this line is a classic of engineering:

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

Any combination of ($\sigma_m, \sigma_a$) that falls on or below this line is considered safe from both fatigue and yielding. Because it uses the lower, more restrictive yield strength as its anchor, the Soderberg criterion is extremely conservative. It builds in a large margin of safety, which is warranted in applications where dimensional stability is paramount and phenomena like progressive plastic deformation (ratcheting) must be avoided.

The Pragmatists: Goodman's Line and Gerber's Curve

For many engineering components, a tiny, imperceptible amount of plastic deformation is not catastrophic. What really matters is avoiding outright fracture. For this more pragmatic approach, we can anchor our safety boundary to the ultimate tensile strength, $S_u$, giving us an anchor point at ($S_u, 0$).

• ​​Goodman Criterion:​​ If we connect ($0, S_e$) to ($S_u, 0$) with a straight line, we get the ​​Goodman line​​. Its equation is: $\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_u} = 1$

• ​​Gerber Criterion:​​ Experimental data often shows that even the Goodman line is a bit too pessimistic for many ductile materials. A curve that bows upward from the Goodman line often provides a better fit to reality. A simple parabola does the trick, giving us the ​​Gerber criterion​​: $\frac{\sigma_a}{S_e} + \left( \frac{\sigma_m}{S_u} \right)^2 = 1$

A Family Portrait

Let's now plot all three criteria on the same Haigh diagram. Since for any ductile metal, $S_y \lt S_u$, the picture becomes crystal clear.

In the previous chapter, we became acquainted with a set of elegant, straight-line relationships that describe how a material’s tolerance for cyclic stress, $\sigma_a$, is diminished by the presence of a steady, or mean, stress, $\sigma_m$. We have drawn these lines—the Soderberg, the Goodman, the Gerber—on a graph. But so what? A line on a graph has never stopped a bridge from collapsing or an axle from snapping. The real magic, the true beauty of these ideas, appears when we carry them out of the abstract world of plots and into the messy, complicated, and fascinating world of engineering design. How do these concepts allow us to build things that work, and work safely? That is our journey for this chapter.

At its heart, engineering design is a conversation with nature, and a key part of that conversation is defining what "failure" means. This is not as simple as it sounds. Imagine you are designing a steel shaft for an engine. If it breaks in two, that is certainly a failure. But what if it just permanently bends a little? The engine might still run, but perhaps it now vibrates violently, or other parts start grinding against each other. For all practical purposes, this is also a failure.

This choice—guarding against permanent bending (yielding) versus guarding against outright fracture—is the fundamental philosophical difference between our fatigue criteria.

The ​​Soderberg relation​​ is the most cautious and prudent of the family. By anchoring its failure line not at the ultimate tensile strength, $\sigma_u$, but at the yield strength, $\sigma_y$, it makes a profound statement: any macroscopic yielding is failure. To a designer using the Soderberg criterion, a component that has permanently changed its shape, even slightly, is a component that has failed. This philosophy is indispensable for high-precision machinery, where even microscopic changes in dimension can be catastrophic, or in any situation where unexpected overloads might occur. If a machine is subjected to a rare but severe jolt, the Soderberg criterion ensures the components return to their original shape, ready for normal service. It is the philosophy of "no surprises."

The ​​Goodman relation​​, on the other hand, is more liberal. By extending the failure line all the way to the ultimate strength, $\sigma_u$, it essentially says, "I'm not so worried about a little bit of local yielding, as long as the part doesn't completely break." This might be perfectly acceptable for many components where small plastic deformations at, say, the root of a thread, don't compromise the overall function.

So you see, the choice between Soderberg and Goodman isn't a matter of which formula is "right." It is a conscious design choice based on the question: what kind of failure am I trying to prevent?. This decision then allows us to calculate a "factor of safety," a number that tells us not just if our design is safe, but how safe it is. Geometrically, it's a simple and beautiful idea: if our operating stress point is $(\sigma_m, \sigma_a)$, the safety factor is simply how much we could scale up both stresses before we hit the failure line.

Interestingly, the Soderberg philosophy finds an echo in the world of Low-Cycle Fatigue (LCF), where large plastic strains are the norm. You might think a yield-based criterion has no place here, but the physics of strain-controlled cycling causes any large mean stress to "relax" away, physically bounded by the material's yielding behavior. A yield-anchored criterion like Soderberg is therefore surprisingly consistent with the physical reality of a stabilized LCF stress-strain loop, whereas a criterion based on ultimate strength could be misled by non-physical stress values that couldn't possibly be maintained by the material.

Our simple models assume we are dealing with a perfectly smooth, polished bar. The real world is rarely so kind. Real components have bolt-holes, shaft shoulders, and sharp corners. These geometric features act as stress concentrators. You can think of a notch as a tiny lever, prying the material apart and amplifying the nominal stress. The local stress at the root of a notch can be several times higher than the stress in the bulk of the part.

To use our fatigue criteria correctly, we must account for this. We can't use the nominal stress; we must use the local stress at the critical point. This is done using a fatigue strength reduction factor, $K_f$, which tells us how much the stress amplitude is effectively magnified by the notch. We then plug these higher, local values of $\sigma_a$ and $\sigma_m$ into our Soderberg equation to see if the material at that tiny, critical spot will survive.

But applied loads aren't the only source of mean stress. Often, stresses are locked into a material during manufacturing, long before it ever sees service. These are called ​​residual stresses​​. When a welder joins two plates, the intense local heating and cooling leaves behind a field of tensile stress near the weld, often as high as the material's yield strength. Conversely, processes like shot peening or case hardening intentionally create a layer of compressive residual stress at the surface.

For the purpose of fatigue analysis, a stable residual stress, $\sigma_r$, acts just like an externally applied mean stress. The total mean stress experienced by the material is $\sigma_m = \sigma_{m,applied} + \sigma_r$. This has staggering implications. The hidden tensile stress from a weld can drastically reduce a component's fatigue life, pushing its operating point dangerously close to the failure line. On the other hand, the "gift" of a compressive residual stress from shot peening can push the operating point into a much safer region, sometimes even making a component seemingly immune to fatigue failure under its service load. This beautiful connection shows how the Soderberg relation is not just a tool for mechanical designers, but also for materials scientists and manufacturing engineers who can manipulate these hidden stresses to create incredibly durable components.

Life is rarely a simple push-pull. The crankshaft in your car is simultaneously bent and twisted. The skin of an airplane wing is pulled and sheared. What happens to our simple one-dimensional model in these ​​multiaxial​​ states? We cannot simply add the stresses. Instead, we must turn to a deeper, more unified view of mechanics.

The key is to recognize that a general state of stress can be decomposed into two parts: a hydrostatic part that tries to change the volume of the material (like deep-sea pressure) and a deviatoric part that tries to change its shape (like shear). For most ductile metals, it is the amplitude of the shape-changing, deviatoric stress that drives fatigue damage. The mean stress effect, however, is primarily governed by the hydrostatic part of the stress.

Modern, more sophisticated fatigue criteria—which are spiritual descendants of the Soderberg and Goodman ideas—are built on this physical separation. They use stress invariants, mathematical quantities that capture the essence of the stress state, to create relationships between the deviatoric stress amplitude and the mean hydrostatic stress. This allows for a much more physically defensible prediction of fatigue life under complex, real-world loading.

Another layer of reality we must confront is ​​uncertainty​​. The yield strength of a material is not a single, fixed number. If you test a hundred "identical" samples from a batch of steel, you will get a hundred slightly different values, forming a statistical distribution. The same is true for the loads a component might see in service.

Modern design is moving beyond the simple, deterministic factor of safety and into the realm of ​​probabilistic design and reliability analysis​​. Instead of asking "Is it safe?", we ask, "What is the probability of failure?". In this framework, the Soderberg and Goodman equations become "limit-state functions." By feeding the statistical distributions of our material properties and loads into these functions, we can calculate a ​​reliability index​​, $\beta$, which is a measure of the probability that the component will survive its intended life. This represents a profound shift, connecting our classical fatigue diagrams to the powerful tools of modern statistics and risk assessment.

We have seen how versatile and powerful the Soderberg relation and its cousins are. They form the foundation of design against fatigue. But every tool has its limits. A master craftsperson knows not only how to use a tool, but also when to use it.

The entire framework we have discussed—the S-N approach—is fundamentally about predicting the life to initiate a crack. It assumes the material starts out smooth and clean, on a microscopic level. But what if it doesn't? What if the component, due to its manufacturing process, already contains a small crack or flaw?

This is the domain of a different, parallel field of mechanics: ​​Linear Elastic Fracture Mechanics (LEFM)​​. The philosophy of LEFM, often called "damage tolerance," is completely different. It assumes a crack is already present and asks two questions:

1. Is the cyclic stress large enough to make this crack grow?
2. If so, how many cycles will it take for the crack to grow to a critical size and cause catastrophic failure?

The deciding factor for crack growth is the stress intensity factor range, $\Delta K$, a quantity that characterizes the severity of the stress field at the crack tip. If $\Delta K$ is below a certain material threshold, $\Delta K_{\text{th}}$, the crack will not grow, and the component is safe. If it is above the threshold, the crack will grow with each cycle.

This leads to a fascinating and critically important paradox. You can have a situation where the operating stresses are so low that the Soderberg relation predicts an infinite life—the stresses are in the "endurance" regime. However, if a small flaw is present, the stress intensity at its tip could be high enough to exceed the growth threshold. The result? The S-N analysis predicts infinite life, while the LEFM analysis correctly predicts a finite life as the crack slowly grows towards its doom.

Therefore, the ultimate application of the Soderberg relation is knowing its own limitations. For many general machine components where the goal is to prevent cracks from ever starting, it is the perfect tool. But for safety-critical structures like aircraft fuselages, nuclear pressure vessels, or bridges, where one must assume that flaws exist, designers must put down the S-N tools and pick up the tools of fracture mechanics.

The simple line on a graph has taken us on quite a journey. We have seen it as a statement of design philosophy, a guide for handling complex geometry, a window into the hidden world of residual stresses, and a foundation for advanced theories of multiaxial and probabilistic design. And finally, by understanding its limits, we see its proper place in the grand, beautiful tapestry of tools that engineers use to build our world, safely and reliably.