# Fatigue Strength: Definition Equation & Coefficient

Instructor: Raghav Mahalingam

Raghav has a graduate degree in Engineering and 20 years of professional experience.

This lesson discusses the concept of fatigue strength and fatigue life using an example of a cantilever beam. A simple method to estimate it for a given material is shown, where the number of cycles to failure is related to the stress amplitude.

## A Little Experiment

Let's start with a small experiment. Take a small wire clip and completely open it up. Be very careful not to hurt yourself on the pointed ends of the clip. You can also use the metal ties that are used to wrap food items in plastic. Hold one end of the wire, and bend the other end back and forth several times, keeping count of the number of times you've gone back and forth. Each time you go back and forth is counted as one cycle. Write down the number of cycles it took for the wire to break. You've just found out the fatigue life of the wire under the bending stress applied by your fingers. The process of the breaking of the wire is called fatigue failure.

## Fatigue Strengh and Fatigue Life

Fatigue strength is defined as the the amplitude of cyclic stress that can be applied to a material without causing fatigue failure. In other words, how much force you can put on an object before it breaks. Fatigue life is the number of cycles the material survives under a given stress level.

Let's begin by looking at what stress and cyclic stress mean. We can start with a simple example of a cantilever beam. There are many examples of cantilever beams that we encounter in real life. A classic example is a diving board, that is attached on one end and is free to move on the other. Cantilever beams are typically shown as in Figure 1.

In the figure, a force is applied on the beam at the tip and denoted by P. In response to the force, the beam bends downwards at the tip, while the fixed end on the left stays put. When the beam bends, it creates a stress in the structure that tries to restore to beam to its original shape. Think back to Newton's Law of Action and Reaction. The action of pushing down on the tip of the beam causes a reaction stress in the structure that tries to restore the beam back to its original shape. The stress developed in the material is related to the deflection (or strain).

Now, if you apply the tip load in the opposite direction, the resultant stress in the structure also changes direction to oppose the load. If you extend this to think about a load that keeps changing direction, i.e., when the tip load is cyclic, the stress in the structure is also cyclic (and hence called cyclic stress). Cyclic stress in the structure eventually causes the material to develop small cracks, which then propagate through the entire structure resulting in complete failure. The development of cracks is a random process. This means that every time you repeat the experiment, you will get a slightly different result (i.e., different number of cycles when the fracture occurs).

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