Datasheets for most materials list a fatigue strength. Depending on the material, this represents either the level of stress below which fatigue does not occur (sometimes called the endurance limit), or the level of stress at which fatigue failure occurs in a given, usually large, number of cycles.
Fatigue strengths listed on datasheets are usually determined by means of fully reversed loading. This means that, during the test, the stress varies from a certain value in tension to an equal and opposite value in compression. This is the how the stress varies in a rotating shaft under a bending load, for example. However, it may not be the way the stress varies in your application. For instance, in your application, the load may repeatedly go from zero to a given value. Or your part may have a certain static tensile load, plus a vibration, which causes the load to oscillate around this value.
In these cases, you should not simply compare the maximum stress in the part to the fatigue strength. Remember, fatigue is all about cyclic stresses -- stresses that vary over time. If the stress doesn't fluctuate, fatigue can't happen. So in addition to the maximum stress, you also need to know the minimum stress at the same location; that is, the range of stresses in a given location over time. Another way of approaching this is to consider an average or steady stress, and an alternating stress, which is the amount by which the stress varies around this value over time.
There are a number of different equations for converting maximum and minimum stress values (or average and alternating stress values) to an equivalent fully reversed stress. The fact that there are so many different equations suggests that none of the equations works perfectly all the time; after all, if one of the equations worked, everyone would simply use that one. You can find these equations in any mechanical engineering textbook. One of the simplest and best known is the Goodman equation. However, I've found that, in most cases, the Walker equation is the most accurate.
As you can see, designing for fatigue is not as simple as designing for overload. This probably helps to explain why fatigue failures are so much more common than overload failures. However, observing these basic principles can help to avoid fatigue failures.