Statistics and Strength
Two identical structures made from the same material and loaded in the same way will not fail at exactly the same load. There will always be small differences in the structures or the materials. Engineers must somehow account for these differences. For structures, a predicted loads or stresses are multiplied by a safety factor. For materials, a large number of tests are performed on a variety of samples to get a statistical distribution of material properties.
Many reference books can show you how to calculate statistical properties given a set of data. What they don't always show, however, is what that data means in a physical sense. Here's a quick example. The two test specimens shown below are identical unidirectional laminates (same material, thickness, and cure schedule), except that one is notched on both sides. Which specimen will fail at a lower stress? The answer is at the bottom of this page, but if you keep reading you should be able to figure it out on your own.
Back to the statistics. Look at any manufacturer's brochure, and you'll see a list of material properties, usually including strength and modulus. Should you use these properties to design a product? I hope you answer no. The brochure will usually show the test method used to generate the property, but it won't tell you how many specimens were tested, under what conditions the specimens were prepared, or which results they are showing. If they tested ten specimens, are they showing an average of the results, the best result, or the worst? Without this information, the data is only good for comparison to other materials, or for very preliminary designs.
The proper way to present material data is to generate tens or even hundreds of data points, fit the data to a statistical distribution, then apply a knock-down factor based on the number of tests. A larger number of tests gives you a higher confidence in the data, and your knock-down factors will therefore be less severe. Properties are then presented as an allowable. One material may have multiple allowables, depending upon the desired confidence level--the higher the confidence, the lower the allowable. In aerospace, the two allowables used are:
- A-basis: 95% confidence that 99% of the samples will exceed the allowable
- B-basis: 95% confidence that 90% of the samples will exceed the allowable
For metals, MIL-HDBK-5E contains A- and B-basis allowables for the alloys typically used in aerospace applications. In fact, for most aerospace projects, the specifications usually require the use of MIL-HDBK-5E allowables.
For composites, the equivalent book is MIL-HDBK-17. Unfortunately, this book doesn't contain much data. There are several reasons for this:
- The project is very young, so there hasn't been much time to generate data.
- It is expensive to generate data, and companies are reluctant to freely share the information.
- New composite materials are being introduced at a rapid rate.
When fitting data to a statistical distribution, you have a choice of curves. Most people are familiar with the normal distribution, or bell curve. Experience has shown, however, that a better fit for composites is the Weibull distribution.
The normal distribution is symmetric, which means that for every weak sample there is a corresponding strong sample. The Weibull distribution, however, is skewed towards the weak side, which means for any set of tests there will be more weak samples than strong samples.
This skewed distribution has physical significance, and is especially apparent in testing of fibers. It is very difficult to test the strength of an individual fiber, so manufacturers will often test them in bundles or tows (see terminology). Tow strength is governed by the weakest fibers. As the number of fibers increases, the Weibull distribution implies the chance of encountering weak fibers increases more than the chance of encountering strong fibers. This means that tested fiber strength will decrease as the number of fibers in the tow increases.
This effect is not just limited to raw fibers, but can also be seen in composites. Kevin Potter, in his book An Introduction to Composite Products, shows that hybrid composites made of two different, finely intermingled fibers, will have a higher strength than would be predicted by simply looking at the proportions of the two fibers. This is because the effective sample size of the weaker fiber is reduced, giving it a higher statistical strength.
You should now have enough information to answer the question posed at the beginning of this column. Because the specimens are unidirectional in the load direction, the fibers essentially carry all of the load, and the specimens are relatively insensitive to stress concentrations. Furthermore, the cut fibers in the notched specimen carry no load, so the number of fibers in the notched specimen is smaller. Thus, the unnotched specimen will fail at a lower stress (but not load), because the statistical strength of the fibers is lower.
I would like to acknowledge Longin Greszczuk for the notched specimen example, which he presented at the UCLA Short Course on Advanced Analysis and Design of Composite Materials and Structures.