1.4.3 Statistical Basis for Design

When a material such as concrete is manufactured to a specified mean strength, tests on samples show that the actual strength deviates from the mean strenght to a varying degree depending on how closely the process is controlled and on the variation in strength of the component materials. It is found that the spread of results approximates to a normal distribution curve. 

Normal distribution curve

The probability of a test result falling between two values of strength such as x1 and x2 is given by the area under the curve between the two values, i.e. the shaded area in the figure. The area under the whole curve is thus equal to unity.

The effect of quality control is as shown, poor control producing a flatter curve. The probability of a test falling below a specified value is clearly greater when the quality of control is reduced.

The equation of the normal distribution curve is:

y= 1/[σ(2π)˄0,5] exp [-(x-x)˄2/(2σ˄2)]

which shows that the curve is fully defined by the mean x and the standard deviation σ of the variable x.

For a set of n values of x, the mean is given by:

X= Σx/n

The standard deviation which is a measure of the dispersion, is the root mean square of the deviations of x from the mean, given by:

σ= [Σ(x-x)˄2/n]˄0,5

In practice, it is not usually possible to obtain all of the values of x that would theoretically be available. For example, it would not be possible to test all the concrete in a structure and it is therefore necessary to obtain estimates of x and σ by sampling. In this case, the best estimate of the mean is still that given by X= Σx/n, but the best estimation for the standard deviation is given by:

σ= [Σ(x-x)˄2/(n-1)]˄0,5

where n is the number of the test results in the sample.

For hand calculations, a more convenient form is 

σ=[(Σx˄2 - nx˄2) / (n - 1)]˄0,5

Statistical distributions can also be obtained to show the variation in strength of other structural materials such as steel reinforcement and prestressing tendons. It is also reasonable to presume that if sufficient statistical data were available, distributions could be defined for the loads carried by a structure. It follows that it is impossible to predict with certainty that the strength of a structural member will always be greater than the load applied to it or that failure will not occur in some other way during the life of the structure. The philosophy of limit state design is to establish limits, based on statistical data, experimental results and engineering experience and judgement, that will ensure an acceptably low probability of failure. At present there is insufficient information to enable distributions of all the structural variables to be defined and it is unlikely that is will ever be possible to formulate general rules for the construction of a statistical model of anything so complicated as an actual structure.