## Worked examples

The worked examples in this chapter illustrate the way in which statistics may be used to determine the characteristic values of various geotechnical parameters: the standard penetration blow count in Thames Gravel (Example 5.1); the undrained strength of London Clay (Example 5.2); and the angle of shearing resistance of Leighton Buzzard Sand (Example 5.3).

Specific parts of the calculations are marked O, ©, ©, etc., where the numbers refer to the notes that accompany each example.

5.7.1 Standard penetration tests in Thames Gravel

Example 5.1 applies statistical analysis to the data presented in Figure 5.14 for the site near Gravesend, in Kent.

In Investigation A, where there is no obvious trend for blow count to change with depth, the statistics are based on the blow counts alone and not on the penetration test depths.

In Investigation B, where there appears to be a slight increase in blow count with depth, the analysis is multi-variate and hence considerably more complicated.

Figure 5.17 shows the outcome of the analysis on the data from Investigation B.

Notes on Example 5.1

O The blow counts quoted here are plotted as the open white symbols on Figure 5.14.

© The coefficient of variation of the sample (VN = 0.067) is below the recommended threshold value (VN = 0.1) and hence the calculation should proceed with an assumed standard deviation based on the threshold.

© The standard deviation is based on the threshold coefficient of variation.

© Student's t-value for the 95% confidence limit has been taken from standard tables for a normal probability density function (see Chapter 2).

© The definition of the statistical coefficient is appropriate for calculating a 95% confident mean value when the variance is unknown and assumes that there is no variation with depth.

© The characteristic value (19.7) is only slightly less than the mean value of N (20.7), reflecting the lack of variability of this data set.

& The blow counts quoted here are plotted as the closed black symbols on Figure 5.14.

© It is assumed that the value of N varies with depth and hence the statistics are more complicated than for Investigation A. Refer to Section 5.5.3 for an explanation of the statistical procedure used here.

© The slope of best-fit regression line gives the rate at which N increases with depth (the slope is negative because depth below ground level has been taken as a negative value).

® The standard error is a measure of the variability of the data set and can be used to provide a simple estimate of the 95% confident mean, as discussed below.

Figure 5.17 shows the outcome of the statistical analysis on data from Investigation B. The analysis, which assumes that the blow count N varies according to a normal distribution, confirms the general trend for N to increase with depth, which is visually obvious from the scatter plot and which an experienced geotechnical engineer might assume.

The line labelled 'best fit' gives values of blow count N that have a 50% probability of being exceeded by the local value of N at the same depth. This line is readily calculated by any spreadsheet program using linear regression.

The curve labelled 'lower characteristic' gives values of N that have a 95% probability of being exceeded by the average value of N over the depths considered (i.e. the 'spatial average'). This is a 95% confident value and is the one most often needed in geotechnical designs which involve a large zone of the ground. The curvature of the lower characteristic line is a consequence of the chosen statistical method and signifies greater uncertainty at the ends of the data set.

SPT blow count, N

SPT blow count, N

The curve labelled 'upper characteristic' gives values of N that have a 95% probability of not being exceeded by the average value of N over the depths considered. This is needed when an upper value is critical to the design.

The line labelled '5% fractile' gives values of N that have a 95% probability of being exceeded by the local value of N at the same depth. We will call this the 'local' characteristic value. It is needed when the ground is unable to redistribute stresses from highly to lightly loaded zones.

The line labelled 'mean less half standard error' is much easier to calculate than the other lines discussed below, being offset from the best-fit line. It also has the distinct benefit of being linear with depth. Since this line is a reasonable approximation to the lower characteristic line, it can be used as a simple means of obtaining the characteristic line through the data.

Example 5.1 Standard penetration tests in Thames Gravel Determination of characteristic blow count

Data from investigation A

Consider the results of a series of standard penetration tests (SPTs) in Thames Gravel. The measured blow counts were:

22, 19, 19, 21, 23, 22, 19, 21, 19, 21, 21, 20, 22. O

Statistical analysis of data

The number of test results is n = 13

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