Judging by the factors of safety used, one of the features of the Eurocodes is an assumption of normal Gaussian statistical variation of the actions of imposed displacements and loads. In the light of recent investigations made by the author, using high quality research observations on live pipe-jacking contracts by Paul Norris under George Milligan's supervision at Oxford University, this assumption deserves wider discussion. The matter was raised at the Meeting of Teachers of Geotechnical Subjects at Birmingham University on 11 September and the main points are summarised here.

This particular research interest centred around determining characteristic values of actions affecting the maximum contact stress between pipes and packers in pipe-jacked tunnelling when the jacking force advances the pipeline. Figure 1 illustrates the generalised arrangement at two abutting pipes and the assumed stress transfer through the relatively compressible packer separating them. One expression for calculating the maximum stress is:

smax = K j{b - FeL p/(E pIp) } (R - X) Where K jis the packer stiffness per unit area; b, the deviation angle between the pipes; F, the jacking force between the pipes; e, the eccentricity of the load from the pipeline axis; L p, E p, I pthe length, Young's modulus and second moment of area respectively of the pipes; and (R-X), the width of the compression zone from maximum to zero stress. Characteristic values are required for the two variable actions of imposed displacement, b, and the force in the pipe, F. The former is a geotechnical engineering feature of the accuracy of tunnel construction while the latter is generated by the sliding resistance between the pipeline and the surrounding soil.

Based on Norris' strain readings from tube extensometers in instrumented pipes installed in working tunnels, the generalised variation of the force in the pipe for three tunnels taken together is shown in Figure 2. Although some high values occur more frequently than normal and the distribution is slightly truncated because negative values cannot occur, most engineers would probably accept representation as a normal distribution.

The 95 percentile is selected as the characteristic value of the force, F k, for a 1 in 20 chance of it being exceeded. Multiplying this by the usual partial factor of safety of1. 5 produces an ultimate limit state design value, F uls , with a 0. 01% theoretical probability (1 in 10,000) ofbeing exceeded for the illustrative distribution.

Given a distribution for deviation angles in Figure 1 Figure 3, calculated from displacement gauges at the joints in four of Norris's tunnels, engineers might similarly accept use of the normal curve.

However, this is actually based on the logarithms of the deviation angles: the distribution is closer to lognormal, shown to natural scale in Figure 4.

Using this lognormal distribution to obtain a characteristic value, b k, at the 95 percentile and multiplying by 1. 5 gives an ultimate limit state design value, b uls , with a 1. 4% probability ofbeing exceeded (1 in 70). The normal distribution is more severely truncated than the curve for the force, but if it had been used in the conventional way the theoretical probability of the ultimate limit state design value being exceeded would be 0. 1% (1 in 1,000) promoting an optimistic sense ofsecurity.

It is tempting to rely on the 'multiplication rule'for combined probabilities: two independent variables, one with a 1 in 10,000 chance ofbeing exceeded and the other with a 1 in 70 chance of being exceeded, result in the chance ofboth being exceeded together of1 in 700,000. Unfortunately the local jacking force is affected by the deviation angle in the tunnel. At the Tunnel Construction and Piling Symposium 1999 in London, I reported correlation coefficients between the two variables ranging from 0. 3 to 0. 8. Chapman and Ward in their 1997 text on Project Risk Management (cited by the ICE, Faculty ofActuaries and Institute ofActuaries in RAMP,1998) recommend that the effects ofcorrelation can be taken into account by weighted proportion using the correlation coefficient, r, translated for F dependent on b as:

Probability(higher b AND Probability(higher b) {r + higher F, given higher b) = (1-r)Probability(higher F)} Accordingly, for a correlation coefficient of0. 8, the probability that values higher than the above ultimate limit state design values ofdeviation angle and force will occur together is 1 in 90. While the effect ofmaterial characteristic strength and partial safety factor needs to be brought into the overall risk evaluation, the risk ofdamage to pipes in the tunnel, with consequent delays, remedial work and costs, appears greater than would usually be expected in design.

Just as the distribution ofvalues of force do not fit exactly the Gaussian curve, so the deviation angles do not fit exactly the lognormal distribution. The fit to the upper tail of the distribution can be improved and other distributions can be considered; several have been, but the lognormal distribution improved to fit the upper tail has so far given the lowest residual errors between predicted and observed probabilities at higher values.

The aim was to raise the issue of the expectation ofnormal distribution ofvariable actions in the codes and to open discussion on the extent to which engineers should be aware ofother possibilities and their consequences. The Eurocodes allow consideration ofdifferent factors where the risks are atypical, and it would seem prudent that they should advise on the possibility that statistical distributions of actions other than normal Gaussian might increase risks, possibly indicating the order ofmagnitude ofprobabilities inherent in the code methods. Even if this is not done, managers ofpipe-jacked tunnelling projects might add the above techniques to a risk-evaluation toolkit to further inform their own decision-making task.

r. f. haslem@livjm. ac. uk

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