Back analysis of tunnel settlement data is a key discipline in the prediction of tunnel settlement assessment. Most assessment methods treat volume loss as an empirical value determined based on case history and engineering judgement.

To get the case history information we have to look at previous projects and back analyse the settlement troughs to obtain key parameters such as the volume loss and the distance to the point of contra-flexure. The challenge is that the data we typically obtain from site is based on a limited number of settlement points and is often noisey due to issues such as survey inaccuracy. For this reason we need a way to try and fit the data to an interpretation.

It is surprising how little information there is on this interpretation process. It is not considered frequently in academia and despite there being a range of different approaches there is little guidance on the applicability of the different approaches. Jones and Clayton^{1} did however do an assessment of some of the different approaches available. Whilst there are limits to the work they undertook it is useful reading for those undertaking back analysis of settlement data.

I think that’s important to understand the implications of this research in a very practical sense to understand the implications of noise in settlement data when undertaking back analysis. To do this I have looked at undertaking some Monte Carlo analyses in a similar way to the approach that Jones and Clayton adopted. In this case however I’ve simply undertaken the back analyses for five different case and presented the volume loss and trough width from the back analyses, the values that might be inferred the back analyses of the data.

In this model I have looked at a settlement array of 9 points evenly distributed across half a settlement trough. The depth of the tunnel is 15m, and the diameter is 3.5m. The settlement is assumed to take the form of a Gaussian curve with a trough width parameter, K, of 0.5 and a volume loss of 1%. This gives a maximum settlement, s_{max}, of approximately 5mm. Two different standard deviations on the survey tolerance have been considered. The first 0.25mm is consistent with the tolerance proposed by Jones^{2}. The second, 0.5mm, is a value that I find consistent with many projects where high precision monitoring of the movements caused by the construction of the tunnel is either not necessary or not possible.

The following table gives the results of 5 different runs using the NMSAE (Non-linear regressions using the sum of the absolute errors) model proposed by Jones and Clayton.

For 0.25mm standard deviation in the measurement results:

Value | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 |
---|---|---|---|---|---|

Best fit volume loss | 0.991% | 0.947% | 1.000% | 0.995% | 0.993% |

Trough width K | 0.505 | 0.562 | 0.504 | 0.503 | 0.512 |

Volume loss determined by the Simpson Rule | 0.993% | 0.939% | 1.046% | 0.968% | 1.102% |

Volume loss determined by the Trapezium Rule | 1.014% | 1.015% | 1.031% | 0.974% | 1.047% |

For 0.5mm standard deviation in the measurement results:

Value | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 |
---|---|---|---|---|---|

Best fit volume loss | 0.989% | 0.980% | 1.028% | 1.106% | 1.109 |

Trough width K | 0.579 | 0.589 | 0.531 | 0.520 | 0.502 |

Volume loss determined by Simpson’s Rule | 1.025% | 1.045% | 1.082% | 1.098% | 1.179% |

Volume loss determined by the Trapezium Rule | 1.027% | 0.974% | 1.045% | 1.050% | 1.202% |

From these results it is relatively clear that a degree of caution needs to be applied when interpretting back analysis of tunnel settlement data unless the results are of the very best quality. For good quality measurements with 0.25mm standard deviation on the survey result there is at least a 10% variation in the results and with 0.5mm the variation is significantly higher.

All of this must be bourne in mind when selecting appropriate values to use for undertaking settlement analysis. Data from back analysis is an essential part of the tool kit that engineers should use to predict settlements for future projects however it is clear that picking a single value for the settlement parameters on the basis of case history is difficult.

Going forward it would be really useful to be able to gain some understanding of the potential variability of in these parameters from a particular back analysis. Should we expect that a broad range of parameters might be reasonable given the data or in fact is the range of potential parameters from a back analysis very tight? I will discuss a technique to acheive this in future posts.

^{1. B. Jones and C. Clayton, “Guidelines for Gaussian curve-fitting to settlement data,” presented at the Underground – the way to the future!: World Tunnel Congress 2013, Geneva, 2013, pp. 1–8.↩}

^{2. B. D. Jones, “Low-volume-loss tunnelling for London ring main extension,” Proceedings of the ICE - Geotechnical Engineering, vol. 163, no. 3, pp. 167–185, Jan. 2010.↩}