## 4 Validation and Discussion
### Database Establishment and Error Analysis  
To thoroughly and quantitively assess the reliability of the proposed method, a database consisting of 22 centrifuge model tests, 10 field tests, and 62 FELA results on the shallow tunnel stability in clays is assembled as shown in Table 1. It needs to be noted that only the results satisfying \(H/D<6\) and \(L/D\geq 0.5\) are selected to meet the requirement of shallow tunnels. The original references of in situ results were not easily available but were summarised in Macklin [32]. For FELA results, only the average values of lower and upper bounds are recorded.  
Figure 5: Comparison of predicted stability numbers with 3D FELA results.  
Figure 6: Sinkhole stability: (**a**) calculation model; (**b**) comparison of predicted stability numbers with FELA results.  
Stability numbers calculated by cavity contraction solutions (Equations (18) and (19)) are plotted versus those measured (tests in the field/laboratory) and simulated results (FELA), as shown in Figure 7 on a logarithmic scale. Figure 7a shown that the cavity contraction method gives a mean of 2.5% overestimation of the stability number than that measured in the field/laboratory. The coefficient of variation (COV) is 0.178, which is mainly caused by the scatter of field test results [32]. This may be caused by the uncertainty of complex geological conditions, stress history, measuring error and so on. In Figure 7b, when comparing with simulated stability numbers calculated by the FELA method, the cavity contraction solutions give a very good estimation of tunnel stability with the COV = 7.4%, which indicates the reliability of cavity contraction in predicting the tunnel heading stability. However, cavity contraction solutions seem to overestimate the tunnel stability at a lower \(N\) (usually for shallow tunnels) but underestimate the tunnel stability at a higher \(N\). The overpredictions are mainly because the term of \((H+0.5D)\) in Equation (1) results in a higher \(N\) for very shallow tunnels. On the other hand, the cavity contraction method is developed under the framework of static analysis and thus tends to give lower bound results for tunnel stability, which is why the underestimation is observed for a larger \(N\). Furthermore, the abovementioned reasons such as the simplified definition of \(k\), the definition of the tunnel collapse, and the axisymmetric assumption, also play a role in the prediction accuracy of tunnel stability.  
Although many factors seem to put a limit on the power of the cavity contraction method, the comparison with measured/simulated results indicates this simple method can still give a rather accurate prediction of tunnel stability in clays.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Author(s)** & **Year** & **Number of Data** & **_L/D_** & **_H/D_** & **Notation** \\ \hline \hline  & 1979 & 6 & plane strain & 1–3 & centrifuge \\  & 1979 & 7 & 0.5–3 & 1.5/3 & model tests \\  & 1999 & 10 & 0.67–5 & 1.65–5.9 & in situ tests \\  & & & & & centrifuge \\  & & & & & model tests \\  & & & & & \\  & & & & & \\ \end{tabular}
\end{table}
Table 1: Database of the tunnel/sinkhole stability number (\(L/D\geq 0.5\)).  
Figure 7: Measured versus predicted stability number: (**a**) model tests; (**b**) FELA results.