## 2 Problem Definition  
It is necessary to make some assumptions and simplifications to investigate the stability of a shallow tunnel/sinkhole. This paper mainly takes the tunnel collapse as an example in the theoretical analysis and then extends this similar method to the sinkhole stability analysis. As schematically shown in Figure 1a [24], a circular traffic tunnel with a diameter of \(D\) is excavated by a tunnel shield, and the vertical distance from the ground surface to the mid-crown of the tunnel is denoted as \(H\). The rigid, permanent linings are installed behind the tunnel face and an unlined zone is usually left with a length of \(L\). A temporary supporting pressure \(\sigma_{\text{T}}\) is required in the unlined zone and there is a uniform surcharge \(\sigma_{\text{S}}\) acting on the ground surface, which could accelerate the tunnel collapse process. The soft clay around the tunnel is assumed to be isotropic and homogenous with a unit weight of \(\gamma\). For convenience, the soft clay is modelled by the linear elastic-perfectly plastic Tresca material to develop the analytical solution, which is a commonly used, simplified constitutive model for soft clay in undrained conditions. It is worthwhile to remark here that the undrained shear strength usually varies with depth, depending on the clay type and site history [31]. There are, however, a large number of cases in which this simplified assumption is adequate to make preliminarily predictions of the tunnel stability [4,32]. Besides, this paper does not aim to investigate the improved or deteriorated tunnel stability with time. For heavily over-consolidated clay, the dissipation of the pore water pressure, which may affect the tunnel stability, is not considered here. For a shallow sinkhole, it is assumed the shape of the sinkhole is a standard sphere with a soil cover of \(H\) and diameter of \(D\), and the collapse behaviour of the sinkhole is modelled by the spherical cavity contraction model.  
Following [PERSON] and [PERSON] [3], the excavation of a circular tunnel could be well modelled by the contraction of a cylindrical or spherical cavity if the axisymmetric assumption is adopted, as shown in Figure 1b. It is assumed that the tunnel excavation is simulated by slowly reducing the internal cavity pressure from \(p_{0}\) (\(p_{0}=\sigma_{S}+\gamma(H+D/2)\)) to \(\sigma_{T}\), and the inner and outer radii of the cavity are \(a=D/2\) and \(b=H+D/2\), respectively. In reality, the in situ stress state is normally non-hydrostatic and the axisymmetric assumption is not strictly correct, but previous studies have shown the simple concept of cavity contraction could predict tunnel stability and ground deformation [3, 25-27]. Therefore, the assumptions of [PERSON] and [PERSON] [3] are still followed and the induced error will be discussed.  
To assess the stability of a tunnel heading/sinkhole, the stability number \(N\) defined by [PERSON] and [PERSON] [33] is adopted here, which is expressed as:  
\[N=\frac{\sigma_{S}+\gamma(H+D/2)-\sigma_{T}}{s_{\rm u}}=\frac{p_{0}-\sigma_{T }}{s_{\rm u}} \tag{1}\]  
where \(s_{\rm u}\) is the undrained shear strength of soft clay. The derivation of \(N\) by the cavity contraction theory will be shown in the following section.