## 1 Introduction  
Ground stability is one of the two major concerns for geotechnical engineering practices, which require that the ground soil mass remains stable under the given or expected loading conditions. This paper focuses on the stability analysis of two typical underground openings (i.e., shield tunnels and sinkholes), aiming to present simple methods for the safety assessment during their design and construction.  
In order to maintain the stability during shield tunnelling in clays, temporary supports provided by compressed air or slurry [1, 2] are usually necessary. For shield tunnelling in clay, the construction process is usually sufficiently rapid that the clay behaviour around the heading is often regarded as undrained [3, 4]. Since the failure mechanism of a tunnel may be similar to that of a sinkhole, sinkhole stability is also considered in this paper.  
Many studies have investigated the undrained stability of shallow tunnels and sinkholes by model tests [5, 6], the limit analysis method [1, 7, 8, 9, 10, 11, 12, 13, 14, 15], and the displacement finite element/difference method [16, 17]. To investigate the influence of embedment ratio and tunnel heading geometry on tunnel stability, [PERSON] [5] conducted two series of 2D/3D centrifuge model tests in clays, and found that the tunnel heading stability was strongly affected by the tunnel heading geometry. Following [PERSON] [5], [PERSON] and [PERSON] [6] studied the tunnelling-induced ground movement and collapse mechanisms of single and parallel tunnels, and obtained a series of stability results. [PERSON], et al. [1] assessed the stability of shallow tunnels using the lower and upper bound theory under plane strain conditions. Afterwards, many attempts have been made to derive the upper bound (UB) solutions for 2D/3D tunnel stability, with various kinematically admissible velocity fields [11, 12, 13, 16, 18, 19]. [PERSON] [20, 21] developed the finite element limit analysis (FELA)method by linear programming, which made it possible to conduct geotechnical stability analysis without predefining a failure mode. Using the FELA method, the undrained stability of tunnels under plane strain conditions can be easily determined, even taking the variation of the undrained shear strength into consideration [7,8,22]. Later, [PERSON], et al. [9], [PERSON], et al. [10] and [PERSON] [23] revisited these problems using more advanced FELA techniques and smaller gaps between results obtained, and the lower bound and the upper bound methods were derived. Furthermore, the undrained face stability in clay was also analyzed by the finite element method (e.g., [PERSON], et al. [17], [PERSON], et al. [12], [PERSON], et al. [16]), which provided a valuable benchmark for tunnel stability analysis.  
Apart from the abovementioned methods of stability analysis for tunnels/sinkholes, cavity contraction theory--which is a simple and powerful tool--has also been widely applied in tunnel deformation and stability analysis [3,24,25,26,27,28,29,30,24,29,31,32,33,34,35,36,37,38,39,39]. For tunnel stability in undrained clay, [PERSON] [25] proposed a simple equation based on cavity contraction theory in Tresca materials. However, this equation failed to consider the influence of finite cover depth on tunnel stability, which could overestimate the stability for a shallow tunnel. In addition, there is no equation, to the best knowledge of the authors, that can be directly used to account for the stability of a tunnel heading. To fill these gaps, this paper mainly extends the work of [PERSON] [25] and modifies [PERSON]'s equation by developing the cavity contraction theory in a finite Tresca soil mass. Then, a simple equation that could predict the tunnel/sinkhole stability is proposed based on the newly developed cavity contraction solution, and validated after comparison with published results.