## 2 Deformation Allowance of Loess Tunnel and Statistical Monitoring Samples  
### Surrounding Rock Deformation  
The initial tunnel support serves as the primary means of protecting against unstable surrounding rock, while the deformation allowance plays an equally crucial role by effectively reducing the rock pressure exerted on the tunnel. The Constraint Convergence Method (CCM) is widely used to evaluate tunnel convergence and determine the necessary support requirements, taking into account the complex interaction between the surrounding rock and the support system. This method is particularly effective in analyzing these factors under different geological conditions and varying stress environments [20; 21; 22]. CCM establishes a relationship between the internal support force (Pi) and the radial displacement (u) of a circular tunnel, providing valuable insights into the tunnel's structural behavior under different loading conditions. This relationship is visually represented by the ground reaction curve (GRC), which characterizes the interaction between the ground and the tunnel. In addition, the support reaction curve (SRC) illustrates the elastic relationship between the longitudinal support stress and the resulting tunnel convergence. Based on crucial, albeit potentially debated assumptions, tunnel support is presumed to not alter the surrounding rock's response. The tunnel support system is expected to achieve equilibrium at the intersection of the GRC and the SRC [23]. The constitutive relationship of soft rock suggests that the use of flexible tunnel support allows the surrounding rock mass to transition into a plastic state, thereby accommodating the rock's deformation under stress. Different from the load action, the huge plastic property of surrounding rock will be released in some form. The resultant force (\(P_{0}\)) from the tunnel excavation's impact on surrounding rock mainly consists of three parts. Specifically, the process can be described as follows: (I) \(P_{0}\) is initially released through deformation, which is governed by the elastic-plastic behavior of the soft rock; (II) \(P_{0}\) is then supported by the intrinsic strength of the surrounding rock mass before it reaches a state of looseness; (III) however, these two factors alone are generally insufficient to ensure the stability of the surrounding rock, thus requiring the addition of external engineering support. Therefore, the support mechanism for tunnels in soft surrounding rock can be summarized as follows:  
\[P_{0}=P_{d}+P_{r}+P_{s} \tag{1}\]  
\(P_{0}\)--Resultant force of surrounding rock movement to the free area after excavation of tunnel surrounding rock, including gravity, water force, expansion force, tectonic stress, etc. ;  
\(P_{d}\)--Engineering force is typically expressed as deformation, especially in the case of soft rock where plastic properties are primarily released through deformation;  
\(P_{r}\)--Self-bearing capacity of surrounding rock;  
\(P_{s}\)--Engineering support force. In real-world construction, as illustrated in Figure 1, accurately quantifying the surrounding rock pressure presents a significant challenge due to the complex and variable nature of the geological conditions. Typically, when a certain level of displacement occurs, the actual rock pressure often exceeds the initial design assumptions for the support system. This discrepancy can lead to an insufficient support force, compromising the ability to maintain equilibrium between the surrounding rock and the support structure. On the one hand, if support is applied rapidly following tunnel excavation, as depicted at point A in the figure, it requires a significant support force. However, this approach imposes excessive strain on the surrounding rock, as it undergoes only elastic deformation, which is not optimal. Without timely support, or in the absence of any support, the surrounding rock can experience uncontrolled deformation, potentially leading to instability.