## 5 Estimation and Prediction of Tunnel Floor Heave  
### Load Calculation Formula  
Based on the circumcircle theory of tunneling and the principles of active and passive earth pressure, a mechanical model of tunnel floor heave in inclined rock strata has been developed (as illustrated in Figure 17). According to this theory, the tunnel cross-section can be approximated as circular. [PERSON]'s earth pressure theory indicates that the surrounding rock at the tunnel floor generates both active and passive plastic stress. Specifically, the rock mass in zone BCD is subjected to active plastic pressure, while zone ABD is influenced by passive plastic pressure. When the rock mass in the ABD zone reaches a passive plastic state, horizontal ground stress, compounded by the effects of time, causes the tunnel floor at AB to bulge upward, leading to floor heave.  
Figure 16: Comparison diagram of inverted arch uplift between tunnel with secondary lining and tunnel without secondary lining. (**a**) Only initial support. (**b**) Initial support and secondary lining structure.  
Figure 15: Statistical chart of tunnel floor heave under different support conditions. (**a**) Only initial support. (**b**) Initial support and secondary lining structure. (**c**) Initial support with secondary lining structure and inverted arch filling layer.  
The angle between the slip surface CD of the active plastic zone of the tunnel floor and the plane AB of the floor is equal to \(45^{\circ}+\varphi/2\), and the angle between the passive plastic zone and the horizontal plane is \(\theta\) due to the influence of the weak surface AD of the rock contact. According to the theory of a pressure arch, regarding the vertical pressure \(\gamma(h+H)\) from the BC surface to the surrounding rock in the pressure arch, \(h\) is the height of the tunnel, \(H\) is the height of the pressure arch, and \(\gamma\) is the active slip surface subjected to horizontal in situ stress. When the tunnel floor is in the limit equilibrium state, the active plastic body BCD generates a horizontal left active pressure \(F_{1}\) on the right side of BD (BD = \(Z\)). The passive plastic body ABD generates a horizontal rightward active pressure \(F_{2}\) on the left side of BD, where  
\[F_{1}=\frac{1}{2}\Big{[}\gamma(h+H)\text{tan}^{2}\Big{(}45^{\circ}-\frac{ \varphi}{2}\Big{)}+\gamma(h+H+Z_{D})\text{tan}^{2}\Big{(}45^{\circ}-\frac{ \varphi}{2}\Big{)}\Big{]}Z_{D} \tag{10}\]  
\[F_{2}=\frac{1}{2}\gamma Z_{D}^{2}\text{tan}^{2}(90^{\circ}-\theta) \tag{11}\]  
The difference \(F_{12}\) between the active force \(F_{1}\) and the passive force \(F_{2}\) on the BD surface is the actual force that drives the ABD to slip to the left, and the horizontal in situ stress also acts on the BD surface (as shown in Figure 18).  
When the surrounding rock pressure of the tunnel floor is greater than the strength of the surrounding rock (\(\sigma_{c}\)), the stress instability of the tunnel floor occurs; that is, the floor behave occurs. In order to control the floor heave of the tunnel, it is necessary to use the support force (\(P_{\text{S}}\)) to balance the surrounding rock pressure \(P_{0}\) of the tunnel floor and have a certain safety factor, a, to ensure the stability of the tunnel floor; that is,  
\[P_{\text{S}}=a(P_{0}-\sigma_{c}S_{AB}) \tag{12}\]  
In the formula, \(\sigma_{c}\) is the uniaxial compressive strength of rock; \(S_{AB}\) is tunnel floor area per unit length along the axial direction.  
Figure 17: Mechanical model of floor heave of inclined rock tunnel.  
Figure 18: Force analysis of passive plastic zone.