## 5 Analysis of Tunnel Bottom Structure Damage and Countermeasures  
### Types and Mechanism of Tunnel Bottom Drum  
Tunnel floor heave can be classified into four fundamental types based on the mechanics of failure: extrusion flow deformation, water-induced swelling, flexible folding, and shear dislocation. Each type is illustrated in Figure 29.  
The causes and mechanical mechanisms of uplift deformation in tunnel inverted arches can be attributed to two primary factors: lateral load mechanisms and longitudinal load mechanisms. Figure 28: The right side of the damage circle location—depth 100 m, \(K_{0}=1.25\). Figure 27: Left failure circle location—depth 100 m, \(K_{0}=1.25\). **Lateral load mechanism:** The ends of the inverted arch are subjected to vertical loads from the overlying strata and the structure's weight, which are transmitted through the arch foot. Simultaneously, the base of the arch experiences an upward reaction force from the underlying foundation. These forces generate significant internal stresses within the inverted arch and its fill, resulting in substantial lateral tensile stress at the arch's cracking points. **Longitudinal load mechanism:** Along the tunnel's longitudinal axis, the presence of extensive weak surrounding rock near the uplifted sections of the inverted arch contributes to increased surrounding rock pressure, structural internal forces, and displacement, particularly foundation settlement. Moreover, the long-term operation of trains exacerbates uneven settlement between uplifted and settled sections, inducing significant internal forces along the tunnel's longitudinal axis. This phenomenon resembles a \"leverage effect\" or the distribution pressure on an upper cantilever beam, where the structural internal force peaks at the stratigraphic boundary. The transfer of shear forces intensifies the vertical load at the ends of the inverted arch, further increasing the transverse tensile stress at the arch's cracking points. The accumulation of bending moments results in substantial longitudinal compressive stress across the cross-section of the inverted arch. While both Thin Plate Differential Equations and Euler Beam Theory are used to analyze thin structural elements under load, they differ significantly in the type of structures they describe (plates vs. beams), the complexity of their governing equations, and the dimensionality of their applications. Euler Beam Theory is simpler and focuses on one  
Figure 29: Tunnel floor heave model diagram. (**a**) Squeezed flow model of tunnel floor heave. (**b**) Water swelling model of tunnel floor heave. (**c**) Shear-staggered model of tunnel floor heave. (**d**) An upper arch model of tunnel bottom with flexural folds. -dimensional bending, whereas Thin Plate Theory deals with two-dimensional bending and is more complex, accommodating a wider range of loading and boundary conditions. From the above analysis and combined with the Thin Plate Differential Equations [33] and Figure 30, we can obtain:  
\[D\
abla\omega^{4}-\left(N_{x}\frac{\partial^{2}\omega}{\partial x^{2}}+2N_{xy} \frac{\partial^{2}\omega}{\partial y^{2}}+N_{y}\frac{\partial^{2}\omega}{ \partial y^{2}}\right)=0 \tag{4}\]  
where \(\omega\) is the deflection of the bottom plate (m); \(D\) is the bending stiffness of the bottom plate (N-m); \(N_{x}\), \(N_{xy}\), and \(N_{y}\) are the mid-plane stress of the bottom plate (N/m). The stress in the bottom plate is (positive in tension). \[\sigma_{x}=-\frac{p_{x}}{t},\sigma_{y}=-\frac{\mu p_{x}}{t},\tau_{xy}=0 \tag{5}\]  
where \(p_{x}\) is the uniform pressure per unit length of the side of the bottom plate (N-m); \(t\) is the thickness of the bottom plate (m); and \(\mu\) is the Poisson's ratio.