## 2 Stiffness Reduction Method
### Stiffness Reduction Model  
The seam of the PUTCGE had a typical semi-rigid characteristic, the behavior of which lay between hinged and rigid joints. How to determine the rotational stiffness of the joint and its influence on structural behavior is the key to the structural analysis and calculation of this type of prefabricated utility tunnel. Similar problems also exist in the analysis and calculation of shield tunnel lining structures, as mentioned in Section 1. [PERSON] [24] and [PERSON] et al. [30] have both outlined detailed steps of stiffness reduction models for determining the effective bending stiffness coefficient of longitudinal joints in shield tunnels in which the maximum horizontal displacement of the lining ring is treated as the target parameter. The calculation methods of the beam-spring model and the stiffness reduction model are both used to calculate the maximum horizontal displacement. When the two models give the same displacement, the stiffness reduction coefficient in the stiffness reduction method corresponds to the effective bending stiffness coefficient in the shield tunnel. Since the segment joints are basically evenly arranged along the lining  
Figure 1: Cross-section of the double-cell PUTCGE (unit: mm). ring and the PUTCGE has joints only in the middle-height of the sidewalls, the stiffness reduction method for the whole ring in the shield tunnel is not directly applicable to the prefabricated utility tunnel. For a PUTCGE, the internal force analysis and calculation can be carried out by reducing the stiffness of the sidewalls while keeping the stiffness of the top and bottom slabs unchanged, which was our working assumption. Based on its rectangular section characteristics, the bending moment adjustment factor was further introduced to adjust the bending moments at the ends of the sidewalls and at the middle-height of the sidewalls to consider the effect of joint stiffness, as shown in Figures 2 and 3. In Figure 2, \(El_{e}\), \(El_{b}\), \(El_{b}\), and \(El_{b}\) are the flexural rigidities of the external sidewalls, the internal sidewall, the top slab, and bottom slab, respectively; \(q\) is the vertical earth pressure, and the lateral earth pressure is considered to follow a trapezoidal distribution, with the upper and lower earth pressures denoted by \(p_{1}\) and \(p_{2}\), respectively. Figure 2: Structural systems for PUTCGEs: (**a**) uniform stiffness model, (**b**) beam-spring model, and (**c**) stiffness reduction model. Assuming that the width of the undeformed utility tunnel is \(b\), under the action of the loadings, the variation of the width was \(\Delta b_{1}\) when the uniform-stiffness model (without considering joint stiffness) was adopted, as shown in Figure 2a; the variation of the utility tunnel width was \(\Delta b_{2}\) when the beam-spring model (considering rotational stiffness of the joint) was adopted, as shown in Figure 2b. When the stiffness reduction model was adopted, as shown in Figure 2c, the stiffness reduction coefficient (\(\eta\)) of the sidewalls (considering reduction of flexural rigidity) was then defined as:  
\[\eta=\frac{\Delta b_{1}}{\Delta b_{2}} \tag{1}\]  
It should be noted that, under the action of earth pressures, the cross-section of a circular shield tunnel turns into an elliptical shape, in general, and the diameter of the horizontal long axis after deformation is larger than the original diameter [31; 32; 33; 34]. Therefore, within a range of 45 degrees of the central angle from the horizontal diameter direction, the foundation resistance caused by deformation should, generally, be considered. Under the action of earth pressures, the deformed shape of the rectangular utility tunnel is shown by the dotted line in Figure 2 for each model, where the external sidewalls are inwardly convex.