## 5 Conclusions  
A double-cell prefabricated utility tunnel composed of groove-shaped elements (PUTCGE) has typical semi-rigid characteristics in the seam joints connecting different structural components. A semi-rigid joint means that it can transfer bending moments while causing rotational displacements, the behavior of which is generally known to lie between systems comprised of hinged and rigid joints. How to determine the rotational stiffness of a joint and its influence on structural behavior is the key to analyzing such types of structures. Considering similarities in the structural system and loading conditions between shield tunnel lining and PUTCGEs, the methodology associated with the stiffness reduction method as originally adopted for tunnel linings was extended to a PUTCGE in the current study. The concepts of stiffness reduction and bending moment adjustment coefficients of the sidewalls were introduced based on concepts of the classic uniform stiffness model and the beam-spring model. The effects of the joint stiffness ratio, the aspect ratio of width to height, the coefficient of subgrade reaction, and the embedment depth of a PUTCGE on the stiffness reduction and bending moment adjustment coefficients were investigated via FE simulations. Based on the FE simulation results, a fitting formula was proposed for describing the relationships among the main parameters that influence the structural behavior. The data from a full-scale physical PUTCGE model test presented in the literature were used to check the stiffness reduction method proposed in this paper. The main conclusions are as follows:  
* The stiffness reduction coefficient of the sidewalls was found to increase with the increase in the joint stiffness ratio, while the bending moment adjustment coefficient was found to decrease with the increase in the joint stiffness ratio; \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Items** & **Experimental** & **Calculated** & **(Calculated Value–Experimental Value)** \\  & **Value (kN\(\cdot\)m)** & **Value (kN\(\cdot\)m)** & **Value/Experimental Value** \\ \hline Bending moment at the end of the sidewalls & 64.3 & 60.5 & \(-6.0\%\) \\ Bending moment at middle-height of the sidewalls & 26.3 & 23.5 & \(-10.6\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of calculated values and experimental values. * The stiffness reduction coefficient of the sidewalls was found to decrease with the increase in the aspect ratio of width to height, while the bending moment adjustment coefficient was not much-affected by aspect;
* Neither the coefficient of subgrade reaction nor the embedment depth was found to have a significant influence on either the stiffness reduction coefficient or the bending moment adjustment coefficient;
* A fitting formula was proposed for describing the stiffness reduction coefficient or the bending moment adjustment coefficient as functions of the joint stiffness ratio and the aspect ratio;
* The stiffness reduction method proposed for PUTCGEs was validated through comparison with a full-scale physical test model with an error not exceeding 10.6%. Q.W. : Conceptualization, funding acquisition, model development, validation, and original drafting; G.G. : writing--review and editing; J.L.H. : review and editing. All authors have read and agreed to the published version of the manuscript. This research was funded by the Science and Technology Program Projects in Nantong, China (grant numbers: MS22020025 and MSZ21046), and by Xi'an Jiaotong-Liverpool University (grant numbers: RDF 18-01-23, PGRS1906002 and REF-20-01-01). Not applicable. The data used to support the findings of this study are available upon request from the corresponding author. The authors would like to acknowledge the support provided by the Science and Technology Program Projects in Nantong, China (MS22020025 and MSZ21046), and by Xi'an Jiaotong-Liverpool University (RDF 18-01-23, PGRS1906002 and REF-20-01-01).