# Bearing Capacities and Failure Behaviors of F-Type Socket Joint in Rectangular Pipe Jacking Tunnel
## 2 Full-Scale Experiments
### Analysis of Test Data
#### 2.3.2 Stress Deformation of Pipe Joint  
The force conditions of the pipe joint are shown in Figure 7, where F is the upper jack reaction force, and the \(k_{i}x_{j}(i=j=1,2,3)\) term is the equivalent spring reaction force under the foundation, where \(k_{1}=5202\) kN/m and \(k_{2}=k_{3}=3468\) kN/m. According to the force balance \(\sum M=0\), the joint bending moment \(M\) can be obtained, and the joint bending moment changes as shown in Figure 8.  
From Figure 8, it can be found that under the same load displacement, the larger the foundation stiffness, the greater the moment of the pipe joint. The joint will be deformed under the upper load, and the amount of joint opening will change as shown in Figure 9.  
As shown in Figure 9, the joint tension is closely related to the joint moment, the larger the joint bending moment, the larger the corresponding joint tension.  
In addition, under the 9 springs, when the loading reached 15 mm, the trend of joint opening distance changes slowed down, and at the same time, it was observed that microcracks started to appear at the middle of the top and bottom plate of the pipe joint; with  
Figure 5: Concrete failure characteristics. (**a**) Concrete cracks in the side walls. (**b**) Concrete cracks in the roof slab. (**c**) Concrete cracks in the subgrade. (**d**) Crushed concrete in the roof slab.  
Figure 6: Steel ring warping.  
the increase of loading displacement, the micro-cracks continued to expand and penetrate, indicating that the upper applied displacement was not only borne by the deformation of the foundation spring and the deformation of the joint, but also partly borne by the deformation of the pipe joint, and thus the rate of increase of the joint tension slowed down.  
Similarly, under 6 springs, when the loading reached 20 mm, small cracks started to appear at the middle of the top and bottom of the pipe joint, and the rate of change of the joint opening slowed down. At 4 springs, when the load reached 30 mm, a small crack started to appear at the middle of the top and bottom plate of the pipe joint and the rate of change of the joint tension slowed down.  
The change of joint bending stiffness can be further obtained from the change of bending moment and tension at the joint, as shown in Figure 10.  
Figure 8: Joint bending moment.  
Figure 7: Force diagram of pipe joint. (**a**) 9 springs (\(K_{\mathrm{o}}=10.16\times 10^{3}\)\(\mathrm{kN}\cdot\mathrm{m}^{-3}\)). (**b**) 6 springs (\(K_{\mathrm{o}}=6.77\times 10^{3}\)\(\mathrm{kN}\cdot\mathrm{m}^{-3}\) ). (**c**) 4 springs (\(K_{\mathrm{o}}=4.52\times 10^{3}\)\(\mathrm{kN}\cdot\mathrm{m}^{-3}\) ).  
Figure 9: Joint opening variety.  
As can be seen from Figure 10, the joint bending stiffness can be roughly divided into two stages. When the tensioning amount was less than 0.007 rad, the joint itself did not have large deformation, the joint stiffness was mainly controlled by the strength of the joint itself and the difference of the joint bending stiffness under each equivalent foundation was small. When the tensioning amount was greater than 0.007 rad, micro-cracks appeared at the axillary corners of the side walls of the pipe joint, and as the upper loading displacement continued to increase, the micro-cracks expanded further and penetrated, and the pipe joint itself started to deform more, leading to the slowdown of the joint tensioning amount and further increasing the slope of the curve.