## 2 Field Testing
### Comparison of Measured and Calculated Earth Pressure  
At present, the common methods for calculating the dynamic earth pressure over the buried pipeline are the [PERSON] formula and the distributing angle method. Based on the elastic mechanics equation, [PERSON] [25] obtained the steady-state response of semi-infinite space under vertical point load using a three-dimensional Fourier transform and extended it to the steady-state response of semi-infinite space under uniform rectangular load. The stress response expression is as follows:  
\[\sigma_{x}=-\frac{T}{\pi^{2}}\int_{0}^{\frac{\pi}{2}}\frac{\cot\theta}{H}\bigg{\{} \bigg{(}1-\frac{1}{2}\alpha_{2}^{2}\cos^{2}\theta\bigg{)}\bigg{(}1+\frac{1}{2} \alpha_{2}^{2}-\alpha_{1}^{2}\bigg{)}v_{1}-\gamma_{1}\gamma_{2}v_{2}\bigg{\}}d\theta \tag{1}\]  
\[\sigma_{y}=-\frac{T}{\pi^{2}}\int_{0}^{\frac{\pi}{2}}\frac{1}{\text{Hsin}\theta \cos\theta}\Big{\{}\Big{(}1-\frac{1}{2}\alpha_{2}^{2}\cos^{2}\theta\Big{)} \Big{[}\sin^{2}\theta+\Big{(}\frac{1}{2}\alpha_{2}^{2}-\alpha_{1}^{2}\Big{)} \cos^{2}\theta\Big{]}v_{1} \tag{2}\]  
\[\sigma_{z}=\frac{T}{\pi^{2}}\int_{0}^{\frac{\pi}{2}}\frac{1}{\text{Hsin} \theta\cos\theta}\Bigg{\{}\bigg{(}1-\frac{1}{2}\alpha_{2}^{2}\cos^{2}\theta \bigg{)}^{2}v_{1}-\gamma_{1}\gamma_{2}v_{2}\bigg{\}}\text{d}\theta \tag{3}\]  
Figure 8: Absolute values of the axial strain of (**a**) pipe side and (**b**) pipe roof.  
\[\left\{\begin{array}{c}\alpha_{2}=\beta\alpha_{1}=\frac{\gamma}{c_{2}}\\ H=\left(1-\frac{1}{2}\alpha_{2}^{2}\cos^{2}\theta\right)^{2}-\gamma_{1}\gamma_{2 }\\ u_{1,2}=\log\left[\frac{(\arccos\theta-\text{bsin}\theta)^{2}+\gamma_{1}^{2} \text{,}2^{2}}{(\arccos\theta+\text{bsin}\theta)^{2}+\gamma_{1}^{2}\text{,}2^ {2}}\right]\end{array}\right. \tag{4}\]  
where \(c_{2}\) is the shear wave velocity in semi-infinite space.  
The distributing angle method [26] assumes that the surface load on the pavement is transmitted downward along a specific diffusion angle and that the upper load is uniformly distributed in the isometric plane after transmission. According to the above assumptions, the vertical pressure generated from the excitation force generated by the vibratory roller vibration wheel at the top of the pipe can be obtained from:  
\[q_{\text{ex}}=\frac{k_{d}Q_{\text{ex}}}{(L+1.4H)(B+1.4H)} \tag{5}\]  
where \(q_{\text{ex}}\) is the vertical pressure produced by the vibrating wheel on the pipe top (kN/m), \(Q_{\text{ex}}\) is the contact force between the vibratory roller and the soil (kN), \(L\) is the effective contact width between the vibrating wheel and the soil (m), \(B\) is the length of the vibration wheel (m), \(H\) is the pipe-jacking soil-filled height, and \(k_{d}\) is the dynamic coefficient.  
The earth pressure values obtained from the G Eason formula, the distribution angle method, and the field test during the ninth compaction are provided in Figure 9. It is observed from the figure that the variation law of the dynamic earth pressure amplitude in the subgrade with the depth calculated using the G Eason formula and the distribution angle method are consistent with the measured values. When the depth of the measuring point is less than 0.6 m, the dynamic earth pressure amplitude calculated by the G Eason formula is closer to the field measured value. In addition, it can be seen that the dynamic earth pressure amplitude at each depth measurement point calculated by the distribution angle method (when the buried depth is greater than 0.7 m) is the closest to the measured value. However, in the subgrade surface area, the distribution angle method underestimates the magnitude of the dynamic earth pressure produced by the vibratory roller. Therefore, in engineering, the [PERSON] formula can be used to predict the dynamic earth pressure amplitude above the pipe when the depth of the measuring point is less than 0.6 m. When the depth of the measuring point is greater than 0.7 m, the distribution angle method should be used to predict the dynamic earth pressure amplitude.