# The Vibration Response Characteristics of Neighboring Tunnels Induced by Shield Construction
## 2 Basic Theory
### Microscopic Concrete Particle Vibration Energy State  
The force state of concrete is much more complex than uniaxial compression. If we take the segment adjacent to the cutter plate in shield construction as an example, its force includes axial propulsion jack pressure, earth pressure, and active earth pressure [28], in addition to the construction-induced power load transmitted through the jack connection. The force state of concrete particles in the concrete segment is shown in Figure 1.  
The earth pressure is replaced by parameters such as soil density and burial depth. In order to satisfy the energy analysis of the microstates, assuming that the concrete particles are in the form of \(1\times 1\times 1\), the active earth pressure and soil pressure can be expressed by Equations (5) and (6), as follows:  
\[\sigma_{\text{horizontal}}=\gamma\ h\ K_{a} \tag{5}\]  
\[\sigma_{\text{vertical}}=\gamma\ h \tag{6}\]  
where \(\gamma\) is the unit gravity density of the soil; \(h\) is the burial depth; and \(K_{a}\) is the active earth pressure coefficient, which can usually be calculated based on Coulomb earth pressure theory or Rankine earth pressure theory. Assuming that the total thrust of the shield machine is shared equally by the propelling jacks, the propelling jack pressure can be expressed by Equation (7), as follows:  
\[\sigma_{\text{axial}}=\sigma_{\text{all}}/(n\ A) \tag{7}\]  
Figure 1: Force state of concrete particles in pipe sheets. where \(\sigma_{\rm all}\) is the total thrust of the shield machine; \(n\) is the number of propelling jacks; and \(A\) is the contact area between the jacks and the concrete pipe sheet, which can be calculated according to the cross-sectional area of the jacks. For the analysis of the effect of its vibration load, specifically for the aspect calculation, we assume that the sinusoidal vibration wave is applied in the axial direction and the velocity amplitude of the vibration wave is \(v(t)\)=\(Acos(2\pi ft)\); therefore, we integrate over the velocity to obtain the displacement \(u(t)\), i.e., Equation (8), as follows:  
\[u(t)=\int v(t)dt=\int Acos(2\pi ft)dt=\frac{A}{2\pi ft}sin(2\pi ft) \tag{8}\]  
where \(A\) is the velocity amplitude and \(f\) is the vibration frequency. Strain \(\varepsilon(t)\), on the other hand, is the ratio of displacement to length \(L\), which is calculated using Equation (9), as follows:  
\[\varepsilon(t)=\frac{u(t)}{\rm L}=\frac{A}{2\pi fL}sin(2\pi ft) \tag{9}\]  
The stress \(\sigma d(t)\) can be obtained from the strain using Hooke's law, as detailed in Equation (10) below:  
\[\sigma_{d}(t)=E\varepsilon(t)=E\;\frac{A}{2\pi fL}sin(2\pi ft) \tag{10}\]  
Therefore, the stress state of concrete under a triaxial stress state is shown in Equation (11), as follows:  
\[\sigma_{1}=\sigma_{\rm axial}+\sigma_{d}(t)\sigma_{2}=\gamma\;h\;K_{a}\sigma_ {3}=\gamma\;h \tag{11}\]  
The strain energy density \(U(t)\) per unit volume after substituting the total stress state is shown in Equation (12), as follows:  
\[ \begin{array}{c}U(t)=\frac{1}{2E}((\sigma_{\rm axial}+\sigma_{d}(t))^{2}+( \gamma\;h\;K_{a})^{2}+(\gamma\;h)^{2}-2v(\sigma_{\rm axial}+\sigma_{d}(t))( \gamma\;h)-2v(\sigma_{\rm axial}+\\ \sigma_{d}(t))(\gamma\;h\;K_{a})-2v(\gamma\;h)(\gamma\;h\;K_{a})\end{array}