# Experimental Study on a Granular Material-Filled Lining in a High Ground-Stress Soft-Rock Tunnel
## 4 Study of the Mechanical Properties of Different Filling Materials
### Analysis of the Energy Consumption of the Compressible Granular Filling Material
#### 5.3.2 Calculation of the Discharge and Energy Release of Compressible Granular Filling Material  
The pressure relief effect of the granular material filling the lining support is analyzed from an energy perspective. \(W_{unload}\) is equal to the difference value between the product of the deflection of the supported beam and the corresponding load before and after sand unloading. The theoretical calculation process is as follows:  
The bending moment of the steel channel beam at any point under uniform load \(q\) is:  
\[M(x)=\frac{qlx}{2}-\frac{qx^{2}}{2}=\frac{q(lx-x^{2})}{2} \tag{14}\]  
where \(M(x)\) denotes the bending moment at \(x\) from the end of the beam; \(q\) denotes the uniform load acting on the beam after homogenization of the granular filling, and \(l\) represents the length of the beam. Then, the approximate differential equation for the deflection curve is:  
\[E1w^{\prime\prime}=-M(x)=-\frac{qlx}{2}+\frac{qx^{2}}{2} \tag{15}\]  
where \(E\) denotes the elastic modulus of the steel beam, \(I\) denotes the moment of inertia of the channel, and \(w\) indicates the deflection of the beam. Figure 26: Energy absorption diagram of each filling material. The deflection equation is obtained by integrating the bending moment equation twice:  
\[EIw=\int{(-M(x)dx)dx}+Ax+B=-\frac{qlx^{3}}{12}+\frac{qx^{4}}{24}+Ax+B \tag{16}\]  
where \(A\) and \(B\) denote constants. Using boundary conditions:  
\[w=0\text{ when }x=0,\ x=l,\]  
Solution \(A=\frac{q^{3}}{24}\), \(B=0\). Then, the deflection curve equation of the supported beam under uniform load is:  
\[w(x)=\frac{qx(x^{3}-2 lx^{2}+l^{3})}{24 EI} \tag{17}\]  
It can be seen from the test that when the compressible granular fillings are discharged, the uniform load \(q\) decreases with the change in time \(t\). Assuming that the load before discharge is \(q_{0}\) and the load after discharge is \(q(t)\), then the energy \(W_{unload}\) released by compressible granular fillings is unloaded as follows:  
\[W_{unload}=W^{\prime}_{fc}=\int_{0}^{l}{(q_{0}-q(t))w(x)dx} \tag{18}\]  
Substituting it into Equation (17), Equation (18) is  
\[\frac{(q_{0}-q(t))^{2}}{24 EI}\bigg{(}\frac{x^{5}}{5}-\frac{lx^{4}}{2}+\frac{ l^{3}x^{2}}{2}\bigg{)}_{0}^{l}=\frac{(q_{0}-q(t))^{2}l^{5}}{120 EI}\]  
The pressure value measured in the test is transformed into a uniform load. The energy discharge value of the compressible granular filling material unloading process can be determined using the theoretical calculation Equation (18), according to the difference between the loading force at the top and the pressure at the bottom of the supported beam as measured in the test. This is multiplied by the settlement value caused by the discharge of compressible granular filling material. The energy discharge value in the test process was calculated. Finally, the energy discharge value calculated theoretically and the energy discharge value measured by test were compared and analyzed, as shown in Figure 27.  
Figure 27: Diagram of the unloading pressure and energy release of the sand-filled lining. As seen in Figure 27, with increasing sand discharge time, the pressure at the bottom of the supported beam shows a linear decline trend. In addition, the released energy shows an increasing linear trend. The pressure relief rate at the bottom of the compressible granular filling material decreases first and then rises. The reason is that the compressive compactness of sand is the largest. The friction force is also the largest when the sand first starts to unload. The pressure relief rate in the early stage is generally low, and more sand is discharged when the pressure relief valve has just been opened. With the sand filling the pressure relief valve, the pressure relief rate decreases, so the pressure relief rate in the first 4 s shows a downward trend. As more sand escapes from the supported beam, the sand compactness and friction begin to decrease. The rate of relief pressure increases, so it shows an increasing trend.