# Bearing Capacity of a Concrete Grouting Pad on the Working Surface of a Highway Tunnel Shaft
## 5 Design Method and Engineering Application of Grouting Pad Thickness  
### Design Method of Grouting Pad Thickness  
To design the thickness of a flat-bottomed grouting pad, China's coal mine vertical shafts often use the calculation method shown in Equation (12) [29]:  
\[B_{0}=P_{0}r_{2}/[\sigma_{\rm c}]+0.3r_{2} \tag{12}\]  
where \(B_{0}\) is the thickness of the grouting pad, m; \(P_{0}\) is the termination pressure of the grouting orifice, MPa; \([\sigma_{\rm c}]\) is the permissible uniaxial compressive strength of the grouting pad concrete, MPa.  
Equation (12) shows that the formula belongs to the elastic design method, which prohibits plastic deformation inside the grouting pad. However, deformation of the grouting pad is allowed to occur during field grouting construction due to the appearance of the plastic region of the grouting pad. Therefore, this paper proposes the design method for the plastic limit of the shaft grouting pad.  
Separate the cross-sectional ultimate bending moment \(M_{\rm P}\) variable in Equation (7), which can be given as Equation (13):  
\[M_{\rm P}=\frac{q_{\rm u}(2+b)\Big{[}2c^{2}(c/r_{1})^{1+b}-(3+b)a^{2}+r_{1}^{ 2}(b+1)\Big{]}}{(2b^{2}+6+8b)} \tag{13}\]  
According to the formula for calculating the bending normal stress of a cross-section in material mechanics [30], the ultimate bending moment inside a circular plate can be obtained, given as Equation (14):  
\[M_{\rm P}=\sigma_{\rm s}W_{x}=\frac{1}{6}(2r_{2})B_{0}^{2}\sigma_{\rm s}= \frac{r_{2}B_{0}^{2}\sigma_{\rm s}}{3} \tag{14}\]  
where \(\sigma_{\rm s}\) takes the design value of tensile strength of the grouting pad, MPa; \(W_{x}\) is the grouting pad section modulus, m\({}^{3}\).  
To solve for \(B_{0}\), substitute Equation (14) into Equation (13), replace the plastic ultimate load \(q_{\rm u}\) with \(P\), and define the safety factor \(\gamma\); the solution can be given as Equation (15):  
\[B_{0}=\sqrt{\frac{3P\gamma b(2+b)\Big{[}2c^{2}(c/r_{1})^{1+b}-(3+b)c^{2}+r_{1} ^{2}(b+1)\Big{]}}{r_{2}(2b^{2}+6+8b)\sigma_{\rm s}}} \tag{15}\]where \(P\) is the mean grouting pressure of the working surface and single hole, MPa; \(\gamma\) is the safety coefficient. Considering the importance of the engineering service cycle, it is recommended that the mine shaft \(\gamma\) should be 1.2 and the tunnel shaft \(\gamma\) should be 1.5.  
Equation (7) can be used to calculate the bearing capacity of the constructed grouting pad, which guides the grouting construction. Equation (15) is used for the design of grouting pad thickness and material selection based on grouting pressure and water pressure. Both equations ensure a scientific and reasonable grouting of the working surface.  
In order to facilitate better implementation by design and construction personnel, a specific design process is given, as shown in Figure 14. Firstly, determine the strength theory \(b\) value, shaft radius \(r_{2}\), grouting diffusion radius \(c\) (three parameters), so as to substitute into Formula (8) to calculate the \(r_{1}\) value and, at the same time, to determine the value of the working surface of the shaft water pressure value \(p_{w}\) and \(p_{az}\) to calculate the value of \(P\),as well as concrete material \(\sigma_{\mathrm{s}}\) (\(f_{\mathrm{t}}\)) and the safety factor \(\gamma\). Secondly, the above seven parameters are substituted into Formula (15) to calculate the thickness of the grouting pad \(B_{0}\). Finally, the site construction is carried out according to the designed thickness of the grouting pad.