## 5 Engineering Application of the Proposed Method
### Adaptive Analysis of the Proposed Analytical Model in Qingdao Jiaozhou Bay Subsea Tunnel
The GRCs under different conditions evaluated by the analytical models are plotted in Figure 14. Obviously, the maximum displacement \(u\) would reach 591 mm without reinforcement, which is far beyond the control standards (namely, \([u]=150\) mm), and, thus, the construction safety was seriously threatened. Hence, the reinforcement measure must be taken into account for controlling the displacement (objective I). Figure 14: _Cont._  
The demonstrated results in Figure 14 show that the Poisson's ratio \(\mu^{\prime}\) of the reinforced ground had no fairly obvious influence on the displacement, while the larger the other four physical parameters (except for the permeability coefficient) were, the smaller the corresponding displacement, and the influence degree of each parameter on the displacement differed. The elastic modulus \(E^{\prime}\) had a significant impact on the elastic displacement, while it impacted weakly on the final value when the plastic displacement occurred. The influence of the internal friction angle \(\varphi^{\prime}\) and cohesive force \(c^{\prime}\) on the displacement can be divided into three levels: when the surrounding rock was elastic, variations of both \(\varphi^{\prime}\) and \(c^{\prime}\) had no influence on the displacement; when the plastic zones appeared and did not run through the reinforced ground and the natural ground, both parameters had relatively weak influence on the displacement; and when the plastic zones in the reinforced ground and the natural ground were connected, they significantly impacted on the displacement. However, the influence of the permeability coefficient \(k^{\prime}\) on the displacement was ruleless. It can be explained, according to Equation (10), that when \(k^{\prime}\) decreases, the seepage force in the natural ground decreases while that in the reinforced ground increases. Therefore, the reasonable permeability coefficient cannot be determined by the displacement objective approach. Excluding the factors \(\mu^{\prime}\) and \(k^{\prime}\), when the values of other parameters increase to a certain level, the control over the displacement is no longer significant, so there would be an economic, reasonable, and optimal value for the design of reinforced parameters. According to Equation (14), the sensitivity and optimum value of each parameter can be obtained as displayed in Table 5. By implementing the analytical models, the tunnel radial displacement under the optimal solution's condition is evaluated to be 110 mm, which meets the displacement control requirements. The surplus values in Table 5 were calculated by substituting \([u]\) into the corresponding configuration. Taking \(\varphi^{\prime}\) as an example, one only has to introduce \(u=150\) mm to Equation (A18). By doing this, the critical value of \(\varphi^{\prime}\) was calculated as \(\varphi^{\prime}=30^{\degree}\) while the other parameters were inputted into the model as the optimized values; hence, the surplus value would be approximately equal to \(1^{\circ}\). As shown in Table 5, \(\varphi^{\prime}\) had the most significant influence on the displacement, followed by \(c^{\prime}\), \(t_{a}\), and \(E^{\prime}\), and the former three parameters were much more sensitive than the latter one. By comparing with the factual reinforced parameters in the project passing through f\({}_{3-2}\), it was observed that the suggested values were basically consistent with the actual engineering parameters, while the factual values tended to be relatively conservative. Figure 14: Effects of the reinforcement parameters on the tunnel radial displacement: (**a**) reinforcement thickness, (**b**) reinforcement elastic modulus, (**c**) reinforcement internal friction angle, (**d**) reinforcement cohesive force, (**e**) reinforcement Poisson’s ratio, (**f**) reinforcement permeability coefficient.