# Research on the Stress Characteristics of Initial Tunnel Supports Based on Active Load Adjustment
## 2 Principle of Optimal Adjustment of Tunnel Support Load
### Load Adjustment Distribution Ratio  
Under actual conditions, setting adjustment points that are too dense is unlikely. According to the form and quantity of steel arch frames comprising a tunnel's cross section combined with the convenience of field implementation and other factors, 15 adjusting units were finally determined and arranged in a cross section for a standard three-lane highway tunnel, and the specific distribution positions are presented in Figure 2.  
The above 15 adjusting units correspond to three different load adjustment ratios. The internal force of tunnel support structure can be obtained based on symmetric hingeless arches. This is calculated using the elastic center method, and the analysis and calculation diagram is shown in Figure 3.  
\[\delta_{11}X_{1}+\Delta_{1p}=0\] \[\delta_{22}X_{2}+\Delta_{2p}=0 \tag{1}\] \[\delta_{33}X_{3}+\Delta_{3p}=0\]  
\[y_{s}=\frac{\int\frac{y}{EI}ds}{\int\frac{1}{EI}ds} \tag{2}\]  
\[\delta_{11}=\int\frac{M_{1}^{2}}{EI}ds=\int\frac{1}{EI}ds\] \[\delta_{22}=\int\frac{M_{2}^{2}}{EI}ds+\int\frac{F_{2}^{2}}{EI}ds =\int\frac{(y-y_{s})^{2}}{EI}ds+\int\frac{cos^{2}q}{EA}ds\] \[\delta_{33}=\int\frac{M_{3}^{2}}{EI}ds=\int\frac{x^{2}}{EI}ds\] \[\Delta_{1p}=\int\frac{M_{1}M_{2}}{EI}ds\] Figure 3.0 Figure 3.1  
Figure 3.2 Distribution of adjustment load position. After redundant unknown forces \(X_{1}\), \(X_{2}\), and \(X_{3}\) were calculated from the force method equation, the equilibrium conditions of the isolator or the internal force superposition formula can be used to yield the following formulas:  
\[\left. \begin{array}{l}M=X_{1}+X_{2}(y-y_{s})+X_{3}x+M_{\rm P}\\ F_{\rm Q}=X_{2}\sin\varphi+X_{3}\cos\varphi+F_{\rm QP}\\ F_{\rm N}=-X_{2}\cos\varphi+X_{3}\sin\varphi+F_{\rm NP}\end{array} \right\} \tag{4}\]  
where \(M_{\rm P}\), \(F_{\rm QP}\), and \(F_{\rm NP}\) denote the bending moment, shear force, and axial force of the section of the basic structure under load, respectively. The internal force of the structure layer was obtained. Due to the symmetry of the structure, load, and constraint, the stress of the lining structure was also symmetrical. Thus, the left half was selected for analysis. Lastly, the distribution of tunnel bending moment caused by the adjustment of load was examined, as shown in Figure 4, under the condition that all eight adjustment loads were of unit 1. By adjusting the size and proportion of loads at different points, the bending moment of tunnel lining structure was the smallest when the adjusting proportion was distributed as illustrated in the Figure 5.  
Figure 4: Bending moment distribution caused by unit adjustment load. Figure 3: Computational analysis model. The above analysis suggests that when the load of the regulating unit of the arch is 1, the arch shoulder and the side wall correspond to 1.2 and 1.4, respectively, and a better load optimization adjustment effect can be achieved, as shown in Figure 8.  
To analyze the effect of the tunnel load adjusting unit on the support force of the tunnel, the shallow buried standard three-lane highway tunnel under grade V surrounding rock was selected as the analysis object. Tables 1 and 2 list specific calculation parameters. Lastly, Figure 9 presents the bending moment of tunnel lining structure before and after implementation. The above result showed that the stress state of the tunnel lining structure changed significantly before and after the implementation of load adjustments. Under the control of appropriate load adjustment ratios, the bending moment of the part above the arch foot of the tunnel structure was significantly reduced, with a maximum reduction of 78%. The structural form of the tunnel initial support in the cross section was similar to the arch's structure.