## 2 Analytical Model and Equations
### Theoretical Verification of Automatic Correction  
Through the introduction of the main control in the previous section, this section will further prove the feasibility of active control. Considering that the sum of the deformation potential energy of the structural element is equal to the work done by the external force acting on the structure, the following formula can be obtained:  
\[\delta U_{Total}+\delta W=0 \tag{11}\]  
The force on the structure is deduced by the variational method:  
\[\sum_{j=1}^{n}D\delta=\{P\}+\sum_{j=1}^{n}L_{0}, \tag{12}\]  
Figure 1: The impact of tower deviation on segment elevation. \[L_{0}=\left\{ \begin{array}{c}(l-l_{0})\frac{EA}{L}c+\left(K_{io}-K_{jo}\right) \frac{EA}{I}s\\ (l-l_{0})\frac{EA}{I}s-\left(K_{io}-K_{jo}\right)\frac{EI}{I}c\\ -K_{jo}EI\\ -(l-l_{0})\frac{EA}{I}c-\left(K_{io}-K_{jo}\right)\frac{EI}{I}s\\ -(l-l_{0})\frac{EA}{I}s+\left(K_{io}-K_{jo}\right)\frac{EI}{I}s\\ K_{jo}EI\\ \end{array} \right\} \tag{13}\]  
In the above formula: \(\delta U_{Total}\) is the potential energy of structural deformation; \(\delta W\) is the potential energy of structural deformation; \(K_{io}\), \(K_{jo}\) are curvature of member \(i\) and \(j\); \(E\), \(A\), \(l\) are the elastic modulus, cross-sectional area, length of the rod; \(L_{0}\) is the generalized additional load. According to Equation (13) above, it can be seen that the bridge structure state that determines the phased construction is mainly related to boundary conditions, external load and structural system. Thereby, the structure state can be changed by changing any of the conditions, but it is difficult to change the structure system and boundary conditions of the tower that have been installed and constructed. Moreover, the target state of the tower can be adjusted by changing the external load [29]. According to the general engineering practice research, tower deflection is mainly caused by the horizontal force in the longitudinal direction of the bridge. The vertical force in the resultant force of the cable system is small relatively to the dead weight of the tower, thus the control of horizontal force is the key to tower deviation correction. Under the impact of the lifting weight, the tower is affected by the horizontal force P in the direction of the river. The stress state of the tower with only horizontal force during the lifting process of the arch rib is established, as shown in Figure 2.  
The initial state is shown in Figure 2a. The B end is the bank, and the C end is the middle of the river. The A end of the cantilever beam is constrained by the wind cable AB and AC. The inclined angle is assumed to be \(\alpha\) and \(\beta\), and the pretension to be \(T_{1}\) and \(T_{2}\), respectively. When end A is subjected to horizontal force P (Figure 2b), the end A has a tendency to move to the right to make section AB stretch and load, the size is \(N_{1}\) and end A has a tendency to move to the left to shorten section AC and unload, and the size is \(N_{2}\). At this point, the internal forces of the wind cables on both sides are (\(T_{1}+N_{1}\)) and (\(T_{2}-N_{2}\)), respectively. \[T_{1}\sin\alpha=T_{2}\sin\beta, \tag{14}\]  
Assuming that the shear force at the A end is to the right, with the size of \(Q\), then the system re-reaches balance after active regulation. Based on the balance condition \(\sum F_{X}=0\), the following relation can be reached:  
\[(T_{1}+N_{1})\sin\alpha=Q+(T_{2}-N_{2})\sin\beta+P, \tag{15}\]  
Figure 2: The stress state of wind cable cantilever beam under horizontal force only. (**a**) Initial stress state; (**b**) Force state under horizontal force. Assuming that the regulating horizontal force is \(Q_{0}\), after substituting Equation (14) into Equation (15), the top shear force is: \[Q=(Q_{0}+N_{1}\sin\alpha+N_{2}\sin\beta)-P,\] (16) The top shear force is discussed as follows: 1. When \(Q_{0}=0\), \(N_{1}=N_{2}=0\), i.e., it is in the state of no pretension. According to Equation (16), \(Q=-P\), indicating that the direction of \(Q\) is left. 2.