## 4 Solutions to the Influence Line Equations of the Main Beam
### Solution to the Beam Equation on Elastic Foundation Under the Concentrated Load
#### 4.3.2 Solution of the Compatibility Equation  
For the hybrid bridge, the longitudinal constraints on the top of the main tower, including the main tower, stay cables, and main cables of the side span, can be simulated with an equivalent longitudinal stiffness \(K_{tx}\) at the two main cable saddles, as shown in Figure 7.  
When the equivalent longitudinal stiffness \(K_{tx}\) is used, the Equation (34) can be rewritten as follows:  
\[\frac{H_{p}}{K_{tx,1}}+\frac{H_{p}}{K_{tx,2}}+\frac{H_{p}L_{\phi}}{E_{c}A_{c} }+a_{t}\Delta_{T}L_{t}-\int_{-\frac{L}{2}}^{\frac{L}{2}}y(x)^{\prime}w(x,\xi) ^{\prime}dx=0 \tag{36}\]  
Integrating the last term of the left side of Equation (36) and consider the initial conditions \(w\left(-\frac{L}{2},\,\xi\right)=0\) and \(w\left(\frac{L}{2},\,\xi\right)=0\), it gives the following:  
Figure 7: Load diagram of the main cable of the hybrid bridge system. Under the action of the concentrated load, by substituting Equation (33), the deflection curve \(\
u(x,\xi)=\
u_{1}(x,\xi)+\
u_{2}(x,\xi)\) of the main beam of the hybrid system is as follows:  
\[v(x,\xi)=v_{1}(x,\xi)+H_{p}(y_{0}^{\prime\prime}\int_{-\frac{L_{xd}}{2}}^{\frac {L_{xd}}{2}}+y_{1}^{\prime\prime}\int_{\frac{L_{xd}}{2}}^{\frac{L_{xd}}{2}+L_{ j_{c}}}+y_{3}^{\prime\prime}\int_{-(\frac{L_{xd}}{2}+L_{j_{c}})}^{-\frac{L_{xd}}{2}})v_{ 1}(x,t)dt \tag{41}\]  
Among them, the integral of deflection \(\
u_{1}(x,t)\) relative to the load position \(t\) can be written in the following form:  
\[\int v_{1}(x,t)dt=\int A(t)dtf_{1}(x)+\int B(t)dtf_{2}(x)+\int C(t)dtf_{3}(x)+ \int D(t)dtf_{4}(x) \tag{42}\]  
Double integrating the \(\
u_{1}(x,t)\) relative to the load position and longitudinal coordinate, it gives the following:  
\[\iint v_{1}(x,t)dtdx=\int A(t)dt\int f_{1}(x)dx+\int B(t)dt\int f_{2}(x)dx+\int C (t)dt\int f_{3}(x)dx+\int D(t)dt\int f_{4}(x)dx \tag{43}\]  
Based on the Equation (41), the derivative of the deflection of the main beam, the moment, and shear force of the main beam for the hybrid system under the concentrated load can be computed as follows:  
\[v(x,\xi)^{\prime}=v_{1}(x,\xi)^{\prime}+H_{p}(y_{0}^{\prime\prime}\int_{-\frac {L_{xd}}{2}}^{\frac{L_{xd}}{2}}+y_{1}^{\prime\prime}\int_{\frac{L_{xd}}{2}}^{ \frac{L_{xd}}{2}+L_{j_{c}}}+y_{3}^{\prime\prime}\int_{-(\frac{L_{xd}}{2}+L_{ j_{c}})}^{-\frac{L_{xd}}{2}})v_{1}(x,t)^{\prime}dt \tag{44}\]  
\[M(x,\xi)=M_{1}(x,\xi)+H_{p}(y_{0}^{\prime\prime}\int_{-\frac{L_{xd}}{2}}^{ \frac{L_{xd}}{2}}+y_{1}^{\prime\prime}\int_{\frac{L_{xd}}{2}}^{\frac{L_{xd}}{2 }+L_{j_{c}}}+y_{3}^{\prime\prime}\int_{-(\frac{L_{xd}}{2}+L_{j_{c}})}^{-\frac{ L_{xd}}{2}})M_{1}(x,t)dt \tag{45}\]  
\[Q(x,\xi)=Q_{1}(x,\xi)+H_{p}(y_{0}^{\prime\prime}\int_{-\frac{L_{xd}}{2}}^{ \frac{L_{xd}}{2}}+y_{1}^{\prime\prime}\int_{\frac{L_{xd}}{2}}^{\frac{L_{xd}}{2 }+L_{j_{c}}}+y_{3}^{\prime\prime}\int_{-(\frac{L_{xd}}{2}+L_{j_{c}})}^{-\frac{ L_{xd}}{2}})Q_{1}(x,t)dt \tag{46}\]  
where \(\
u_{1}(x,\xi),\
u_{1}(x,\xi)^{\prime}\), \(M_{1}(x,\xi)\), and \(Q_{1}(x,\xi)\) are the deflection, derivative of deflection, moment, and shear force of the beam supported on the elastic foundation. They form the mechanical characteristics vector \(T_{1}=(\
u_{1}(x,\xi),\
u_{1}(x,\xi)^{\prime}\), \(M_{1}(x,\xi)\), \(Q_{1}(x,\xi))^{T}\). Based on Section 4.2, the coefficient vector \(S=\big{(}A\ B\ C\ D\big{)}^{T}\) can be solved. Then, the mechanical characteristic vector \(T\) can be solved as \(T=KS\). Equations (41)-(46) can be rewritten as the matrix form. \[T=K[S+H_{p}(y_{0}^{\prime\prime}\int_{-\frac{L_{xd}}{2}}^{\frac{L_{xd}}{2}}+y _{1}^{\prime\prime}\int_{\frac{L_{xd}}{2}}^{\frac{L_{xd}}{2}+L_{j_{c}}}+y_{3} ^{\prime\prime}\int_{-(\frac{L_{xd}}{2}+L_{j_{c}})}^{-\frac{L_{xd}}{2}})Sdt] \tag{47}\]  
Or  
\[\left\{\begin{aligned}  
catenary. The schematic of the hybrid bridge system in Figure 5 is employed in this section. For the FEM model, the initial stage is determined based on the completion stage of the bridge under the permanent dead load to calculate the stress-free stress and coordinates of the main cable and hangers.