## 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
As shown in Figure 5a, the parameters of the hybrid bridge are as follows. The lengths of the pure suspension section \(L_{xd}\), the hybrid section at each side \(L_{jc}\), the cable-stayed section at each side \(L_{xl}\), the auxiliary span \(L_{f}\), and the side span \(L_{b}\) are 364 m, 168 m, 350 m, 406 m, and 112 m, respectively. The rise-to-span ratio of the main cable is 1/6.5. Two main cables are used. The equivalent weight density of the main cable is \(\gamma_{c}=82\) kN/m\({}^{3}\). The total area of two main cables is \(A_{c}=0.866\) m\({}^{2}\), and hence, the main cable's own dead load is \(q_{c}=71\) kN/m, and the elastic modulus of the main cables is \(E_{c}=2.0\times 10^{5}\) MPa. For the pure suspension section, the load density bear by hangers is \(q_{h0}=577\) kN/m, and the corresponding total load density of the main cables is \(q_{0}=q_{c}+q_{h0}=641\) kN/m. For the hybrid section, the load density taken by hangers is \(q_{h1}=415\) kN/m, and the corresponding total load density of the main cables is \(q_{1}=q_{c}+q_{h1}=486\) kN/m. Then, based on the parabolic approximation for the main cables, the horizontal load component of the main cables due to the dead load is \(H_{q}=518\),175 kN. The bending stiffness of the main beam is \(EI=2.1\times 10^{9}\) kN/m. For the cable-stayed section and the auxiliary span and the side span, the elastic stiffness of the main beam provided by the cables is \(k=400\) kN/m\({}^{2}\). In the hybrid section, the elastic stiffness of the main beam provided by the cables is \(k=200\) kN/m\({}^{2}\). The axial load sustained by the main beam N is \(1.5\times 10^{5}\) kN, \(4.0\times 10^{5}\) kN, \(4.0\times 10^{5}\) kN, and \(1.5\times 10^{5}\) kN in the side span, auxiliary span, cable-stayed section, and hybrid section, respectively. In the pure suspension section, the axial force of the main beam is neglected. The longitudinal constraint stiffness of the theoretical top of the main cables is \(K_{tx}=2.5\times 10^{5}\) kN/m. Considering a unit load of 1 kN applied in the middle-span, the deflection and moment of the elastic main beam are computed using both the numerical model and the proposed model. Comparisons of the deflection curve and moment are shown in Figure 8a and 8b, respectively. As seen in Figure 8a, the deflection curve computed using the proposed model matches well with that by the numerical model. The maximum deflection of the main beam at the middle-span is \(1.53\times 10^{-4}\) m and \(1.45\times 10^{-4}\) m based on the numerical model and the proposed model, respectively, and the difference is \(-5\%\), which is small. Similarly, as seen in Figure 8b, the moment computed using the proposed model also matches well with that using the numerical model. The maximum moment of the main beam at the middle-span is 30.86 kN-m and 30.73 kN-m based on the numerical model and the proposed model, respectively, and the difference is \(-0.4\%\), which is also small. Therefore, when a concentrated unit load is applied at the middle span, the proposed method can give an accurate estimation of the deflection and moment of the main beam. \[\frac{H_{p}}{K_{tx,1}}+\frac{H_{p}}{K_{tx,2}}+\frac{H_{p}L_{\phi}}{E_{c}A_{c} }+a_{t}\Delta_{T}L_{t}-\int_{-\frac{L}{2}}^{\frac{L}{2}}y(x)^{\prime}w(x,\xi) ^{\prime}dx=0 \tag{36}\]0 Comparison of the deflection curve and moment diagram of the main beam under a concentrated unit load applied at the middle-span.