## 4 Solutions to the Influence Line Equations of the Main Beam
### Solution to the Beam Equation on Elastic Foundation Under the Concentrated Load  
Based on the derivation of the differential equations for various sections of the hybrid bridge system, the coefficients of the differential equations are all assumed to be constant, i.e., the cross-sectional properties, bending stiffness, elastic support stiffness, and axial force in each section of the main beam are constant. In fact, the stiffness and axial force of the elastic support of the main beam are constantly changing at different positions, which makes the equilibrium differential equation become a homogeneous differential equation with variable coefficients, and the general solution of the equation cannot be obtained. In order to solve the differential equations with variable coefficients, the main beam is divided into several sections. In each section, the bending stiffness and axial force of the main beam and the stiffness of the elastic support of the cable are taken as constants. Joints should also be formed at the point where the concentrated load is applied, and the beam section is divided into left and right sections to ensure that there is no additional load in the main beam section, and the additional load is all applied to the joints. As shown in Figure 6, the main beam is discretized into \(n\) sections, the left-end ordinate of the section is \(x_{i}\), the right-end ordinate is \(x_{i+1}\), and \(n+1\) nodes are formed on the main beam, and the ordinate is in the sequence \(x_{1},x_{2},\cdots,x_{i},x_{i+1},\cdots,x_{n},x_{n+1}\). The eigenvalue matrix of the section is \(K_{i}(x)\), and the inverse matrix is \(K_{i}^{-1}(x)\). There are two main methods to solve the equation: the Transfer Matrix Method [32] and the Large Matrix Method. The Transfer Matrix Method is used to establish the relationship between the coefficient vector of any beam segment and the coefficient vector of a particular beam segment via matrix transfer, thus reducing the number of unknowns. Its advantage is that the unknown quantity is less, while its disadvantage is that the element value of the eigenvector of each beam segment is quite different, which will lead to the accumulation of errors in the transmission process, thus affecting the calculation accuracy. The Large Matrix Method is similar to the matrix displacement method in structural mechanics. Its characteristic is that there is an unknown coefficient vector in each beam segment, and the relationship between the unknown coefficients between two adjacent beam segments is determined by the joint boundary conditions at the common nodes of the adjacent beam segments. Its advantage is that there is no need to transfer between eigenvectors and the error caused by data accuracy is small. The disadvantage is that there are relatively many unknowns and the size of the matrix is large. In this study, the Transfer Matrix Method is employed, which is described as follows. It is assumed that a concentrated load is applied at the location \(x_{k}\) and a vertical constraint is set at the location \(x_{m}\). Then, the characteristic mechanical vectors at the common endpoint of the main beam are equal except for the two positions \(x_{m}\) and \(x_{k}\). According to the principle that the mechanical characteristic quantities of the adjacent beam segments of the joint \(x_{i}\) are equal, the following equation is obtained:  
Figure 6: Schematic diagram of the main beam sections. where \(D_{t,1}=\prod\limits_{i=t}^{2}\left\{K_{i}^{-1}(x_{i})K_{i-1}(x_{i})\right\}\),\(D_{t,m}=\prod\limits_{i=t}^{m+1}\left\{K_{i}^{-1}(x_{i})K_{i-1}(x_{i})\right\}\), \(D_{t,k}=\prod\limits_{i=t}^{k+1}\left\{K_{i}^{-1}(x_{i})K_{i-1}(x_{i})\right\}\), and let \(D_{t,t}=E\). According to the constraint equations of the left and right ends of the main beam, four linear equations with five unknowns can be formed. Due to the vertical constraint at the node \(x_{m}\), an unknown quantity \(R_{m}\) is added, and an independent constraint equation is added.