# Assessment of Modal Characteristics of Steel-Concrete Composite Grider Bridge with V-Shaped Piers
## 5 Analysis of Structural Self-Vibration Characteristics
### Influence of Main Beam Stiffness on Self- Vibration Characteristics  
As the frequency of structural self-vibration is positively related to the stiffness of the structure, both the girder and V-shaped piers are the main load-bearing members of the structure, and changes in the stiffness of the girder and V-shaped piers have a great influence on the self-vibration characteristics of the structure. By maintaining the beam cross-section and changing its stiffness by varying the modulus of elasticity of the material, the structural self-vibration frequency and the variation of vibration pattern are analyzed. The main beam flexural stiffness is varied by \(-10\%\), \(-5\%\), \(5\%\) and \(10\%\), respectively, to analyze the changes in the self-vibration frequencies and vibration modes of the structure. The effect of the change in the main beam flexural stiffness on the self-vibration frequencies and vibration characteristics of each modal is shown in Tables 4-6 and Figures 8-10. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & \multicolumn{4}{c}{**Self-Vibration Frequency**} \\
**Order** & \(-\)**0.2** & \(-\)**0.1** & **0** & **0.1** & **0.2** \\ \hline
1 & 1.40 & 1.40 & 1.40 & 1.41 & 1.41 \\
2 & 1.91 & 1.91 & 1.91 & 1.91 & 1.91 \\
3 & 3.73 & 3.73 & 3.73 & 3.73 & 3.73 \\
4 & 3.95 & 3.95 & 3.95 & 3.95 & 3.94 \\
5 & 4.36 & 4.37 & 4.37 & 4.37 & 4.37 \\
6 & 4.79 & 4.79 & 4.79 & 4.79 & 4.79 \\
7 & 5.28 & 5.28 & 5.28 & 5.28 & 5.28 \\
8 & 6.18 & 6.18 & 6.18 & 6.18 & 6.18 \\
9 & 6.66 & 6.66 & 6.66 & 6.66 & 6.66 \\
10 & 7.68 & 7.68 & 7.68 & 7.68 & 7.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Self-vibration frequencies (Hz) for different beam heights of the original structure. Figure 7: Self-vibration frequencies of the original structure at different beam heights. By analyzing the data, it can be seen that as the stiffness of the main beam increases, the fundamental frequency of the structure changes to a small extent, and the vibration pattern of each order remains unchanged. The first two orders of self-vibration frequencies are affected by the stiffness of the main beam to a small extent, and are mainly negatively correlated. The third-order self-vibration and the second-order vertical vibration frequency are affected by the stiffness of the main beam more significantly, and are positively correlated. The first-order vertical vibration frequency increases by 4.14% when the stiffness of the main beam is increased from \(-10\%\) to \(10\%\). The fifth-order self-vibration frequency decreases, but the sixth- to tenth-order self-vibration frequency increases to a greater extent, with the seventh-order self-vibration frequency increasing by 3.75%. This indicates that the increase in the stiffness of the main beam causes a significant increase in the vertical stiffness of the bridge, including the V-shaped piers, and has little effect on the transverse stiffness. The effect of the stiffness changes of the main beam of Structure I is similar to that of the original structure, with the first-order vertical self-vibration frequency increasing by 5.59% when the stiffness of the main beam is increased from \(-10\%\) to \(10\%\). This indicates that the increase in the stiffness of the main beam has a greater effect on the vertical stiffness of Structure 1 compared to the original structure. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Model** & \multicolumn{4}{c}{**Self-Vibration Frequency**} \\
**Order** & \(-\)**10\%** & \(-\)**5\%** & **0** & **5\%** & **10\%** \\ \hline
1 & 1.41 & 1.40 & 1.40 & 1.40 & 1.40 \\
2 & 1.88 & 1.91 & 1.91 & 1.94 & 1