# Vibration Response Characteristics of Prefabricated Frame Structures Around the Subway
## 3 Finite Element Model of the Baseline Structure  
By modulating joint rotational spring stiffness, diverse semirigid connection configurations can be parametrically represented, allowing systematic analysis of structural vibration behavior in subway operational conditions. Defining a relative rotational stiffness parameter \(\gamma\), the relationship between rotational stiffness \(K_{\mathrm{r}}\) and \(\gamma\) is established as:  
\[K_{\mathrm{r}}=\gamma\cdot K_{\mathrm{r,max}} \tag{2}\]  
where \(K_{\mathrm{r,max}}\) denotes the maximum rotational stiffness of semirigid joints, typically set as \(i_{\mathrm{b}}\) (\(i_{\mathrm{b}}\) being the beam's linear stiffness) in reinforced concrete frames. The parameter \(\gamma\) physically characterizes the normalized joint rotational stiffness (current/maximum), constrained within \(0<\gamma\leq 1\). This dimensionless index effectively differentiates semirigid joint categories through \(\gamma\)-value variations.  
Following the establishment of joint modeling approaches for prefabricated frames, a standardized reference structure conforming to design codes was required to serve as the baseline model for subway-induced vibration analysis. Fundamental site characteristics included Site Classification II, Seismic Design Category 1, seismic precautionary intensity 7 (0.1 g peak ground acceleration), site-specific characteristic period \(T_{\mathrm{g}}=0.35\) s, and terrain roughness Class B. Material specifications comprised C30 concrete for structural elements, HRB400-grade steel for longitudinal reinforcement, and HPB300-grade steel for slab distribution bars. Loading criteria were as follows: floor live load 2.0 kN/m\({}^{2}\), floor dead load 3.5 kN/m\({}^{2}\), roof live load 2.0 kN/m\({}^{2}\), occupied roof dead load 5.5 kN/m\({}^{2}\), beam linear load 8.5 kN/m, parquet linear load 3.0 kN/m, basic wind pressure 0.4 kN/m\({}^{2}\). The structural layout featured a 4 \(\times\) 4 bay configuration (Figure 3) with 5000 mm column spacing, 3600 mm story height, and 5 stories. The structural design parameters were input into the PKPM2025R2.3 software to develop the baseline frame structure following Design Code [9].  
The joint design employed four M20 (Grade 8.8) high-strength friction-type bolts in a dual-row symmetric arrangement, with 12 mm splice plates maintaining material compatibility with parent components. Nodal translational stiffness was computed as \(K_{\mathrm{v}}=2.8\times 10^{9}\) N/m, while rotational stiffness values assigned based on \(\gamma\) variations (Table 2) enabled simulation of diverse semirigid connection configurations. The finite element elevation schematic of the baseline structural model is presented in Figure 4.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Vibration Mode** & **1 st Mode** & **2 nd Mode** & **3 rd Mode** \\ \hline Vibration direction & Translational & Translational & \\ Motion—Y & Motion—X & Rotation \\ Simulated frequency Hz & 4.278 & 5.058 & 6.573 \\ Experiment frequency Hz & 4.319 & 5.258 & 6.573 \\ Error & 0.95\% & 3.80\% & 2.15\% \\ \hline \end{tabular}
\end{table}
Table 1: Comparison of the results of vibration mode frequencies between simulation and test.