Vibration Response Characteristics of Prefabricated Frame Structures Around the Subway

Abstract

Prefabricated structures have gained wider application. However, there is little research on the vibration response of prefabricated frame structures in subway environments. Prolonged metro-induced vibrations may severely degrade human comfort levels for nearby residents, interfere with the operation of precision instruments, and accelerate structural fatigue damage. Consequently, it is imperative to investigate the vibration response patterns of prefabricated frame structures under metro operational conditions. Structural vibration responses demonstrated greater sensitivity to column sections and slab thickness than beam dimensions when using semirigid connections, though marginal effects emerged with parameter increases. Enhanced vibration thresholds require supplementary vibration reduction measures. Increasing total spans proved more effective in vibration reduction than adding stories, with vibration transmission exhibiting notable edge effects. Related research can provide reference for the structural design of prefabricated frame structures around the subway.

1. Introduction

To meet the demands of urbanization, subway mileage has been increasing annually, and prefabricated structures have gained wider application. However, subway operations have exacerbated environmental vibration issues [1]. Existing studies on the vibration characteristics of prefabricated frame structures in subway environments are scarce. Long-term vibration exposure can severely degrade human comfort in adjacent areas, interfere with sensitive equipment operation, and accelerate structural fatigue failure [2]. Thus, understanding vibration response patterns of such structures under subway-induced excitations is imperative. Prefabricated concrete frame joints are categorized into three types according to their load-bearing performance: rigid connections, pinned connections, and semirigid connections. Current research on prefabricated frames with semirigid connections has yielded two primary outcomes: (1) developing simplified yet accurate semirigid joint models describing M-θ relationships through extensive experimental data fitting and (2) investigating how semirigid joints influence global structural mechanical behavior [3]. However, most seismic performance studies of prefabricated structures remain at the component level. Evaluating the seismic behavior of prefabricated frames with semirigid joints requires finite element analysis of complete structures, demanding substantial computational resources. Under limited computational capacity, researchers frequently adopt simplified joint models for holistic structural simulations. A common approach involves implementing zero-length spring elements in finite element platforms to represent joint behavior. In addition to modeling moment transfer through rotational springs, the axial stiffness of semirigid joints constitutes an essential modeling parameter. Zhao et al. (2004) [4] experimentally investigated the seismic performance of precast concrete beam–column joints. Under multiple seismic scenarios, significant crack propagation occurred at beam–slab connections, demonstrating that joint models must account for both rotational and axial stiffness. Tian et al. (2008) [5] demonstrated via pseudodynamic testing that the axial stiffness of bolted semirigid connections between beams and columns is governed by bolt shear stiffness, where initial axial stiffness closely matches the shear stiffness of bolt cross-sections. Based on previous experimental studies, Zhu et al. (2017) [6] developed a bilinear semirigid joint model employing separate axial and rotational springs to represent translational and bending stiffness, with SAP2000 verification confirming its engineering applicability. Bournas et al. (2013) [7] conducted quasidynamic tests on a three-story concrete frame, revealing that reduced joint stiffness prolongs structural natural periods and significantly alters lateral displacements under varying stiffness conditions. Tosa et al. (2016) [8] compared structural responses under seismic action with varying joint stiffnesses, demonstrating significant influence of joint stiffness on dynamic behavior. Although existing research addresses semirigid joint impacts on prefabricated frames during earthquakes, the distinct spectral characteristics, duration, and magnitude of subway-induced vibrations warrant dedicated studies on vibration response patterns of semirigid prefabricated frame systems under subway operational conditions.

Variations in beam cross-sections, column sections, slab thicknesses, total stories, and spans alter the vibration response of prefabricated frame structures under subway environments. Structural parameter changes and semirigid joint modifications exhibit coupled effects on vibration behavior. To isolate the effects of structural parameters on global vibration responses, the analyses herein compared dynamic behaviors between baseline structures with $\gamma$ = 0.5 semirigid joints (simulating bolted connections) and ideal rigid joints under variations of beam/column sections, slab thicknesses, total stories, and spans. This systematic investigation aims to establish the influence patterns of structural parameters on vibration responses of prefabricated frame structures in subway environments.

ABAQUS, a powerful finite element software for engineering simulation developed by Dassault Systèmes, is well-suited for analyzing the vibration propagation patterns of the ground induced by subway operations. This type of analysis requires multiple solution methods, multiphysics coupling, precise boundary condition settings, accurate load simulations, and robust postprocessing capabilities—all of which are strengths of ABAQUS. Therefore, ABAQUS was adopted for numerical analysis of structural vibration responses in this paper.

2. Modeling Method and Verification of Prefabricated Frame Structures with Semirigid Joints

This study focuses on prefabricated concrete frame structures with semirigid joints, necessitating the selection of appropriate semirigid joint configurations as the research subject.

2.1. Selection of Joint Model for Prefabricated Frame Structures

Semirigid joints typically employ connection types such as bolt-welded hybrid connections, prestressed prefabricated connections, and bolted connections. Given that upper structures predominantly remain within the elastic range under subway-induced vibrations, linear spring elements were adopted in linear elastic simulations to represent the rotational stiffness effects of semirigid joints on the global stiffness matrix, as schematically shown in Figure 1.

At the semirigid joint, the relationship among the shear force $F_{\mathrm{v}}$, bending moment $M$, joint shear deformation $V_j$ and joint rotation angle ${\theta }_r$ is as follows:

(1)$F_{\mathrm{v}}=K_{\mathrm{v}}{V}_j, M=K_{\mathrm{r}}{\theta }_r$

Equation (1) can be employed to simulate the influence of semirigid joints on beam element stiffness matrices. The ABAQUS software offers solutions for modifying 3D beam element stiffness matrices via semirigid connections, with its Interaction module’s Connector tool enabling integrated stiffness modeling. Within the Connector tool, the Bushing component enables separate specification of translational stiffness components ($K_{\mathrm{a}}$, $K_{\mathrm{v}\mathrm{x}}$, $K_{\mathrm{v}\mathrm{y}}$) and rotational stiffness parameters ($K_{\mathrm{r}\mathrm{x}}$, $K_{\mathrm{r}\mathrm{y}}$, $K_{\mathrm{r}\mathrm{z}}$). As the axial stiffness $K_{\mathrm{a}}$ does not contribute to the beam-end stiffness matrix in semirigid connections, it is appropriately designated as rigid in the model configuration.

2.2. Verification of the Model

To validate the rationality of the modeling method for prefabricated concrete frame structures in this study, based on parameters from [5] establishing the prefabricated frame structure model, and considering that superstructure responses under subway vibrations generally remain elastic, only simulated vibration frequencies were compared with experimental measurements. A 1/5-scale experimental specimen was constructed as a three-story, three-bay precast RC frame consisting of inverted L-shaped edge beams, inverted T-shaped middle beams, bracket columns, and prefabricated slab components. Slab-to-beam connections employed bolted joints through slab ribs, with beam–column connections similarly bolted and subsequently grouted with concrete to fill joints. The structural configuration is illustrated in Figure 2a.

A uniform mesh size of 0.05 m was adopted, employing B31 elements for beam/column/rib members and S4R shell elements for slabs. To account for slab–rib interaction, binding constraints were implemented between components. The finite element model configuration is illustrated in Figure 2b,c. Comparative results of simulated and experimental mode shapes (vibration directions, frequencies, and errors) are presented in Table 1. The maximum frequency error of 3.8% demonstrated the modeling method’s satisfactory accuracy for semirigid prefabricated RC frames, validating its applicability for subway vibration analysis within elastic ranges.

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:

(2)$K_{\mathrm{r}}=\gamma ·K_{\mathrm{r},\mathrm{m}\mathrm{a}\mathrm{x}}$

where $K_{\mathrm{r},\mathrm{m}\mathrm{a}\mathrm{x}}$ 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 $0<\gamma \le 1$. 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/m², floor dead load 3.5 kN/m², roof live load 2.0 kN/m², occupied roof dead load 5.5 kN/m², beam linear load 8.5 kN/m, parapet linear load 3.0 kN/m, basic wind pressure 0.4 kN/m². The structural layout featured a 4 × 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 × 10⁹ 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.

4. Multipoint Vibration Excitation Input Method

This section systematically evaluates vibration excitation methods by contrasting tunnel–soil–structure coupled systems with base-uniform excitation approaches, aiming to establish an effective multipoint input strategy for subway-induced vibrations. The baseline precast concrete frame structure ($\gamma$ = 0.5) was integrated with the site vibration model using coupling techniques, forming a 3D tunnel–soil–structure interaction system depicted in Figure 5. Infinite element boundaries were added around the model. These infinite element boundaries can absorb the vibrations propagating to the model boundaries, thereby simulating the soil mass extending to infinity.

A 6 m diameter shield-constructed tunnel was modeled, featuring 0.4 m thick C50-grade lining segments, C30 concrete track bed, and T60 rail profiles. For the targeted 1–100 Hz frequency spectrum (${\omega }_1=6$.28 rad/s, ${\omega }_2=6$28 rad/s), site-specific Rayleigh damping parameters were calculated: $\alpha$ = 0.124 and $\beta$ = 3.15 × 10⁻⁵. Geotechnical parameters are provided in Table 3. The model employed C3D8R elements for soil, lining segments, and track bed (0.5–2.5 m mesh size, 507,288 elements), B31 beam elements for rails (3000 elements), and CIN3D8 infinite elements for boundary conditions. The loading protocol implemented wheel forces through ABAQUS DLOAD subroutine using 2.5 ms time steps.

Figure 6 compares the one-third octave band results between the computational model and experimental measurements from Reference [10]. The model’s vibration responses exhibited consistent growth-attenuation trends with field data across center frequencies. Prior to 40 Hz, vibration acceleration magnitudes exhibited frequency-proportional amplification, whereas post-40 Hz spectral components displayed graded high-frequency attenuation. Both model and measurements showed peak energy at 40 Hz, with a 4.8% discrepancy. Within the 10–50 Hz dominant frequency range for subway-induced environmental vibrations, the model demonstrated superior simulation fidelity.

Vertical vibrations were extracted from roof corner nodes in both tunnel–soil–structure coupled models and base-consistent input baseline structures. The spectral distribution characteristics of vertical vibrations showed remarkable consistency between the coupling model and the base-uniform excitation method, maintaining amplitude discrepancies within 5% across frequency bands. This indicated comparable computational mechanisms between both methods. Given the superior computational efficiency of the base-uniform excitation method over tunnel–soil–structure coupling modeling, it was adopted for subsequent structural vibration analyses.

5. Vibration Response Characteristics of Prefabricated Frame Structures Around Subways

Vibration transmission from subway operations exhibits vertical predominance at receiving sites. Human vibration sensitivity predominantly resides within a 1–80 Hz spectrum, demonstrating heightened physiological responsiveness to acceleration fluctuations. As subway-induced vertical vibration acceleration spectra coincide with human sensitivity ranges, the Standard for Allowable Vibration of Construction Engineering (GB50868-2013) [11] establishes weighted acceleration level criteria for occupant comfort. Table 4 presents the regulatory thresholds for vertical vibrations.

(3)${\mathrm{V}\mathrm{L}}_{\mathrm{z},\mathrm{m}\mathrm{a}\mathrm{x}}=10\lg \left(\sum {10}^{{\displaystyle \frac{\left({\mathrm{V}\mathrm{A}\mathrm{L}}_i+{\alpha }_i\right)}{10}}}\right)$

where ${\mathrm{V}\mathrm{L}}_{\mathrm{z},\mathrm{m}\mathrm{a}\mathrm{x}}$ denotes maximum Z-vibration level, ${\mathrm{V}\mathrm{A}\mathrm{L}}_i$ represents 1/3-octave band vibration acceleration levels at center frequencies, and ${\alpha }_i$ indicates Z-weighting factors per 1/3-octave band.

5.1. Influence of Beam Cross-Section Variation on the Structural Vibration Response

The baseline structure from Section 2 served as the basis for variable beam cross-section analysis, maintaining constant parameters for other components while varying beam dimensions to achieve different height–width ratios. Component parameters for different models are specified in Table 5.

With identical design specifications as the Section 2 baseline model, reinforcement detailing was generated through the PKPM2025R2.3 software, with beam reinforcement configurations illustrated in Figure 7.

Joints retained four M20 (Grade 8.8) high-strength friction-grip bolts in a double-row symmetric configuration, with 12 mm splice plates matching base material properties. Translational joint stiffness was maintained at $K_v$ = 2.8 × 10⁹ N/m. With $\gamma$ = 0.5, rotational stiffness values for various beam sections are presented in Table 6. Semirigid joint behavior in L1-S/L2-S/L3-S was modeled using Bushing connectors, whereas ideal rigid connections for L1-R/L2-R/L3-R were achieved through ABAQUS’ Beam connector element in the Interaction module.

Vibration time-history signals along the X/Y/Z directions generated by an 80 km/h operating six-car Type B train above tunnel central axis were implemented in the six models using the base-uniform excitation method. Vibration time-history data at midspan nodes of edge beams across all floors were extracted. After FFT processing, weighted acceleration levels at monitoring points were calculated using Equation (3) and ISO 2631-1 (https://www.iso.org/standards-catalogue/browse-by-ics.html) weighting factors, with results presented in Figure 8.

Overall, structures with semirigid joints exhibited 3–4 dB higher weighted acceleration levels than rigidly connected structures. Regardless of joint rigidity configurations, structural vibration responses under subway environments showed consistent distribution patterns across floors, with slight amplification from base to top levels. The top floor exhibited 0.5–1 dB higher weighted acceleration levels than the first floor, analogous to vibration behavior in fixed-base cantilever column models. Vibration energy transmission from base excitation to free ends induces cumulative energy dissipation through structural oscillations, manifesting as amplified vibration magnitudes. The limited story count in the model attenuated significant amplification at upper levels.

According to Code [11] regarding permissible weighted acceleration levels for human comfort under repetitive vibrations, structures with semirigid joints exceeded residential nighttime limits across all floors. Semirigid structures with 250 × 500 mm beam sections exceeded daytime residential limits at certain floors, rendering this structural form unsuitable for residential use without effective vibration mitigation. While rigid-jointed structures complied with residential day/night limits, they exceeded stringent vibration zone requirements. This demonstrates that beam section modifications inadequately control vibration levels within stringent zone limits, requiring supplementary mitigation strategies.

From a beam-cross-sectional perspective, increasing beam height or enhancing the height-to-width ratio (while maintaining height) marginally reduces structural weighted acceleration levels, regardless of joint rigidity configurations. Height augmentation substantially increases the sectional moment of inertia, thereby enhancing beam flexural rigidity. This global stiffness improvement benefits vibration acceleration level control. Maintaining beam height while increasing aspect ratio (equivalent to width reduction) yields lower acceleration levels, attributed to modified stiffness relationships. Narrower beams marginally decrease flexural stiffness, reducing beam–column stiffness ratios, and studies have confirmed that higher stiffness ratios amplify structural displacement responses [12], so diminished stiffness ratios suppress overall structural vibration responses. For the parametric range investigated, aspect ratio adjustment achieved superior vibration control to that of height increase, with more significant effects observed in rigid connection systems. In semirigid configurations, vibration attenuation differences between height and aspect ratio modifications remained statistically insignificant.

To further identify vibration response patterns of different beam sections under subway environments, vertical acceleration spectral data at midedge beam nodes of top floors were analyzed, as shown in Figure 9.

All structural configurations exhibited similar spectral characteristics under identical base excitations, regardless of semirigid joint implementation or beam cross-sectional variations. However, rigid-jointed structures demonstrated dominant frequencies between 20 and 35 Hz, whereas semirigid configurations exhibited primary frequency concentrations within 10–15 Hz. This frequency disparity originated from joint stiffness effects on global stiffness matrices—increased nodal rigidity elevates structural stiffness and consequently enhances natural frequencies. Broadband subway vibrations potentially excite higher-order modes, explaining the elevated frequency ranges in rigid-jointed structures.

Structural acceleration amplitude responses varied across beam cross-sections, generally exhibiting elevated dominant frequency bands with increasing aspect ratio. Previous analyses have revealed that beam aspect ratio modifications alter beam–column stiffness ratios, consequently affecting structural vibration characteristics. Beyond influencing global stiffness matrices, aspect ratio variations (under constant cross-sectional area conditions) modify beam mass distributions, consequently altering system mass matrices. As mass matrix constitutes a critical vibration determinant, the observed frequency escalation under aspect ratio enhancement aligns with theoretical expectations for the parametric variations investigated.

The aspect-ratio-induced frequency amplification effect was more pronounced in rigid-jointed structures. For semirigid systems, reduced rotational stiffness weakens low-frequency vibration attenuation because of semirigid joints’ influence on the first two natural frequencies, resulting in complex aspect-ratio-dependent frequency modulation patterns.

In addition to evaluating human comfort through structural acceleration responses, investigating vibration kinetic energy transmission via velocity parameters [13] is crucial for understanding transient structural impacts induced by peak velocities in the time domain. X-/Y-direction velocity time-history diagrams at central roof locations are shown in Figure 10, with corresponding time-domain peak velocities extracted in Table 7.

The two-axis horizontal peak velocities at roof centers of all models remained below residential building thresholds, indicating that the beam sections met structural safety requirements for residential applications. However, horizontal peak velocities exceeded permissible levels for nonresidential and nonindustrial structures. Rigid-jointed systems exhibit 8.47% lower peak velocities than semirigid counterparts, suggesting reduced long-term subway vibration impacts on rigid-connected structures. Beam section enlargement significantly reduces Y-axis peak velocities while marginally decreasing X-axis responses. Height-constant aspect ratio enhancement moderately decreases biaxial velocities, though less effectively than direct height increases, therefore, beam height enhancement proves optimal for mitigating subway vibration effects on structural health and ensuring long-term safety.

Beyond time-domain analysis of subway vibration impacts, frequency-domain velocity investigations are required to identify critical frequency bands prone to energy accumulation and structural fatigue damage. Combined directional velocity time-history data at midedge beam nodes of top floors were extracted and processed via FFT, yielding composite velocity spectral data shown in Figure 11.

Structural velocity spectra were predominantly concentrated in low-frequency ranges across all configurations. Under identical excitation inputs, composite velocity spectral patterns exhibited similar low-frequency multipeak characteristics regardless of joint types or beam cross-sections. This phenomenon originated from subway vibration energy concentration in low frequencies, where structural fundamental frequencies intersecting with dominant subway spectral components induce resonant peaks. For rigid connection configurations, each 0.2 aspect ratio augmentation produced 1–2 Hz upward shifts in structural fundamental frequencies, with negligible velocity amplitude variations at characteristic frequencies.

Structures with semirigid connections maintained stable fundamental frequencies while demonstrating distinctive narrowband resonance patterns in low-frequency velocity spectra. This behavior arose from the concentrated vibration modes in semirigid systems, where reduced rotational stiffness diminishes low-frequency energy dissipation capacity, prolonging vibration duration (time-domain) and manifesting as narrowband spectral resonances (frequency-domain). Aspect ratio modifications without height changes showed negligible spectral effects in semirigid systems, while height increases marginally elevated fundamental frequencies, indicating limited beam parameter influence on velocity spectra.

5.2. Influence of Column Cross-Section Variation on the Structural Vibration Response

Variations in column cross-section significantly impact overall structural stiffness, consequently inducing distinct vibration characteristics in subway environments. The baseline structure from Section 2 served as the analytical basis for column cross-section variations, with structural layout details shown in Figure 3. While preserving original component specifications, column sectional sizes were systematically altered, with parametric configurations tabulated in Table 8.

Consistently with Section 2′s design criteria, structural parameters were input into PKPM2025R2.3 to generate reinforcement layouts for all models. Uniform column reinforcement patterns based on ground-floor central column designs were implemented across models, with cross-sectional reinforcement details illustrated in Figure 12. Joints utilized four M20 (Grade 8.8) high-strength friction-grip bolts in a dual-row symmetric arrangement, with 12 mm splice plates matching parent material properties. Translational stiffness $K_{\mathrm{v}}$ = 2.8 × 10⁹ N/m and rotational stiffness $K_{\mathrm{r}}$ = 1.83 × 10⁸ N·m/rad were assigned, with structural $\gamma$ maintained at 0.5. Bushing connectors simulated semirigid joints for the Z1-S/Z2-S/Z3-S models, while Beam connectors represented ideal rigid joints in the Z1-R/Z2-R/Z3-R series.

Triaxial vibration signals (X/Y/Z directions) generated by an 80 km/h six-car metro train above the tunnel’s central axis were applied to the six models through a base-uniform excitation protocol. After signal processing of midspan edge beam nodal vibrations, frequency-weighted acceleration levels across floor monitoring points were obtained as shown in Figure 13.

Globally, frames with rigid connections demonstrated weighted acceleration magnitudes reduced by 2–4 dB per floor relative to semirigid systems under equivalent column sectional dimensions. Rigid connection models achieved up to 2 dB vibration reduction through column section enlargement, while semirigid systems attained approximately 1 dB attenuation via similar modifications. This disparity arose from rigid joints creating fully fixed connections that enhance structural integrity, enabling complete transfer of column bending stiffness to global system rigidity. The restricted rotational capacity of semirigid connections impedes full lateral stiffness realization, as column stiffness improvements are partially dissipated through joint flexibility rather than contributing to holistic structural enhancement.

According to Standard [11] vibration thresholds, structures with 450 mm × 450 mm or 500 mm × 500 mm column sections employing semirigid joints exceeded residential nighttime acceleration limits. Increasing column dimensions effectively constrained weighted acceleration levels within nighttime residential limits, though significant gaps persisted relative to daytime thresholds. Sustained occupant comfort would require either joint stiffness augmentation or integrated vibration control strategies for residential applications. Comparative analysis with Figure 8 demonstrates that column section modifications induced greater weighted acceleration level variations than equivalent beam section adjustments. This effect was amplified in rigid-jointed systems, indicating columns’ higher participation factors in vibration energy transmission—column parameter alterations provoked more substantial system response deviations.

To further analyze the pattern of the structural vibration response changes under the subway environment caused by the variation of column cross-sections, the vertical acceleration time-history data at the middle nodes of the top-side beams of models with different column cross-sections were extracted and converted into spectral data. The results are shown in Figure 14.

Spectral waveform characteristics remained relatively stable when varying column dimensions and joint configurations under equivalent excitation conditions. Column section modifications in semirigid systems shifted the dominant vibration energy to 10–15 Hz bandwidths, whereas rigid-jointed configurations preserved 30 Hz-centered spectral distributions despite parametric changes. This indicates that rigid joints minimized modal sensitivity to column changes, while semirigid configurations demonstrated heightened mode shape susceptibility to column variations, significantly affecting spectral peak characteristics. Semirigid-jointed structures exhibited pronounced upward shifts in dominant vibration frequencies with column section enlargement. This frequency escalation stemmed from enhanced global stiffness outweighing mass effects during column enlargement, as natural frequencies are proportional to stiffness and inversely related to mass. Rigid joints’ inherent high stiffness established elevated baseline frequencies, making subsequent column changes induce negligible frequency variations. However, Figure 14b reveals amplitude attenuation at characteristic frequencies with column enlargement, indicating that increased column dimensions primarily affected vibration energy distribution across frequency bands in rigid systems.

To further study the influence of the variation of column cross-sections on the structural vibration velocity in the time domain, the X- and Y-direction velocity time histories at the central position of the top floor of each structure were extracted, as shown in Figure 15. The peak velocities in the time domain were extracted and are shown in Table 9.

Time-domain analyses revealed that even the maximum column section within the parametric range exceeded nonresidential/nonindustrial vibration velocity limits. While converting to rigid connections and enlarging column sections effectively reduced peak velocities, stricter compliance would demand additional mitigation measures—continuous column enlargement exhibited diminishing returns. Column enlargement demonstrated superior Y-direction peak velocity reduction efficacy (max 11.22%) compared with X-direction effects. As Table 7 indicates, column modifications outperformed beam section changes in peak velocity mitigation. This superiority persisted in semirigid systems, where column enlargement provided greater velocity reduction than beam modifications. Thus, column parameter optimization delivers superior cost–benefit ratios in vibration control design.

To understand the kinetic energy accumulation brought to each frequency band of the structure by the subway vibration under different column cross-sections, the mixed-direction acceleration time-history data at the middle nodes of the side beams on the top floor of each structure were extracted. After being processed by FFT, the mixed-direction acceleration spectral data were obtained, as shown in Figure 16.

Structural velocity responses were predominantly concentrated in low-frequency ranges across column section variations. Identical subway excitations produced similar spectral waveforms, with column size adjustments primarily modifying peak responses in specific frequency bands. Converting semirigid to rigid joints significantly altered spectral distributions by modifying vibration mode excitations, whereas column changes minimally affected mode shapes while adjusting sub-band peak velocities. Theoretical analysis predicts that column enlargement increases stiffness and mass, with competing effects on natural frequency. Concurrent damping ratio elevation from material volume growth further influences spectral peak characteristics. In semirigid systems, column expansion caused marginal frequency shifts (0.5–1 Hz) but significantly reduced peak velocities and bandwidths, narrowing spectral energy concentration ranges. Rigid-jointed structures experienced 2–3 Hz base frequency elevation with column upsizing, accompanied by spectral broadening. This reflects competing effects: mass increase enhances low-frequency inertial responses, while stiffness growth inadequately suppresses high-frequency components, creating widened spectral distributions.

5.3. Influence of Slab Thickness Variation on the Structural Vibration Response

The baseline structure in Section 2 was adopted as the analysis object for the variation of slab thickness. The structural plan layout is shown in Figure 3. Keeping the parameters of other components in the reference structure unchanged, only the size of the slab thickness was varied. The values of component parameters for different models are shown in Table 10.

Structural design parameters remained consistent with Section 2’s baseline model. Design information was input into the PKPM2025R2.3 software to generate reinforcement layouts for various models. Uniform slab reinforcement patterns (Φ8@200 bidirectional bottom layer) were implemented for comparative analysis. Automated design validation ensured consistent reinforcement compliance. Connections employed four M20 (Grade 8.8) friction-grip bolts in a dual-row symmetric configuration, with 12 mm splice plates matching base material properties. Translational stiffness $K_{\mathrm{v}}$ = 2.8 × 10⁹ N/m and rotational stiffness $K_{\mathrm{r}}$ = 1.83 × 10⁸ N·m/rad were assigned, maintaining $\gamma$ = 0.5. Semirigid behavior in B1-S/B2-S/B3-S was modeled using Bushing connectors, whereas B1-R/B2-R/B3-R achieved full rigidity through Beam connector implementation. A uniform material damping ratio of ξ = 0.02 was applied across all components. Triaxial vibration data (X/Y/Z) from six-car Type B trains at 80 km/h along the tunnel axis were input via base-uniform excitation into all six models. Vibration histories from midedge beam nodes were processed to obtain floor-specific weighted acceleration levels shown in Figure 17.

From an occupant comfort perspective, rigid-jointed frame structures maintained weighted acceleration levels within residential day–night limits across all slab thicknesses yet still exceeded stringent zone vibration thresholds. Each 10 mm slab thickness increase yielded a 0.5 dB acceleration reduction, indicating that 120 mm slabs did not reach the critical thickness where stiffness–mass effects neutralize. Further thickening may sustain gradual vibration attenuation. Reference [14] identified 140 mm as a critical thickness for stiffness–mass equilibrium in rigid systems. Though project-specific variations exist, exceeding 120 mm in this study may have achieved 74 dB compliance but risked stress concentration and cost escalation. Standard practice limits slabs to 100–120 mm, necessitating alternative controls beyond 120 mm.

Semirigid-jointed structures with tested slab thicknesses complied with daytime residential limits but violated nighttime criteria. It is found that when the slab thickness increased from 100 to 110 mm, the overall acceleration vibration level of the structure decreases by 1 dB while decreasing by 0.25 dB when the slab thickness increased from 110 to 120 mm. This diminishing-returns pattern suggests 120 mm as a critical thickness for semirigid systems beyond which vibration levels may plateau or rebound. Continuous slab thickening failed to achieve nighttime compliance in semirigid systems, necessitating supplemental vibration mitigation for residential applications. Semirigid systems demonstrated greater vibration reduction per thickness increment but reached critical thickness earlier than their rigid counterparts.

To further analyze the pattern of structural vibration response changes under subway environment caused by slab thickness variation, vertical acceleration time-history data at middle nodes of top-side beams in different models were extracted and converted into spectral data, as shown in Figure 18.

Structures with varying slab thicknesses and joint connection types exhibited similar vibration spectral waveforms under identical excitation inputs, demonstrating low-frequency multimodal characteristics in the frequency domain. Rigid-jointed structures generally exhibited higher dominant frequencies than their semirigid counterparts. Increasing slab thickness did not significantly alter these dominant frequencies. For semirigid systems, increasing slab thickness from 100 mm to 110 mm raised the dominant frequency by 4 Hz, caused slight bandwidth expansion, and reduced peak amplitude by 10.01%. From 110 mm to 120 mm, dominant frequency remained stable while peak amplitude decreased by 8.66%. Approaching the critical thickness, frequency sensitivity diminished, with primary effects shifting to peak amplitude reduction. In rigid-jointed structures, each 10 mm slab increase elevated the dominant frequency by 2 Hz and reduced peak response by 10%, indicating stable vibration modulation within 100–120 mm range and sub-critical-thickness conditions.

To further study the influence of slab thickness variation on structural vibration velocity in the time domain, the X- and Y-direction velocity time histories at the central position of the top floor in each structure were extracted, as shown in Figure 19. The peak velocities in the time domain are listed in Table 11.

Overall, slab thickness variations demonstrated limited influence on horizontal velocity peaks. Y-direction peak velocities at monitoring points exceeded nonresidential/nonindustrial building thresholds despite slab thickening. In rigid-jointed structures, each 10 mm slab thickening achieved 4.5% Y-direction peak velocity reduction, while X-direction responses showed negligible changes. Y-direction peak velocity reduced during 110 to 120 mm thickening in semirigid configurations, revealing critical-thickness-induced performance degradation in velocity modulation. Comparative analysis with Table 9 demonstrates higher velocity sensitivity to column section changes versus slab thickness modifications, indicating inferior cost–benefit ratios for slab-based Y-velocity control.

To understand the kinetic energy accumulation in each frequency band of the structure caused by subway vibrations under different slab thicknesses, mixed-direction acceleration time-history data at the middle nodes of the side beams on the top floor of each structure were extracted. After processing via FFT, the mixed-direction acceleration spectral data were obtained, as shown in Figure 20.

From the composite velocity spectra at monitoring points of frame structures with varying slab thicknesses, altering slab thickness did not significantly modify the spectral waveform of structural vibrations. The primary effects of slab thickness variation manifested in bandwidth, peak amplitudes, and fundamental frequency characteristics of velocity responses. In semirigid jointed structures, the fundamental vibration frequency demonstrated progressive elevation with slab thickening. However, this frequency modulation effect diminished significantly when slab thickness reached 120 mm. Slab thickening minimally affected primary frequency peaks but significantly suppressed secondary peak amplitudes, indicating effective mitigation of higher-mode vibrations while preserving dominant mode characteristics. Rigid-jointed structures demonstrated distinct spectral evolution patterns: each 10 mm thickness increment induced consistent 2 Hz frequency elevation and amplitude reduction across all spectral peaks, suggesting comprehensive vibration energy suppression through multimodal stiffness enhancement.

5.4. Influence of the Total Structural Stories Variation on the Structural Vibration Response

To understand the vertical vibration propagation law of subway vibrations in structures, the vibration response of frame structures with semirigid joints of different total stories under subway environmental conditions were studied. The structural plan layout was the same as the baseline structure in Figure 3, and the basic design specifications were consistent with those of the baseline structure in Section 2. Only the total number of stories was changed: based on the 5-story baseline structure, 8-story and 11-story frame structure models with semirigid joints were designed, with a difference of 3 stories between each model. Since the change in the number of stories correspondingly altered the internal forces on beams, columns, and slabs, the parameters for the beams, columns, and slabs of the 8-story and 11-story models were selected according to the relevant requirements of Code [9], as shown in Table 12.

The design specifications and sectional parameters were incorporated into the PKPM2025R2.3 software to derive reinforcement configurations for various models. To streamline structural analysis in the eight-story FEM model: beam reinforcement in floors 1–5 replicated the 1–1 section beam detailing from the first floor (Figure 7), while column reinforcement employed the first-floor central column cross-section. For floors 6–8, beam reinforcement adopted the sixth-floor 1–1 section configuration, with column reinforcement corresponding to the sixth-floor central column cross-section. In the 11-story FEM, the same simplification logic applied to beam–column reinforcement for floors 1–5 and 6–7. For floors 9–11, beam reinforcement followed the 1–1 section beam configuration from the ninth floor, with column reinforcement matching the ninth-floor central column cross-section. Slab reinforcement employed Φ8@200 arranged bidirectionally.

Calculation verification demonstrated that joints utilized four M20 (grade 8.8) high-strength friction-grip bolts in a dual-row symmetric configuration, accompanied by 12 mm splice plates with material properties identical to parent components, fulfilling all structural joint design criteria. Translational stiffness $K_{\mathrm{v}}$ = 2.8 × 10⁹ N/m and rotational stiffness $K_{\mathrm{r}}$ = 1.83 × 10⁸ N·m/rad were assigned, maintaining $\gamma$ = 0.5. Schematic diagrams of finite element models with different story configurations are shown in Figure 4 and Figure 21.

Triaxial vibration data (X/Y/Z) from six-car Type B trains at 80 km/h along the tunnel axis were input via base-uniform excitation into all models. Vibration histories from midedge beam nodes were processed to obtain floor-specific weighted acceleration levels shown in Figure 22.

Overall, the weighted acceleration levels across floors in semirigid jointed structures exhibited a decreasing trend with increasing total stories. This primarily stemmed from enhanced structural mass at higher story counts—with fixed foundation column quantities, similar vibration energy inputs distribute across larger masses, resulting in lower global acceleration magnitudes under equivalent excitations. Under identical subway operational and site conditions, five-story structures exceeded daytime residential limits at certain floors. At eight stories, levels complied with daytime thresholds but violated nighttime limits. When extended to 11 stories, all floors satisfied nighttime criteria. Thus, structural story count must be integrated into comfort evaluations for subway-adjacent buildings. Low-rise structures (n < 11) showed monotonically increasing vibration levels with height. Beyond the critical height (n = 11 in this study), levels first decreased and then increased because of axial wave propagation in vertical members. Vibration energy reflects at the roof level without transmission medium, causing wave superposition and top-floor amplification [15]. As the story count increased, enlarged lower-floor member cross-sections enhanced stiffness and mass. This dual effect transformed vibration distribution patterns from monotonic growth to initial attenuation followed by amplification when reaching critical heights.

Semirigid joints induced stronger reflection of axial and bending waves at connections than rigid joints, resulting in more pronounced floor-level vibration amplification effects in semirigid framed structures. To investigate vibration response patterns under varying story counts in subway environments, vertical acceleration time histories at midedge beam nodes of top floors were analyzed through FFT processing and one-third octave band level calculations, with results presented in Figure 23.

In general, the peak distributions of the one-third octave band curves for each structural layer were quite close. The dominant frequency of the 5-story structure was approximately 16–20 Hz, that of the 8-story structure was around 12.5–20 Hz, and that of the 11-story structure was approximately 25–31.5 Hz. This was attributed to the increased stiffness of lower structural components required as the total number of stories increased, which altered the structure’s sensitive frequency range. The energy of ground vibrations caused by metro operation is concentrated in the 30–60 Hz range, which is prone to resonating with the structural modes within this frequency band. Regardless of the total number of stories, the low-frequency components (below 10 Hz) experienced only slight attenuation as vibration waves propagated through the structure. In contrast, the high-frequency components (above 40 Hz) showed more significant attenuation during propagation, with a high-frequency amplitude reduction of over 10 dB between the top and the first floor. The five-/eight-story models shared similar spectral characteristics: high-frequency components attenuated rapidly while low frequencies persisted, indicating wave-reflection-induced energy accumulation at upper floors. The 11-story structure exhibited middle-floor (4–7F) vibration attenuation within the 16–20 Hz range, contrasting with lower-/higher-floor amplification. This “saddle-shaped” distribution arises from wave reflection dynamics: roof-reflected waves (affecting up to 7F) destructively interfere with incident waves at specific middle floors.

To further investigate the influence of variations in the total number of stories on the time-domain vibration velocity of the structure, the X-/Y-direction velocity time histories at the center of the top floor of each structure were extracted and are shown in Figure 24, while the peak time-domain velocities are listed in Table 13.

Progressive heightening induced systematic Y-direction velocity attenuation (5% to 7% per three stories) at the roof center, demonstrating logarithmic decay characteristics in horizontal vibration components. This phenomenon originates from subway vibration transmission through vertical members: dominant axial waves experience progressive attenuation with height, while reflected waves at the roof level induce secondary horizontal vibrations. The cumulative axial wave attenuation ultimately governs peak velocity reduction at upper floors. While Y-direction peaks complied with residential limits, all configurations exceeded stringent zone thresholds. Continued heightening proved ineffective for strict compliance, necessitating supplemental vibration control measures for critical applications.

To assess the frequency-dependent energy accumulation caused by metro vibrations in structures with different total story numbers, mixed-direction acceleration time histories were collected from midspan nodes of edge beams at the first-, intermediate-, and top-floor levels. The data were processed via FFT to derive the corresponding spectral data, as illustrated in Figure 25.

Across structural configurations with varying total stories, mixed-direction velocity responses were predominantly concentrated within the 10–15 Hz frequency range. While the dominant frequency exhibited slight upward shifts with increasing story counts, the spectral energy distribution remained anchored in low-frequency domains. The frequency bandwidth of mixed-direction velocity responses progressively narrowed with structural heightening, resulting from heightened modal frequency concentration and simplified vibration patterns, ultimately manifesting as multipeak narrowband spectral characteristics. Spectral characteristics diverged across vertical monitoring positions: ascending floors exhibited (1) downward frequency migration, (2) high-frequency component attenuation, and (3) multipeak emergence in the sub-5 Hz range. This spectral evolution originated from wave reflection dynamics at the structural apex, where superposition of incident and reflected waves generates multiple resonance peaks.

5.5. Influence of the Total Structural Spans Variation on the Structural Vibration Response

This section investigates the vibration responses of frame structures with different total spans and semirigid joints under subway vibration environment. The basic design data for all structures were consistent with the baseline structure described in Section 2, with only the total number of spans altered. Based on the four-span baseline structure shown in Figure 4, six-span and eight-span frame models with semirigid joints were designed, with their structural layout plans illustrated in Figure 26.

Design parameters for models with varying total spans are listed in Table 14. The PKPM2025R2.3 software processed the input design criteria and cross-sectional data to produce differentiated reinforcement schematics. Translational joint stiffness was assigned as $=2$.8 × 10⁹ N/m. A Rayleigh damping model was implemented with a material damping ratio ξ = 0.02.

Increasing total spans amplified foundation vibration inputs. For structures perpendicular to tunnel axes, subway vibrations experienced partial attenuation in outer spans of wider configurations [15]. To enhance simulation accuracy, S2/S3 models were divided into Region 1 and Region 2 (Figure 27). Region 1 received triaxial vibration data from six-car Type B trains at 80 km/h above the tunnel centerline, while Region 2 incorporated vibration inputs 20 m laterally offset from the centerline. While span expansion enhanced global stiffness in semirigid systems, it simultaneously intensified standing wave superposition effects, requiring critical attention. Room numbering at top floors (Figure 27) facilitated analysis. Vertical one-third octave band levels at room centers are presented in Figure 28.

Generally, dominant vibration frequencies escalated with span counts: 16 Hz for four-span vs. 25 Hz for six-/eight-span models across monitored rooms. The six- to eight-span transition showed minimal frequency shifts, indicating that span-induced stiffness enhancement dominated over modal reorganization in semirigid systems. Beyond critical span thresholds, structural frequency stabilization occurred at 25 Hz, demonstrating diminishing returns of span expansion on dynamic properties. Similar spectral profiles across rooms suggested consistent lateral vibration transmission without frequency-selective attenuation, contrasting with vertical propagation patterns. Continuous column integrity enables progressive vertical vibration attenuation through wave transmission rather than reflection [16]. Semirigid beam–column connections create impedance discontinuities, causing wave reflection and constrained lateral propagation ranges. Each column acts as a vibration input source for adjacent horizontal members, creating uniform spectral distributions across floor plans. Expanded spans increase vibration input areas without elevating peak levels but cause broadband amplitude escalation, enhancing human perceptibility.

Vibration time-history data from room centers at top floors were processed to obtain floor-specific weighted acceleration levels shown in Figure 29.

Edge effects persisted across all span configurations, with peripheral rooms exhibiting higher vibration levels than central areas due to wave reflection dynamics in horizontal members. Wave reflection at structural boundaries creates localized amplification through constructive interference between incident and reflected waves [17]. Central rooms maintained compliance with stringent vibration thresholds, while edge rooms exceeded these limits, significantly impacting occupant comfort. Increased span counts introduced competing effects of higher vibration inputs vs. enhanced stiffness, ultimately reducing maximum weighted acceleration levels through global stiffening.

Long-term safety evaluation included X-/Y-direction velocity histories (Figure 30) and peak velocities (Table 15) at roof centers across span configurations.

Each two-span increment achieved 15% Y-direction peak velocity reduction at roof centers through enhanced lateral system rigidity. Comparative analysis with Table 12 reveals greater horizontal velocity sensitivity to span vs. vertical parameter changes, validating span optimization as critical for metro-adjacent structural safety. Unlike acceleration metrics showing diminishing returns, Y-velocity maintained linear attenuation, enabling nonresidential structures to achieve compliance through progressive span expansion.

6. Conclusions

Through systematic parametric studies of beam/column cross-sections, slab thickness, and total stories/spans under metro type B train running at 80 km/h vibrations, the following conclusions were drawn regarding semirigid and rigid connection systems:

(1). Beam section modifications demonstrated complex vibration control mechanisms influenced by both height and aspect ratio. Increasing beam height or aspect ratio (at constant height) achieved moderate acceleration reduction. While velocity spectral characteristics remained stable, peak velocity attenuation through height enhancement benefited long-term structural safety. Rigid-jointed systems showed greater sensitivity to beam parameter changes than semirigid configurations.

(2). Column section enlargement demonstrated superior acceleration control efficacy to that of beam modifications. However, semirigid systems exhibited weaker column size effects than their rigid counterparts, often exceeding code-specified acceleration limits. Joint stiffness enhancement should precede column sizing optimization. Column parameter adjustments achieved greater peak velocity reduction than beam changes, establishing column design as a primary velocity-control strategy.

(3). Slab thickening achieved limited vibration reduction with critical thickness boundaries. Semirigid systems reached critical thickness thresholds earlier than rigid counterparts, making slab-based control unreliable for code compliance. While providing modest peak velocity reduction, slab optimization offered lower cost-effectiveness than beam/column modifications.

(4). Structural heightening induced vibration attenuation with amplified roof-level responses, which were particularly pronounced in semirigid systems. Beyond a critical height (n = 11 stories), vibration distribution transitioned to initial attenuation followed by amplification. Frequency escalation impacts human comfort thresholds. Although reducing time-domain peak velocities, multispan nonresidential structures still require supplementary controls for strict compliance.

(5). Span expansion significantly altered dominant frequencies and intensified edge effects in semirigid systems. Horizontal peak velocity reduction through span optimization provides effective safety assurance for metro-adjacent structures.

Based on the shaking table test of prefabricated structures, this paper proposes a modeling method for concrete prefabricated frame structures to study the vibration response of structures on the site caused by subway operation. The relevant research can provide references for subway route planning, environmental vibration impact assessment of new structures, and design of control measures.

Author Contributions

Methodology, Y.Y.; Software, Z.H.; Writing—original draft, Z.H.; Writing—review & editing, Y.Y. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1 Simplified mechanical model of prefabricated joints.

Figure 2 Schematic diagrams of the test and numerical models: (a) test model; (b) numerical model; (c) spring model joints.

Figure 3 Plan layout diagram of the baseline structure.

Figure 4 Elevation diagram of the numerical baseline model.

Figure 5 Three-dimensional tunnel–soil–structure interaction system.

Figure 6 One-third octave band results between the computational model and experimental measurement.

Figure 7 Detailed drawings of the variable beam sections: (a) L1-S, L1-R; (b) L2-S, L2-R; (c) L3-S, L3-R.

Figure 8 Weighted acceleration levels of the middle nodes of the side beams on each floor in different beam section models. Note: The gray dashed lines represent the allowable vertical vibration values for residential areas at night and during the day, respectively.

Figure 9 Acceleration spectrum of the middle nodes of the side beams on the top floor in different beam section models: (a) semirigid-jointed systems; (b) rigid-jointed systems.

Figure 10 Time-history diagrams of horizontal biaxial velocity at the center of the top-floor slab in different models: (a) L1-S; (b) L2-S; (c) L3-S; (d) L1-R; (e) L2-R; (f) L3-R.

Figure 11 Mixed-direction velocity spectrum of the middle nodes of the side beams on the top floor in different beam section models: (a) semirigid-jointed systems; (b) rigid-jointed systems.

Figure 12 Detailed drawings of the variable column sections: (a) Z1-S, Z1-R; (b) Z2-S, Z2-R; (c) Z3-S, Z3-R.

Figure 13 Frequency-weighted acceleration levels of the middle nodes of the side beams on each floor in different column section models.

Figure 14 Acceleration spectra of the middle nodes of the side beams on the top floor in different column section models: (a) semirigid-jointed systems; (b) rigid-jointed systems.

Figure 15 Time-history diagrams of horizontal biaxial velocity at the center of the top-floor slab in different models: (a) Z1-S; (b) Z2-S; (c) Z3-S; (d) Z1-R; (e) Z2-R; (f) Z3-R.

Figure 16 Mixed-direction velocity spectrum of the middle nodes of the side beams on the top floor in different column section models: (a) semirigid-jointed systems; (b) rigid-jointed systems.

Figure 17 Weighted acceleration levels of the middle nodes of the side beams on each floor in different slab thickness models.

Figure 18 Acceleration spectrum of the middle nodes of the top-floor side beams in different slab thickness models: (a) semirigid-jointed systems; (b) rigid-jointed systems.

Figure 19 Time-history diagrams of horizontal biaxial velocity at the central position of the top-floor slab in different models: (a) B1-S; (b) B2-S; (c) B3-S; (d) B1-R; (e) B2-R; (f) B3-R.

Figure 20 Mixed-direction velocity spectrum of the middle nodes of the top-floor side beams in different slab-thickness models: (a) semirigid-jointed systems; (b) rigid-jointed systems.

Figure 21 Schematic elevation diagrams of finite elements for models with different total numbers of stories: (a) 8-story model; (b) 11-story model.

Figure 22 Weighted acceleration levels of the middle nodes of the side beams on each floor of models with different total numbers of stories.

Figure 23 One-third octave vibration levels at the measuring points on each floor of models with different numbers of stories: (a) the 5-story model; (b) the 8-story model; (c) the 11-story model.

Figure 24 Horizontal biaxial velocity time-history diagrams at the central positions of the top-floor slabs of different models: (a) the 5-story model; (b) the 8-story model; (c) the 11-story model.

Figure 25 Mixed-direction velocity frequency spectrum diagrams at the measuring points on each floor of models with different numbers of stories: (a) the 5-story model; (b) the 8-story model; (c) the 11-story model.

Figure 26 Plan layout diagrams of structures with different total numbers of spans: (a) the 6-span model; (b) the 8-span model.

Figure 27 Room numbers on the top-floor slabs of models with different total numbers of spans: (a) S1; (b) S2; (c) S3.

Figure 28 One-third octave vibration levels of each room with different total numbers of spans: (a) S1; (b) S2; (c) S3.

Figure 29 Weighted acceleration levels of each room on the top-floor slabs of models with different total numbers of spans: (a) S1; (b) S2; (c) S3.

Figure 30 Horizontal biaxial velocity time-history diagrams at the central positions of the top-floor slabs of different models: (a) S1; (b) S2; (c) S3.

Table 1

Comparison of the results of vibration mode frequencies between simulation and test.

Vibration Mode	1st Mode	2nd Mode	3rd Mode
Vibration direction	TranslationalMotion—Y	TranslationalMotion—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%

Table 2

Values of rotational stiffness under different $\gamma$.

$\gamma$	1.0	0.9	0.8	0.7	0.6	0.5	0.4	0.3	0.2	0.1
$K_r$N·m/rad	3.66 × 108	3.29 × 108	2.93 × 108	2.56 × 108	2.19 × 108	1.83 × 108	1.46 × 108	1.09 × 108	7.32 × 107	3.66 × 107

Table 3

Soil layer parameters.

SoilLayer	Thickness m	Dynamic ElasticModulus Mpa	Poisson’sRatio	Densitykg/m3	RayleighParameters
Layer 1	10	141	0.443	1650	$\alpha$ = 0.124$\beta$ = 3.15 × 10−5
Layer 2	12	175.3	0.439	1940
Layer 3	18	353	0.439	1985

Table 4

Allowable vibration weighted acceleration levels for human comfort in buildings under repetitive vibration action (dB) [11].

Area	Strict Area	Residential Area	Office Area	Production Line
Daytime	74	80	86	92
Nighttime	74	77	86	92

Note: The strict vibration zone refers to work areas with strict vibration requirements, such as operating rooms in hospitals.

Table 5

Design parameters of the variable-beam-section model components.

Models	L1-S	L1-R	L2-S	L2-R	L3-S	L3-R
γ	0.5	Rigid	0.5	Rigid	0.5	Rigid
Number of stories	5	5	5	5	5	5
Number of spans	4	4	4	4	4	4
Slab thickness mm	100	100	100	100	100	100
Column sectionmm × mm	500 × 500	500 × 500	500 × 500	500 × 500	500 × 500	500 × 500
Beam sectionmm × mm	250 × 500	250 × 500	250 × 550	250 × 550	200 × 500	200 × 500
Aspect ratio	2	2	2.2	2.2	2.5	2.5

Table 6

Values of nodal rotational stiffness for different beam section models.

Model	L1-S	L2-S	L3-S
$K_r$ N·m/rad	1.83 × 108	2.41 × 108	1.51 × 108

Table 7

Peak values of horizontal biaxial velocity at the center of the top-floor slab in different models.

Model	L1-S	L2-S	L3-S	L1-R	L2-R	L3-R
X mm/s	1.635	1.574	1.598	1.484	1.347	1.375
Y mm/s	3.592	3.314	3.437	2.976	2.724	2.793

Table 8

Design parameters of components of the variable column section model.

Models	Z1-S	Z1-R	Z2-S	Z2-R	Z3-S	Z3-R
γ	0.5	Rigid	0.5	Rigid	0.5	Rigid
Number of stories	5	5	5	5	5	5
Number of spans	4	4	4	4	4	4
Slab thickness mm	100	100	100	100	100	100
Column sectionmm × mm	450 × 450	450 × 450	500 × 500	500 × 500	550 × 550	550 × 550
Beam sectionmm × mm	250 × 500	250 × 500	250 × 500	250 × 500	250 × 500	250 × 500

Table 9

Peak values of horizontal biaxial velocity at the central position of the top-floor in different models.

Models	Z1-S	Z2-S	Z3-S	Z1-R	Z2-R	Z3-R
X mm/s	1.697	1.635	1.506	1.519	1.484	1.311
Y mm/s	3.692	3.592	3.254	3.187	2.976	2.624

Table 10

Design parameters of components in the variable-slab-thickness models.

Models	B1-S	B1-R	B2-S	B2-R	B3-S	B3-R
γ	0.5	Rigid	0.5	Rigid	0.5	Rigid
Number of stories	5	5	5	5	5	5
Number of spans	4	4	4	4	4	4
Slab thickness mm	100	100	110	110	120	120
Column sectionmm × mm	500 × 500	500 × 500	500 × 500	500 × 500	500 × 500	500 × 500
Beam sectionmm × mm	250 × 500	250 × 500	250 × 500	250 × 500	250 × 500	250 × 500

Table 11

Peak values of horizontal biaxial velocity at the central position of the top-floor slab in different models.

Models	B1-S	B2-S	B3-S	B1-R	B2-R	B3-R
X mm/s	1.635	1.541	1.497	1.484	1.417	1.379
Y mm/s	3.592	3.451	3.407	2.976	2.842	2.729

Table 12

Parameter values of components in models with different total stories.

Models		Beam Section mm2	Column Section mm2	Slab Thickness mm
8-story model	1~5 story	250 × 500	550 × 550	100
	6~8 story	250 × 500	500 × 500	100
11-story model	1~5 story	250 × 600	600 × 600	110
	5~8 story	250 × 600	550 × 550	110
	9~11 story	250 × 600	500 × 500	110

Table 13

Peak values of horizontal biaxial velocity at the central position of the top-floor slab in different models mm/s.

Direction	Five-Story Model		Eight-Story Model		Eleven-Story Model
	X	Y	X	Y	X	Y
Peakvelocity	1.653	3.592	1.371	3.334	1.175	3.176

Table 14

Design parameters of components in models with different total numbers of spans.

Model	Γ	Number of Stories	Number of Spans	Slab Thickness mm	Column Sectionmm × mm	Beam Sectionmm × mm
S1	0.5	5	4	100	500 × 500	250 × 500
S2	0.5	5	6	100	500 × 500	250 × 500
S3	0.5	5	8	100	500 × 500	250 × 500

Table 15

Horizontal biaxial velocity peaks at the central positions of the top-floor slabs of different models.

Direction	S1		S2		S3
	X	Y	X	Y	X	Y
Peak velocity	1.653	3.592	1.241	3.139	1.020	2.612

References

1. Ma, M.; Xu, L.H.; Du, L.L.; Ma, M.; Xu, L.H.; Du, L.L.; Zhang, Q.; Huang, W. Prediction of building vibration induced by metro trains running in a curved tunnel. J. Vib. Control; 2021; 27, pp. 515-528. [DOI: https://dx.doi.org/10.1177/1077546320930910]

2. Metrikine, A.V.; Vrouwenvelder, A.C.W.M. Surface ground vibration due to a moving train in a tunnel: Two dimensional model. J. Sound Vib.; 2000; 234, pp. 43-66. [DOI: https://dx.doi.org/10.1006/jsvi.1999.2853]

3. Liang, X.; Wang, Y. Research progress on prefabricated semi-rigid connection nodes of steel structures. Prog. Build. Steel Struct.; 2022; 24, pp. 1-14.

4. Zhao, B.; Lu, X.; Liu, L. Experimental Study on the Seismic Performance of Prefabricated Concrete Structure Beam-Column Assemblies in a Fully Assembled Manner. Earthq. Eng. Eng. Vib.; 2005; 1, pp. 81-87.

5. Tian, Y.; Lv, X.; Zhao, B. Experimental study on vibration table of precast concrete frame structure. Struct. Eng.; 2008; 1, pp. 66-71.

6. Zhu, H.; Zhao, B.; Lu, X. Numerical simulation of shaking table test on a fabricated precast concrete frame structure. Eng. J. Wuhan Univ.; 2017; 50, pp. 815-822.

7. Bournas, D.A.; Negro, P.; Molina, F.J. Pseudodynamic tests on a full-scale 3-storey precast concrete building: Behavior of the mechanical connections and floor diaphragms. Eng. Struct.; 2013; 57, pp. 609-627. [DOI: https://dx.doi.org/10.1016/j.engstruct.2013.05.046]

8. Tosa, F.V.; Salha, R. The Behavior of Different Precast Concrete Structures Under Seismic Action. Procedia Technol.; 2016; 22, pp. 235-242. [DOI: https://dx.doi.org/10.1016/j.protcy.2016.01.073]

9.GB50010-2010 Code for Design of Concrete Structures; China Architecture & Building Press: Beijing, China, 2011.

10. Chen, J. Research on the Coherence and Prediction of Vibration and Secondary Noise of Buildings Along the Subway Line. Ph.D. Dissertation; Southwest Jiaotong University: Chengdu, China, 2019.

11.GB50868-2013 Allowable Vibration Standards for Construction Projects; China Planning Press: Beijing, China, 2013.

12. Coulier, P.; Lombaert, G.; Degrande, G. The influence of source–receiver interaction on the numerical prediction of railway induced vibrations. J. Sound Vib.; 2014; 333, pp. 2520-2538. [DOI: https://dx.doi.org/10.1016/j.jsv.2014.01.017]

13. Vladimir, S.; Ilya, T. To the Question of Vibration Levels Prediction Inside Residential Buildings Caused by Underground Traffic. Procedia Eng.; 2017; 176, pp. 371-380. [DOI: https://dx.doi.org/10.1016/j.proeng.2017.02.334]

14. Zhao, D.; Ding, Y. The influence of subway operation on the vibration of high-rise buildings varying with the horizontal distance between building and subway tunnel and its calculation and analysis method. Build. Struct.; 2023; 53, pp. 708-715.

15. Zeng, Q.; He, L.; Wu, Y.; Zhang, W.; Liu, F.; Chen, H.; Wang, J.; Li, X.; Zhao, M.; Sun, Y. Distribution law and influencing factors of vibration inside high-rise residential buildings adjacent to subway. Noise Vib. Control; 2024; 44, pp. 242-248.

16. Zou, C.; Moore, J.A.; Sanayei, M.; Wang, Y. Impedance model for estimating train-induced building vibrations. Eng. Struct.; 2018; 172, pp. 739-750. [DOI: https://dx.doi.org/10.1016/j.engstruct.2018.06.032]

17. Qu, S.; Yang, J.J.; Zhu, S.Y.; Wang, L.; Liu, X.; Chen, W.; Zhang, P.; Li, H.; Sun, J.; Zhou, F. Experimental study on ground vibration induced by double-line subway trains and road traffic. Transp. Geotech.; 2021; 29, pp. 100564.1-100564.14. [DOI: https://dx.doi.org/10.1016/j.trgeo.2021.100564]