Vibration and Wave Propagation in High-Rise Industrial Buildings

Abstract

Investigations and conclusions. In the Guangdong–Hong Kong–Macao Greater Bay Area, several high-rise industrial buildings exceeding 100 meters in height are under construction. These structures uniquely combine industrial production facilities and office spaces within a single architectural entity. This study investigates the vibration-related comfort challenges arising from the transmission of vibrational waves across different sections of these towering complexes. Using a real-world, under-construction high-rise industrial building as a reference, a detailed structural model was developed with advanced finite-element software. Human-induced vibratory loads were applied on a designated floor, and the resulting vibration time-history data were analyzed to understand wave propagation characteristics. To validate the model’s accuracy, a combination of on-site experimental tests and theoretical calculations was conducted. Vibration-time-history data were extracted from a specific building level and analyzed in both the time and frequency domains. Comparative examination of experimental results, theoretical computations, and finite-element simulations confirmed the precision of the finite-element model. The study concludes that vibration-wave propagation in high-rise industrial structures follows a discernible pattern, and a linear regression equation encapsulating these dynamics was formulated.

1. Introduction

In the Guangdong–Hong Kong–Macao Greater Bay Area, the skyline is increasingly dominated by high-rise industrial buildings, many of which exceed 100 m in height. These structures are characterized by their dual-purpose design, integrating industrial production facilities and office spaces within a single architectural entity. This innovative approach to industrial-building design is driven by the need to maximize land-use efficiency in a densely populated and economically vibrant region. However, the coexistence of industrial and office functions within the same building presents unique challenges, particularly in terms of vibration-related comfort.

Vibration-related comfort issues in high-rise industrial buildings primarily stem from human activities and equipment-induced vibrations. While the industrial zones are subjected to significant vibrations due to heavy machinery and equipment, the office areas experience relatively lower levels of human-induced vibrations. Despite this, occupants in office spaces, accustomed to quieter environments, exhibit heightened sensitivity to vibrations. Conversely, workers in the factory zones, accustomed to a noisier and more dynamic environment, may perceive vibrations differently. This disparity in vibration perception between different functional areas within the same building underscores the necessity of a comprehensive study on vibration-related comfort.

Previous research has extensively addressed vibration-related comfort in various structural contexts. For instance, Ebrahimpour and Sack (1989) introduced a simulation formula for jumping and landing scenarios [1]. Blakeborough and Williams (2003) demonstrated that footfall testing using force platforms produced results consistent with those of traditional hammer tests [2]. Setareh (2010) evaluated the vibration-related comfort of steel-reinforced concrete composite floors by comparing peak and root mean square acceleration values from walking tests [3,4]. In 2014, Davis, Liu, and Murray proposed a straightforward methodology for assessing the comfort of low-frequency steel-reinforced concrete composite floors under pedestrian loads [5,6,7]. Zhou Xuhong, Li Jiang, and Liu Jiepeng (2016) used finite-element simulation to validate on-site footfall test results for a prestressed concrete rigid-frame cable-beam floor [8,9,10]. Further, Liu Jiepeng, Zheng Xing, and Li Jiang (2019) conducted empirical investigations on prefabricated concrete composite slabs, analyzing peak and root mean square acceleration values from footfall and walking tests [11]. Most recently, Chen Jun, Zhou Gejie, and Wang Pengcheng (2022) conducted running-load tests using high-precision wireless insole sensors, developing a Fourier-series model based on the experimental findings [12].

Despite these advancements, there remains a significant research gap in understanding the propagation of vibration waves within high-rise industrial buildings that combine industrial and office functions. Specifically, there is a need for studies that address the differential impacts of vibrations across diverse functional zones within such buildings. This study aims to fill this gap by developing a finite-element model to simulate vibrations produced by human activities within high-rise industrial buildings. The primary focus is to examine the intricate patterns of vibration propagation both across floors on the same level and between different levels within a building.

To ensure the accuracy and reliability of the finite-element model, a comprehensive approach combining on-site experimental tests and theoretical calculations was employed. This dual-methodology approach integrates experimental data with theoretical insights, providing a robust validation of the model’s capability to accurately replicate the dynamics of vibration transmission in building structures. By addressing these objectives, this research seeks to enhance our understanding of vibration-related comfort in high-rise industrial buildings, ultimately contributing to the design and construction of structures that ensure both structural integrity and occupant comfort.

2. Model Overview

This paper examines a towering industrial edifice situated in the Guangdong–Hong Kong–Macau Greater Bay Area, which currently serves as the hub for a novel-energy-automobile industrial project. Architecturally, this structure is an amalgamation of steel and concrete, fashioned into a combined-frame design. It encompasses two subterranean levels and twelve above-ground stories, culminating in a total height of 84.02 m. Functionally, the building is partitioned into two distinct zones: Area A (indicated by the red box in Figure 1), designated as the factory space, and Area B shown by the blue box in Figure 1, comprising the office segment. The layout from the first to the tenth floor is characterized by a planar structure that integrates both these zones. In contrast, the upper stories, spanning the 10th to 12th floors, are exclusively dedicated to Area A. For a comprehensive understanding of the building’s structural blueprint and its overall model, readers are directed to refer to Figure 1.

The structural composition of the beams in this high-rise industrial building includes Q355B steel, renowned for its robustness. The precise cross-sectional dimensions of the beam are meticulously detailed in

Table 1. Dimensions of beam.

Part Number	$\mathit{H}\times \mathit{B}\times {\mathit{T}}_{\mathit{f}}\times {\mathit{T}}_{\mathit{W}}$	Steel Designation
GKL1	$H700\times 200\times 12\times 16$	Q355B
GKL2	$H800\times 250\times 14\times 16$	Q355B

. Additionally, the columns were engineered as steel-concrete-composite structures and also incorporate Q355B steel for enhanced structural integrity. These columns are uniquely characterized by their steel-pipe cores, which are filled with C60 grade concrete. The steel pipes are noteworthy for their substantial thickness of 10 mm and considerable diameter of 700 mm. Complementing these elements, the flooring system was designed as a truss-reinforced floor slab. This slab is composed of concrete with a strength grade of C30, ensuring durability, and measures 180 mm in thickness.

3. Finite-Element Simulation

In this research, we established a comprehensive finite-element model to simulate global structural dynamics, focusing on modal analysis of the entire factory building and specific single-floor vibration behaviors. The validity and accuracy of this model were rigorously confirmed by conducting a series of comparative evaluations. These evaluations involved juxtaposing the simulation outputs with data obtained from on-site testing and theoretical calculations. Such a meticulous verification process not only ascertains the fidelity of the finite-element model but also establishes a solid foundation for subsequent explorations. In particular, this paves the way for an in-depth investigation into the propagation patterns of vibration wave; thus, finite-element simulation (FES) techniques are utilized in the ensuing sections of this study.

3.1. Model Development

In the finite-element representation of the high-rise industrial building’s overall structure, a strategic approach was adopted to depict various components accurately. The beams and columns, fundamental to the structural integrity, were modeled using line elements. This choice efficiently captures their linear geometry and load-bearing characteristics. In contrast, the slabs and walls, which require a more complex representation due to their planar nature, were depicted using shell elements. These elements are suitable for simulating the multi-dimensional stress and strain behaviors inherent to such structures. For a comprehensive visualization of this finite-element modeling, readers are referred to Figure 2. This figure provides an illustrative depiction of the overall finite-element model, elucidating the structural layout and the representations of each element.

To accurately simulate the human-induced load within the structural model, we have employed the Bachmann model, a well-regarded approach in the field of structural dynamics. Central to this methodology is the utilization of a single-step footfall load curve, a detailed representation of which is provided in Figure 3. Recognizing the necessity of applying this load at critical points within the structure, we specifically targeted locations on the floor that are most susceptible to significant deflections. Accordingly, the human-induced load was methodically applied along a predefined path, delineated by an arrow [13], on the floor within Area A, as depicted in the diagram. Further, to capture the resulting vibrational behavior of the structure, the extraction point for the acceleration time-history curve was determined. This crucial aspect of the simulation is visually represented and elaborately explained in Figure 4.

3.2. Results of Finite-Element Simulation

3.2.1. Floor-Vibration Modes

Initially, the study entails a modal analysis of the single-layer floor slab, a crucial step in understanding its dynamic behavior. As an illustrative example, we conducted an analysis of the 12th-floor slab. This analysis provided insights into the first six vertical mode shapes and their corresponding natural frequencies. For a detailed visual representation of these mode shapes and frequencies, refer to Figure 5. This figure serves to elucidate the vibrational characteristics of the floor slab, offering a clear depiction of the dynamic response patterns observed during the analysis.

In the course of our analysis, we meticulously extracted the first-order vertical natural frequencies of the floor slabs for each individual floor. These crucial data, pivotal for understanding the dynamic behavior of the building’s floors, have been comprehensively tabulated. For a detailed overview of these natural frequencies, readers are directed to consult

Table 2. Fundamental frequencies associated with the vertical vibrations of different floors.

Floor	f (Hz)
1st	8.27
2nd	8.17
3rd	8.24
4th	7.69
5th	7.86
6th	8.15
7th	8.22
8th	8.22
9th	8.22
10th	7.86
11th	7.63
12th	7.67

. This table provides an organized and accessible presentation of the extracted frequencies, thereby facilitating a clear understanding of the vibrational characteristics of different levels of the building.

3.2.2. Results of Vertical-Vibration Peak Acceleration

To comprehensively analyze the vertical-vibration response of the floor slab, a detailed examination of the acceleration time-history curves was undertaken. This involved extracting these curves from designated test areas on each floor slab. The resultant data provide in-depth insights into the dynamic vibrational behavior of the slabs under varying conditions. For a visual representation of these acceleration time-history curves, as extracted from the floor slabs of different levels, Figure 6 serves as a crucial reference. This figure offers a clear and structured depiction of the vibrational responses, thereby facilitating an enhanced understanding of the floor slabs’ dynamics. The load-model curve is shown in Figure 3, and the diagram illustrating the loading path and the acceleration extraction points is shown in Figure 4

The study meticulously arranges the data pertaining to the vertical peak acceleration of the floor slab from each floor. This arrangement is crucial for understanding the varying degrees of vibrational intensity across different levels of the building. The comprehensive compilation of these data is presented in

Table 3. Vertical-vibration peak acceleration.

Floor	Peak Acceleration (m·s−2)
1st	0.0894
2nd	0.0911
3rd	0.0944
4th	0.0951
5th	0.0913
6th	0.0939
7th	0.0950
8th	0.0932
9th	0.0911
10th	0.0946
11th	0.0875
12th	0.0926

. This table methodically outlines the peak acceleration values, offering a clear and organized overview that is instrumental for analyzing the dynamic behavior of the floor slabs under different conditions.

3.3. Experimental Testing and Verification

3.3.1. Experimental Overview

To ascertain the fidelity of the finite-element model, extensive on-site testing was conducted on the high-rise industrial and office dual-use building. The testing apparatus comprised the Donghua 1A401E sensor, depicted in Figure 7, alongside the Donghua DH8303 dynamic signal-testing and -analysis system, illustrated in Figure 8. The cornerstone of this testing regime was the Donghua DHDAS dynamic signal-acquisition and -analysis system. This sophisticated system played a pivotal role in recording the acceleration time-history curve at various measurement points. It was calibrated with a sensitivity of 5.183 mV/m/s² to ensure precise data collection. This testing methodology not only validates the model but also provides invaluable empirical data for a comprehensive understanding of the building’s dynamic behavior.

In best practices in structural testing [14,15], sensors are optimally positioned at locations where significant deflections are anticipated due to external excitations. In this study, the sensor was strategically installed at the mid-span of the floor slab, a site typically prone to larger deflections. The precise positioning of the sensor is highlighted within a red box in Figure 9. Additionally, the testing points were carefully chosen at the intersections of the long and short spans of the test floor slab. This placement ensures the comprehensive capture of the dynamic response of the slab, as duly indicated in the figure. Such meticulous positioning of sensors is crucial for accurately gauging the vibrational characteristics of the structure under test conditions.

In this experimental setup, we employed the pulse-excitation method, specifically utilizing jump excitation on the floor slab to initiate its vertical vibration [16]. This method was chosen for its effectiveness in simulating realistic dynamic-loading conditions. The acceleration signals generated from the floor slab within the test area were captured using an accelerometer. Subsequently, these signals underwent a comprehensive frequency-spectrum analysis.

The crucial step involved exporting the acceleration time-history curve data. These data were then converted into a spectrum curve through the application of Fast Fourier Transformation (FFT) [17]. This transformation facilitated the extraction of the first-order vertical natural frequency of the floor vibration at that specific floor level.

Furthermore, to assess the floor’s compliance with vibration-related comfort standards, running excitation was applied to the floor slab. This approach aimed to stimulate vertical vibrations. The peak acceleration of these vibrations, as recorded by the accelerometer, was meticulously analyzed. This analysis provided critical insights into whether the floor slab meets the established criteria for vibration-related comfort.

The testing procedure implemented in this study comprised two distinct phases, each tailored to gather specific vibration data from the floor slab. The process was meticulously planned and executed as follows:

• Jump-Stimulation-Data Collection: Prior to the initiation of data collection, a test subject with a body weight of 80 kg was positioned, standing still, at the designated measurement point. Upon commencement of data collection, the subject proceeded to jump and then land naturally, maintaining bodily stability throughout. The data-collection phase concluded once the test data returned to a stable state, ensuring that the transient effects of the jumping action were adequately captured.
• Running-Excitation-Data Collection: Before this phase began, the designated test personnel underwent training to control their running frequency consistently at 2 Hz. The data collection commenced with a test subject, also weighing 80 kg, running along a predetermined route marked by an arrow within the test area, which is marked by a red box. The subject maintained a constant speed of 2 m/s [18,19], as delineated in Figure 10. The data-collection phase concluded once the test data stabilized, marking the end of the running-induced vibration response.

Refer to Figure 11 for an illustrative depiction of the on-site-test setup.

Each phase of this testing process was designed to elicit specific vibrational responses from the floor slab, thereby enabling a comprehensive analysis of its dynamic behavior under varying excitation conditions.

3.3.2. Results of Modal Analysis of Slab Vibration

The study meticulously captured the vertical acceleration response of the floor slabs spanning from the 1st to the 12th floor under the conditions of jump excitation. These responses provide critical insights into the dynamic behavior of the slabs when they are subjected to transient loads. For a comprehensive visualization of this response, readers are directed to Figure 12. This figure offers a detailed depiction of the acceleration response across various floors, enhancing the understanding of the building’s vibrational characteristics under jump-induced excitations.

Furthermore, the acceleration data were subjected to fast Fourier transformation (FFT) to obtain spectrum curves. These curves were instrumental in analyzing the frequency content of the vibrations, offering a deeper understanding of the vibrational dynamics. Figure 13 presents these spectrum curves post-FFT conversion, providing an analytical perspective on the vibrational frequencies elicited by the jump excitation.

The study systematically tabulated the vertical natural frequencies of the floor slabs from the 1st to the 12th floors. This organization of data is crucial for comprehending the inherent vibrational characteristics of each level under varying conditions. The collected frequencies provide valuable insights into the dynamic behavior of the building’s structure. For a detailed and organized presentation of these vertical natural frequencies, readers are referred to

Table 4. Vertical first–order frequency.

Floor	f (Hz)		Error (%)
	FES	Test
1st	8.27	8.49	2.66
2nd	8.17	7.99	2.20
3rd	8.24	7.88	4.37
4th	7.69	7.99	3.90
5th	7.86	7.99	1.65
6th	8.15	7.73	5.15
7th	8.22	7.91	3.77
8th	8.22	7.68	6.57
9th	8.22	7.785	5.29
10th	7.86	7.655	2.61
11th	7.63	7.99	4.72
12th	7.67	7.495	2.28
Average Error			3.76

. This table is designed to offer a clear and concise overview of the frequency data, facilitating an in-depth understanding of the structural dynamics across different floors of the building.

The analysis reveals that the first-order vertical natural frequency derived from FFT transformation closely aligns with the corresponding frequency obtained through finite-element simulation. The calculated average error between these two sets of frequencies stands at 3.76%. This modest discrepancy is indicative of the finite-element model’s substantial reliability. Such a level of accuracy underscores the model’s effectiveness in replicating the dynamic behavior of the structure under study. Consequently, this validation lends credence to the idea of the model’s applicability to analyzing and predicting the vibrational characteristics of similar structures.

3.3.3. Results of Vertical Vibration Acceleration Testing of Slabs

The study comprehensively captured the vertical acceleration time-history curves for the floor slabs ranging from the 1st to the 12th floors under running excitation conditions. These curves are instrumental in understanding the dynamic response of the building’s floors when they are subjected to continuous dynamic loading. For a detailed visual representation of these acceleration responses, Figure 14 is provided. This figure offers an in-depth view of the vibrational behavior of each floor slab, illustrating how the slabs react to the running-induced excitations. The data presented in Figure 14 are essential for analyzing the structural dynamics and assessing the vibrational impact of running loads on the building’s floors.

According to Figure 14, there are noticeable discrepancies between the experimental and finite-element results. The main reasons for these discrepancies are as follows:

• Environmental vibrations or external forces at the site can introduce errors into the data.
• The sensors used are very sensitive, and changes in environmental temperature and humidity on the test day can cause errors in the test data.
• Errors generated by the signal-acquisition device due to prolonged field testing usually require external power. Our power-supply equipment, due to battery-level differences, can lead to voltage differences. These voltage changes, which are known to cause errors based on multiple tests and communications with the equipment manufacturer, are unavoidable.
• Errors in load application by testing personnel. Although efforts have been made to coordinate and standardize the sources of human-induced vibration, humans are not machines, and it is impossible to avoid errors that lead to changes in the signal line type. Finite-element simulations, being computed, typically have smaller errors in each result and thus more consistent line types. We believe that this is also a source of the differences between the experimental and finite-element results.

To eliminate or reduce these differences, the collected data were filtered to avoid irrelevant signal interference and we conducted numerous field tests. However, field-test conditions can never be as ideal as laboratory conditions, and we could not control the environmental conditions for each test. Under these circumstances, we believe that the results obtained are sufficient for further research.

The vertical peak accelerations for the 1st to 12th floors have been meticulously compiled and are summarized in

Table 5. Comparison of FES and Test Vertical vibration peak accelerations.

Floor	Peak Acceleration (m·s−2)		Error (%)
	FES	Test
1st	0.0894	0.0870	2.76
2nd	0.0911	0.0889	2.47
3rd	0.0944	0.0904	4.42
4th	0.0951	0.0977	2.66
5th	0.0913	0.0941	2.98
6th	0.0939	0.0925	1.51
7th	0.0950	0.0930	2.15
8th	0.0932	0.0992	6.05
9th	0.0911	0.0702	22.94
10th	0.0946	0.0922	2.60
11th	0.0875	0.0984	11.08
12th	0.0926	0.0891	3.93
Average Error			5.46

. This table presents a concise yet comprehensive overview of peak acceleration values, crucial for evaluating the dynamic response of each floor under various loading conditions. The data in Table 5 provide essential insights into the vibrational behavior of the building’s structure, offering a clear understanding of how each floor responds to different excitations. Such a detailed compilation is invaluable for both analyzing structural integrity and assessing levels of vibrational comfort across different levels of the building.

The comparative analysis of the vertical peak acceleration data obtained from the actual testing and that predicted by the finite-element simulation revealed an average error of 5.46%. This relatively small margin of discrepancy underscores the model’s reliability in accurately simulating the dynamic behavior of the building’s structure. The consistency between the empirical test results and the simulation data creates a significant degree of confidence in the model’s validity. Such a level of precision indicates that the finite-element model is a dependable tool for predicting the vibrational characteristics of similar structures, making it valuable for both theoretical and practical applications in structural engineering.

3.4. Theoretical Verification

3.4.1. Computational Theory

To further substantiate the accuracy of the finite-element model, theoretical calculations were rigorously conducted on the high-rise industrial-and-office dual-purpose building. The methodology for determining the first natural frequency of the floor relied on the formula provided in Appendix A of the ‘Technical Standard for Vibration Comfort of Building Floors’ JGJT441-2019 [14]. This standard offers a recognized and authoritative framework for assessing vibrational characteristics in building structures. Employing this standard as a benchmark, the theoretical calculations provided a robust means by which to evaluate the fidelity of the finite-element model. The alignment of these theoretical results with the model’s predictions would significantly reinforce the model’s validity and reliability in simulating structural vibrations. The formula is as follows:

$$f_1={\displaystyle \frac{C_f}{\sqrt{\Delta }}}$$

(1)

where $f_1$ is the first vertical natural frequency in Hz, ∆ is the maximum deformation of the beam-type floor in mm, and $C_f$ is the frequency coefficient of the beam-type floor, which is taken as 20 in this study.

According to the “Technical Standard for Vibration Comfort of Building Floors” JGJT441-2019 [14], for floor structures primarily excited by walking, the floor’s vibration response can be calculated based on the excitation from a single person walking. The peak acceleration of the vibration can be calculated using the following formula:

$$\begin{array}{l}{\alpha }_{\mathrm{p}}={\displaystyle \frac{F_p}{\xi W}}g\\ F_p=0.29e^{-0.35f_1}\end{array}$$

(2)

where ${\alpha }_{\mathrm{p}}$ is the peak vertical acceleration of the vibration in m/s²; $F_p$ is the force generated by walking when the floor structure resonates in kN; $\xi$ is the damping ratio and can be taken as 0.02 to 0.05, with 0.05 being used in this study; $g$ is the acceleration due to gravity and is equal to 9.8 m/s²; $f_1$ is the first vertical natural frequency of the floor; and $W$ is the effective weight of the vibration, which can be calculated using the formula given in Appendix B of JGJT441-2019 “Technical Standard for Vibration Comfort of Building Floors” [14] for steel-concrete-composite floors.

3.4.2. Results of Theoretical Calculations

The calculated results for the first vertical natural frequencies and peak vertical accelerations of vibration across the 1st to 12th floors have been comprehensively tabulated. The compilation of these calculations is presented in

Table 6. Calculation results of the first–order vertical natural frequency and vibration peak acceleration.

Floor	f (Hz)	${\mathit{F}}_{\mathit{P}}$ (kN)	$\mathit{\xi }$	$\mathit{W}$ (kN)	Peak Acceleration (m·s−2)
1st	8.27	3.70	0.05	7502.74	0.0965
2nd	8.17	3.74	0.05	7806.51	0.0939
3rd	8.24	3.74	0.05	7806.51	0.0939
4th	7.69	3.74	0.05	7806.51	0.0939
5th	7.86	3.74	0.05	7806.51	0.0939
6th	8.15	3.74	0.05	7806.51	0.0939
7th	8.22	3.70	0.05	7502.74	0.0965
8th	8.22	3.70	0.05	7502.74	0.0965
9th	8.22	3.44	0.05	7265.31	0.0929
10th	7.86	3.70	0.05	7502.74	0.0965
11th	7.63	3.74	0.05	7500.31	0.0978
12th	7.67	3.44	0.05	7265.31	0.0928

. This table methodically outlines the calculated frequencies and accelerations, offering a clear and organized overview that is instrumental for analyzing the structural dynamics of the building.

3.4.3. Comparison of Theoretical Calculation and FES Results

The study meticulously compares the first vertical natural frequencies derived from theoretical calculations with those obtained from the finite-element simulations. The results of this comparison are systematically presented in

Table 7. Vertical first–order natural frequency.

Floor	f (Hz)		Error (%)
	FES	Theory
1st	8.27	7.85	5.08
2nd	8.17	7.92	3.06
3rd	8.24	7.92	3.88
4th	7.69	7.92	2.99
5th	7.86	7.92	0.76
6th	8.15	7.92	2.82
7th	8.22	7.85	4.50
8th	8.22	7.85	4.50
9th	8.22	7.925	3.59
10th	7.86	7.85	0.13
11th	7.63	7.425	2.69
12th	7.67	7.425	3.19
Average Error			3.10

.

Based on the results from Table 4 and Table 7, we can see that the discrepancies between the finite-element results and both the experimental and theoretical results are consistently small, which indicates that the experimental and theoretical calculation methods are suitable for validating the accuracy of the finite-element model. Regarding the error analysis, we believe that there are two main causes of error:

The vertical first-order natural frequency obtained from the field test was derived from FFT transformations of the collected data. Inevitably, the data-collection process introduced some errors, which were then reflected in the results of the FFT transformations.

The finite-element model was constructed based on structural drawings. However, due to the presence of various building materials on site, the presence of construction workers, and differences in construction progress, there were inevitable discrepancies between the finite-element model and the actual model. Therefore, errors were unavoidable.

For a comprehensive analysis, the study juxtaposes the peak vertical accelerations derived from theoretical calculations with those ascertained through finite-element simulations. The detailed results of this comparative assessment are showcased in

Table 8. Comparison of FES and Theoretical Vertical vibration peak accelerations.

Floor	Peak Acceleration (m·s−2)		Error (%)
	FES	Theory
1st	0.0894	0.0965	7.15
2nd	0.0911	0.0939	2.98
3rd	0.0944	0.0939	0.53
4th	0.0951	0.0939	1.28
5th	0.0913	0.0939	2.77
6th	0.0939	0.0939	0
7th	0.0950	0.0965	1.55
8th	0.0932	0.0965	3.42
9th	0.0911	0.0929	1.94
10th	0.0946	0.0965	1.97
11th	0.0875	0.0978	10.53
12th	0.0926	0.0928	0.22
Average Error			2.86

. This table facilitates an in-depth examination of the acceleration values obtained from both methodologies, providing a critical insight into the correlation and discrepancies between the theoretical and simulated data.

Table 7 and Table 8 provide a compelling demonstration of the consistency and reliability of the finite-element model used in this study. The first-order vertical natural frequencies, as determined by both simulation and theoretical calculations, consistently exceed 3 Hz. Remarkably, the average error between these two sets of frequencies is only 3.19%. Additionally, the comparison of the vertical vibration peak acceleration data reveals an even lower average error of 2.86%. These minimal discrepancies are indicative of the finite-element model’s substantial accuracy and its strong alignment with theoretical expectations. Consequently, it is reasonable to assert that the results obtained from the finite-element model are reliably accurate.

4. Propagation of Vibrational Waves Caused by Human Activities

To investigate the propagation of vibration waves across a single floor slab, a specific approach was adopted in this study. Floor slab No. 1 in Zone A of each level was designated as the excitation source. In contrast, floor slabs No. 2 to No. 6 on each level were identified as the vibrating floor slabs, where the effects of the propagated vibrations were to be analyzed. For a detailed examination of these effects, the vibration acceleration time-history curves were extracted from the midpoints of floor slabs No. 1 to No. 6. The segmentation and layout of these floor slabs are clearly illustrated in Figure 15. This figure provides a visual guide to the division of the floor slabs and their roles in the experiment. Additionally, Figure 16 presents the vibration acceleration time-history curves for each floor slab.

The vertical peak acceleration of each layer under excitation is summarized in

Table 9. Peak Vertical vibration accelerations on the Same Floor from 1st to 12th Levels.

Floor	Peak Acceleration (m·s−2)
	No. 1	No. 2	No. 3	No. 4	No. 5	No. 6
1st	0.0911	0.0747	0.0592	0.0365	0.0210	0.0081
2nd	0.0912	0.0684	0.0465	0.0294	0.0195	0.0070
3rd	0.0941	0.0792	0.0556	0.0297	0.0181	0.0068
4th	0.0923	0.0689	0.0396	0.0271	0.0121	0.0051
5th	0.0960	0.0780	0.0492	0.0306	0.0119	0.0059
6th	0.0922	0.0766	0.0473	0.0243	0.0109	0.0044
7th	0.0926	0.0717	0.0488	0.0340	0.0171	0.0063
8th	0.092	0.0730	0.0471	0.0327	0.0163	0.0062
9th	0.0941	0.0762	0.0558	0.0307	0.0157	0.0055
10th	0.0895	0.0693	0.0439	0.0272	0.0189	0.0074
11th	0.0908	0.0754	0.0443	0.0298	0.0175	0.0065
12th	0.0881	0.0720	0.0444	0.0278	0.0190	0.0072

.

According to Table 9, it can be observed that within the same floor and span, each floorboard’s peak acceleration decreased in a regular pattern as the distance from the source of excitation increased. Additionally, between different floors, the peak vibration acceleration generally increased as the floor levels became higher. This increase can potentially be attributed to the “whip effect,” where, under the same excitation conditions, the peak vibration acceleration increases with the floor height. The third and fifth floors do not follow this pattern due to variations in the main and secondary beam steel between floors, which led to a relatively lower overall vertical stiffness and thus to greater peak accelerations on these floors. Conversely, the lower peak accelerations on floors 10–12 are due to their design function as sunroom viewing layers, which have no equipment or personnel arrangements, resulting in reduced floor thickness and material strength.

According to the linear regression of the peak acceleration values at the midpoint of each floor on stories 1–12 and the distance from the excitation source point, the linear relationship between the distance and peak acceleration of each floor was obtained, as shown in Figure 17. The linear equation is assumed to be as follows:

$$\alpha ={\beta }_0+{\beta }_{1}s$$

(3)

where $\alpha$ is the vertical vibration peak acceleration in m/s²; ${\beta }_0$, ${\beta }_1$ is regression coefficient; and  s is the distance from the excitation source in floor span in m.

The regression coefficients of the vibration-wave-propagation linear equations for each floor slab were obtained from Figure 17, as shown in

Table 10. Vertical vibration peak acceleration.

$\mathit{h}$	${\mathit{\beta }}_0$	${\mathit{\beta }}_{0\mathit{i}}-{\mathit{\beta }}_{0\mathit{i}-1}$	${\mathit{\beta }}_1$	${\mathit{\beta }}_{1\mathit{i}}-{\mathit{\beta }}_{1\mathit{i}-1}$
1	0.09155		−0.00212
2	0.08585	−0.00570	−0.00207	0.00005
3	0.09378	0.00793	−0.00229	−0.00022
4	0.08550	−0.00828	−0.00219	0.0001
5	0.09338	0.00788	−0.00236	−0.00017
6	0.09013	−0.00325	−0.00233	−0.00003
7	0.08907	−0.00106	−0.00216	0.00017
8	0.08878	−0.00029	−0.00217	−0.00001
9	0.09315	0.00437	−0.00230	−0.00013
10	0.08442	−0.00873	−0.00205	0.00025
11	0.08801	0.00359	−0.00216	−0.00011
12	0.08491	−0.0031	−0.00205	0.00011
	${\Delta }_0$	−0.0006	${\Delta }_1$	0.0000064

. To derive the vibration-wave-propagation linear equations applicable to each floor slab, the following equation was assumed:

$$\alpha =({\beta }_0+{\Delta }_{0}i)+({\beta }_1+{\Delta }_{1}i)s$$

(4)

where ${\Delta }_0$, ${\Delta }_1$ is the average coefficient of adjacent-layer difference and $i$ is the number of Floors.

Equations (3) and (4) are applicable to predicting vibration waves in the floor slabs of large-span industrial buildings where the arrangement and structural form of the building are most similar to the structure described in this article for more accurate predictions. Given the specific conditions of different industrial buildings, factors that might cause prediction errors include the overall damping ratio of the floor slab and the overall thickness of the floor slab (related to slab thickness, primary and secondary beam structure, and the strength of the material used).

The linear regression equation for the propagation of vibration waves in the same layer can be obtained from Table 10, as follows:

$$\alpha =(0.09155-0.0006i)-(0.00212-0.0000064h)s$$

(5)

It is worth mentioning that the application of machine learning algorithms is now widespread. Through the input of a large amount of data into existing models for training, these models can be continuously improved to make them more accurate and realistic [20,21,22,23,24]. This allows for effective predictions of vibration-wave propagation in such structures or similar types of structures, which is a potential future extension of this research.

Furthermore, research on wave-propagation methods is relevant to applications in structural-health monitoring and seismic interferometry techniques [25]. In future studies, it could be beneficial to feasibly integrate these technologies, comparing them such that they complement each other. Such an approach could provide valuable insights into vibration-related research in the field of building structures.

5. Conclusions

This paper conducted a finite-element analysis of a high-rise industrial building. The accuracy of the finite-element model was verified through on-site testing and theoretical calculations. Based on the results of the finite-element analysis, the following conclusions were drawn:

• Vibration waves can propagate regularly along the floor slab. Within the same floor, as the distance between the vibration point and the vibration source increases, the energy of the propagated vibration wave decreases, resulting in lower peak accelerations;
• Given the same vibration source, the higher the floor where the vibration source is located, the greater the peak acceleration due to the presence of the whip effect;
• Many factors influence the propagation of vibration waves, including the overall stiffness of the floor and the damping ratio, which affect the magnitude of the peak acceleration;
• The propagation of vibration waves in similar structures or identical types of structures can be predicted by Equations (3) and (4), as proposed in this paper..