A design-oriented method for response prediction of light-weight timber floors under bouncing excitation

Abstract

Owing to the light weight and high fundamental frequency, timber floors exhibit impulse-like responses under human-induced excitation, which is different with the resonance-like responses for heavy concrete structures. The vibration serviceability of timber floors thus needs to be considered in a different manner. Many design codes for timber structures have required that the static displacement or dynamic response under human excitation should be limited within a threshold for the purpose of serviceability, while failing to provide appropriate method for predicting structural responses considering various affecting factors. Inspired by the idea of response spectrum, this paper proposed a design-oriented method for the peak acceleration prediction of high-frequency floors under human bouncing excitation. The prediction can be obtained for any desired confidence level. Statistical analysis shows that the acceleration responses are mostly dependent on structural fundamental frequency, structural damping ratio, and excitation frequency, which are considered in the proposed mathematical model. The application procedure and the experimental assessment of the proposed model are provided, showing the decent applicability of the proposed method.

Keywords

timber floors vibration serviceability response spectrum design-oriented bouncing excitation

Introduction

Timber has many advantages and thus has been adopted as building materials worldwide. For example, a timber structure usually has lower self-weight compared to concrete and steel structures, making it suitable to be used in some seismic active areas because it experiences smaller seismic forces during earthquakes. Modern timber structures, including timber-frame structures and glued-laminated timber structures, have structural members that can be prefabricated for higher assembly efficiency (Li et al., 2019). Timber is also recognized as a type of eco-friendly material from the perspective of recyclability and carbon fixation. The above advantages have greatly enhanced the development of timber materials to be utilized for both public buildings and residential buildings in recent decades.

Despite its advantages, the light-weight characteristic of the timber usually results in large acceleration responses when subject to human activities (Hassan and Girhammar, 2013; Huang et al., 2021a; Weckendorf et al., 2016), causing the so-called vibration serviceability problem (Kullaa and Talia, 1998; Ohlsson, 1982). Excessive vibration of the structure may result in the discomfort of structure’s occupants, which is usually related to structural responses, especially acceleration responses. Therefore, to prevent unpleasant vibration, the structural serviceability design, that is, to predict the level of human-induced vibration, is necessary during the design stage of the structure. Additional devices may be needed if the predicted vibration level is not acceptable (Huang et al., 2021b). For timber structures, although there are many design codes that have proposed requirements on static displacement or dynamic responses to ensure the serviceability of the floor (BSI, 2008; Dolan et al., 1999; ISO, 2007; Ohlsson, 1988; Onysko, 1986; Smith and Chui, 1988; Toratti and Talia, 2006), most of them failed to provide a reasonable and accurate method to calculate such responses, mainly because the complexity of the affecting factors of structural responses. The structural responses are actually related to many factors including structural frequency, damping ratio, excitation frequency, movement amplitude of human body, etc (Wang et al., 2022). Moreover, the human-induced excitation is also characterized by its stochastic feature, which results in the randomness of the structural responses (Wang et al., 2020). A method for structural response prediction under human-induced load that is capable to consider various affecting factors is thus strongly desired.

The response spectrum model presents to be a useful tool for structural dynamic response prediction for many types of dynamic loads including seismic (United States Atomic Energy Commission, 1973), wind (Solari, 1989), and vehicular excitation (Wang and Nagayama, 2022). The idea of response spectrum was also introduced to the research field of human-induced vibration prediction to predict structural response under bouncing excitation (Chen et al., 2016, 2021), which, however, is limited to low frequency structures below 10 Hz. Reports show that structures with high fundamental frequencies, represented by timber floors, behave completely different compared with structures that have low fundamental frequencies (Hu, 2000; Onysko, 1986). Because of their high fundamental frequencies and high damping ratios, the responses are usually characterized by a series of impulses corresponding to the continuous steps of occupants, instead of being dominated by near-resonant vibration (Zhang and Xu, 2020). The features of an impulse, i.e., peak or root-mean-square value, are usually not much affected by previous or following impulses. Therefore, the structural responses need to be evaluated based on features of each impulse rather than those of the entire continuous response time history.

In this paper, a response spectrum model is proposed for the structural vibration serviceability design of high-frequency floors. This model aims to predict structural peak acceleration responses under bouncing excitation from basic parameters of the structure and the bouncing excitation. The remainder of this paper is organized as follows. The characteristics of human-induced vibration of high-frequency structures are first described, explaining the reason why these structures need to be carefully addressed from the perspective of vibration serviceability. A database containing sufficient measured bouncing load time histories is then introduced, which is adopted later for the proposal of the response spectrum model. An experimental test on a timber floor constructed in laboratory validated the applicability of the proposed response spectrum model. Finally, some concluding remarks are made.

Characteristics of high-frequency structural responses

It has been well recognized that structures with low or high fundamental frequencies behave differently under human-induced load. Figure 1(a) shows the acceleration responses measured on a concrete floor (fundamental frequency: 3.52 Hz) under 1.75 bouncing excitation (Zeng et al., 2022). The responses are found to be near-resonant, indicating that the response is mainly dominated by the one of the harmonics of the excitation. When the same type of excitation is exerted on a timber structure (fundamental frequency: 18.23 Hz, details to be introduced in following sections), the measured responses are characterized by a series of impulses, as shown by Figure 1(b). As illustrated, the characteristics of one impulse response are mainly affected by its corresponding load cycle, and are not much related with other cycles, because the impulse soon decays before the next load cycle appears. The reason behind such phenomenon is that the exciting frequency is far away from the fundamental frequency of the structure. This finding is in accordance with some previous results reported in the literature (Murray et al., 1997). Owing to the different behaviors between structures with low and high fundamental frequencies, their design and evaluation on vibration serviceability need to be treated in different manners.

Figure 1.

Acceleration of two representative structures with different fundamental frequencies. (a) A concrete floor of 3.52 Hz, (b) A timber floor of 18 Hz.

Peak acceleration is one of the simple and popular descriptors for vibration serviceability (Hamm et al., 2010; Lou et al., 2020), whose definition is expressed as equation (1)

a_{p e a k} = \max (| a | (t) |), 0 < t ⩽ T

(1)

where a_peak is the peak acceleration of one specific acceleration time history a(t) and T is the length of the time history defined in advance. For most low frequency structures, T is usually defined as the entire time history of the excitation. However, for the responses characterized by Figure 1(b), it is more reasonable to consider the vibration index for each impulse, because these impulses are separated with each other and reflects the stochastic feature of each load cycle. Although there are many descriptors for human comfort evaluation (Zhang et al., 2013), the structural peak acceleration response for each cycle is analyzed and then to be predicted using the proposed response spectrum model in this paper for its simplicity and wide acceptance.

Database of bouncing load cycles

To propose a design-oriented response spectrum model, it is widely acknowledged that directly measured dynamic loads need to be used instead of synthetic ones. To obtain real bouncing loads, experimental tests were conducted at Tongji University. Sensor-attached insoles were equipped by test participants to directly record the ground reaction force at a sampling frequency of 100 Hz. The accuracy and reliability of the sensor-attached insoles have been validated through previous analyses (Hurkmans et al., 2006; Wang et al., 2019).The experimental setup was shown by Figure 2.

Figure 2.

Experimental setup for bouncing excitation.

A time history of bouncing load, which was induced by a 51.8 kg female test participant at a frequency of 1.5 Hz, is shown by Figure 3. The static weight of the test participant has been removed to provide a zero-mean dynamic load. It is observed that this typical time history consists a series of load cycles, corresponding to the up-and-down movement of the human body during bouncing motion. With the human body moves up and down, the dynamic load variates around zero. Slight differences were also observed among cycles, which is known as intra-subject variability. In total, 25 volunteers participated in the test, giving 173 continuous time histories of bouncing excitation with different bouncing frequencies including 1.5 Hz, 1.9 Hz, 2.3 Hz, 2.7 Hz, 3.1 Hz, and 3.5 Hz.

Figure 3.

A representative time history of bouncing load.

As stated in the previous sections, owing to the high fundamental frequencies of timber floors, the structural acceleration responses are characterized by a series of impulses (see Figure 1). The structural responses of high-frequency floors are to be evaluated based on load cycles instead of the continuous time history. Therefore, all the recorded time histories were separated into cycles, giving in total 2587 cycles, which were adopted in the response spectrum analysis described in the next section. Details for these cycles are listed in Table 1.

Table 1.

Number of load cycles for each bouncing frequency.

Frequency	1.5 Hz	1.9 Hz	2.3 Hz	2.7 Hz	3.1 Hz	3.5 Hz
Number	445	443	450	449	415	385
In total	2587

Development of the response spectrum model

Equation of motion of a generalized single-degree-of-freedom system

For most high-frequency structures, the human-induced responses are usually dominated by only one vibration mode that corresponds to the fundamental frequency of the structure (Smith and Chui, 1988). The structure can thus be represented by a single-degree-of-freedom (SDOF) system. Assuming unit mass of the structure, the equation of motion under a bouncing load cycle is expressed as

\ddot{z} (t) + 2 ξ ω_{n} \dot{z} (t) + ω_{n}^{2} z (t) = F (t) / G

(2)

in which z is the displacement response of the SDOF system, ξ is the damping ratio, ω_n is the natural frequency of the system, F(t) represents one cycle of the bouncing load, and G indicates the static weight of the bouncing person.

Mathematical model of the design spectrum

Figure 4(a) shows one cycle of the bouncing load extracted from the continuous time history shown in Figure 3. Replacing the term F(t) by the time history of this cycle, equation (2) can be solved through numerical methods. In this study, the Newmark method is adopted to calculate the response. Once the acceleration response is calculated, its maximum value defined by equation (1) is defined. This procedure is repeated for different fundamental frequencies to formulate a spectrum. For a structural damping ratio of 0.01, the response spectrum of the load cycle in Figure 4(a) is calculated and plotted in Figure 4(b), showing that the maximum acceleration of the SDOF system becomes smaller with the increase of the fundamental frequency in a non-linear manner. This phenomenon is in accordance with the theory of structural dynamics that the structural response is inversely proportional to the stiffness. Note that the lower cut-off frequency is selected to be 10 Hz, corresponding to the high-frequency feature of most light-weight timber floors (Johnson, 1994).

Figure 4.

A representative bouncing load cycle and its corresponding response spectrum. (a) Load cycle (b) Response spectrum.

All the cycles in the database with bouncing frequencies of 2.0 Hz are processed in the above procedure, and the results of the response spectrum are plotted in Figure 5. Each gray dot in Figure 5 represents the peak acceleration response for the SDOF system, whose frequency is indicated by the horizontal axis. Although the similar decreasing trend as Figure 4(b) is also observed in Figure 5, these response values are found to be very much scattered, indicating that the vibration amplitudes should be treated in a stochastic manner.

Figure 5.

Response spectrum curves (damping ratio 0.01, bouncing frequency 1.9 Hz).

To further illustrate this decreasing trend, the frequency axis of Figure 5 is divided into 15 sections with an interval of 1 Hz. For each section, the value of 95% and 75% percentile has been obtained and plotted together in Figure 5.

Based on the above observations, a design-oriented response spectrum is proposed herein. An exponential function was used to describe this descending curve, which is expressed by equation (3)

a (f_{n}) = a_{10} {(\frac{10}{f_{n}})}^{λ}

(3)

where a₁₀ indicates the spectrum value when the structural frequency f_n equals to 10 Hz, and λ is the exponential index. In this expression, a₁₀ and λ are two parameters to be optimized.

Value of a₁₀

The parameter a₁₀ is determined through a statistical analysis. The values of the gray dots at 10 Hz in Figure 5 are extracted, and the distribution of these values is examined, as shown by Figure 6. As observed, these values are found to follow a logarithmic Gaussian distribution, whose probability density function (PDF) is shown by equation (4)

p (x) = \frac{1}{x σ \sqrt{2 π}} e^{- \frac{{(\ln x - μ)}^{2}}{2 σ^{2}}}

(4)

where μ and σ denotes the location and the scale parameter. For the distribution shown in Figure 6, μ is fitted to be −1.249 and σ is fitted to be 0.393, respectively. Note that these values are fitted for the case of 0.01 damping ratio and 2 Hz bouncing frequency. To investigate the dependency on damping ratio and bouncing frequency, the logarithmic Gaussian distribution is also fitted for the a₁₀ values of different values of damping ratio and bouncing frequency, and the parameters of μ and σ are shown in Tables 2 and 3 and plotted in Figures 7 and 8, respectively.

Figure 6.

Logarithmic Gaussian distribution for a₁₀ (damping ratio 0.01, bouncing frequency 1.9 Hz).

Table 2.

Parameter μ for different combinations of damping ratio and bouncing frequency.

μ		Bouncing frequency
μ		1.5 Hz	1.9 Hz	2.3 Hz	2.7 Hz	3.1 Hz	3.5 Hz
Damping ratio	0.01	−1.268	−1.173	−0.993	−0.867	−0.789	−0.615
	0.02	−1.345	−1.249	−1.068	−0.923	−0.938	−0.663
	0.03	−1.412	−1.317	−1.134	−0.975	−0.883	−0.708
	0.04	−1.469	−1.377	−1.193	−1.022	−0.924	−0.750
	0.05	−1.519	−1.430	−1.246	−1.065	−0.962	−0.788

Table 3.

Parameter σ for different combinations of damping ratio and bouncing frequency.

σ		Bouncing frequency
σ		1.5 Hz	1.9 Hz	2.3 Hz	2.7 Hz	3.1 Hz	3.5 Hz
Damping ratio	0.01	0.393	0.409	0.339	0.344	0.448	0.471
	0.02	0.387	0.401	0.333	0.347	0.446	0.470
	0.03	0.384	0.395	0.329	0.349	0.444	0.469
	0.04	0.382	0.389	0.326	0.352	0.442	0.467
	0.05	0.381	0.384	0.325	0.355	0.439	0.464

Figure 7.

Dependency of μ on bouncing frequency and damping ratio. (a) Bouncing frequency (b) Damping ratio.

Figure 8.

Dependency of σ on bouncing frequency and damping ratio. (a) Bouncing frequency (b) Damping ratio.

From the data shown by Figure 7 and Table 2, it is observed that the parameter μ is linearly related to bouncing frequency and damping ratio, which is in accordance with basic theory of structural dynamics. On one hand, when the bouncing frequency becomes larger, the structural response tends to be higher, because larger bouncing frequency usually results in larger amplitude of the load cycle. On the other hand, as the damping ratio increases, the structural response becomes smaller. However, the impact from damping ratio is not as high as that from bouncing frequency, because maximum acceleration under impulse excitation is usually less sensitive to the damping ratio. Based on these observations, the relationship between μ and bouncing frequency f_b and damping ratio ξ is fitted through a least-square manner as equation (5)

μ = 0.35 f_{b} - 5.44 ξ - 1.80

(5)

For the scale parameter σ, the dependency on bouncing frequency and damping ratio is not apparent, as shown by Figure 8(a) and (b). With the change of bouncing frequency and damping ratio, the values of σ do not change much but vary around 0.4, with a highest value of 0.47 and a lowest value of 0.33, indicating that the randomness of the load cycles is similar for different bouncing frequencies and damping ratios. Therefore, in this study, a fixed value of 0.4 is adopted for the scale parameter σ, as expressed by equation (6).

σ = 0.4

(6)

To this point, the distribution parameter μ and σ for a₁₀ can be decided from equations (5) and (6) given that the bouncing frequency and damping ratio are pre-determined. Under a logarithmic Gaussian distribution, the value of a₁₀ for any predetermined confidence level P can be calculated through its cumulative distribution function (CDF) that involves solving the integral equation given by equation (7)

P = \frac{1}{σ \sqrt{2 π}} \int_{0}^{a_{10}} \frac{1}{t} e^{- \frac{{(\ln t - μ)}^{2}}{2 σ^{2}}} d t

(7)

Value of λ

As shown by equation (3), there are in total two parameters, a₁₀ and λ, that are necessary to decide the shape of the design spectrum. The parameter a₁₀ has been determined by equations (5)–(7) from the bouncing frequency, damping ratio, and confidence level. In this subsection, the dependency of λ against other parameters is analyzed.

For each combination of bouncing frequency, damping ratio, and confidence level, the shape of the spectrum is calculated (similar to the one shown by Figure 5), and the value of λ is optimized in a least-square manner. The relationship between λ and other parameters are listed in Tables 4–6 and are plotted in Figure 9. Note that the listed value represents the average value of λ when the corresponding parameter is fixed.

Table 4.

Average value of λ for different bouncing frequencies.

Bouncing frequency	1.5 Hz	1.9 Hz	2.3 Hz	2.7 Hz	3.1 Hz	3.5 Hz
λ	1.06	1.20	1.27	1.32	1.60	1.73

Table 5.

Average value of λ for different damping ratios.

Damping ratio	0.01	0.02	0.03	0.04	0.05
λ	1.32	1.36	1.37	1.38	1.38

Table 6.

Average value of λ for different confidence levels.

Confidence level	75%	80%	85%	90%	95%
λ	1.38	1.37	1.36	1.35	1.33

Figure 9.

Dependency of λ on various factors. (a) Bouncing frequency, (b) Damping ratio, (c) Confidence level.

Figure 9 shows that the value of λ linearly increases with the bouncing frequency, while being less sensitive to damping ratio and confidence level. When the damping ratio increases from 0.01 to 0.05, λ exhibits an increase of less than 5%. Moreover, the increase of confidence level from 0.75 to 0.95 results in a decrease of λ of only 3.6%. Therefore, the parameter λ is assumed to be only related with bouncing frequency and the relationship between λ and bouncing frequency is assumed to be linear. The fit of this relationship is expressed in equation (8)

λ = 0.33 f_{b} + 0.53

(8)

Application procedure for response prediction

Based on the results of the previous section, the steps to predict structural acceleration responses under bouncing load cycles for high frequency structures can be summarized as follows.

(1) Decide basic parameters for the vibration prediction, including the bouncing frequency f_b, the static weight of the bouncing occupant, the structural frequency f_n, the damping ratio ξ, and modal mass M, the desired confidence level P, and the vibration mode values at the position of the bouncing occupant φ_input and at the prediction point φ_output.

(2) Calculate a₁₀ and λ from equations (5)–(8) using f_b, ξ, and P.

(3) Calculate the design spectrum value from equation (3) according to f_n.

(4) The predicted peak acceleration is calculated by equation (9)

a_{p e a k} = \frac{G φ_{i n p u t} φ_{o u t p u t}}{M} a (f_{n}, f_{b}, ξ, P)

(9)

Experimental validation of the spectrum model on a timber floor

General description

The response spectrum method proposed in this study was tested on a 6-meter wide timber floor with a span of 3 m that was constructed in the structural laboratory of Tongji University. Nine pieces of oriented strand boards (OSBs) were connected on the top of joists made from spruce-pine-fir (SPF) materials to form up the main supporting system of this floor. The section of the floor has a height of 199 mm. The position of the SPF joists and OSB are depicted in Figure 10(a) and the structure after construction is shown in Figure 10(b).

Figure 10.

Overview of the timber floor. (a) Geometrical size of the floor and position of SPF joists and OSB, (b) During construction, (c) After construction.

The dynamic properties, including fundamental frequency and damping ratio, of this timber floor were obtained through an ambient vibration modal test and a natural excitation technique-spare time domain (NExT-STD) algorithm. Results are listed by Table 7. The shape of the dominant vibration mode is shown in Figure 11.

Table 7.

Dynamic properties of the timber floor.

Parameter	Modal mass (kg)	Frequency (Hz)	Damping ratio (%)
Value	335	18.23	4.94

Figure 11.

Dominant mode shape of the timber floor.

Assessment of the response spectrum method

A series of crowd bouncing tests were conducted on the timber structure. The participants of the test were student volunteers from Tongji university. Selection of participants and the test protocol were determined according to the requirements of Tongji Medical Ethics Committee. Because this study investigates the response spectrum method for the prediction single-person bouncing-induced vibration, the measurement data of only one 62.7 kg male participant bouncing was used herein. The participant was asked to bounce on the mid-point of the structure with frequencies guided by a metronome, including 1.5, 2.0, 2.5 Hz, which are common excitation frequencies in real case, especially under the guidance of music (Chen et al., 2019). For each bouncing frequency, two tests that lasted for 40–60 s were conducted. The participant bouncing on the structure is shown by Figure 12(a). 15 accelerometers with 200 Hz sampling frequency were attached on the surface on the structure, whose layout is given by Figure 12(b).

Figure 12.

Test participant and layout of accelerometers. (a) Test participant (b) Layout of accelerometers.

For each bouncing frequency, a representative measured time history at mid-point, i.e., the accelerometer indexed as H in Figure 12(b), is plotted in Figure 13(a), (c) and (e). The peak values of each bouncing cycles of these time histories were extracted and plotted in Figure 13(b), (d) and (f), together with the predicted values using the response spectrum method proposed in this paper. Results show that for 95% confidence level, only a few cycles have peak values that exceeds the predicted ones. When the confidence level is released to 75%, more cycles can exceed the predicted value while the majority of the data are still smaller than the prediction, thus proving the feasibility and practicality of the proposed method.

Figure 13.

Comparison of predicted and measured structural acceleration. (a) Measured time history of 1.5 Hz, (b) Prediction of 1.5 Hz, (c) Measured time history of 2.0 Hz, (d) Prediction of 2.0 Hz, (e) Measured time history of 2.5 Hz, (f) Prediction of 2.5 Hz.

In this section, the proposed response spectrum model was tested on a joist-type of timber floor. However, this method is not limited by the structural type of the floor. For different structural types, e.g., a cross-laminated timber (CLT) floor or a timber-concrete composite (TCC) floor, the method remains suitable if the structural dynamic properties are available.

Summary and conclusions

In this paper, a design-oriented response spectrum model to predict human bouncing induced response is proposed for light-weight floors, whose fundamental frequency is usually higher than 10 Hz. Different from those previously proposed response spectrum models, the method proposed in this paper predicts the structural responses based on each load cycle, instead of the entire time history. The response spectrum value for different combination of structural frequency, damping ratio, and excitation frequency is calculated following the definition of the response spectrum. Parameters have been optimized to form a mathematical design spectrum that can be used to predict peak values of bouncing induced excitation. The propose response spectrum model is employed to predict the structural responses of a laboratory constructed timber floor.

The following conclusions are drawn from this study.

(1) The structural responses of timber floors need to be evaluated on the basis of each load cycle instead of the entire time history.

(2) The shape of the design spectrum can be approximately represented by an exponential function, with its parameter related to properties of both the structure and the excitation.

(3) Providing basic parameters of the structure and the excitation, the peak acceleration values can be predicted using the proposed model at any desired confidence level.

(4) The proposed spectrum is tested to be effective and applicable to any type of high-frequency floor as long as its dynamic properties are available.

This paper focuses on the structural vibration under single-person excitation. For the case of crowd excitation, the vibration serviceability presents to be a more complex problem (Wang et al., 2021). Due to the light weight nature of the timber floor, the structural dynamic properties might be largely affected by the crowd, especially the modal mass and damping ratio. Also, the synchronization effect of the crowd may have an influence on the floor vibration prediction, which is to be further investigated in future study.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant number: 52008306, 52178151) and Shanghai Sailing Program (Grant number: 20YF1451300).

ORCID iDs

Haoqi Wang

Jun Chen

References

British Standard (2008) Guide to evaluation of human exposure to vibration in buildings (BS 6472-1).

Chen

Ren

Racic

(2021) Prediction of floor responses to crowd bouncing loads by responses reduction factor and spectrum method. Advances in Structural Engineering 24(7): 1427–1438.

Chen

Wang

Racic

, et al. (2016) Acceleration response spectrum for prediction of structural vibration due to individual bouncing. Mechanical Systems and Signal Processing 76(77): 394–408.

Chen

Tan

Van Nimmen

, et al. (2019) Data-driven synchronization analysis of a bouncing crowd. Shock and Vibration 2019: 8528763.

Dolan

Murray

Johnson

, et al. (1999) Preventing annoying wood floor vibrations. Journal of Structural Engineering 125(1): 19–24.

Hamm

Richter

Winter

. (2010) Floor vibrations–new results. Proceedings of 11th World Conference on Timber Engineering (WCTE2010), Riva del Garda, 20-24 June 2010.

Hassan

OAB

Girhammar

(2013) Assessment of footfall-induced vibrations in timber and lightweight composite floors. International Journal of Structural Stability and Dynamics 13(2): 1–26.

(2000) Serviceability Design Criteria for Commercial and Multi-Family Floors. Votoria: CanadianCanadian Forestry Service.

Huang

Lin

Zhang

, et al. (2021a) Performance of the hollow-core cross-laminated timber (HC-CLT) floor under human-induced vibration. Structures 32: 1481–1491.

10.

Huang

Wang

Chang

(2021b) Reducing human-induced vibration of cross- laminated timber floor—application of multi-tuned mass damper system. Structural Control and Health Monitoring 28(2): e2656.

11.

Hurkmans

HLP

Bussmann

JBJ

Selles

, et al. (2006) Validity of the Pedar Mobile system for vertical force measurement during a seven-hour period. Journal of Biomechancis 39: 110–118.

12.

International Organization for Standardization (2007) Bases for Design of Structures- Serviceability of Buildings and Walkways against Vibrations. Switzerland: ISO.

13.

Johnson

(1994) Vibration Acceptability in Wood Floor Systems. Blacksburg: Virginia Polytechnic Institute and State University.

14.

Kullaa

Talja

(1998) Vibration tests for lightweight steel joist floors–subjective perceptions of vibrations and comparisons with design criteria. Finnish R&D Conference on Steel Structure, Lappeenranta: Finland

15.

Zhou

, et al. (2019) Modern timber construction technology and engineering applications in China. Proceedings - Institution of Civil Engineers: Civil Engineering 172(5): 17–27.

16.

Lou

Huang

, et al. (2020) Technical Standard for Human Comfort of the Floor Vibration. JGJ/T 411-2019 Ministry of Housing and Urban-Rural Development of China. Beijing: China Architecture Publishing and Media Co., Ltd.

17.

Murray

Allen

Ungar

(1997) Steel Design Guide Series 11: Floor Vibrations Due to Human Activity. Chicago: American Institute of Steel Construction.

18.

Ohlsson

(1982) Floor Vibrations and Human Discomfort. Gothenburg: Chalmers University of Technology, Division of Steel and Timber Structures.

19.

Ohlsson

(1988) Springiness and Human-Induced Floor Vibrations: A Design Guide. Stockholm: Swedish Council for Building Research.

20.

Onysko

(1986) Serviceability criteria for residential floors based on a field study of consumer response. 1986.

21.

Smith

Chui

(1988) Design of lightweight wooden floors to avoid human discomfort. Canadian Journal of Civil Engineering 15(2): 254–262.

22.

Solari

(1989) Wind response spectrum. Journal of Engineering Mechanics 115(9): 2057–2073.

23.

Toratti

Talja

(2006) Classification of human induced floor vibrations. Building Acoustics 13(3): 211–221.

24.

United States Atomic Energy Commission (1973). Design response spectra for seismic design of nuclear power plants. Regulatory guide no. 1.60.

25.

Wang

Chang

Yan

, et al. (2021) Predicting the human-induced vibration of cross laminated timber floor under multi-person loadings. Structures 29: 65–78.

26.

Wang

Chen

Nagayama

(2019) Parameter identification of spring-mass-damper model for bouncing people. Journal of Sound and Vibration 456: 13–29.

27.

Wang

Nagayama

(2022) Response spectrum model of vehicle dynamic load for the prediction of bridge vibration level due to single vehicle-passage. Engineering Structures 260: 114180.

28.

Wang

Zhang

Chen

(2022) Prediction of structural responses induced by single-person jumping through a physical principle based on transfer functions. Advances in Structural Engineering 25(1): 146–157.

29.

Wang

Chen

Yokoyama

, et al. (2020) Spectral model for crowd walking load. Journal of Structural Engineering 146(3): 04019220.

30.

Weckendorf

Toratti

Smith

, et al. (2016) Vibration serviceability performance of timber floors. European Journal of Wood and Wood Products 74(3): 353–367.

31.

Zeng

Wang

Chen

(2022) Dynamic reliability analysis of large-span structures under crowd bouncing excitation. Buildings 12(232): 12030332.

32.

Zhang

Rasmussen

Jorissen

, et al. (2013) Comparison of vibrational comfort assessment criteria for design of timber floors among the European countries. Engineering Structures 52: 592–607.

33.

Zhang

(2020) Human-induced vibration of cold-formed steel floor systems: parametric studies. Advances in Structural Engineering 23(10): 2030–2043.

A design-oriented method for response prediction of light-weight timber floors under bouncing excitation

Abstract

Keywords

Introduction

Characteristics of high-frequency structural responses

Database of bouncing load cycles

Development of the response spectrum model

Equation of motion of a generalized single-degree-of-freedom system

Mathematical model of the design spectrum

Value of a10

Value of λ

Application procedure for response prediction

Experimental validation of the spectrum model on a timber floor

General description

Assessment of the response spectrum method

Summary and conclusions

Footnotes

Declaration of conflicting interests

Funding

ORCID iDs

References

Value of a₁₀