Prediction of floor responses to crowd bouncing loads by response reduction factor and spectrum method

Abstract

Bouncing is a typical rhythmic crowd activity in entertaining venues, such as concert halls and stadia. When the activity’s frequency is close to the natural frequency of the occupied structure, the corresponding bouncing loads can cause intense structural vibrations resulting in vibration serviceability problems, even structural damage. This study suggests a method for prediction of vibration response due to crowd bouncing by a response reduction factor (RRF) in conjunction with a previously established response spectrum approach pertinent to a single person bouncing. The RRF is defined as a ratio between structural responses with and without taking into account synchronization of body movements of individuals in a bouncing crowd. The variations of RRF with number of persons, structural frequency, bouncing frequency and structural damping ratios have been studied using experimental records of crowd bouncing loads. Based on the findings a practical design curve for RRF has been proposed. Application of the proposed method has been validated on numerical simulations and field measurements of a long-span floor subjected to crowd bouncing loads.

Keywords

crowd synchronization human-induced loads long-span floor vibration serviceability assessment

Introduction

Bouncing means repeated up-down movements of a human body in the vertical direction in the fashion similar to jumping but the feet remain on the ground all the time. Bouncing is a typical rhythmic crowd activity of spectators celebrating or cheering during concert events or sport games. While music beat is a logical audio stimulus at concerts, spectators at sport matches are also often strongly stimulated to bounce to background music or chants. The corresponding dynamic forces exerted on the occupied structure, such as long-span floors and cantilever grandstands, may cause significant vibrations. If the force’s frequency is equal or close to the structure’s natural frequency, resonant or near-resonant large-amplitude structure vibrations will occur. When the vibration level is over threshold to human perception, it may lead to vibration serviceability problems that make people feel uncomfortable, panic and even cause casualties (Bachmann, 1987; Glackin, 2000; Parker, 2003; Reynolds and Pavic, 2006; Rogers, 2000). Although jumping can generate larger force amplitudes, in design of entertaining venues bouncing is more often taken as the relevant loading case scenario due to its lower energy consumption (Jones et al., 2011). Namely, moderate bouncing can last significantly longer than intense jumping, causing potentially more chronic vibration serviceability issues. Popular design guidelines have recognized this aspect in design of entertaining venues (Dansk Standard DS 410, 2008; IStructE/DCLG/DCMS Joint Working Group, 2008).

A reliable and practical load model is a prerequisite for vibration serviceability assessment. There have been numerous experimental and theoretical studies of bouncing loads. The vast majority of these studies focus on modelling of individual loading. Historically, individual bouncing time histories have been measured directly by a force plate. Normalized by the body weight, the individual records were used typically to fit Fourier series, whose amplitude coefficients are widely known as dynamic load factors (DLFs) (Duarte and Ji, 2009; Racic and Chen, 2015; Wang et al., 2016; Yao et al., 2004). When a crowd bounces to a music beat, perfect synchronization among each individual is unlikely to happen. This is because a group of people is hardly able to keep the same bouncing rhythm even over a short time (Racic et al., 2010). Less perfect synchronization means less intensive crowd DLF, thus lower vibration response.

Experimentally measured loading due to groups and crowds bouncing are a key prerequisite for developing fundamental knowledge of their effect on structural response. However, simultaneously and directly measured individual bouncing forces in groups of various sizes are very rare and limited. This is because each individual in a group needs a costly measuring instrument, such as commercial force plates. Parkhouse and Ewins (2006) addressed this problem by aligning the starting time instants of many individual bouncing load records, assuming that they were bouncing together in a group. Then DLFs for group size of 5, 10, 20, 50, 100, and 200 persons were derived and tabulated for design purpose. Comer et al. (2013) used bespoke force plates to obtain the force signals of 15 participants bouncing to a selection of popular songs with different dominant beats. Higher level of synchronization, yet only moderate in its absolute value, was observed for songs with main beats in the range of 2–3 Hz.

Group and crowd forces can also be measured indirectly, by monitoring movements/trajectories of each individual in a crowd via various non-contact technologies, such as motion capture (Chen et al., 2019), video analysis (Celik et al., 2018), 3D inertial motion-tracking techniques (Van Nimmen et al., 2016) and smart-phones (Chen et al., 2016). The bouncing force of each person can then be estimated through a bio-mechanical model. However, such models still need further research before they can start being used reliably in the civil engineering context.

Moreover, indirect measurements cannot reproduce correctly the force amplitudes. Although the indirect records can be used to study timing of group synchronization (Chen et al., 2019), there are still problems with marker occlusions for larger groups, limited number of wireless sensors in the network and synchronization between smartphones.

At the design stage, a structure’s vibration serviceability is generally assessed by two means: (1) limiting a lower threshold value for the fundamental frequency, for example, ≥3 Hz (BS 6399-1:1996, 1996), or (2) limiting an upper amplitude of its acceleration response, for example, ≤5 cm/s² for a residual building (ISO 10137:2007, 2007). Practical application shows that the latter is better since it involves the dynamic properties of both the load and the structure. In design practice vibration serviceability assessment typically takes several steps: (1) create an analysis model, for example, a finite element model of the structure; (2) apply design load to the model; (3) calculate the vibration response; (4) compare results against the acceptance criteria, and (5) if not satisfied, change structural design parameters and repeat all the steps again. This procedure is generally time-consuming especially when structural design has several alternatives and frequent modifications. To tackle this issue, the authors have proposed a design-oriented acceleration response spectrum approach for predicting vibration response of a structure due to individual bouncing (Chen et al., 2016). This approach, similar to the response spectrum method widely used in earthquake engineering, is convenient for obtaining the maximum acceleration response of a structure subjected to individual bouncing. However, multiple people rather than individuals bouncing is a more relevant loading case scenario pertinent to design of entertain venues. Therefore, it is necessary to extend the existing spectrum approach to crowd bouncing loading case, which is the key aim of the present study.

Section 2 describes crowd bouncing load records. Section 3 first defines so called response reduction factor (RRF), then investigates the variation of RRF with crowd size, floor frequency and structural damping experimental records. Vibration acceleration response due to crowd bouncing can then be predicted by RRF in conjunction with the response spectrum method already derived for individual occupants. The full procedure is elaborated in Section 4 and verified in Section 5 based on numerical simulations and vibrations responses of a real floor structure. Finally, key findings of this study and limitations of the proposed method are summarized and elaborated in Section 6.

Experimental records of crowd bouncing loads

The dataset used in this study was collected by a grandstand simulator shown in Figure 1 (Comer et al., 2013). The simulator has four rows and five seats per row. Apart from the seats in the front row, all remaining seats have a force plate at the feet level. The distance between seats is 0.5 m (left and right direction) and 1.0 m (front and back direction). The simulator was so designed that its natural frequency was far away from the normal bouncing frequency range. Fifteen persons participated in the experiment. They were arranged in a configuration as 3 (rows) × 5 (columns) configuration, as numbered in Figure 1. Each person stood on a force plate, which recorded his/her bouncing load time history. All force plates were connected to one data acquisition system to assure that the corresponding bouncing loads were measured simultaneously.

Figure 1.

Test rig for crowd bouncing.

The experiment collected bouncing records of 15 participants bouncing together at four frequencies: 1.5, 2.0, 2.5, and 3.0 Hz, guided by a metronome. For each frequency, the force measurements lasted 40 s with sampling rate 1000 Hz. Figure 2(a) to (d) shows the recorded 15 bouncing force time histories, in thin and gray lines, for each of the four test frequencies. Note that all the forces were normalized by its corresponding test subject’s body weight. For each test case, the mean force time history of all the 15 person’s records was calculated and normalized by the average weight of the 15 persons, which was 720.4 N. The results are shown in thick and solid line in upper subplot of Figure 2(a) to (d), together with its Fourier amplitude spectrum in the lower subplot. It is apparent that the amplitudes of the average force is lower than any individual’s bouncing load, especially for higher bouncing frequency cases of 2.5 Hz and 3.0 Hz. This observation once again highlights the difference between individual and crowd bouncing loads, and manifests the significance of experimental investigation on crowd synchronization effects.

Figure 2.

Crowd bouncing load records (in each subplot, the thin gray lines are original load records, the thick read line is the average, whose Fourier spectrum shows in the lower plot): (a) 1.5 Hz bounce, (b) 2.0 Hz bounce, (c) 2.5 Hz bounce, and (d) 3.0 Hz bounce.

Load derivation of response reduction factor for crowd bouncing

Definition of response reduction factor

As already mentioned in Introduction, the current knowledge on load models and procedures for structural response predications due to individual bouncing loads are satisfactory for everyday design practice. A similar simplified method, which is suitable for manual yet reliable calculations and simple computerization of vibration response due to crowd loading is urgently needed. To address this issue, the present study introduces response reduction factor (RRF). It is designed to scale the existing vibration response spectra pertinent to individual bouncing loading to obtain vibration response spectra pertinent to bouncing crowds. The RRF is defined as a ratio between vibration responses with and without considering the synchronization effect within a bouncing crowd. For a bouncing crowd of N persons, RRF can be mathematically defined by equation (1).

β = \frac{A_{RMS}^{10 s}}{\sum_{i = 1}^{N} a_{RMS, i}^{10 s}}

(1)

Here, $a_{RMS, i}^{10 s}$ is the peak value of 10-second running root mean square (10s-RMS) due to the ith person’s bouncing load P_i (t) only (Figure 3(a)); $A_{RMS}^{10 s}$ is the 10s-RMS value due to the resultant of the bouncing forces of all crowd members (Figure 3(b)). It is clear that the closer $β$ to unity the better is the synchronization of the bouncing crowd. Similarly, other response quantities such as peak value, RMS of the whole response time history, and 1s-RMS, etc. can be directly introduced into equation (1) to define the corresponding response reduction factor $β$ .

Figure 3.

Load input simulation of single degree of freedom structure: (a) only the ith person’s bouncing load, (b) the bouncing loads of all crowd members.

Determination of response reduction factor

Value of RRF depends on structure’s frequency and damping ratio, the crowd size and bouncing frequency. For a certain bouncing frequency among the four tested values, that is, 1.5, 2.0, 2.5, and 3.0 Hz, the following steps were taken to derive the RRF.

Vary the crowd size N from 2 to 15, and for each size randomly selected N records from the experiments. For instance, N = 2, there will be in total $C_{15}^{2} =$ 105 combinations can be formed using the experimental records.

Apply each load combinations to a SDOF system with unit mass to calculate $β$ as defined in equation (1). During the calculation, vary the SDOF’s natural frequency f_s from 0.05 Hz to 15 Hz with a step of 0.05 Hz, and vary its damping ratio $ζ$ from 0.01 to 0.05 with a step of 0.01.

For each pair of f_s and $ζ$ , one will have in total $C_{15}^{N}$ (because there are $C_{15}^{N}$ load combinations) $β$ values. Their average is then taken as the representative $β$ for the corresponding (f_s, $N$ ) and damping ratio $ζ$ .

For each damping ratio, the final result of $β$ is presented in 3D plot as $β$ (f_s, $N$ ). Each plot consists of 300(f_s) × 15(N) = 4500 points, and each point requires $C_{15}^{N}$ times of calculation. As an example, for damping ratio $ζ = 0.02$ and bouncing frequency 1.5 Hz, such a 3D plot of $β$ is given in Figure 4(a).

Repeat the above steps for another bouncing frequency to obtain its corresponding $β$ . The results for bouncing frequencies 2.0, 2.5, and 3.0 Hz are shown in Figure 4(b) to (d), all for the same damping ratio 0.02.

Figure 4.

Relationship between response reduction factor and crowd size and structure’s natural frequency (damping ratio 0.02): (a) 1.5 Hz bounce, (b) 2.0 Hz bounce, (c) 2.5 Hz bounce, and (d) 3.0 Hz bounce.

Design curve of response reduction factor

Twenty 3D plots of $β$ (four test bouncing frequencies times five damping ratios) have been obtained so far. From the point of view of a structural designer, the fitting curve of $β$ should be as simple as possible. Note from Figure 4 that with the increase of crowd size N, the RRF decreases and tends to be stable. Therefore, we can conservatively take N = 15 to represent a normal bouncing crowd. For N = 15 and damping ratio 2%, Figure 5 compares the RRF curves for four bouncing frequencies 1.5, 2.0, 2.5 and 3.0 Hz. It can be seen that the four curves have similar variation trends, that is, $β$ value decreases with the increase of structure’s frequency and it has peaks at the harmonics of bouncing frequency. The mean curve of all the four curves are then calculated and taken as the representative RRF curve for this damping ratio, which is shown in Figure 5 as a thick solid line.

Figure 5.

Response reduction factor curve of 15 people bouncing (damping ratio 0.02).

The same procedure has been applied to other damping ratios. All obtained five representative RRF curves are plotted in Figure 6. Interestingly, none of these curves is very sensitive to structural damping ratios. A possible explanation could be that RRF itself is a ratio of responses. All curves show similar trend that decreases with the increase of structural frequency the RRF values. Moreover, the higher the damping ratio the smoother the curve. Based on this observation, an average of all the five RRF curves can be taken as the design curve.

Figure 6.

Comparison of representative response reduction factor curve of different damping ratio.

The previously suggested response spectrum approach can give not only 10-s RMS value but also other popular design parameters as peak value and RMS value of the structure response. To stay consistent, the similar derivation procedure has been used to determine the design RRF curves for peak value and RMS value, the results are depicted in Figure 7 together with that of 10s-RMS.

Figure 7.

Representative value of response reduction factor curve for three design parameters.

Again, no significant difference can be observed between RRF curves of the three design parameters. Thus, a final design RRF curve, which is applicable to all three design parameters, can be obtained by fitting the mean of the three curves in Figure 7. The fitting result is shown in Figure 8 and the mathematical expression is given by equation (2).

β = 0.325 + 0.602 e^{- 0.381 f}

(2)

where f is the structural frequency having unit in Hz. After the above simplification process, the final design RRF given by equation (2) depends on structural frequency only, and it is applicable for crowd size N ≥ 15, damping ratio 1% to 5% and design parameters as 10s-RMS, peak value and RMS value.

Figure 8.

Design curve of response reduction factor.

Procedure for using reduction factor in vibration response prediction

The next section presents key aspects of the already established spectral approach to calculate vibration response due to a single individual bouncing. This knowledge is needed to follow its extension to the vibration response calculation due to multiple people bouncing elaborated in Section 4.2.

Response spectrum method for individual bouncing load

The acceleration response spectrum due to an individual bouncing is shown in Figure 9, where the x-axis is the structural frequency and the y-axis is 10s-RMS acceleration response.

Figure 9.

The response spectrum for individual bouncing load.

The mathematical expression of the response spectrum is given by equation (3).

a (f, ζ, P) = {\begin{matrix} 0 f \in (0, 0.5 Hz] \\ \frac{f - 0.5}{1.5} a_{rms}^{1 st} (ζ, P) f \in (0.5, 2.0 Hz] \\ a_{rms}^{1 st} (ζ, P) f \in (2.0, 3.5 Hz] \\ \frac{f - 3.5}{0.5} (a_{rms}^{2 nd} (ζ, P) - a_{rms}^{1 st} (ζ, P)) \\ + a_{rms}^{1 st} (ζ, P) f \in (3.5, 4.0 Hz] \\ a_{rms}^{2 nd} (ζ, P) f \in (4.0, 7.0 Hz] \\ {(\frac{7}{f})}^{2.96} a_{rms}^{2 nd} (ζ, P) f \in (7.0, 15.0 Hz] \end{matrix}

(3)

where $a (f, ζ, P)$ is nominal 10s-RMS acceleration (dimensionless), f is the structure’s frequency, $ζ$ is the structural damping ratio and P is a user-selected percentile indicating expected probability of occurrence (e.g. P = 75% or P = 95%). The above equation gives 10s-RMS (nominal) acceleration response, which can be further converted into other design parameters such as peak, RMS or 1s-RMS acceleration response. Conversion equations between different design parameters are elaborated in Chen et al. (2016).

Prediction of structural response due to crowd bouncing

Taking a floor structure subjected to a bouncing crowd as an example (Figure 10), the acceleration response can be calculated using equations (4) or (5) for two starting conditions typical in design practice (1) the size of the crowd N and location of each person are known; (2) the density of the crowd (i.e. $ρ$ person per unit area) is known and is distributed uniformly on the floor:

a_{k} = β (f_{k}) ϕ_{k}^{s} \frac{\sum_{i = 1}^{N} ϕ_{k}^{i} G_{i}}{M_{k}} a (f_{k}, ζ_{k}, P)

(4)

a_{k} = β (f_{k}) ϕ_{k}^{s} \frac{\int_{A} ρ ϕ_{k} (x, y) ds \cdot \bar{G}}{M_{k}} a (f_{k}, ζ_{k}, P)

(5)

where $a (f_{k}, ζ_{k}$ , P) is the response spectrum value, which can be determined by equation (3) with inputs as the structure’s kth natural frequency f_k, damping ratio $ζ_{k}$ and an expected probability of occurrence P; M_k is the modal mass of the kth vibration mode, $ϕ_{k}^{s}$ is normal coordinate of the k-th mode shape at the response predication point, $β (f_{k})$ is the response reduction factor for f_k which can be evaluated by equation (2). $G_{i}$ and $ϕ_{k}^{i}$ are the ith person’s weight and the modal coordinate of the location where he/she stands. For uniformly distributed bouncing crowd (equation (5)), $\int_{A} ϕ_{k} (x, y)$ is the integration of modal coordinate over the crowd’s standing area A, $ρ$ is the crowd density, $\bar{G}$ is the average weight of the crowd. The k-th vibration mode shape, either determined from numerical calculation or field measurement, should be normalized by its maximum value before introducing into the two equations. All parameters are dimensionless expect for $G_{i}$ ( $\bar{G}$ ), M_k, and $ρ$ , whose units are, respectively, Newton, kg, and person/m². Therefore, the resulting acceleration a_k is in m/s². The maximum value among all considered vibration modes is taken as the prediction result. For instance, if accelerations of the first five vibration modes are predicted, the final result is the maximum of a₁,…, a₅, which are calculated by equation (4) or (5).

Figure 10.

Crowd bouncing on a floor.

Verification of the proposed approach

The vibration prediction approach presented in Section 4 is verified in this section using numerical simulations (Section 5.1) and field measurements (Section 5.2) of several floor structures.

Numerical simulation

The structure adopted in this section is a 12×12 m virtual concrete floor simply-supported at four edges (Figure 11). The first six natural frequencies of this structure are calculated as 2.50, 6.24, 6.24, 9.96, 12.48, and 12.48 Hz. The corresponding modal mass for each mode is listed in Table 1. Fifteen bouncing force records measured at 2.5 Hz as described in Section 2 were used to simulate the resonant vibration response of the floor. The forces were arranged on the structure as shown in Figure 11.

Figure 11.

Numerical example of a simply-supported floor: (a) numerical model of the floor and (b) dimensions.

Table 1.

Calculate steps of acceleration response of crowd bouncing (P = 0.75, $ζ = 0.05$ , $\bar{G}$ = 720.4N).

Order of mode	1	2	3	4	5	6
Modal Frequency (Hz)	2.50	6.24	6.24	9.96	12.48	12.48
Modal Mass (kg)	10,421.2	9609.8	9609.8	10,411.8	8884.2	5236.5
$a (f_{k}, ζ_{k}$ , P)	5.12	1.52	1.52	0.54	0.28	0.28
Effective modal coordinate	0.96	0	0	0	−0.03	0.81
Coordinate at floor center	1	0	0	0	0	1
Response reduction factor	0.557	0.381	0.381	0.339	0.330	0.330
Modal response (m/s²)	2.84	0	0	0	0	0.15
Maximum response (m/s²)	2.84

The acceleration at the center of the floor due to the applied bouncing forces is numerically calculated and the result is shown in Figure 12. The structural damping ratio used in the calculation was 5%, which is the value recommend in Chinese design code for concrete structure. The calculated peak 10s-RMS value is 2.48 m/s². With the response spectrum and RRF, the acceleration response of the first six mode can be calculated and the results are shown in Table 1. For P = 0.75, the predicted 10s-RMS value by the proposed approach is 2.84 m/s², which is slightly higher than the numerical value 2.48 m/s².

Figure 12.

Floor acceleration responses due to 15 persons bouncing.

Field tests on a long-span floor

A series of crowd jumping tests were conducted on an existing floor consisting 5×3 bays, each having dimension 12×12 m, as shown in Figure 13.

Figure 13.

Experimental site layout.

Hammer tests were performed on the floor to determine its dynamic properties. Nine accelerometers (type: LC0132T, produced by Lance Co.) were utilized and moved together from Line 1 to Line 6 during the test as shown in Figure 14 (right to left). For each line, the hammer hit two locations (Point A and B as shown in Figure 14) in sequence. The hammer forces and all acceleration signals were recorded by the same data acquisition system.

Figure 14.

Sensor arrangement during hammer tests.

The eigensystem realization algorithm (ERA), which is a built-in method in the acquisition system, was adopted to process the records. The floor’s natural frequencies, damping ratios and modal masses (corresponding to mode shape normalized by its maximum modal coordinate), are listed in Table 2. The first four mode shapes were determined by fitting 54 measurement points (6 lines×9 sensors) and are illustrated in Figure 15.

Table 2.

The dynamic properties of the floor.

Vibrationmode	Naturalfrequency [Hz]	Modal mass[×104 kg]	Dampingratio [%]
1	5.35	3.0	1.57
2	8.17	14.32	2.04
3	9.29	27.37	1.14
4	10.54	3.06	1.21

Figure 15.

Measured vibration modes of the tested floor: (a) first mode, (b) second mode, (c) third mode, and (d) fourth mode.

Fifteen volunteers participated in the experiment. They were all healthy adults and students of Tongji University. Selection criteria of the volunteers and the test protocol satisfied the requirements of the Medical Ethics Committee of Tongji University. Before the experiment, the test purpose and procedure were introduced to all participants in detail. Then, each test subject was numbered and his/her body weight was measured. The average body weight of the participants is 696.2 N. Seven crowd size cases were selected in the test, which were N = 1, 2, 4, 8, 9, 12, and 15 persons. For each case, 12 bouncing frequencies 1.5, 1.8, 1.9, 2.0, 2.3, 2.5, 2.7, 2.8, 2.9, 3.0, 3.1, and 3.5 Hz were tested. The bouncing frequency was guided by a metronome and played via loud speakers. Each bouncing test lasted for more than 30 s and was repeated twice. Nine accelerometers were utilized to collect acceleration responses at different locations of the floor. The measured responses at the floor center for three cases, that is, 1.5, 2.7, and 3.5 Hz are shown in Figure 16. The thick solid line is the running 10s RMS curve on which the maximum value is marked out by a circle.

Figure 16.

Acceleration response measured at the center of the floor: (a) 1.5 Hz, (b) 2.7 Hz, and (c) 3.5 Hz.

For the case 15 persons bouncing at 2.7 Hz in a three (row) by five (column) configuration, the predicted vibration responses are shown in Table 3. Two level of probability of occurrence P = 75% and P = 95% are considered in the calculation.

Table 3.

Predicated acceleration responses by RRF and response spectrum.

Rank of mode	1	2	3	4
Modal frequency	5.35	8.17	9.29	10.54
Modal mass (10⁴ kg)	3.00	14.32	27.37	3.06
Modal damping ratio (%)	1.57	2.04	1.14	1.21
Modal coordinate	0.9470	−0.5652	0.4742	−0.0054
Spectrum value for P = 75% (m/s²)	3.0877	1.6649	1.6253	1.0784
Spectrum value for P = 95% (m/s²)	4.3591	2.3195	2.3338	1.5435
RRF (N = 15)	0.4034	0.3518	0.3425	0.3359
Response of each mode for P = 75% (m/s²)	0.3592	0.0117	0.0039	0.0000
The maximum response for P = 75% (m/s²)	0.3592
Response of each mode for P = 95% (m/s²)	0.5071	0.0163	0.0057	−0.0001
The maximum response for P = 95% (m/s²)	0.5071

Figure 17 compares the predicted 10s-RMS results by the proposed approach (using two percentiles P = 75% and P = 95%) with that measured at all test cases. As expected, the largest measured vibration amplitude occurs at bouncing frequency 2.7 Hz which is almost half of the floor’s first natural frequency 5.35 Hz. Figure 17 shows that all measured responses are below the predicated value at P = 75%. Since human-induced vibration is by its nature a serviceability issue rather than a safety issue, the occurrence level of P = 75% seems reasonable for serviceability limit state assessment of floor subjected to crowd bouncing.

Figure 17.

Comparison of predicted results and measured ones.

Conclusion

This study proposes a response reduction factor (RRF) that can be used together with the previously established response spectrum approach to predict acceleration response of a structure due to a crowd bouncing. For a SDOF system and a bouncing crowd having N persons, the RRF is defined as the ratio of the system’s maximum acceleration response due to the resultant of all individual’s force with the summation of maximum response of the system due to each person’s force individually. By definition, the RRF links the response of a structure to single person bouncing with that due to crowd bouncing. The ‘relevant value’ of the response can be represented by popular design parameters such as peak value of 10-s running root-mean-square value, root-mean-square value or the peak value. The mathematical expression of RRF is derived using experimental load records of 15 persons bouncing at four frequencies as 1.5, 2.0, 2.5, and 3.0 Hz. A simple deterministic curve rather than a probabilistic model of RRF is suggested in order to avoid complex numerical calculations in everyday structural design. The proposed response prediction approach was verified by numerical simulations and field tests of an existing floor. The suggested method gives satisfied result in both cases.

Although convenient and user-friendly, the proposed approach has the following obvious limitations. The derivation procedure assumes that the RRF value reduces and becomes stable with the increase of crowd size. Further verification of this assumption is needed using force measurements from larger crowd sizes. However, such a dataset has not been published yet to the best of authors’ knowledge. But once such a database is available, the derivation procedure suggested in this study can be used to derive an updated version of the RRF design curve. Moreover, the effect of human-structure-interaction on vibration response has not been taken into account.

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: The authors would like to acknowledge the financial support provided by National Natural Science Foundation of China (51778465) and the State Key Laboratory for Disaster Reduction of Civil Engineering (SLDRCE19-B-22).

ORCID iD

Jun Chen

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