1. Introduction
Currently, the modern assembly structures (e.g., large-span floor slabs, footbridges, grandstands, etc.), gradually exhibit the characteristics of low natural frequencies and low damping ratios, due to the extensive use of lightweight, high-strength building materials [
1,
2,
3,
4,
5]. Accordingly, structural vibration serviceability issues [
6,
7] as well as safety risks [
8,
9] caused by crowd activities have been increasingly emerging, which may further lead to huge economic losses and even heavy casualties in extreme cases. One typical example of this was the London Millennium Bridge in 2000, in which severe lateral vibrations of the bridge were induced by pedestrian use [
10]. The bridge was therefore closed to traffic for three days after the grand opening, and it cost GBP 5 million to solve the vibration problem. Additionally, at a sports concert given in Vigo, Spain in 2018, the wooden stands collapsed due to the audience’s rhythmic activities, with over 300 people injured [
11]. Jumping is usually considered to generate the most significant vertical loads in contrast to other common activities [
12,
13], in particular when the crowd is coordinated by an external beat. This will be the focus of this research.
Load models are necessary, both in the design stage and the as-built stage to evaluate the vibration performance of structures subjected to crowd jumping activities [
14]. Considering that jumping activities are approximate periodic, Fourier series models are mostly used to represent jumping loads [
13,
15,
16,
17,
18,
19], where the dynamic load factor (DLF) is the key parameter to characterize the load magnitude and crowd synchronization. When crowds jump to an external beat, it is unlikely to achieve perfect synchronization because of both inter- and intra-subject variability [
20], and the crowd behave differently to the beat of different dominant frequencies [
17]. Therefore, the DLF is a stochastic variable and related to the crowd size and guidance frequency, where the guidance frequency denotes the dominant frequency of the metronome or music that guides the crowd to jump. Massive investigations on jumping loads have been conducted [
13,
14,
15,
16,
17,
18,
19], and DLF models have been established accordingly. The DLF of these models is identified by the peaks [
16] or energies [
17] from the corresponding Fourier amplitude spectrum. The intra-subject variability of a jumping activity caused the load energy to spread around the jumping frequency and its multiplier [
21], where the jumping frequency denotes the dominant frequency of a person’s jumping load time history. When a crowd jumps at a given guidance frequency, most people can follow the beat, and their jumping frequency is equal to the guidance frequency. As for a few people who cannot follow the beat, their jumping frequency is unequal to the guidance frequency. The DLF obtained by identifying the peaks can underestimate structural responses because it overlooks the energy near the jumping frequency and its multiplier [
22]. The DLF obtained by identifying the energy can overestimate structural responses because it assumes that the energy is all concentrated at the jumping frequency and its multiplier. When using the resonance assumption to calculate structural responses, the energy at the jumping frequency and its multiplier contributes more to structural responses than the energy around them. Moreover, most of the DLF models lack the data support of large crowd jumping tests. For one thing, the DLF model is developed based on the individual jumping load records aligned by the time instant of the beat [
17,
19], in which the influence of human–human interaction is not involved. For another, the DLF is modeled through crowd jumping loads identified by structural responses [
15], in which the jumping load of each person could not be recorded. Moreover, owing to the limitation of test conditions, the model ignores the change of the DLF with the guidance frequency. In addition, only the mean value or a certain quantile value of the DLF is provided, which is not applicable for the structural design of different reliabilities.
The accurate calculation of structural responses is the main purpose of load modeling. Hence, the modeling parameters could be obtained based on the principle of equal structural responses. This idea has been applied to the modeling of crowd walking loads, in which the number of equivalent synchronized pedestrians is obtained using this principle [
23,
24]. Similarly, the DLF can be identified with the equal structural response, hereafter termed as the equivalent DLF. This is the main highlight of this study. It can overcome the shortcomings of the existing DLF modeled by identifying the peaks or energies from the corresponding Fourier amplitude spectrum. Moreover, structural response could be calculated in the form of Fourier series using the equivalent DLF, which is further more convenient for application to structural design and evaluation than other complicated stochastic models of jumping loads [
12,
25,
26,
27,
28,
29]. In addition, to further apply the model to structural reliability design, the equivalent DLF of different non-exceedance probability is established. This is another main highlight of this study. It can overcome the shortcomings of the existing DLF, which only provides its mean value or a certain quantile value. The paper begins with the formula of the equivalent DLF calculation, which adopts the frequency response function (FRF) to weight and integrate the power spectral density (PSD) matrix of crowd jumping loads. This approach makes the root mean square value of structural acceleration responses equal. The formula of simulating PSD matrices of crowd jumping loads is then provided. This is followed by the modeling of the equivalent DLF with four parameters, i.e., structural damping ratio, guidance frequency, crowd size, and non-exceedance probabilities. The Engineering application of the equivalent DLF is followed. Afterwards, discussions on this work are provided.
2. Formula of the Equivalent Dynamic Load Factor
The Fourier series-based model for crowd jumping loads,
F(
t), is usually expressed as [
15]:
where
W is the total body weight of the jumping crowd,
rj and
φj are the DLF and phase angle of the
jth harmonic,
fgu is the guidance frequency, and
nh is the number of load harmonics considered.
The values of
rj for jumping loads are commonly determined by two methods. In the first method, hereafter termed as the peak method, the maximum value of the
jth harmonic region in the Fourier amplitude spectrum is adopted as
rj, as depicted in
Figure 1, and the values are listed in
Table 1 for further comparison. In the above, the
jth harmonic region is ((
j − 0.5)
fgu, (
j + 0.5)
fgu). As for the second method, hereafter termed as the energy method, the total energy of the
jth harmonic region in the Fourier amplitude spectrum is adopted as
rj, as calculated by:
where
Ajh is the amplitude of the
hth sinusoid in the
jth harmonic region and
ns is the number of the sinusoids. The values of
rj calculated by the energy method for the Fourier amplitude spectrum in
Figure 1 are provided in
Table 1. The peak method only considers the energy at the jumping frequency and its multiplier, and the energy in between is ignored, which can underestimate structural responses. The energy method assumes that all the energy of a jumping load is concentrated at the jumping frequency and its multiplier, which overestimates structural responses when the resonance assumption is made.
A novel DLF calculating method that can accurately estimate the root mean square value of structural acceleration responses is therefore proposed. Considering a single degree-of-freedom system with a unit mass subjected to crowd jumping loads, as shown in
Figure 2, the equation of motion of the system is expressed as:
where
ζs,
ωs, and
us(
t) are the damping ratio, circular frequency, and displacement of the system, respectively,
Wl is the body weight of the
lth jumping person, and
xl(
t) is the ground reaction force time history of the
lth jumping person divided by
Wl, which is dimensionless. Therefore,
xl(
t) reflects the change of center when an individual jumps, and it is the most direct variable to reflect the jumping pattern. The FRF
H(
f;
fs,
ζs) related to the input,
, and output,
, is deduced from Equation (3):
where
fs = 2π
ωs is the cyclic frequency of the structure and i is the imaginary unit. Then, the auto-PSD [
21]
G(
f) of
can be obtained by the stochastic vibration theory [
30]:
where
Glk(
f) is the
lth row and
kth column element of the PSD matrix of crowd jumping loads. If
l ≠
k,
Glk(
f) is the cross-PSD of
xl(
t) and
xk(
t). If
l =
k,
Gll(
f) is the auto-PSD of
xl(
t). Integrating
G(
f) within the frequency range yields the root mean square value
arms of
:
If the crowd jumping load, i.e., the right term of Equation (3), is modeled by the Fourier series, as expressed by Equation (1), the auto-PSD
GF(
f) of
is:
where
is the Dirac delta function. Then, as the derivation procedure of Equations (3)–(5),
arms of
can be calculated by:
The resonance scenario is commonly assumed for structural response estimation to consider the severest case [
31], where structural frequency is assumed to be equal to the guidance frequency or its multiplier. Consequently, structural responses induced by the
jth harmonic of jumping load is much larger than the rest of the harmonics:
Making
arms calculated by Equations (6) and (9) equal and assuming a resonance scenario in Equation (6) yields:
If the weight of every jumping person is assumed to be equal, Equation (10) will be further simplified as:
In the above,
H(
jfgu;
jfgu,
ζs) is the maximum of
H(
f;
jfgu,
ζs), so it is indicated that the equivalent DLF for crowd jumping loads can be identified using the FRF to weight and integrate the PSD matrix of crowd jumping loads, hereafter termed as the FRF-weighting method. The equivalent DLF is related to the structural damping ratio because Equation (11) is derived from the principle of equal structural responses. Then, the equivalent DLF of the Fourier amplitude spectrum in
Figure 1 calculated by Equation (11) is provided in
Table 1. It is observed from
Table 1 that the DLF obtained by the peak method and energy method are quite different, especially for high-order harmonics, because the degree of energy diffusion increases as the harmonic order increases [
21]. The maximum-normalized FRF is adopted in the FRF-weighting method to weight the contribution of different sinusoids to structural responses, and the equivalent DLF obtained by this method is between the DLF obtained by the peak method and the DLF obtained by the energy method. Moreover, the equivalent DLF obtained by the FRF-weighting method is strongly related to structural damping ratio. When the structural damping ratio is small, the bandwidth of the FRF is narrow, so the energy around the jumping frequency and its multiplier contributes less to structural responses. Therefore, the equivalent DLF is small and close to the DLF obtained by the peak method. When the structural damping ratio is large, the bandwidth of the FRF is wide and the energy around the jumping frequency and its multiplier contributes more to structural responses. Therefore, the equivalent DLF is large and close to the DLF obtained by the energy method.
4. Modeling of the Equivalent Dynamic Load Factor
The equivalent DLF is related to the structural damping ratio, guidance frequency, crowd size, and non-exceedance probability, so the modeling of the equivalent DLF is a multivariable regression problem. The working condition, where the crowd size is equal to 50, the structural damping ratio is equal to 0.005, and the non-exceedance probability is equal to 0.5, is defined as the standard working condition. First, the equivalent DLF model of the standard working condition is built. Afterwards, the conversion models for other crowd size, structural damping ratio, and non-exceedance probability are established.
4.1. Modeling of the Equivalent DLF for the Standard Working Condition
According to the definition of the standard working condition, the median of 1000 simulation data with a structural damping ratio equal to 0.005 and crowd size equal to 50 are adopted to build the model. The crowd reacts differently to different beats. This is also proven by the probability distribution of the jumping frequency, the probability distribution of the time lag shift, and the values of the coherency function magnitude, all of which are related to the guidance frequency. Therefore, the equivalent DLF of the standard working condition is related to the guidance frequency, as depicted in
Figure 3, from which it is observed that the equivalent DLF increases first and then decreases with the guidance frequency. This is because it is hard for a crowd to keep up with the beat, and the jumping activity is exhausted when the guidance frequency is high or low. Particularly, the equivalent DLF at the low guidance frequency is only about one-third of the maximum value at the moderate guidance frequency. A fourth-order polynomial is used to model the equivalent DLF for the standard working condition, as depicted in
Figure 3, in which Equation (17) is used for the first harmonic and Equation (18) is used for the second harmonic.
4.2. Conversion Model for Structural Damping Ratio
According to Equation (11), the structural damping ratio affects the equivalent DLF by changing the magnitude of the FRF. The bandwidth of the FRF increases as the structural damping ratio increases, so the contribution for the energy between different harmonics to structural responses increases. As a result, the equivalent DLF will increase with the structural damping ratio. Moreover, the influence degree of the structural damping ratio on the equivalent DLF is related to the PSD matrix. In particular, the impact degree increases as the bandwidth of the sum of the PSD matrix’s elements increases, because more energies are distributed between the jumping frequency and its multiplier, and these energies are weighted by the FRF to contribute for structural responses. First, the bandwidth increases as the jumping frequency increases, which is illustrated by the expression of the normalized PSD, i.e., Equation (15). Moreover, this bandwidth increases as the dispersion degree of the jumping frequency increases according to Equation (16). The dispersion degree of the jumping frequency is large at the low guidance frequency [
32], because many people cannot keep up with the beat. Consequently, the influence degree of the structural damping ratio on the equivalent DLF for different guidance frequencies is complicated. To further yield the influence degree, the median of the 1000 simulations with a crowd size equal to 50 is used to plot
Figure 4, where the equivalent DLF is normalized by the equivalent DLF with a structural damping ration equal to 0.005.
It is observed from
Figure 4 that the influence of the structural damping ratio on the equivalent DLF is not greatly affected by the guidance frequency in general. Only at the low guidance frequency, such as 1.5 and 1.6 Hz, the influence is different and is slightly greater than the other guidance frequencies. Therefore, the influence of the guidance frequency is overlooked in the conversion model of Equations (19) and (20) for the structural damping ratio. In addition, the crowd size and non-exceedance probability are also overlooked in the conversion model of Equations (19) and (20), because these two variables are not related to the influence of the structural damping ratio on the equivalent DLF according to Equation (11). Then, by fitting the average curve of different guidance frequencies in
Figure 4, power functions, as depicted in
Figure 5, are adopted to establish the conversion model, as shown in Equations (19) and (20).
where
c1(ζ
s) is the conversion model for the first harmonic and
c2(ζ
s) is the conversion model for the second
harmonic.
4.3. Conversion Model for Crowd Size
Owing to the inter-subject variability, the equivalent DLF is not proportional to the crowd size. In existing models, the crowd DLF divided by the crowd size is defined as the coordination factor to quantify the crowd synchronization [
15], because the coordination factor tends to be a constant as the crowd size increases, which makes the modeling of the crowd DLF easy. Learning from this idea, the coordination factor,
coi, in this section is defined as:
where the subscript
j denotes the harmonic order,
cj is the conversion model for the crowd size, and
rj(
np) is the equivalent DLF with crowd size
np. To further investigate the law of the coordination factor, the median of the 1000 simulations with structural damping ratio equal to 0.005 is used to plot
Figure 6, where the ordinate is obtained by Equation (21).
It is observed from
Figure 6 that the influence of the crowd size on the coordination factor is affected by the guidance frequency when the crowd size is small. Nevertheless, as the crowd size increases, the relationships between the coordination factor and crowd size for different guidance frequencies tend to coincide, and practical structures are commonly subjected to jumping loads with large crowd. Therefore, the influence of the guidance frequency is overlooked in the modeling of the coordination factor. In addition, the structural damping ratio and non-exceedance probability are also overlooked in the coordination factor model, because these two variables are not related to crowd synchronization. Then, by fitting the average curve of different guidance frequencies in
Figure 6, power functions, as depicted in
Figure 7, are adopted to establish the model of coordination factor:
According to Equation (21), the conversion model for the crowd size can be obtained by:
4.4. Conversion Model for Non-Exceedance Probability
Due to the randomness of crowd jumping loads, the equivalent DLF is a random variable. Therefore, the probability distribution of the equivalent DLF is the key to building the conversion model for non-exceedance probability. Some histograms with crowd size equal to 50 and structural damping ratio equal to 0.005 are provided in
Figure 8 and
Figure 9. It is observed that the equivalent DLF for both the first harmonic and second harmonic follows a normal distribution, and similar histograms are found for other crowd sizes and structural damping ratios.
The equivalent DLF
rj of a certain non-exceedance probability
p can be deduced by:
where
F−1( ) is the inverse function of the cumulative distribution function for the standard normal distribution,
is the average of the equivalent DLF, and
cvj is the coefficient of variation of the equivalent DLF. According to Equations (17)–(20) and Equations (24) and (25), the average of the equivalent DLF can be calculated by:
Introducing Equation (27) into Equation 26) and reorganizing the equation yields:
Therefore, the coefficient of variation of the equivalent DLF is the key to building the conversion model for the non-exceedance probability. First, the simulations with crowd size equal to 50 and structural damping ratio equal to 0.005 are taken to investigate the
cvj of different guidance frequencies. To further determine the number of Monte Carlo simulations, the value of
cvj of a different number of Monte Carlo simulations is calculated, as shown in
Table 4 and
Table 5. It can be observed that when the number of Monte Carlo simulations exceeds 600,
cvj tends to be stabilized. Therefore, 1000 is adopted as the number of Monte Carlo simulation in
Section 3.2, and the simulated results are depicted in
Figure 10. It is observed from
Figure 10a that the trend of the coefficient of variation for the first harmonic is inverse to the trend in
Figure 3a, because the randomness of the crowd jumping load is weak when the crowd synchronization is good. A fourth-order polynomial is then used to fit the coefficient of variation for the first harmonic of different guidance frequencies:
As for the second harmonic in
Figure 10b, the difference of the coefficient of variation for different guidance frequency is small, so the average of the coefficient of variation for different guidance frequency is adopted:
Simulations with a structural damping ratio equal to 0.005 are taken to investigate the relationship between the coefficient of variation and crowd size, and the relationships of some guidance frequencies are illustrated in
Figure 11. It is observed from
Figure 11 that the coefficient of variation tends to be a constant as the crowd size increases. Because practical structures are commonly subjected to large crowd jumping loads, the influence of the crowd size on the coefficient of variation is overlooked. Then, the simulations with a crowd size equal to 50 are taken to investigate the relationship between coefficient of variation and structural damping ratio, and the relationships of some guidance frequencies are illustrated in
Figure 12. It is observed from
Figure 12 that the coefficient of variation is only slightly affected by the structural damping ratio, so the influence of the structural damping ratio on the coefficient of variation is ignored.
4.5. Model of the Equivalent DLF
To date, the model of the equivalent DLF for the standard working condition (Equations (17) and (18)) and the conversion model for structural damping ratio (Equations (19) and (20)), crowd size (Equations (24) and (25)), and non-exceedance probability (Equations (28)–(30)) have been established. Consequently, the equivalent DLF can be calculated by:
5. Engineering Application of the Equivalent DLF
The obtained equivalent DLF
rj can be used to estimate structural responses under crowd jumping activities in the design stage or as-built stage. First, structural dynamic properties can be obtained by dynamic tests or finite element models. Normally, structural responses are controlled by the resonant mode, where it is assumed the
sth vibration mode is resonant, i.e.,
fs =
jfgu. The crowd size,
np, can be determined by structural functional requirements, and the non-exceedance probability,
p, can be determined by structural reliability requirements. Then, introducing the values of
j,
fgu,
np, and
p and the damping ratio ζ
s,
rj can be calculated by Equations (31) and (32). The root mean square value of structural acceleration
arms can then be calculated by:
where
Ms is the modal mass of the resonant mode, normalized by the maximum modal coordinate.
6. Discussions
The principal contribution of this paper is to develop an equivalent DLF model for crowd jumping loads. The FRF is applied to weight and integrate the PSD matrix to calculate the equivalent DLF. The structural damping ratio is introduced as an independent variable in the model, so that the root mean square of the structural responses can be accurately calculated, which overcomes the shortcomings of the existing DLF [
13,
15,
16,
17,
18,
19] overestimating or underestimating structural responses. Moreover, the Fourier series model based on the developed equivalent DLF is more convenient and efficient to calculate structural responses than complicated stochastic models [
12,
25,
26,
27,
28,
29]. In addition, by introducing the non-exceedance probability as an independent variable, a normal distribution-based model is built to obtain the equivalent DLF of different reliabilities, which could avoid using massive Monte Carlo simulations for structural reliability assessment and design. However, to better predict structural responses in real scenes, some limitations and extensions can be described as follows:
(1) Practical engineering structures, such as concert halls and grandstands, are commonly occupied by crowds of a large size. A crowd that is in constant contact with the structure, such as a standing crowd or a sitting crowd, can influence structural dynamic properties [
34,
35,
36], causing an increase of the structural damping ratio in particular. Therefore, when Equations (31) and (32) are adopted to obtain the equivalent DLF, the influence of crowd on the structural damping ratio should be considered.
(2) Crowd jumping loads are multi-point excitations, because different people are at different positions in the structure. The Fourier series model based on the developed equivalent DLF simplifies multi-point excitation to single-point excitation, which cannot consider the modal value of each person. As a result, the modal value of each jumping person is assumed to be 1 to predict structural responses, as shown in Equation (33), which indicates that every jumping person is located at the maximum vibration mode, and it can overestimate structural responses. To reduce the errors in structural response calculation, a representative value to characterize each jumping person’s modal value needs to be further investigated.