Human-Induced Vibration Serviceability: From Dynamic Load Measurement towards the Performance-Based Structural Design

Abstract

Since the well-known Millennium bridge accident happened at the beginning of this century, both researchers and engineers realized that the human-induced vibration may lead to unaffordable consequences. Although such vibrations hardly threaten the safety of the structure, the large vibration may affect the functionalities of the structure, causing the serviceability problem. The first study on the human-induced vibration serviceability problem started from the measurement of human-induced load, with many mathematical models proposed. The strong randomness of the measured data led to the investigation on the randomness feature of the load. With the research going deeper, the phenomenon of human–structure interaction was found, which attracted the researchers to study the randomness of the human body dynamic properties that may affect the structural response. Once the interaction mechanism and the system parameters became available, random vibration analysis methods could be proposed to calculate human-induced random vibration, providing the foundation of the reliability analysis from the perspective of vibration serviceability. Such reliability is highly related to subjective feelings of the human body, which has also been deeply studied in the literature. Furthermore, the purpose of studying the dynamic reliability is to conduct the reliability-based structural design. This paper provides a review of the research on human-induced vibration serviceability following the above logic, from the first attempt on load measurement towards the modern techniques for performance-based vibration serviceability design.

1. Introduction

Nowadays, the vibration serviceability problem induced by human activities must be considered during the structural design stage, especially for structures with large spans, such as footbridges, sport stadia, and open-plan floors in high-rise buildings. However, several decades ago, the problem of human-induced vibration did not gain as much attention from researchers and engineers as it did during the past two decades. The reasons behind this phenomenon are roughly explained via the following aspects. Firstly, the development of new materials with light weight and higher strength has made it possible for engineers to design structures with less materials. The newly designed structures thus usually have less mass than those previously designed structures. Basic laws of mechanics show that the vibration becomes larger when the mass of the structure becomes less. Secondly, modern people usually have higher requirements on the open space of public buildings from both functional and aesthetic perspectives. Under such requirements, the span of the structures becomes much larger than before, which results in low fundamental frequencies. If this fundamental frequency falls in the range of human pacing frequency, excessive vibration may occur. Such unpleasant vibrations may lead to serious economic losses and adverse social impact if not well-controlled.

To avoid the negative effects caused by human-induced activities on large-span structures, vibration serviceability design must be conducted during the structural design stage. The simplest way to address the vibration serviceability assessment in design codes is to set a lower limit of the fundamental frequency of the structure. However, this method is not always applicable, especially when dealing with long-span footbridges that have fundamental frequencies that are always lower than the limit. Currently, most design codes require the engineers to calculate structural responses under human-induced excitation, which must be controlled under a predetermined value to ensure vibration serviceability of the structure. To fulfill this requirement, calculation methods are suggested by design codes, which usually consider crowd size, the type of activities, and structural dynamic properties. The calculated structural response is then compared with a threshold and must not exceed this pre-determined value. For more complex structures, a human-induced load model is usually applied to a detailed analytical model or a finite element model (FEM) of the structure to obtain the human-induced vibration.

The above-mentioned process for vibration serviceability design has provided the structural engineers with a basic framework to conduct human-induced vibration serviceability design. However, in the engineering practice, structures still exhibit excessive vibration even after the serviceability design is conducted. The first reason behind this phenomenon lies in the fact that the excitation generated by human activities is stochastic, which is usually neglected by the current design codes. For light-weight structures, the human–structure interaction ensures even more random sources for the system because the dynamic properties of the human bodies, including their frequencies and damping ratios, are not deterministic. The second reason is that the vibration serviceability is mostly determined by the feelings of the occupants on the structure, which is highly subjective and thus can hardly be evaluated by a deterministic value of the structural responses.

To illustrate the above statement, a calculation method suggested by the American Institute of Steel Construction (AISC) [1] is listed here in Equation (1)

$$a_{\mathrm{p}}=\frac{p_{0}g\mathrm{e}\mathrm{x}\mathrm{p}\left(-0.35f_{\mathrm{n}}\right)}{\beta W}⩽a_0$$

(1)

where a_p is the predicted structural acceleration under human-induced load, f_n, β, W is the natural frequency, damping ratio, and weight of the structure, respectively, p₀ is a deterministic pre-defined load factor, g is the gravitational acceleration, and a₀ is the acceleration limit. It is found that the parameters in the above equation are deterministic, therefore neglecting all the randomness in both the excitation and the structure. Moreover, the acceleration limit a₀ is only related to the function of the structure, failing to consider the fuzziness of the subjective feelings of the occupants. A structure is considered to be “serviceable” if the requirement represented by Equation (1) is satisfied. If a_p is calculated to be larger than a₀, the structure is unacceptable from the perspective of vibration serviceability. In this manner, the structural vibration serviceability becomes a “pass-or-not-pass” problem. Any inaccuracy in the parameters of Equation (1) or in the equation itself may lead to an incorrect judgement of the vibration serviceability, as is found in many engineering cases.

However, because of the randomness in the human–structure system and the fuzziness of the occupants’ subjective feelings, the vibration serviceability should be analyzed through reliability-based methods. The vibration serviceability of a structure is no longer a “pass-or-not-pass” problem but is evaluated by a certain degree of probability. Such probability includes the uncertainties in human-induced dynamic load, the human–structure interaction, the structural dynamic properties, and the subjective feelings of the structural occupants. To achieve the reliability-based vibration serviceability design, the following aspects are necessary, including (a) The random model of human-induced load, or the method to generate random load time history; (b) The random description of human body dynamic properties, aiming at proper description of human–structure interaction; (c) The appropriate random vibration analysis method to calculate and describe the randomness of the human-induced structural vibration; and (d) A quantification method to describe the subjective feelings of structural occupants, as well as the affecting factors when vibration takes place.

This literature review contains the following parts. Section 2 presents how researchers started to include randomness in the structural response calculation by studying the randomness in human-induced dynamic load, and by providing random load models. Since the human–structure interaction has a significant effect on structural responses, Section 3 aims to describe the methods to obtain dynamic parameters of human body models, as well as their probability distribution. Once the randomness of the excitation becomes clear, the analytical methods to obtain random responses are necessary, which are introduced in Section 4. An overview of how to describe subjective feelings of human bodies is presented in Section 5. In Section 6, some reliability-based structural design strategies under external loads are reviewed, although they do not have much direct implementation on vibration serviceability problems. Finally, Section 7 summarizes the key findings of this review. This review includes the entire process of structural vibration serviceability, from the perspective of excitation, analytical method, assessment, and design strategies.

2. Human-Induced Load

2.1. Measurement of Human-Induced Load

The research on vibration serviceability problem originated from the research on the human-induced load, which is the excitation source of structural vibration. Such research started from the work by Harper in 1961. Harper obtained the first test data of human walking load by force plates [2]. The measured time history showed that the vertical human walking load is similar to an M-shape curve, as can be observed from Figure 1, which shows a three-dimensional force plate. The use of force plates to measure human-induced load was then examined by many other researchers, including Galbraith et al. [3], Matsumoto et al. [4], Ohlsson et al. [5], Ebrahimpour et al. [6,7], Kerr et al. [8,9], Kasperski et al. [10], Racic et al. [11], and Chen et al. [12].

Since the force plate is always fixed on the rigid ground, the drawback of using a force plate to measure human-induced load is quite obvious. The number of measured steps is limited to the number of force plates. In usual cases, only a few steps of pedestrian load can be measured. To solve this problem, researchers applied sensor-equipped insoles to achieve continuous measurement, as shown by Figure 2a. In the measurement, the insoles are attached under the feet of the pedestrian. The recorded signals are then wirelessly transmitted to the device nearby. A typical continuous time history of human walking load generated by a 75-kg male test participant walking at a frequency of 1.5 Hz, and its corresponding Fourier spectrum, are depicted in Figure 2b.

The force plates and the sensor-equipped insoles are the most commonly adopted methods for single-person load tests. The force plates are more suitable for the case of human bouncing and jumping, where the position of the human body remains unchanged during the test. For walking load measurement, the sensor-equipped insoles are more often adopted. If the synchronization effect of crowd excitation is to be analyzed, the 3-D motion capture system can be used. In the crowd test, reflective markers are attached on the skin of the human body. Its moving trajectory is recorded by an infrared camera, as shown by Figure 3a. More recently, the development of computer vision techniques has also been adopted in crowd measurement, including DIC, PIV, and the optical flow method. The GRF can be obtained through a pre-calibrated relation from the recorded human body movement (Figure 3b).

2.2. Stochastic Human-Induced Load Model in Time Domain

These measurement data recorded by techniques mentioned in the previous section have made it possible to extract a model for the human-induced load. Since the human excitation is usually near-periodic (as shown by Figure 2b), a Fourier-series model was first proposed by researchers with the form of

$$F\left(t\right)=G+G\sum _{i=1}^n{\alpha }_i\mathrm{s}\mathrm{i}\mathrm{n}\left(2i\pi f_{p}t-{\phi }_i\right)$$

(2)

where F(t) is the vertical dynamic load induced by human walking, G is the static weight of the human body, f_p is the pacing frequency, and α_i is the i-th Fourier series coefficient of the model, which is also known as the dynamic load factor (DLF), and φ_i is the corresponding phase angle.

Based on the Fourier series model, researchers studied the values for DLFs and phase angles. Researchers found that DLFs are usually related to pacing frequencies and the proposed corresponding relation. Suggested values of DLFs were also demonstrated. In these early attempts, the values of DLFs were usually considered as deterministic. Since this paper mainly focuses on the random feature of human–structure systems, the detailed summarization of these deterministic suggested values is not listed here but can be found in previous review papers [14,15].

Researchers then realized that the human-induce load has a strong feature of randomness, which is usually expressed by “inter-subject variability” and “intra-subject variability”. These variabilities are reflected in the randomness of the DLF. The works by Kerr [9], Young [16], and Chen [12] have shown that the DLF values are strongly scattered and can hardly be represented by a deterministic value, as observed in Figure 4, which shows the first four orders of DLFs.

Brownjohn et al. [17] pointed out through their research that it is more reasonable to regard walking load as a narrow band random process. On the basis of summarizing the previous work, Zivanovic et al. [18] regarded step frequency, step length, and walking load amplitude as normal distributions and provided their probability distribution rules. Among them, the distribution of dynamic load factors of the first five main harmonics and sub-harmonics is proposed based on the research of Kerr [9] and Brownjohn [17] et al. It is listed in

Table 1. Probabilistic description of DLFs [18].

Harmonic	DLF
	Mean Value μDLF	Standard Deviation σDLF
1	−0.26 fp3 + 1.32 fp2 – 1.76 fp + 0.76	0.16μDLF
2	0.07	0.03
3	0.05	0.02
4	0.05	0.02
5	0.03	0.015

.

Based on a large number of experimental data that include 73 test subjects and 4814 single footfall records obtained from the force plates, Ding and Chen proposed a random Fourier series model in which the variation coefficient of weight, pacing frequency, DLF, and phase angle are introduced to reflect both the inter-subject and intra-subject variability of the pedestrian load [19]. They concluded that the variability of the pedestrian load has a large influence on the structural responses and thus cannot be neglected in engineering practice [20]. Such random Fourier series models have become the foundation of the analysis on the crowd-induced responses [21].

2.3. Stochastic Human-Induced Load Model in Frequency Domain

When the measured walking load time history is transformed into the frequency domain, the amplitude spectrum or power spectrum is obtained. It is shown that the peaks of the spectrum not only occur at the harmonics of the pacing frequency but distribute along the frequency axis, showing that the walking load is not strictly periodic. The random properties of the walking load can be described more comprehensively by expressing it in the form of power spectrum or amplitude spectrum.

In 1982, Ohlsson introduced the idea of the frequency domain model of walking load in his doctoral thesis [5]. Based on the periodic assumption of walking load, the measured single-step time histories were extended to a continuous load. The power spectrum of the walking load is thus obtained. Since Ohlsson’s research focused on the properties of high-frequency floors, the proposed power spectrum model includes the frequency components in the range of 6 to 50 Hz. In 1994, Eriksson focused on the study of low-frequency floors and obtained the power spectrum of walking load with a frequency lower than 6 Hz. Eriksson further pointed out that each peak in the spectrum has a certain bandwidth, which means that the walking load is not completely periodic but a narrow band random process. Therefore, the DLFs, which are defined from the Fourier-series model in the time domain, do not accurately describe the features of the walking load.

Brownjohn et al. used a force-measuring treadmill to measure the continuous walking load time histories of three people and obtained the corresponding power spectrum model [17]. Assuming that the pacing frequency follows a normal distribution with a mean value of 2 Hz and a standard deviation of 0.2 Hz, the power spectrum of the response of a crowd composed of two people under walking load was also provided. It is worth noting that calculating structural responses in the frequency domain is one of the basic methods for random vibration, which will be further discussed in Section 4.

3. Human–Structure Interaction

When studying the dynamic properties of the human excitation, researchers found that the human-induced load measured on a rigid floor seems to be different from the one measured on a moving structure, which triggers the research on the human–structure interaction phenomenon. The research attempts in this field include the effect on human-induced load and structural dynamic properties. Later, researchers realized that the reason why the human–interaction effect occurs is because the human bodies on the structure have their own mechanical properties, including mass, stiffness, and damping, thus forming a coupling system with the structure they occupy. Therefore, the study on the dynamic properties of the human body attracts the attention of researchers. This section provides a review on the human–structure interaction effect from the literature.

3.1. The Influence of Human–Structure Interaction on the Human-Induced Load

In 2004 and 2006, Yao et al. designed a flexible platform to study the similarities and differences between the loads caused by people jumping and bouncing on a flexible structure and those caused by people moving on rigid ground [22,23]. They found that the human jumping and bouncing load on the flexible structure was smaller than that measured on the rigid ground and was related to the frequency and amplitude of platform vibration. In 2016, Dang et al. used a force measuring treadmill to measure the ground reaction force induced by human walking. The study concluded that the vibration of the structure would reduce the force. Such reduction becomes more obvious when the pacing frequency is closer to the structural fundamental frequency [24]. Similar results were also reached in 2018 by Ahmadi et al., who measured the single-person walking load on a rigid ground and flexible bridge using sensor-equipped insoles, showing that the first-order DLF on the flexible bridge floor was smaller than that on rigid ground. When the pacing frequency was closer to the structural frequency, or when the human–structure mass ratio decreased, the difference on DLF became smaller [25].

In addition to the load amplitude and DLF, the phase of the human-induced load may also be affected by the structural vibration. In 1993, Fujino et al. analyzed the reasons of excessive lateral vibration on the T-Bridge in Japan. They found that although the initial pacing frequency and the phase of the people in the crowd were random, they tended to converge to an identical value with the increase of the lateral vibration of the bridge [26]. Such a phenomenon was described as a “lock-in”. In 2005, Strogatz et al. proposed a theoretical model describing the phase adjustment of crowd-walking frequency in view of the lateral locking phenomenon on the Millennium Bridge and derived the structural damping required to overcome this phenomenon [27]. The “lock-in” phenomenon is usually found in the human-induced lateral vibration, which mostly occurred in footbridges. However, the lock-in phenomenon in the vertical direction is very rare. At present, only Yao et al. [23] have observed this phenomenon due to excessive structural vibration in the laboratory under a crowd-jumping case.

Since it is difficult to install instruments to test human-induced loads in vibrating structures, there is a lack of research on the differences of loads on rigid and flexible interfaces. Moreover, the ground reaction force on flexible structures is also related to the dynamic properties of the structures, making it difficult to propose a generalized mathematical load model. Therefore, the current research mainly considers the influence of structural vibration on human-induced load by establishing the motion equation of the human–structure coupled system [28]. Nimmen et al. [29] and Shahabpoor et al. [30] claimed that the vertical ground reaction force under walking excitation consists of two parts: the human body self-drive force, and the force generated by human–structure interactions. They claimed that the self-drive force on flexible structures was consistent with that on the rigid ground because it is difficult for people to feel the vertical vibration of the structure when walking. Based on this assumption, the structural responses considering the human–structure coupling effect can be calculated based on the results on the rigid ground.

3.2. The Influence of Human–Structure Interaction on Structural Dynamic Properties

The structural dynamic properties, including the natural frequency and damping ratio, may be influenced by the presence of crowd. This influence is related to the state of the human body, which means that the influence of the static crowd and moving crowd on the structural dynamic properties may be different.

It is recognized by researchers that static population can affect the dynamic properties of structures. In 1997, Ellis and Ji built a simply supported concrete beam with a fundamental frequency of 18.68 Hz and studied the changes in the dynamic properties of the simply supported beam when a human body was standing, sitting, walking, and jumping [31]. The results show that the fundamental frequency and damping ratio increased simultaneously when the human body was standing and sitting on the beam while remaining unchanged when the person was jumping or walking. In 1999, Falati used simply supported prestressed concrete slabs to study the influence on structural dynamic properties by people standing on the structure. The results showed that when there was a single person standing on the slab, the fundamental frequency of the slab decreased, while the damping ratio increased. The author also conducted field measurements on a real floor and found that when there were three people standing on the floor, the fundamental frequency and damping ratio of the floor did not exhibit significant change, suggesting that the influence of the static person on the structural dynamic properties of the structure varied with situations [32]. Later in 2013, Salyards et al. designed a cantilever plate, which has a fundamental frequency that could be adjusted in the range of 4–8 Hz. Tests on this plate showed that the damping ratio of the structure increased significantly when people stood on the structure with their legs straight, reaching more than nine times the damping ratio of the empty structure. When people stood with their legs relaxed, the damping ratio of the structure increased less, sometimes only two times that of the empty structure. The authors identified that the influence on the structural dynamic properties is related to posture of the people [33].

Although researchers believe that the static people can affect the structural dynamic properties, they do not reach an agreement on whether walking, bouncing, and jumping have the same effect. In 1997, Ellis and Ji argued that jumping does not change the structural dynamic properties because people’s feet are lifted off the ground [31]. A similar statement was also proposed by Jones et al. [34] in 2011. They claimed that the structural dynamic properties will only be changed by people whose feet make contact with the structure during movement, i.e., walking or bouncing. In 2009, Zivanovic et al. tested the dynamic properties of a simply supported footbridge with and without people. It was determined that walking crowds would increase the damping ratio of the structure as the number of people became larger, but the increase level was less than the case of people standing. For the fundamental frequency of the structure, a significant decrease was observed when a standing crowd occupied the structure, while a walking crowd had little effect on the structural frequency [35]. In the same year, Duarte and Ji studied the effect of bouncing on structural dynamic properties. It is determined that the fundamental frequency and damping ratio of the structure increased under certain bouncing frequencies [36]. In 2011, Setareh tested the dynamic properties of a real cantilever stand under crowd bouncing excitation and determined that the bouncing crowd would reduce the structural frequency, increase the first-order damping ratio, and decrease the higher-order damping ratios [37].

From the above review, it is determined that the influence of the human–structure interaction on the structural dynamic properties is a complex problem. In the latest research, it is generally believed that both the static and moving crowds affect the structure dynamic properties, while the influence from the static crowd is larger compared with the moving crowd.

3.3. Dynamic Properties of Human Body

In the early stage of human–structure interaction research, the human body on the structure was simulated as an additional mass on the structure [7,38]. Later, researchers realized that, in addition to the mass, the human body also includes the elastic and energy-consuming properties, which are simulated as a rigid system with a spring connected to a damper. For example, in 2011, Maca and Valasek proposed two-dimensional and three-dimensional multi-rigid body models of the human body [39]. In 2013, Qin et al. proposed a bipedal-inverted pendulum model of human body [40]. Among these human body models, single-freedom spring–mass–damper (SMD) is the most widely used for its simplicity and acceptable accuracy.

In 1987, Foschi and Gupta successfully explained the change of dynamic properties of wood structures affected by the human–structure interaction using the SMD model [41]. In 1997, Ellis and Ji [42] tested the dynamic properties of Twickenham Stadium in UK in the presence of a crowd during a match [31]. New vibration modes appeared when there were crowds on the structure. The authors simplified both the structure and the crowd into single-degree-of-freedom systems, thus proposing a two-degree-of-freedom coupling system representing the crowd and the structure. Such an assumption explained the physical mechanism of the emergence of new vibration modes and deduced that the fundamental frequency of the seated crowd in the stadium is in the range of 5.5–5.8 Hz. In 1999, Brownjohn used the multi-degree-of-freedom human body model given by ISO7962 [43] to derive the dynamic properties of the human body. The first-order modal mass, stiffness, and damping were suggested as 80 kg, 82 kN/m, and 1946 Ns/m, respectively. The feasibility of the parameters of the SMD model was verified by testing the dynamic properties of a prestressed concrete slab with a span of 7 m under the existence of a standing person [44]. In the same year, when studying the influence of non-structural components on the structure, Falati tested the structural dynamic properties with and without a seated crowd and identified the representative parameters of the crowd. The fundamental frequency of the crowd was 10.43 Hz, and the damping ratio is around 0.44–0.45 [32]. Wei and Griffin identified the parameters of 60 seated people and provided the parameter values of each volunteer [45]. In 2003, Matsumoto and Griffin adopted the same method to identify the mechanical model parameters of 12 standing people [46]. Results showed that the mean value of the standing people’s natural frequency was 5.7 Hz, and the standard deviation was 0.57 Hz. The damping ratio was tested to be 0.69 with a standard deviation of 0.07. In 2015, Nimmen et al. adopted an SMD model to represent the standing crowd and approximated the structure as a single degree of the freedom system. The natural frequency and damping ratio of the standing crowd in four different standing postures were tested [47]. The results showed that the SMD parameters of walking pedestrians could be replaced by the parameters when one leg was bent and one leg was stepping forward. In the same year, Lou et al. used a three-dimensional motion capture system to collect the motion trajectory of walking people and recorded the corresponding walking force with a high-precision force measuring plate to identify the SMD parameters and biological force parameters of human walking [48]. In 2017 and 2019, Wang et al. adopted the particle filter technique and proposed an output-only system identification method to extract the human model parameters from measured continuous ground reaction force for human walking and bouncing, respectively [49,50]. The human model parameters were shown to be very much scattered while the trend against the pacing frequency was also found, as shown by Figure 5. Due to the huge differences in human parameters under different circumstances and their strong randomness, there are still relevant studies being carried out continuously.

3.4. Research on Human–Structure Coupling System

As mentioned above, the human–structure coupling phenomenon is usually studied by means of dynamic modeling. Once the human–structure coupling equation is derived, it is possible to analyze the dynamic properties of the coupling system, the ground reaction force between the human and the structure, and the structural responses.

The simplest way to derive the coupling equation is to represent the crowd and the structure by as a single-degree-of-freedom system, as was proposed by Sachse in 2004 [51]. By solving the complex eigenvalues λ_i of the coupling equation, the i-th order natural frequency f_i and damping ratio ξ_i of the coupled system can be obtained by

$$f_i=\frac{1}{2\pi }\left|{\lambda }_i\right|,{\xi }_i=\frac{\left|\mathrm{R}\mathrm{e}\left({\lambda }_i\right)\right|}{{\lambda }_i},i=1,2$$

(3)

In their work, the factors affecting the dynamic properties of the coupled system are discussed. The analysis showed that the low-frequency structure is most likely to exhibit a decrease of natural frequency and an increase of the damping ratio. For high-frequency structures, both the natural frequency and damping ratio of the coupled system will increase. In 2013, Shahabpoor et al. conducted parametric sensitivity analysis on the dynamic properties of the above coupling system [52]. The results showed that the presence of the walking crowd increased the damping ratio of the coupled system. The closer the natural vibration frequency of the walking crowd was to that of the structure, the more the damping ratio of the coupled system increased. When the natural frequency of the pedestrian was greater than that of the structure, the natural frequency of the coupled system will decrease. When the pedestrian frequency is less than the natural vibration frequency of the structure, the natural vibration frequency of the coupled system will increase.

A multi-degree-of-freedom system was also adopted by researchers to study the coupling effect. In 2017, Nimmen et al. proposed an equation of motion for the human–structure coupling system [29]. Each person in the crowd was considered as an SMD model, and the structure included multiple vibration modes. They concluded that the crowd on the structure could greatly increase the damping ratio of the structure and reduce the vibration response of the structure. In 2021, Xiong et al. regarded each person in the crowd as an SMD model to establish a general expression of the human–coupling system using the matrix expression. Based on the assumption that the self-drive force of the human body on the structure is equal to the one on the rigid ground, the authors derived the structural response considering the contribution of multiple modes [53]. The feasibility and accuracy of the proposed method were verified by comparing the experimental results. The models adopted by these researchers are depicted in Figure 6. There are many other similar studies, such as [54,55,56], which are not listed here to avoid lengthiness.

3.5. Notes on the Research of Human–Structure Interaction

In summary, it is observed from the existing research results that the mechanical properties of the crowd have a non-negligible impact on the dynamic properties and vibration response of the structure. At present, the mainstream practice to consider such influence is to simplify the human body into various rigid body models and conduct modeling research on the human–structure coupling system. However, once the randomness of the human–structure parameters is considered, even the simplest SMD model will result in a so-called “dimensional disaster” phenomenon in the random analysis if the number of people is larger than 10, which greatly increases the demand for computing power and calculation difficulty. Therefore, it is important to propose an efficient way to meet the requirement of stochastic analysis in the human–structure coupling problem for structural response calculation, which is to be discussed in latter sections.

4. Stochastic Vibration Analysis

In the review of human-induced loads in Section 2, it is found that the human-induced loads generated by a single human body exhibit narrow band random process properties. In addition, as described in Section 3, the human body parameters represented by a rigid body model show strong variability, which has a probability density function that is studied in [49,50]. Therefore, it is realistic to study human-induced vibration from the perspective of random analysis. When considering the coupling system formed by the human body and the structure, the randomness of the human body will be included in the equation of motion to provide randomness to the coupling system, while the randomness of the load is reflected in the non-homogeneous term at the right end of the equation. It is necessary to use the stochastic dynamic system analysis theory, which is widely used in the field of earthquake resistance and wind resistance to solve the problem.

4.1. Spectrum Analysis Method

Although the structural vibration induced by human excitation may result in vibration serviceability problems, the structures usually still remain in the linear phase. According to the random vibration, the random response can be obtained by a spectrum analysis method.

Assuming there are m excitation points, the s-order equation of motion is expressed by

$$M_l{\ddot{q}}_l+2{\xi }_l{\omega }_l{M}_l{\dot{q}}_l+{\omega }_l^2{M}_l{q}_l=\sum _{s=1}^m{\varphi }_{l,es}{W}_s{x}_s\left(t\right)$$

(4)

in which ϕ_l,es is the mode shape value of lth order at the sth excitation point, x_s(t) is the dimensionless excitation force, and W_s is the weight of the sth person. The self-spectrum of the structural response G_R(f) at the measurement point is expressed by

$$G_R\left(f\right)={\mathsf{\Psi }}_k{\mathbf{H}}_Q^{\mathrm{*}}\left(f\right){\mathsf{\Psi }}_e^{\mathrm{T}}\mathbf{w}{\mathbf{G}}_{\mathrm{X}}\left(f\right){\mathbf{w}}^{\mathrm{T}}{\mathsf{\Psi }}_e{\mathbf{H}}_Q\left(f\right){\mathsf{\Psi }}_k^T$$

(5)

where w is the diagonal matrix of the crowd, G_X(f) is the cross-spectrum matrix of the excitation with the form of

$${\mathbf{G}}_{\mathrm{X}}\left(f\right)=\left(\begin{array}{ccc}{G}_{11}\left(f\right)& \cdots & G_{1m}\left(f\right)\\ ⋮& \ddots & ⋮\\ G_{m1}\left(f\right)& \cdots & G_{mm}\left(f\right)\end{array}\right)$$

(6)

$${\mathsf{\Psi }}_{\mathrm{k}}=\left({\varphi }_{1,k}\cdots {\varphi }_{\begin{array}{c}n,k\end{array}}\right)$$

(7)

$${\mathsf{\Psi }}_{\mathrm{e}}=\left(\begin{array}{ccc}{\varphi }_{1,e1}& \cdots & {\varphi }_{n,e1}\\ ⋮& \ddots & ⋮\\ {\varphi }_{1,em}& \cdots & {\varphi }_{n,em}\end{array}\right)$$

(8)

$${\mathbf{H}}_{\mathrm{Q}}\left(f\right)=\left(\begin{array}{ccc}{H}_1\left(f\right)& & 0\\ & \ddots & \\ 0& & H_n\left(f\right)\end{array}\right)$$

(9)

where H_i(f) is the frequency response function of the ith mode. Once G_R(f) is obtained, the root–mean–square value of the structural response can be calculated through

$$a_{\mathrm{r}\mathrm{m}\mathrm{s}}^2={\int }_0^{\mathrm{\infty }}{G}_R\left(f\right)\mathrm{d}f$$

(10)

From the above equations, it is observed that the key to the spectrum analysis method is to determine the cross-spectrum matrix of the excitation. As stated in Section 2, the research on the power spectrum model of human-induced loads began in 1982, when Ohlsson introduced the frequency domain method into vibration serviceability analysis in his doctoral thesis, proposing the self-spectrum of human walking load in the range of 6–50 Hz [5]. In 1994, Eriksson [57], a student of Ohlsson, carried out the research on low-frequency floor slabs and constructed the low-frequency self-spectrum using the measured time history of a continuous single person walking load. In the same year, Mouring and Ellingwood attempted to use an inverse algorithm to obtain the self-spectrum of pedestrian loads from the acceleration time history of pedestrian bridges [58]. In 2012, inspired by the spectrum model in seismic engineering, Krenk proposed a two-parameter spectrum model that only considers the first order walking load [59].

In terms of cross-spectrum, Brownjohn et al. measured the continuous vertical walking force time history of three testers using a force measuring Treadmill in 2004 [17]. Based on the measurement data, a self-spectrum of vertical walking load was proposed. After introducing the coherence coefficient, which is similar to the one used in the wind-induced load, they proposed the cross-spectrum model of a vertical walking load for the first time. In 2012, Piccardo and Tubino proposed an equivalent spectral model for unconstrained pedestrian loads. By using the Delta function to simulate the correlation between pedestrians during walking, they obtained the expression of the cross-spectrum function [60]. In 2016, Ferrarotti and Tubino extended the above work to a high-density crowd state and proposed a universal equivalent spectral model [61]. The model used a coherence function similar to [17], which shows that the coherence between pedestrians in the crowd decreases rapidly until zero as their distance increases. Other works regarding human-induced random vibration calculations using the spectrum analysis method include [62,63,64].

Due to a lack of measurement data, the above work, especially the modeling of cross-spectrum models, is mostly at the level of numerical simulation. The breakthrough took place in 2018 when the 3-D motion capture technology was creatively introduced to crowd load experiment, making it possible to obtain a large number of real crowd load data. Based on these data, the cross-spectrum model of crowd jumping [53,65], bouncing [66], and walking load [67,68] has been proposed. A structural response calculation method, considering the interaction between humans and structures, was thus proposed, which solved the problem response calculation in the frequency domain.

4.2. Random Structural Analysis

By using the spectrum method described in Section 4.1, the structural random vibration can be obtained in the frequency domain. However, this method is limited to the case where human excitation is represented by an external force and can be expressed by a spectrum. In the case where human–structure interactions is strong and the uncertainties of the dynamic properties of both the human body and the structure need to be considered, the random structural analysis becomes necessary. In the late 1960s, owing to the popularity of probability theory, researchers started to pay attention to the influence of the uncertainties in the structural analysis and gradually developed three basic methods based on stochastic simulation, stochastic perturbation, and orthogonal expansion. These methods are briefly introduced herein.

The stochastic simulation method mainly uses the Monte Carlo method to sample random sources and perform statistical calculations on a large number of deterministic analyses to obtain random structural response or reliability [69]. Despite its advantage of easy implementation, it has the characteristic of random convergence and thus cannot accurately capture the full probability information of random response. Moreover, the calculation accuracy is inversely proportional to the square root of the calculation times. Although some researchers tried to develop techniques in an attempt to improve the computational efficiency and accuracy [70], the corresponding improvements are often accompanied by the sacrifice of applicability. At present, the stochastic simulation method is often used in the case where the computational demand is small.

The stochastic perturbation method mainly expands the structural response into the form of a series of basic random variables of the structure and transforms the random response analysis into the problem of solving a series of recursive equations [71]. Compared with stochastic simulation, the calculation amount is greatly reduced, but it is only suitable for problems with small randomness. Although using higher-order perturbation can improve its applicability, it will bring unbearable computational cost. At the same time, the inherent “duration term” problem in the perturbation solution will seriously affect the numerical solution accuracy, which limits the development and application of this method.

The orthogonal expansion method expands the structural response as a linear combination of orthogonal polynomial bases to solve the structural response [72]. Since orthogonal polynomials have the property of mean square convergence, it is also suitable for the case of strong randomness. Correspondingly, the disadvantage of this method is that the order of the extended system equation is much higher than that of the original system equation, which only shows satisfactory results for linear systems. Therefore, when the dimension of random variables is high, the computation amount of this method increases sharply to an unacceptable level. Although researchers [73] proposed numerical strategies to improve the solution efficiency, it is still difficult to apply in actual engineering structures.

The most widely used random structural analysis among the three methods mentioned above is the random stochastic simulation method for its simplicity and acceptable accuracy. For example, based on their proposed random load model in the time domain, Zivanovic et al. numerically calculated the structural response of a footbridge considering the randomness of the human load, showing the feasibility of the stochastic simulation method in vibration serviceability analysis. This method was also experimentally validated through a field test on a real footbridge [18]. In 2016, Venuti et al. combined the social force model with the human–structure coupling equation. Using the random simulation method, they studied the influence of the randomness of human parameters and load parameters [74]. The structural response of a 100-m-span footbridge with a fundamental frequency of 2 Hz and a damping ratio of 0.5% was numerically calculated. The results showed that the maximum acceleration response does not strictly follow the normal distribution but is closer to the Weibull distribution.

In 2018, Tubino et al. simplified the human body and structure as a single degree of freedom system, which established a coupled vibration analysis equation for pedestrian bridges and used random simulation methods to calculate the mean value and the standard deviation of the coupled system under different numbers of people, damping ratios, crowd–structure mass, and frequency ratios [75].

4.3. Probability Density Evolution Method

Researchers have always attempted to understand the random vibration problem from the perspective of the probability density evolution. The most widely adopted Monte–Carlo method has an extremely high requirement on the computational cost and is thus not always applicable. Therefore, a numerical technique that does not require a high computational cost and maintains high accuracy is strongly desired by researchers and engineers.

To reduce the computational cost while keeping the high accuracy on the analytical results of structure random responses, Li and Chen proposed the generalized density evolution equation (GDEE) based on the principle of the conservation of probability [76]. In the past 20 years, a probability density evolution method (PDEM) has been developed, which has made a breakthrough in this field. Compared with the classical probability density evolution equations (Liouville, FPK, and D-P), GDEE can consider the initial conditions, external excitation, and structural parameters in a unified manner, which allows for the probability density solution, with a complete structural response, to be obtained.

If Θ = (Θ₁, …, Θ_s)^T contains all the random sources in the system and Z = (Z₁, …, Z_m)^T contains all the structural responses quantities of interest, (Z, Θ) can be regarded as a probabilistically conservative system with a description of random events based on the principle of conservation of probability. This idea is shown by Figure 7.

The GDEE satisfying the probability density function at time t can be derived as [77]

$$\frac{\partial p_{\mathbf{Z}\mathsf{\Theta }}\left(\mathbf{z},\mathsf{\theta },t\right)}{\partial t}+\sum _{j=1}^m{\dot{Z}}_j\left(\mathsf{\theta },t\right)\frac{\partial p_{\mathbf{Z}\mathsf{\Theta }}\left(\mathbf{z},\mathsf{\theta },t\right)}{\partial z_j}=0$$

(11)

The purpose of solving the random vibration problem is to obtain the probability density function of the response. Since human comfort is related to the acceleration responses of the structure, the physical quantity Z in the above equation is usually represented by structural acceleration for the vibration serviceability problem.

For most practical engineering problems, the numerical solution of the probability density function needs to be obtained by a numerical method. The numerical solution of GDEE is divided into four steps:

(1)

Select the representative point set θ_q, divide the probability space Ω_Θ characterized by all random variables Θ, and determine the corresponding assigned probability P_q, as denoted by

$${\theta }_q={\left(\begin{array}{cccc}{\theta }_{q,1}& {\theta }_{q,2}& \cdots & {\theta }_{q,N_{\theta }}\end{array}\right)}^{\mathrm{T}}, q=1,2,\cdots ,n_{pt}$$

(12)

$$P_q={\int }_{{\Omega }_q}{p}_{\mathsf{\Theta }}\left(\mathsf{\theta }\right)\mathrm{d}\mathsf{\theta }$$

(13)

(2)

For each representative point, carry out the deterministic analysis to obtain the velocity time history of the response quantity of interest.

(3)

Numerically solve the GDEE to obtain the integral of p_ZΘ (z,θ,t) on each probability subfield.

(4)

The probability density function of the concerned physical quantity is obtained by summing the probability subfields, as denoted by

$$p_Z\left(z,t\right)={\int }_{{\Omega }_{\mathsf{\Theta }}}{p}_{Z\mathsf{\Theta }}\left(z,\mathsf{\theta },t\right)\mathrm{d}\mathsf{\theta }=\sum _{q=1}^{n_{\mathrm{p}\mathrm{t}}}{\int }_{{\Omega }_q}{p}_{Z\mathsf{\Theta }}\left(z,\mathsf{\theta },t\right)\mathrm{d}\mathsf{\theta }=\sum _{q=1}^{n_{\mathrm{p}\mathrm{t}}}{p}_q\left(z,t\right)$$

(14)

The above steps are the solution idea of GDEE point evolution [78]. On the one hand, the selection of representative point set will directly affect the accuracy and efficiency of probability density evolution analysis. Therefore, a series of methods have been developed to obtain a more uniform and high-quality point set [79,80,81,82,83]. On the other hand, some numerical methods to solve GDEE have also been suggested for the common case where the solution of only one physical quantity is of interest [76,84,85].

Currently, the PDEM-based method has been successfully adopted in the random vibration analysis under the excitation of earthquake, wind, and traffic, among others. The first attempt to use PDEM-based method for human-induced response calculate is achieved by Ding and Chen [20], who tried to analyze the effect of the randomness in human-induced load on the structural response. By comparing the 10-s RMS acceleration response of a 30-m span and 4.6-Hz frequency concrete slab, it is found that the negligence of human load parameters will strongly underestimate the floor vibration.

The success of the PDEM-based method also enhanced the development of reliability analysis. In 2020, Jia et al. conducted sensitivity analysis, random vibration acceleration response analysis, and dynamic reliability analysis on the pedestrian bridge coupling system using the PDEM [86]. A method for the reduction of dimensions for high-dimensional systems was proposed. The reliability of a 30-m span and a 0.85-Hz lateral fundamental frequency was evaluated and discussed. In 2022, Zeng et al. used PDEM to build up the framework for analyzing the dynamic reliability of large-span structures under crowd excitation from the perspective of vibration serviceability [87].

4.4. Notes on Stochastic Analysis and Reliability Calculatinon in Human-Induced Vibration Problems

From the review of this section, it is seen that the Monte Carlo simulation is still the most-used method because of its simplicity for implementation. At present, there is still a lack of clear understanding of how the random human model parameters affect the structural responses. Therefore, research on human–structure stochastic systems from the perspective of full probability based on probability density evolution theory and sensitivity analysis method is helpful to deepen the understanding on the properties of the system and the stochastic human-induced responses.

5. Vibration Serviceability Assessment

One of the purposes of human-induced vibration analysis is to evaluate the comfort level of the structural occupants. The groundbreaking work on occupants’ comfort was conducted in 1931 by Reiher and Meister [88]. The study asked participants to use the words “imperceptible”, “just perceptible”, “clearly perceptible”, “perceptible”, “annoying”, “unpleasant”, “intolerable”, and other six descriptive indicators to express their feelings. According to the test results, a curve representing the relation between the vibration amplitude and occupant perception is determined. Their methodologies and results have been widely adopted and developed by later researchers and have laid the foundation for relevant research on vibration comfort evaluation.

5.1. Evaluation Index for Human Comfort

Since the relationship between vibration amplitude and a human body’s perception is unclear, various representative values of acceleration time history have been used as indicators for vibration comfort level. Common indicators for a vibration comfort level include the acceleration peak value (PV), weighted root–mean–square value (RMS), weighted root–mean–quad value (RMQ), vibration dose value (VDV), and maximum transient vibration value (MTVV). The definitions of the above indicators are listed below [89]

$$\mathrm{P}\mathrm{V}=\mathrm{m}\mathrm{a}\mathrm{x}\left(\left|a_{\mathrm{w}}\left(t\right)\right|\right)$$

(15)

$$\mathrm{R}\mathrm{M}{\mathrm{S}}_{\mathrm{w}}={\left\{\frac{1}{T}{\int }_0^T{\left[a_{\mathrm{w}}\left(t\right)\right]}^2\mathrm{d}t\right\}}^{\frac{1}{2}}$$

(16)

$$\mathrm{R}\mathrm{M}{\mathrm{Q}}_{\mathrm{w}}={\left\{\frac{1}{T}{\int }_0^T{\left[a_{\mathrm{w}}\left(t\right)\right]}^4\mathrm{d}t\right\}}^{\frac{1}{4}}$$

(17)

$$\mathrm{V}\mathrm{D}\mathrm{V}={\left\{\frac{1}{T}{\int }_0^T{\left[a_{\mathrm{w}}\left(t\right)\right]}^4\mathrm{d}t\right\}}^{\frac{1}{4}}$$

(18)

$$\mathrm{M}\mathrm{T}\mathrm{V}\mathrm{V}=\mathrm{m}\mathrm{a}\mathrm{x}\left[{\left\{\frac{1}{\tau }{\int }_{t_0-\tau }^{t_0}{\left[a_{\mathrm{w}}\left(t\right)\right]}^2\mathrm{d}t\right\}}^{\frac{1}{2}}\right]$$

(19)

where a_w(t) is the structural acceleration time history induced by human excitation. The above indicators are usually adopted according to different situations. For example, PV can be easily affected by accidental factors and is usually used to define the perceptible vibration limit. The RMS value represents the energy of the vibration that the human body is subjected to and is strongly related to the length of the adopted time history T, which is usually set as 1 s or 10 s. The vibration dose value is a measure of the cumulative effect of long-term vibration on human comfort, often 4 to 8 h, which is difficult to measure.

5.2. Frequency-Independent Standard

To prevent vibration serviceability problems, many design codes have suggested limit values for the structural vibration. These limit values are mainly divided into two categories, i.e., the frequency-independent and the frequency-dependent standard. The former category has the advantage of simplicity, while the latter one could consider the influence of different frequency components but is more complicated to use.

The European specification EN 1990 provides vibration comfort limits applicable to footbridges in its appendix [42]. When the fundamental frequency of the structure is fewer than 5 Hz in the vertical direction and fewer than 2.5 Hz in the horizontal or torsional direction, the acceleration peak value shall not exceed the limit listed in

Table 2. EN 1990 vibration comfort limits for bridge floors [42].

Application Situation	Maximum Acceleration (m/s2)
Vertical vibration	0.7
Horizontal vibration produced by normal use	0.2
Abnormal congestion state	0.4

.

The research fund for coal and steel (RFCS) proposed vibration design codes for both building floors [90] and footbridges [91]. For building floors, the RMS value of structural acceleration is adopted as the evaluation index, while the acceleration peak value is used for footbridges. The vibration limits with different functions under the pedestrian load are provided by

Table 3. Limits on RMS acceleration for floors with different functions (unit: cm/s²) [90].

	Strict Region	Nurse	Education	Resident	Working	Conference
Acceptable	0.1	0.8	0.8	3.2	3.2	3.2
Unacceptable	>0.2	>3.2	>3.2	>12.8	>12.8	>12.8
	Shop	Hotel	Prison	Factory	Sport
Acceptability	3.2	3.2	3.2	12.8	12.8
Unacceptability	>12.8	>12.8	>12.8	>51.2	>51.2

and

Table 4. Limits on acceleration peak value for footbridges (unit: m/s²) [91].

Comfort Level	Sense of Comfort	Vertical	Lateral
CL1	Best	<0.5	<0.1
CL2	Intermediate	0.5–1.0	0.1–0.3
CL3	Poor	1.0–2.5	0.3–0.8
CL4	Intolerable	>2.5	>0.8

for floors and footbridges, respectively. The limit values for footbridges are also adopted by EN03 in Germany [92] and Setra in France [93].

5.3. Frequency-Dependent Standard

The frequency-dependent standards consider the influence of different frequency components in vibration signals in the form of continuous curves or segment coefficients. Representative indicators include the ISO 2631 [94,95], ISO 10137 [96], AISC DG 11 [1], and BS 6841 [97], among others. Figure 8 shows the relation between the weight value W_m and the vibration frequency extracted from ISO 2631 [95]. A higher value in this curve indicates that the vibration under the corresponding frequency is more likely to induce human discomfort.

5.4. Consideration of the Uncertainties in the Subjective Feelings of the Structural Occupants

It can be seen from the summary that the vibration comfort is usually evaluated by a representative value of the structural acceleration response, and the current standards are mainly divided into simple and easy-to-use frequency-independent ones and more refined frequency-dependent ones. Although it is straightforward to evaluate the vibration serviceability according to the given limit value, it is difficult to consider the uncertainties of the human body’s own feelings. Song used probability theory and fuzzy mathematics to express the uncertainty of human body’s own sensory ability and comfort level, which is the first attempt to consider the uncertainties in the subjective feelings of the occupants [98]. More advanced criteria for reliability evaluation of vibration serviceability based on fuzzy mathematical method is a future research topic for the reliability-based design of human-induced structural vibration.

6. Reliability-Based Vibration Serviceability Design and Control

To prevent vibration serviceability problems, the vibration serviceability design needs to be carried out during the structural design stage. In this process, there are roughly two types of strategies, including (a) Changing the dynamic properties of the main structure, e.g., mass or frequency; and (b) Attaching non-structural components, like dampers, to improve the damping effect of the system. In the engineering practice, the first strategy is rarely adopted because the cost to change the structural mass or stiffness is usually very high. Therefore, using the vibration control techniques to reduce excessive structural vibration induced by human excitation becomes the first choice for engineers [99].

The vibration control methods are usually divided into active control [100], semi-active control [101], and passive control [102]. For vibration serviceability problems, the active control method is rarely used because of its requirement on the significant external energy. Without such an energy input, the active control will not function [103]. The errors in the input commands for active dampers may also lead to instability. For semi-active dampers, although a much lower cost is required compared to the active control methods [104], external energy is still necessary. On the other hand, the passive control methods, represented by tuned–mass–dampers (TMD) and multiple tuned–mass–dampers (MTMD), have been widely adopted in the vibration control of footbridges [105], large-span floors [106], and other structures subjective to human excitation. In these applications, a determinative index, usually the structural peak acceleration, is adopted as a key value to be reduced or minimized. The position, mass, and other parameters of the TMD or MTMD are thus optimized. Representative works include but are not limited to [62,107,108,109,110,111].

With the development of design theory, the randomness and reliability are more and more considered in the process of structural design, which aims at the development to the performance-based design. For reliability-based control strategies, the randomness of the system, including the eternal load and structure, is considered, and the purpose is to maximize the dynamic reliability or minimize the financial cost while keeping the reliability larger than a threshold value. This trend has been affecting the research field of vibration control under seismic excitation [112,113] and wind excitation [114,115], where the modern techniques represented by topology optimization have been utilized.

From the review above, it is seen that the reliability-based vibration serviceability design is still at its initial stage. With the continuous development of advanced algorithms and knowledge on the uncertainties of the human–structure system, the vibration serviceability design is to develop towards the performance-based design with higher accuracy to ensure the human comfort under external excitation.

7. Summary and Conclusions

This paper has reviewed more than 100 references under the topic of human-induced vibration serviceability problems. These references cover the topics of human-induced load measurement, load model, human body dynamic parameters and the interaction mechanism, random vibration analysis methods, reliability analysis, human comfort evaluation, and the performance-based vibration control methods. From the selected references, the history of the development of human-induced vibration serviceability problems is clearly observed, i.e., from the initial measurement to the state-of-the-art optimization techniques.

Compared to the research on other types of dynamic excitation like earthquakes and wind, the research on human-induced vibration is not as developed. The first reason is because this research field was not fully established until the well-known accident of Millennium Bridge happened, which is much later than the field of seismic and wind engineering. The second reason is that the mechanism of the human–structure system is complicated. Excitation is characterized by strong randomness, which is sometimes related to the biomechanical features of the human body that are not clearly represented by a physical model. Owing to the efforts of the researchers during the past two decades, the feature of the human-induced load, as well as its randomness, has been thoroughly studied, and the mechanism of the human–structure system was also investigated. Development in the advanced computational methods for human-induced structural responses, including response spectrum methods, has been proposed by researchers and included in some design codes for vibration serviceability [116,117]. Some of these methods can also consider the human–structure interaction effect [118,119].

Although researchers have been studying the human-induced vibration serviceability problem for decades and have made significant achievements, there are still many problems to be addressed to achieve the performance-based structural design. First of all, the human-induced vibration serviceability problem has not been fully studied from the perspective of dynamic reliability, which is the foundation of the performance-based structural design. However, because each human body has its own dynamic properties, the crowd–structure system is regarded as a high-dimensional random system that has a high requirement on the computational power. Therefore, advanced computational methods suitable for such a crowd-structure system are desired. Second, the uncertainties in the human comfort, which is the target of the vibration serviceability design, are well-acknowledged to be related to many affecting factors and are far from being clearly investigated. Development in the fuzzy mathematics and fuzzy reliability analysis may provide a tool for the evaluation on the subjective feelings on the human-induced structural vibration. Third, the design methodologies that directly treat the vibration serviceability as the controlling factors are still lacking. The recently developed topology optimization technique has a potential for the vibration serviceability design, which may also consider the dynamic reliability as the design goal. Moreover, the effect of other types of loads (e.g., ground motion and wind) on the human-induced vibration serviceability also must be further investigated [120]. The performance-based vibration serviceability design can be achieved only when the above problems are clearly addressed.