1. Introduction
With the progress of building technology and the application of high-strength materials, “Large span, low self weight and low damping” is the development direction of building structures, and various types of long-span floor forms have been invented [
1,
2]. The Vierendeel sandwich plate is a new type of structure that is widely used in long-span industrial and public buildings. Crowd aggregation is inevitable during normal use of floors. A floor is characterized by light weight and small vertical stiffness. A large vibration response is easily produced under a pedestrian load. At the least, it will cause people’s discomfort, and at the worst, it will lead to fatigue damage of the floor and reduce the service life of the structure [
3,
4]. Historically, the Millennium Bridge in London, Techno Mart building in Korea, and other projects had to be stopped due to excessive human-induced-vibration response. It can be seen that the structural vibration caused by a human-induced load has become a problem that must be considered in the design of long-span structures.
A hollow sandwich plate is a bidirectional stress hollow structure that is composed of a surface concrete slab, top and bottom chords, and shear connectors. Compared with the general frame structure, bidirectional stress, good integrity, and the use of less steel are its characteristics. The structure of a Vierendeel sandwich plate is shown in
Figure 1. It has a wide application prospect in large-span multistory buildings [
5]. As a large-span, lightweight floor structure, a steel Vierendeel sandwich plate may be sensitive to vibration due to its own structural characteristics. It is necessary to deeply study the influence of human-induced loads on Vierendeel sandwich plates, as well as the comfort evaluation method.
At present, the human-induced-load model and comfort evaluation standards are the main research directions of domestic and foreign scholars on the human-induced vibration of long-span floors. For the study of loads, the load model is mainly established on the basis of the single-step drop test. Based on a large number of tests, a variety of periodic walking load models have been proposed by researchers [
6,
7,
8]. On this basis, Chen Jun et al. used the probability density evolution method to analyze the impact of load randomness on the vibration response of a floor. It was considered that the randomness of the pedestrian load had a significant influence on the vibration response of the floor, and the randomness of the pedestrian load should be considered in the comfort evaluation of the floor [
9]. In terms of comfort assessment, the current project was mainly implemented with reference to some national or industrial standards. Standards set by the American Institute of Steel Construction [
10] and Precast/Prestressed Concrete Institute [
11], as well as in the UK Concrete Society’s
Technical Report No. 43 [
12] and the Concrete Centre’s
CCP-016 [
13] are widely used. Human-induced loads are divided into general walking loads and rhythmic loads under AISC and PCI standards, and different acceleration limits and frequency limits are given.
CCP-016 states that when the first natural vibration frequency of a floor is lower than 4.2 times the fundamental frequency of the pedestrian load, resonance will occur; and when the first natural frequency of the floor is greater than 4.2 times the fundamental frequency of the pedestrian load, the floor vibration is mainly caused by effective impact. Two methods that can be used to evaluate the structural comfort are proposed in the UK standards. One is the evaluation method based on the response factor, which assumes that the floor vibration is continuous and of the same amplitude. Another evaluation method is based on the vibration dose value, which considers a possible pause in the vibration process and the impact of different vibration amplitudes on human comfort, and can be used for long-term evaluation of the comfort degree. In a comfort evaluation, the analytical or numerical calculation results are usually directly compared with the standard values used in the engineering community to determine the comfort level. However, the span of a long-span floor can reach tens of meters, and pedestrian comfort is affected by the spatial and temporal distribution characteristics of the stimulated points and the feeling points. Therefore, it is of great significance to study the human-induced-vibration response distribution of long-span floors for reasonable evaluation of comfort performance.
Many scholars have studied the vibration comfort of structures. The most basic research on comfort is the human motivation model. The human-induced motivation mode is the most basic research of comfort. In 1961, Harper completed the earliest walking load test with a force-measuring plate, and stated that the walking load curve was M-shaped [
6]. Subsequently, several researchers tested the load of human walking using the direct or indirect method [
7,
14] and analyzed the influence of walking speed, shoe type, ground characteristics, and other factors on the load model. Chen et al. adopted optical motion-capture technology in which the reflective marks of key parts of the human body were captured by high-speed infrared cameras [
15]. This was used to identify the human motion space trajectory to obtain human walking parameters. At present, the time-domain model of load is mainly used in the comfort analysis of structures. The Fourier series model was the most commonly used in existing pedestrian load model research. The dynamic load factor is included in the Fourier series model, and the load mode is directly affected by the value of the dynamic load factor. Therefore, the dynamic load factor has been studied by many researchers [
8,
16]. In conclusion, it is generally believed that the vertical first-order dynamic load factor is around 0.3~0.5. In terms of the comfort analysis method, the time-domain analysis method based on finite elements is an effective method. Zhu et al. considered the interaction between pedestrians and structures, and used an ANSYS finite element software simulation to study the structural vibration comfort of a two-story cantilevered steel truss floor deck in the Gansu Science and Technology Museum as the engineering background [
17]. Cao et al. conducted an experimental study of the human-induced vibration of a large-span composite floor based on a single-person foot-load model in order to meet the comfort requirements and control the floor [
18]. Wang et al. carried out an human-induced-vibration test and an ANSYS finite element analysis of a large-scale glulam arch bridge model in order to study the human-induced-vibration characteristics of a wooden-structure pedestrian bridge [
19]. Peak acceleration is used for comfort performance evaluation. Based on the walking route method, the vibration comfort performance of a steel Vierendeel sandwich plate was analyzed by Jiang et al. under different walking routes [
20]. Based on the existing load model, a finite element numerical calculation method was adopted in the above research to analyze the maximum response of different types of structures under human-induced excitation and to evaluate the comfort. However, for long-span floors, the vibration responses of different positions on the floor will differ greatly under human-induced excitation. If the maximum response is simply used to evaluate the comfort of the entire floor, the evaluation will inevitably be too conservative. Therefore, for large-span floor structures such as a Vierendeel sandwich plate, it is necessary to analyze the vibration-response-distribution characteristics of the floor under human-induced excitation in order to provide a basis for the reasonable evaluation of comfort performance.
For the evaluation of comfort, there are mainly two methods: one is to limit the natural vertical vibration frequency of the floor above a certain value, which is called the frequency threshold method. Another method requires that the dynamic response (such as acceleration and speed) of the floor under a given human-induced load does not exceed a certain limit, which is called the dynamic response threshold method. The dynamic response threshold method is widely used because it considers multiple factors of floor vibration and can better evaluate comfort. In the evaluation, the vibration acceleration response is often used as the index. Under the British BS 5400 standard [
21], the peak acceleration of the structure is used as the pedestrian comfort limit index, and a function of the vertical first-order frequency of the floor is given as the acceleration limit. Similar acceleration limits are also given in the European EN 1990 standard [
22]. The International Organization for Standardization stipulates in its ISO 10137 that when the acceleration response is less than the vibration comfort limit, the comfort is considered to meet the requirements; otherwise, the comfort is considered to not meet the requirements [
23]. The German EN03 standard states that pedestrian comfort cannot be simply divided into comfort and discomfort. The vibration comfort level should be divided in detail according to the natural vibration frequency and structural acceleration response [
24]. The Chinese specifications GB50010-2010 [
25] and JGJ3-2010 [
26] refer to ISO standards, and the peak acceleration limits for floors with different natural frequencies are given. Among the above standards, the evaluation of peak acceleration is adopted in BS 5400, EN 1990, EN03, and Chinese standards, and the evaluation of peak acceleration and RMS acceleration is adopted in ISO 10137. In the application of standards, Fiore et al. proposed a practical probabilistic method for evaluating bridge reliability based on a histogram. Useful estimates of the probability of exceeding the predefined human sensitivity limit were provided by histograms [
27]. The above published standards provides references for the evaluation of the human-induced-vibration comfort of large-span Vierendeel sandwich plates.
In this paper, the steel Vierendeel sandwich plate at the Guizhou Museum was taken as the research object. According to the functional characteristics of steel Vierendeel sandwich plates, the response characteristics of the floor with time and space were analyzed using a time-domain method, and the effects of different factors on the acceleration-response distribution were studied. A corresponding distribution mathematical model was constructed, and a comfort evaluation method based on the floor area comfort assurance rate was proposed.
4. Comfort Evaluation Method Based on Comfort Assurance Rate
At present, when evaluating the comfort degree, the peak response evaluation criterion is adopted in the codes of various countries; that is, the response of the floor under a human-induced load is not greater than the specified value. However, through the analysis conducted in this paper, it was found that for the large-span steel Vierendeel sandwich plate structure, the response peak distribution was funnel-shaped (
Figure 7). The area with a large response only accounted for a small part of the total area of the floor. The maximum value was located in the center of the floor, and decayed sharply to the surrounding areas. In addition, the value at each point of the floor was also the maximum value on the acceleration response time history curve, and the duration of the maximum value accounted for a very small proportion of the entire response process, as shown in
Figure 20. In addition, the maximum value on the acceleration response time history curve was taken as the value of each point of the floor, and the duration of the maximum value accounted for a very small proportion of the entire response process, as shown in
Figure 20. Therefore, we found that the current peak acceleration evaluation scheme commonly used in engineering is too conservative. This paper attempted to establish a comfort evaluation method based on the floor area comfort assurance rate.
To reflect the proportion of the area with an acceleration response on the floor that was less than a certain value in the total floor area, we introduced coefficient
λ, which we defined as the floor comfort assurance rate. According to the definition, it can be calculated according to Equation (10):
where
r is the radius of the circular area when the peak acceleration response on the floor was greater than a and
L is the side length (or span) of the floor, as shown in
Figure 21.
Referring to the unified standard reliability design of building structures (GB50068-2018) [
34], we took the peak acceleration
a0.95 corresponding to
λ = 95% as the representative value of the floor acceleration response and compared it with the specification limit (Equation (11)) used for comfort evaluation:
where
a is the allowable value of the specification for human-induced-vibration acceleration; the value can be taken from Refs. [
35,
36].
We substituted
λ = 95% into Equation (10) to obtain:
We substituted Equation (12) into Equations (5) and (6) to obtain the representative value of the acceleration response under resonance and nonresonance, which could be calculated according to Equations (13) and (14), respectively:
By substituting the fitting values of the parameters in
Table 8 into Equations (13) and (14), the representative value of the acceleration response of the floor could be obtained. The comfort performance of the floor was evaluated using Equation (11).
For the numerical example in this study, the evaluation method used was compared with the maximum evaluation method; the relationship between the values of
a0.95 and
amax are shown in
Table 10.
It can be seen in
Table 10 that the percentage of
a0.95 in
amax was affected by the span, concrete slab thickness, and whether resonance occurred. Overall, it was between 54.73% and 86.51%. In the case of nonresonance, the proportion of
a0.95 in
amax was between 54.73% and 78.78%. In the case of resonance, the proportion of
a0.95 in
amax was between 75.77% and 86.51%. It can be seen in the above analysis that the representative value of the evaluation could be greatly reduced by using
a0.95 to avoid being too conservative.