Dynamic Response Measurement and Finite Element Analysis of Large-Span Pedestrian Corridor

Abstract

The natural frequency of the long-span steel structure corridor is close to the pedestrian step frequency, which makes it very easy to cause resonance. This paper aims to study crowd-induced vibration control of long-span steel pedestrian corridors with different dynamic characteristics by combining methods of site measurement and numerical simulation. First, based on the steel structure corridor project of a multi-tower structure, the field modal test and the acceleration response under pedestrian load excitation are measured, and the dynamic characteristics and acceleration response under different frequency pedestrian loads are studied. Then, the finite element model of the large-span corridor is established, and the results of the measured and numerical simulation are compared and analyzed. Finally, with the relevant norms, a reasonable evaluation of pedestrian comfort is carried out. The results show that this paper’s measured and finite element results have a certain accuracy. The damping characteristics of humans can absorb the vibration energy of the structure to reduce the vibration acceleration of the structure, and the results are conservative when human action is not considered. After installing the TMD system, the acceleration response of the corridor is significantly reduced, and the vibration reduction effect reaches 54%, which meets the comfort control requirements of the large-span corridor under pedestrian load excitation. The research results and methods in this paper can have particular engineering practical values for carrying out field measurements and comfort control in similar projects and provide a reference for engineering designers.

1. Introduction

In recent years, with the continuous development of high-rise technology, more and more attention has been paid to the functional requirements of buildings to ensure safety and quality. The frequency control problem is becoming increasingly prominent for long-span structures, which has become an important factor affecting the structure. The main problems are size and the architectural effect. Many projects at home and abroad have adopted a long-span multi-tower structure, and corridors connect the towers. On the one hand, its role is to connect different buildings by setting up corridors to form a relatively whole structure, making the traffic between buildings more convenient and pedestrians more comfortable and faster; on the other hand, due to the unique shape of the multi-tower structure, it brings a solid visual impact and makes the building more distinctive. The corridor’s design is the same as the design of the urban bridge, and more and more attention is being paid to buildings’ safety, durability, and comfort [1]. With the continuous development of material properties, the span of corridors is also increasing. Its natural frequency is also significantly improved and close to the pedestrian step frequency, which can easily cause corridor resonance. Resonance will adversely affect the safety and use of the structure [2]. The fundamental frequency has been used to calculate the flexural rigidity of thin-walled box-girder bridge members. An accurate fundamental frequency value is also a proper indicator for axial load estimation in steel beams, bridge cables, and hangers [3,4]. The larger-span bridge has a significant vibration, which is harmful to the safe use of the project [5]. Some bridges in Japan and France also show apparent resonance. Such as Japan’s T and M bridges [6] and France’s Sofrino bridges [7]. It can be seen that understanding the dynamic characteristics of large-span structures is of great significance for their safe use.

Domestic and foreign scholars have conducted a lot of research on the human-induced vibration of footbridges and their vibration reduction methods. Xu first applied MTMD to single-degree-of-freedom systems to reduce the resonance caused by broadband random excitation and proposed the idea of MTMD with linear frequency distribution [8]. Fu et al. used the force platform to give the vertical component of the single-step excitation load and the horizontal cross-bridge component drop foot curve. They considered that the vertical component curve had two peaks and a trough [9]. Andriacchi carried out experimental research. The research shows that with the increase in step frequency, the vertical excitation load under continuous walking increases obviously, and the step frequency significantly affects the amplitude and waveform of the vertical excitation load time history curve [10]. Galbraith tested the vertical component of the excitation load caused by different movement modes of pedestrians, from slow walking to running, and pointed out that the running excitation load had only one peak [11]. Wheeler conducted a more detailed study of the walking load, systematically summarized the results of excitation load tests from slow walking to running, and summarized the relationship between walking load parameters (peak value, time, etc.) and the step frequency. It is pointed out that, in general, the amplitude, stride, and pace of the walking load increase with an increase in the step frequency [12,13]. Matsumoto verified through experiments that the step frequency of normal walking obeys a normal distribution with an expectation of 2.0 Hz and a standard deviation of 0.173 Hz [14,15]. Bachmann believes that the frequency range of a pedestrian walking is 1.6 to 2.4 Hz, running is 2.0 to 3.5 Hz, bouncing is 1.8 to 3.4 Hz, and the horizontal body swing is 0.4 to 0.7 Hz at rest [16].

In addition, the phenomenon of the human–bridge interaction during the vibration of the footbridge is almost inevitable because the human is itself a complex control system with an automatic control function. In general, the problem of the pedestrian–bridge vibration interaction needs to be considered in two aspects [17,18]. First, the appearance of pedestrians on the bridge deck changes the dynamic characteristics, such as the damping and natural frequency of the pedestrian bridge. Mourning believed that even if the primary natural frequency of the footbridge is not in the normal walking frequency range (1.8~2.2 Hz), the synchronization effect of the crowd excitation load should also be considered [19]. In practical use, some footbridges have problems such as excessive vibration and other performance problems, which need to be solved by people. People feeling the vibration is a complex problem related to the intensity of the structural vibration, the environment in which people live, and the sensitivity of the people. Many countries in the world have detailed regulations and descriptions on structural comforts, such as the International Organization for Standardization ISO10137-2007 [20], the American steel structure design guide AISC-11 [21], the British steel concrete bridge design specification BS5400 [22], the European specification Eurocode: Basis of Structural Design [23], and the Swedish specification Bro200, etc. [24].

In summary, many scholars’ research on pedestrian vibration control is biased towards pedestrian bridges, and there is a lack of definition of failure modes and corresponding thresholds [25,26]. There are few studies on the vibration control of corridors, and many cases lack reference to measured data. Currently, the research on the walking ‘synchronization’ phenomenon is mainly based on the observation results of footbridge vibration examples, which are still based on preliminary perceptual knowledge. TMD vibration control is only used in a few footbridge structures in China, which is the content of thematic research. The evaluation of its application effect needs to be further explored and determined. Therefore, it is of great practical value and significance to discuss the design parameters and vibration reduction effect of the TMD and how to maximize the vibration reduction performance of the vibration reduction device. This paper introduces the vertical vibration test of an actual multi-tower structure corridor equipped with a TMD and compares the field test results with finite element analysis. Finally, the comfort evaluation is given according to the Chinese code. It has specific practical value for the project’s vibration control and comfort evaluation, which can be used as a reference for engineering designers.

2. Experimental Setup

2.1. Test Corridor and TMD Parameter

The measured project has a total of 7 steel structure corridors, distributed between the tower’s second and fourth floors, as a connection between different towers, and are only for pedestrians to walk across. The corridor of this test is one of the most significant spans. The total length of the corridor is 36.7 m, and the effective width of the pedestrian passage is 2.5 m. The section is a similar steel box girder structure composed of steel plates. The material is Q345 steel. The elastic modulus is 206 GPa, the Poisson’s ratio is 0.3, and the damping ratio is 0.02. The actual view of the corridor is shown in Figure 1. Different from conventional footbridges or corridors, they are relatively regular shapes. The corridors tested in this test are narrow in the middle of the span and wide at both ends. This situation mostly occurs in major shopping malls, mainly to increase tourists. Here, we only give the cross-section of the central part of the corridor as plotted in Figure 2.

In this paper, the TMD system is installed in the corridor span. The working principle of the TMD is mainly to add a spring-mass damping system to the structure. The resonance phenomenon is reduced by changing the frequency of the additional system to make it close to the frequency of the main structure. In general, the additional mass positively correlates with the damping effect but will also increase the cost. When ignoring the damping of the central system (usually, the damping is minimal), the optimal parameters of the additional TMD system [27] are shown in Equation (1):

$${\xi }_{opt}=\sqrt{\frac{3\mu }{8{\left(1+\mu \right)}^3}}\hspace{1em}\hspace{1em}{f}_{opt}=\frac{f}{1+\mu }$$

(1)

where ξ_opt is the optimal damping ratio; μ is the mass ratio of TMD to the main structure; f_opt is the natural frequency of TMD; and f is the main structure frequency.

After the optimization analysis, when the mass of the damper reaches 2 t, the damping effect of the corridor is the best. The TMD is installed in the middle span of the corridor. The TMD design parameters are shown in

Table 1. TMD design parameters.

Quality (kg)	Vibration Frequency (Hz)	Spring Stiffness (N/m)	Damping Coefficient (N·s/m)	Damper Stroke (mm)
2000	1.90	285,034	2388	±50

.

2.2. Pedestrian Incentive Model

Pedestrians walking in the corridor will produce three vertical, horizontal, and vertical force directions. This paper only considers the vertical pedestrian load that may cause vertical resonance. The pedestrian load can be regarded as a cyclic load, and the calculation formula is as follows [28]:

$$F\left(t\right)=P_0\left[1+{\displaystyle \sum _{i=1}^n{\alpha }_i\sin \left(2\pi i{f}_{s}t-{\phi }_i\right)}\right]$$

(2)

where P₀ is the per capita weight, usually P₀ = 700 N; α_i is the dynamic factor of the i order loading excitation; f_s is the walking frequency (Hz); t is time; and φ_i is the phase angle of the i order loading excitation; to simplify the calculation, only the influence of the first three orders is generally considered. This equation is Equation (3):

$$F\left(t\right)=P_0\left[1+{\displaystyle \sum _{i=1}^3{\alpha }_i\sin \left(2\pi i{f}_{s}t-{\phi }_i\right)}\right]$$

(3)

where $\alpha i=0.4+0.25\left(f_s-2\right)$; ${\alpha }_2={\alpha }_3=0.1$; ${\phi }_1=0$; and ${\phi }_2={\phi }_3=\pi /2$.

There are two main differences between jumping and walking. One is that jumping impacts the floor more, which dynamic factors can reflect. The second is that when the feet leave the deck, there is no force for a short time. The load-time curves of jumping and walking are placed in Figure 3. The calculation method of the structural response under the crowd-jumping load of the power spectrum model is widely used in the comfort design of structures [29].

$$F\left(t\right)=\{\begin{array}{l}{k}_{p}G\sin \left(\pi t/t_p\right);0\le t\le t_p\\ 0;t_p\le t\le T_p\end{array}$$

(4)

$$k_p=\pi /\left(2\alpha \right)$$

(5)

where G is human body weight; k_p is the jump dynamic factor; and t_p and T_p represent the load contact time and bounce period, respectively. To simplify the calculation, usually α takes 1, k_p is calculated to be 1.5, so that Equation (5) can be simplified to Equation (6):

$$F\left(t\right)=1.5G\sin \left(2\pi f_{s}t\right)$$

(6)

The simulation of random crowd walking assumes a continuous and stable flow of people on the bridge deck. Pedestrians are evenly distributed on the corridor, with the same step frequency, but the phase angle is randomly distributed. According to the probability theory, the continuous and stable crowd load on the corridor is equivalent to the Np synchronous walking load [29]. When the crowd density is less than 1, the equivalent number of people is calculated as shown in Equation (7):

$$N_p=\sqrt{n}$$

(7)

2.3. Test Scheme

This test consists of two parts. One is to use the random vibration method to test the structure’s dynamic characteristics for modal identification, as shown in Figure 4. The second is based on the response test under the excitation of crowd load, which adopts the principle of the equivalent number of people to simulate the actual walking conditions, as drawn in Figure 5. For a single-span simply supported beam, the maximum structural vibration occurs in the midspan, so the measuring point is generally arranged in the midspan and 1/4 span, and the specific measuring points MP1~MP3 are arranged as plotted in Figure 6. Because the corridors are mainly based on a low-frequency natural frequency vibration, the vibration response frequency of concern is generally not more than 50 Hz, so the 941-B sensor of the Institute of Engineering Mechanics of China Earthquake Administration is selected, and its frequency band range is 0.25~80 Hz, which meets the test requirements. The acquisition instrument uses the UT89 series dynamic acquisition system.

The vibration of the large-span corridor under pedestrian excitation has excellent randomness. The reason is that the random distribution of pedestrians, crowd activity, stride, and the time interval through the corridor are random. Suppose the natural frequency is within the bandwidth of the crowd excitation. In that case, resonance will occur, the response is significant, and it is easy to exceed the comfort limit, causing human discomfort and even catastrophic events. To simulate the various pedestrian loads encountered during the corridor service as much as possible, we designed 10 cases, including continuous walking, running, and jumping, as listed in

Table 2. Definition of pedestrian load cases.

Type	Cases	Frequency (Hz)	Area (m2)	Equivalent Number	Single Quality (kN)	Single Point Load Amplitude (kN)
Walking	S1	1.60	250	21	0.7	0.056
	S2	1.90	250	21	0.7	0.056
	S3	2.20	250	21	0.7	0.056
	S4	2.50	250	21	0.7	0.056
Running	S5	2.70	250	21	0.7	0.056
	S6	3.00	250	21	0.7	0.056
	S7	3.30	250	21	0.7	0.056
Jumping	S8	1.80	250	10	0.7	0.027
	S9	2.70	250	10	0.7	0.027
	S10	3.30	250	10	0.7	0.027

. Cases S1~S4 are crowd walking loads, and the influence of different frequencies on the vibration characteristics of the corridor is considered. Cases S5~S7 and S8~S10 are running and jumping pedestrian loads, respectively, and the influence of different frequencies on the dynamic response of long-span corridors is also considered.

3. Analysis of Test Results

3.1. Ambient Vibration Test

To determine the essential dynamic characteristics of the corridor, a vibration pickup is arranged in the middle of the corridor (MP2). The vibration pickup is connected to the acquisition system through a cable, and the obtained data are post-processed in Matlab [2]. During the measurement, various dynamic loads on the corridor are controlled to make it in a state of unmanned walking, and the influence of other construction dynamic loads is minimized. The structural response signal of environmental excitation is used to identify the modal parameters of the system. Finally, the singular value is used for the frequency domain decomposition to obtain the natural frequency in the vertical direction. Figure 7 shows the acceleration time history curve collected by environmental excitation for 100 s. Figure 8 is the acceleration frequency spectrum of the corridor modal test. From the diagram, we can see that the first-order vertical fundamental frequency of the corridor is 1.95 Hz, which does not meet the “Technical Specification for Urban Pedestrian Bridges and Pedestrian Tunnels” (CJJ69-1995) [30]. The first-order vertical fundamental frequency of the pedestrian bridge must be greater than 3 Hz to avoid resonance problems.

3.2. Pedestrian Incentive Test

To measure the vibration acceleration response of the corridor under a pedestrian load, according to the equivalent number of people calculated above, 21 people were arranged to step in place at the midspan of the corridor according to the designed working frequency, and the vibration pickups were arranged in MP1, MP2, and MP3, respectively. The test conditions of the corridor cover the frequency that may produce resonance. Considering that in the actual situation, the mall corridor will not have a large crowd density, the test is arranged with a smaller crowd density of 0.5, which not only considers safety but also increases the integrity of the test data. In addition, the consistency of pedestrian stride frequency in this test has been trained before the test, and good stride frequency consistency can be achieved under the control of the metronome. Pedestrians’ walking, running, and jumping on the corridor may lead to the vibration of the corridor. Excessive vibration will lead to the physiological and psychological discomfort of pedestrians. In engineering, acceleration is usually used as the comfort control index of the structure. Technical specification for concrete structures of tall buildings (JGJ3-2010) [31] has listed the vertical vibration acceleration of the floor as the control target and pointed out that the vertical natural frequency of the floor should not be less than 3 Hz.

Table 3. Acceleration limit.

Structured Environment	Peak Acceleration Limit (m/s2)
	Not More Than 2 Hz	No Less Than 4 Hz
Residential, office buildings	0.07	0.05
Shopping malls and indoor corridors	0.22	0.15

shows the acceleration limits of vertical vibration.

The peak acceleration measured at each test position under different cases is shown in

Table 4. The peak acceleration measured at different positions.

Measuring Points	Cases	Frequency (Hz)	Peak Acceleration (m/s2)
MP1	S1	1.6	0.050
	S2	1.9	0.040
	S3	2.2	0.035
	S4	2.5	0.064
MP2	S1	1.6	0.104
	S2	1.9	0.167
	S3	2.2	0.164
	S4	2.5	0.112
MP3	S1	1.6	0.040
	S2	1.9	0.065
	S3	2.2	0.064
	S4	2.5	0.085

. It can be found that the measured maximum vertical vibration acceleration occurs in the midspan (MP2), and the maximum acceleration under case S2 reaches 0.167 m/s². Figure 9 shows the acceleration time-history curves of different measuring points when the pedestrian load frequency is 1.9 Hz (case S2). The measured acceleration time-history curve when the pedestrian load and the natural frequency of the corridor are the same. The maximum acceleration occurs in the middle of the span, and the whole time history curve presents a relatively stable state. On the other hand, it also verifies the consistency of the step frequency of the testers participating in the test. From the time history curve, the acceleration of the corridor only suddenly increased to 0.167 m/s² at about 23 s, and the acceleration of other points was significantly less than the limit value, which met the requirements of comfort as a whole.

The peak acceleration of different cases at different positions is drawn in Figure 10. The analysis shows that the acceleration of the corridor at the midspan position is much larger than that at both ends, and only the midspan is greatly affected by the excitation frequency. Under the condition of resonance, the acceleration will increase sharply, and the acceleration of both ends of the corridor under the action of pedestrian load with different frequencies is floating up and down at 0.05 m/s², which is less affected by the excitation frequency.

4. Finite Element Analysis

4.1. Model Verification

Due to the pedestrian corridor being simply supported on the pillars at the two ends, the computational models can be modelled as uniform, simple span beams as adopted [32]. Thus, the beam element is adopted to simulate the pedestrian corridor grid in the SAP2000 software V21 platform. SAP2000 software V21 is used in the model according to the size and parameters of the corridor in Figure 1 above. The main girder of the bridge deck adopts a beam element, and the beam end constraint is considered to be a simple support [2,32]. The model has 67 elements and 68 nodes. There are 14,104 grids in this paper. A rigid connection constrains the beam and boom, releasing the boom end. The two-way rotation degree of freedom around its long axis constrains the three-way displacement. The first-order vibration mode of the corridor obtained via the modal analysis is plotted in Figure 11. It can be found that the first-order vibration mode is a vertical symmetrical bending mode, and the midspan amplitude is the largest, which is very consistent with the measured 1.95 Hz. Such a consistent conclusion can be obtained because, on the one hand, the quality and stiffness of the corridor can be effectively obtained through numerical simulation. The corridor has just completed the construction of the main structure, and the decoration on the bridge deck has not yet begun to be installed, so the decoration has little effect on the overall quality and stiffness, which can be ignored. On the other hand, the pedestrians and construction around the corridor are strictly controlled during the field test.

4.2. Analysis of Finite Element Results

According to the previous working conditions and TMD parameters, the models without the TMD (WOT) and with the TMD (WT) in the middle of the span are calculated in SAP2000, respectively, and the calculation results are extracted as

Table 5. Peak acceleration comparison (m/s²).

Type	S1			S2			S3			S4			S5
	WOT	WT	Result	WOT	WT	Result	WOT	WT	Result	WOT	WT	Result	WOT	WT	Result
MP1	0.06	0.04	33%	0.16	0.06	56%	0.04	0.04	0%	0.06	0.06	0%	0.06	0.06	0%
MP2	0.10	0.09	10%	0.40	0.18	55%	0.17	0.16	6%	0.15	0.14	7%	0.15	0.14	7%
MP3	0.05	0.04	20%	0.15	0.07	53%	0.06	0.06	0%	0.05	0.05	0%	0.05	0.05	0%
Type	S6			S7			S8			S9			S10
	WOT	WT	Result	WOT	WT	Result	WOT	WT	Result	WOT	WT	Result	WOT	WT	Result
MP1	0.06	0.06	0%	0.06	0.05	17%	0.07	0.06	14%	0.04	0.04	0%	0.06	0.06	0%
MP2	0.13	0.13	0%	0.12	0.12	0%	0.18	0.15	17%	0.09	0.09	0%	0.09	0.09	0%
MP3	0.06	0.05	17%	0.06	0.06	0%	0.07	0.05	29%	0.03	0.03	0%	0.05	0.05	0%

. From the results, we can see that the vertical acceleration response of the corridor has been significantly improved after the installation of the TMD. The peak acceleration has decreased from the maximum 0.40 m/s² to 0.18 m/s², and the maximum vibration reduction effect has reached 54%, which meets the comfort limit requirements, thus avoiding the discomfort or panic of pedestrians on the corridor. Figure 12 compares vertical acceleration and the power spectrum of MP2 in the middle of the corridor under the excitation of pedestrian load in resonance conditions. From the diagram, it can be seen that the TMD system begins to play a role in about 2 s after the corridor is excited by the pedestrian, which quickly reduces the acceleration response of the corridor.

4.3. Comparative Analysis of Test and Finite Element Results

The comparison between the simulation results and the measured values of acceleration under each frequency excitation is listed in

Table 6. Comparison between simulation and test (m/s²).

Type	S1			S2			S3			S4
	Test	FEM	Error	Test	FEM	Error	Test	FEM	Error	Test	FEM	Error
MP1	0.042	0.04	5%	0.051	0.06	−15%	0.035	0.04	−14.2%	0.064	0.06	6%
MP2	0.101	0.09	10.9%	0.167	0.18	−8%	0.164	0.160	2%	0.122	0.13	−6.5%
MP3	0.04	0.04	0%	0.065	0.07	−8%	0.064	0.06	6%	0.085	0.08	5.9%

. It can be seen from the table that the maximum value of the result difference appears under the action of MP2 (case S2), and the actual value difference is only 0.035. Located at the left end of the corridor, not close to the midspan, it has little effect on evaluating comfort. In addition, the difference between the simulated and measured values of the corridor at the midspan position (MP2) is slight, and the average error is 10%. The midspan of the corridor is the main area where the acceleration exceeds the limit so that the simulation results can have a certain guarantee rate for the evaluation of comfort. The reasons for the difference between the simulation results and the measured values are analyzed. In addition to the differences in the model’s natural frequency, mass source, damping ratio, and other models, there are also the form and consistency of pedestrian excitation. The sensor at the measuring point in the excitation area is easily affected by the local influence of the incentive person, and the construction personnel and equipment will also impact the test’s accuracy. Therefore, in the case of vibration control, it is best to conduct on-site testing before formulating a control plan.

5. Conclusions

In this paper, the vertical vibration test and comfort evaluation of the pedestrian load excitation of a practical project are evaluated. The corridor model is established using SAP2000 V21 finite element analysis software, and the dynamic response of the corridor under different forms and frequencies of the pedestrian load is compared and analyzed using actual measurements and a simulation. The following conclusions can be drawn.

• Under the action of normal pedestrian walking, the vertical vibration acceleration of the corridor has exceeded the comfort limit of the specification, reaching a maximum of 0.40 m/s², far exceeding the allowable limit of the specification (0.22 m/s²). The main reason for the peak acceleration exceeding the limit caused by the need for acceleration control is that the natural frequency of the corridor is low and falls within the range of the pedestrian walking frequency.
• Under the excitation of the pedestrian load in the resonance condition, the TMD system has a good suppression effect on the vertical vibration in the middle of the corridor span, and the maximum vibration reduction effect reaches 54%, effectively controlling the vertical acceleration response of the corridor below the comfort limit. It is proven that TMD significantly affects the corridor’s comfort control and can be popularized and applied in similar projects.
• The TMD system designed by the first-order natural frequency of the corridor has no apparent effect on the vibration reduction under the excitation of pedestrian loads far away from this frequency. This is because the pedestrian load will not cause structural resonance and will not cause a significant acceleration response. It still meets the requirements of comfort.
• Most similar projects applied numerical model simulation to design TMD vibration reduction schemes. However, there is a certain deviation between the actual dynamic characteristics of the corridor and the numerical simulation. The structure’s natural frequency, mass source, and damping ratio are susceptible to the dynamic response of the corridor under pedestrian load excitation. Under this condition, the preliminary scheme should be determined via a numerical simulation to determine whether the vibration reduction control is needed. Finally, the final control scheme is formulated after the actual dynamic characteristics of the structure are tested on-site.
• The author will carry out a parameter analysis on the dynamic characteristics of the pedestrian corridor based on finite element software and theoretical analysis methods. In the theory of human-induced vibration calculation, how to use the appropriate walking force model in the motion equation is mainly considered. In evaluating pedestrian-induced vibration comfort of pedestrian corridors, it is necessary to establish more reasonable evaluation indicators and systematic evaluation methods.