Human-induced vertical vibration analysis of landscape pedestrian bridges considering assembled crowd effect

Abstract

Modern landscape pedestrian bridges, characterized by distinctive landscape features, cause pedestrians to gather in certain places. This changes the loading distribution on the bridge, consequently impacting the vibration response of the structure. Due to the high sensitivity of pedestrians to vibration, it is essential to explore the effects of assembled crowd on vibration comfort of footbridges. The present study adopts the pedestrian load model of ISO 10137 to conduct the acceleration response analysis under both static and dynamic pedestrian loads. The presence of assembled crowd effect significantly influences the bridge vibration response by altering pedestrian pacing frequency. It increases peak acceleration within the pacing frequency range of 1.8 Hz to 2.0 Hz. However, at 2.1 Hz, pedestrian gathering significantly reduces the peak acceleration. Under different pacing frequencies, the effect of assembled crowd on the comfort of bridge traffic is completely opposite. Besides, the impact of pedestrian flow direction is also considered. The “double peaks” phenomenon in the acceleration response due to bidirectional pedestrian flow may cause the root mean square (RMS) acceleration to surpass the comfort thresholds. However, the peak acceleration is considerably lower compared to unidirectional flow, which is advantageous for traffic comfort. This paper may provide insights into understanding the dynamic performance of landscaped footbridges when assembled crowd occurs and support better designs.

Keywords

Landscape pedestrian bridge human-induced vibration assembled crowd effect vibration serviceability traffic comfort evaluation

1. Introduction

Structural designers and researchers have long been concerned about pedestrian discomfort caused by vibration from bridges (Bachmann, 1992; Pimentel et al., 2001). Incidents of severe public panic induced by the vibrations of pedestrian bridges have been reported worldwide. In 1831, Broughton Bridge in England collapsed due to soldiers marching in unison (Tilly et al., 1984). Later, in 1993, Fujino et al. (Fujino et al., 1993) provided a reasonable explanation for the large transverse vibration of the T-bridge by considering the synchronization effect of the crowd based on the pedestrian load of a single person. In 2000, London’s Millennium Bridge was built, but it swayed violently on its opening day. Studies later revealed that a large number of pedestrians could cause human-bridge resonance, resulting in significant lateral movement (Dallard et al., 2001). Japan’s M-bridge, a pedestrian suspension bridge in a scenic area, has also experienced multiple vibration issues caused by human activity since its construction (Nakamura, 2003). Vertical vibration response and vibration comfort evaluation of the human-bridge system have emerged as critical aspects in the improvement of structural vibration comfort. Current research predominantly focuses on its vertical dynamic characteristics and dynamic response. Sachse et al. (Sachse et al., 2004) demonstrated that the influence mechanisms of static human bodies and moving pedestrians on structural dynamic characteristics are different. Piccardo et al. (Piccardo and Tubino, 2009) analyzed the anthropogenic vibration of pedestrian bridges under the action of different crowds and proposed the concepts of synchronization coefficient as well as equivalent amplification factor. Shahabpoor’s experimental study (Shahabpoor et al., 2017) revealed that pedestrians simultaneously increase the vibration frequency and damping ratio of the coupled system. He and Xie (He and Xie, 2018) showed that as the number of people standing on the bridge increases, the first-order vertical vibration frequency of the coupled human-structure system decreases, while the damping ratio increases.

Modern landscape footbridges are not only intended to serve functional roles within the transportation system but also to harmonize with the surrounding environment, with their structural design increasingly incorporating aesthetic and landscape-oriented elements (Zhang et al., 2017). Given that landscape pedestrian bridges often accommodate both stationary and walking pedestrians, the human-structure coupling vibration effect becomes more complex. Therefore, research on the vertical coupling vibration of bridges considering the coexistence of standing human bodies and moving pedestrians is urgently needed. Previous research has primarily concentrated on conventional pedestrian crossing behaviors, with limited attention given to phenomena such as pedestrian congregation in specific areas of the bridge and its effects on load distribution, dynamic characteristics, pedestrian-induced vibrations, and pedestrian comfort. Zhao (Zhao, 2020) and Hu et al. (Hu et al., 2024) showed that pedestrian gathering increases the modal mass and structural acceleration, causing acceleration to exceed the comfort threshold. However, they did not delve into the specific impacts of varying pedestrian densities on acceleration, nor did they consider parameters such as differing pacing frequencies in pedestrian load calculations, which introduces certain limitations. Additionally, research on short-span curved pedestrian bridges remains limited.

This study selects a curved, irregular steel pedestrian bridge with spans of 25.1 + 38.6 meters as the subject. Based on finite element analysis software and the standard ISO 10137 (ISO 10137, 2007), a comparative analysis is conducted on the bridge’s dynamic characteristics and pedestrian comfort, both with and without assembled crowd effects.

2. Pedestrian behavior characteristics and overview of footbridges

2.1. Pedestrian behavior characteristics survey

To understand the characteristics, behavior patterns, and walking habits of pedestrian on typical landscape pedestrian bridges, a questionnaire was designed and 432 valid responses were collected. The survey covered age, frequency of passage, perception of the environment, and psychological state of pedestrians during crossing. Only pedestrians aged 14 and above were surveyed, with most respondents between 20 and 60, matching the age distribution of primary outdoor activity participants.

In terms of crossing frequency, 162 respondents (37.5%) had visited the bridge only once, primarily tourists, while 198 (45.83%) reported crossing more than once per month, mainly local residents. During the survey, 208 respondents (48.15%) exhibited lingering behaviors, such as taking photos or enjoying the scenery, closely aligning with the percentage who reported crossing “once every 6 months or less.”

Pedestrians on scenic pedestrian bridges can be broadly categorized into two main groups: regular commuters and tourists. Regular commuters, who use the bridge primarily as a thoroughfare, treat it much like a traditional pedestrian bridge and generally do not tend to stay. In contrast, tourists are more inclined to linger in scenic areas on the bridge, which highlights the bridge’s landscape features and distinguishes its usage pattern from that of conventional pedestrian bridges.

The lingering behavior of tourists creates a unique pedestrian load distribution on the bridge deck. This is primarily due to resting platforms and scenic views that encourage tourists to gather in specific areas. This assembled crowd effect is especially pronounced during peak periods (see Figure 1), resulting in a combination of dynamic and stationary pedestrian flow on the bridge. The dynamic flow mainly consists of passing pedestrians, while the stationary flow is formed by tourists who stop to enjoy the scenery. This mixed distribution pattern presents new challenges for bridge design and maintenance, as combined dynamic and static loads can impact pedestrian comfort.

As shown in Figure 1(a), when pedestrian traffic is low, the few stationary tourists have a negligible impact on bridge traffic due to the ample space available. However, as pedestrian volume increases, more people stay on the bridge, often gathering in specific areas of the bridge deck. As illustrated in Figure 1(b), this crowding creates localized high-density areas, with stationary groups occupying space and narrowing the passage for movement. This increase in local pedestrian density disrupts the flow of moving pedestrians. According to Lee (Lee, 2016), in situations with a large number of pedestrians, normal walking flow can be disrupted, reducing the comfort level for crossing. This interference is particularly pronounced in scenic areas, where the movement patterns of tourists and regular commuters may conflict. Therefore, this study examines the impact of high-density pedestrian flows, taking into account the combined distribution of moving and stationary pedestrians and the interactions between gathering tourists and moving pedestrians at varying densities.

2.2. Overview of pedestrian bridge

The main beam of the pedestrian bridge adopts a special-shaped steel structure. The plane line type is a three-circle curve, with a total length of 82.73 m and a span arrangement of (25.1 + 38.6 m). The width of the bridge deck varies along the axis of the bridge, and the net width of the bridge deck is 3.4 ∼ 6.2 m. The middle section of the bridge is provided with a rest platform with a width of 1.5 m. A finite element model of the bridge was developed using ANSYS software, employing BEAM4 elements for the beams and MASS21 elements to simulate the entire structure. The completed finite element model is shown in Figure 2.

In the analysis of this pedestrian bridge, the ISO 10137 pedestrian load model was employed Dynamic calculations were conducted based on this model, yielding the acceleration response of key bridge nodes. The calculation results were analyzed, and the comfort level of this bridge was evaluated.

In this study, to accurately simulate the load and deformation characteristics of the pedestrian bridge structure, all degrees of freedom at both ends of the finite element model were fully constrained to represent fixed supports. Additionally, rigid connections were defined between the auxiliary piers and the main girder, consistent with the actual bridge structure.

3. Pedestrian load modeling and comfort evaluation criteria

In the time domain, mathematical models for human-induced loads can be categorized into two types: deterministic models and stochastic models. Deterministic models assume that human-induced loads follow strict periodicity. In evaluating the performance of pedestrian bridges, deterministic models are widely adopted by many countries and organizations due to their superior engineering practicality.

Li et al. (Li et al., 2021) summarized the comfort evaluation criteria from the ISO 10137 and EN 03 standards. Among them, The ISO 10137 standard defines comfort thresholds for pedestrians in two distinct states: stationary and walking. This standard considers the differing perception of vibrations by pedestrians in these two states. Consequently, in this study, the load model and comfort assessment for human-induced vibrations are based on ISO 10137.

3.1. Single-person load model

Vertical:

F_{V} (t) = Q (1 + \sum_{i = 1}^{k} α_{i, v} \sin (2 π i f t + φ_{i, v}))

(1)

Lateral:

F_{V} (t) = Q (1 + \sum_{i = 1}^{k} α_{i, h} \sin (2 π i f t + φ_{i, h}))

(2)

where:

α_{i, v}

and

α_{i, h}

are the vertical and lateral dynamic load factors for the ith harmonic, respectively;

Q

represents the pedestrian’s weight;

f

is the pedestrian’s pacing frequency; and

φ_{i, v}

and

φ_{i, h}

are the vertical and lateral phase angles for the ith harmonic, respectively.

3.2. Crowd load model

F {(t)}_{N} = N \times F (t) \times C (N)

(3)

where: N represents the number of pedestrians;

F (t)

is the single-person load model; and

C (N)

is the coordination factor, with the expression given by

C (N) = \sqrt{N} / N

The ISO 10137 specifies that, when calculating the dynamic response of a pedestrian bridge, the crowd load should be applied to the bridge in the form of a dynamic load, with a movement speed of 0.75 f (m/s).

3.3. Vibration comfort evaluation index

ISO 10137 establishes a reference curve for the root mean square (RMS) acceleration threshold at different frequencies. For walking pedestrians, the vertical RMS acceleration threshold is 60 times that of the reference curve, while for stationary pedestrians, it is 30 times. The lateral RMS acceleration threshold is set at 60 times that of the reference curve. According to this standard, the relationship between the final RMS acceleration limits and frequency is shown in Figure 3. If the RMS acceleration of the pedestrian bridge’s vibration is below the comfort limit, the bridge is considered to meet comfort requirements.

3.4. Loading methods

In this study, harmonic loads calculated based on equation (3) were applied to perform a dynamic loading analysis on the pedestrian bridge. The analysis accounted for the dynamic process of pedestrians gradually entering the bridge to better simulate real-world conditions. The entire analysis lasted for 90 seconds, with a time step of 0.1 seconds.

Regarding the loading modes, the study simulated two scenarios: unidirectional and bidirectional pedestrian flow. For the unidirectional flow, it was assumed that pedestrians enter the bridge sequentially from the shorter span of the pedestrian bridge. In the case of bidirectional flow, pedestrians entered the bridge simultaneously from both ends. The pedestrian load was modeled as a vertical nodal dynamic load, accurately representing the vertical force exerted by the crowd on the bridge deck.

4. Finite element analysis

4.1. Modal analysis

The Lanczos algorithm was utilized to analyze the intrinsic frequency of the bridge. The structural dynamic characteristics obtained were compared with those from the field environmental vibration test. The comparison of the natural frequency is shown in Table 1. These results validate the reliability of the ANSYS model. It is widely accepted that pedestrian bridges with a fundamental frequency above 5 Hz are unlikely to encounter pedestrian-bridge resonance issues (Xu et al., 2021). Therefore, this study conducts a comfort analysis based on the first mode shape.

Table 1.

Modal analysis results.

Modal step		1	2	3
Frequency (FEM model)/Hz		2.908	7.149	7.670
Frequency (measured)/Hz		3.005	6.694	7.473
Frequency error		3.23%	6.80%	2.64%
Shape participation mass (%)	X	0.10	33.80	0.54
	Y	35.62	3.74	14.81
	Z	1.16	27.39	50.03
Shape description		Vertical bend	Vertical bend and lateral bend	Bending and torsional coupling

4.2. Pedestrian load model and vibration comfort evaluation

4.2.1. Vertical vibration response analysis without pedestrian gathering

In the design of pedestrian bridges, load cases with varying pedestrian densities should be considered. Given that pedestrian bridges in China are typically located in areas with high pedestrian density, and ISO 10137 does not specify load cases for different pedestrian densities, this study adopts crowd densities ranging from 0.5 to 2.0 persons/m², with increments of 0.5 persons/m² for each load case. Bachmann (Bachmann, 1995) suggests that the average step frequency for normal walking is 2 Hz. Therefore, in this study, the pedestrian pacing frequency is set to 2 Hz, and the crowd moves across the bridge at a speed of 0.75 f = 1.5 m/s. The pedestrian weight is assumed to be 700 N, with a dynamic load factor α₁ = 0.37 (f −1.0) and a phase angle φ₁ = 0. Considering an orderly process of pedestrians entering the bridge, the dynamic load cases calculated using equations (1) to (3) are shown in Table 2.

Table 2.

Pedestrian load model.

Load conditions	Pedestrian density (persons/m²)	Number of pedestrians (persons)	Mode of application	Pedestrian load model
ISO-1	0.5	84	The crowd load moves on the bridge at a speed of 1.5 m/s	$F_{V} (t) = \sqrt{84} \times [700 + 259 \sin (12.566 t)]$
ISO-2	1.0	167		$F_{V} (t) = \sqrt{167} \times [700 + 259 \sin (12.566 t)]$
ISO-3	1.5	251		$F_{V} (t) = \sqrt{251} \times [700 + 259 \sin (12.566 t)]$
ISO-4	2.0	334		$F_{V} (t) = \sqrt{334} \times [700 + 259 \sin (12.566 t)]$

Based on the crowd load model described above, loads were applied to the nodes of the pedestrian bridge’s finite element model to perform a human-induced vibration comfort analysis. The first mode primarily exhibits vertical oscillation of the main girder (Figure 4(a)), with the highest amplitude occurring within the main span. Therefore, for subsequent analysis, Node 2047 at the mid-span of the main span (38.6 m) was selected as the reference point (Figure 4(b)).

According to the vertical vibration comfort curve established by ISO 10137 and considering the fundamental frequency of the bridge, the RMS acceleration limit for pedestrian movement is 0.408 m/s². Compared to pedestrian bridge standards in other countries, ISO 10137 imposes a higher vertical acceleration comfort requirement for pedestrian bridges (Zhou et al., 2020). Another most commonly used standards is the vertical acceleration limit set by the Applied Technology Council (ATC) in the United States (Allen and Murray, 1993). This standard specifies different limits based on the usage environment, function, and vibration frequency of the bridge. In this study, the ATC standard is used as the comfort criterion for peak acceleration, with the vertical peak acceleration limit for outdoor pedestrian bridges set as 0.5 m/s².

The formula for calculating the RMS acceleration is shown in equation (4). Since the analysis considers the process of pedestrians entering the bridge, a time delay is accounted for as pedestrians reach each node. Therefore, RMS acceleration is calculated starting from the moment the first pedestrian arrives at the node.

α_{r m s} = \sqrt{\frac{1}{T} \int_{0}^{T} {α_{w}}^{2} π (t) d t}

(4)

where: T represents the vibration duration and

α_{w} (t)

is the vibration acceleration weighted by frequency.

Table 3 presents the RMS acceleration and peak acceleration responses of the pedestrian bridge under various pedestrian load conditions after computation and analysis.

Table 3.

Acceleration calculation results and standard threshold.

Load cases	Peak acceleration (m/s²)	RMS acceleration (m/s²)	ISO threshold value- moving crowd (m/s²)	ATC threshold value (m/s²)
ISO-1	0.167	0.035	0.408	0.5
ISO-2	0.234	0.050	0.408	0.5
ISO-3	0.287	0.061	0.408	0.5
ISO-4	0.332	0.070	0.408	0.5

Data in Table 3 shows that the acceleration values of the pedestrian bridge under different pedestrian load conditions remain below the threshold value, satisfying comfort design requirements. Both peak and RMS acceleration rise with increasing crowd density. The largest increments in peak acceleration and RMS acceleration occur between ISO-1 and ISO-2, at 40.1% and 42.9%, respectively. In contrast, the smallest increments are observed between ISO-3 and ISO-4, at 15.7% and 14.8%. This indicates that as the load conditions increase, the rate of increase in both peak acceleration and RMS acceleration gradually decreases.

4.2.2. Analysis of vertical vibration response considering pedestrian gathering effect

To identify the most unfavorable position for pedestrian gathering effects, this study initially conducted static load simulations under four conditions. The specific load arrangements are illustrated in Figure 5, with each condition analyzed under four different gathering pedestrian densities. Condition A-1 represents a uniformly distributed stationary pedestrian load across the entire bridge and aims to investigate the impact of gathering crowd density on the bridge’s natural frequency. Condition A-2 involves an eccentric longitudinal arrangement of human loads. These analyses indicate that stationary pedestrian load effectively reduces the fundamental frequency. To simulate the most unfavorable case, static pedestrian loads were placed at points of higher amplitude in the first mode shape. Gathered pedestrians typically do not distribute evenly across the bridge width and are more likely to concentrate on one side. Therefore, based on Condition A-2, this study established Conditions A-3 and A-4 to identify the most unfavorable crowding position for the bridge.

In this study, the bridge’s fundamental frequency was used as a reference point for quantifying the reduction rate in natural frequency. At an extreme load density of 2.0 persons/m², the reduction rates in natural frequency for scenarios A-1 and A-2 were similar, indicating that localized pedestrian crowding at the A-2 location has a significant impact on the bridge’s natural frequency. A further comparison between scenarios A-3 and A-4 revealed a greater reduction in natural frequency in A-3. Consequently, scenario A-3 was selected as the critical load configuration for analyzing the effects of pedestrian crowding on the dynamic performance of the structure.

A static pedestrian load, with a density of 2.0 persons/m², was applied at the designated position (A-3) as illustrated in Figure 5. Utilizing this configuration, the four dynamic load conditions detailed in Table 2 were applied to investigate the structural acceleration response, with results shown in Table 4. The comfort threshold for stationary pedestrians is defined as an RMS acceleration not exceeding 0.204 m/s², aligning with Section 4.2.1 criteria.

Table 4.

Acceleration calculation results and standard threshold.

Load cases	Peak acceleration (m/s²)	RMS acceleration (m/s²)	ISO threshold value—stationary crowd (m/s²)	ATC threshold value (m/s²)
ISO-1	0.332	0.092	0.204	0.5
ISO-2	0.469	0.130	0.204	0.5
ISO-3	0.576	0.159	0.204	0.5
ISO-4	0.666	0.184	0.204	0.5

Compared to Table 3, RMS acceleration in Table 4 increases notably across all load conditions, as shown in Figure 6. Under dynamic pedestrian loads, the RMS acceleration for all four conditions remains below the ISO 10137 threshold, satisfying comfort standards. However, peak acceleration in ISO-3 and ISO-4 exceeds the threshold, potentially causing pedestrian discomfort. The largest increases in both peak and RMS acceleration occur from ISO-1 to ISO-2 at 41.3%, while the smallest increases are from ISO-3 to ISO-4 at 15.6% and 15.7%, respectively. When the pedestrian density reaches 1.14 persons/m², peak acceleration reaches the threshold of 0.5 m/s².

4.3. Influence of various factors on acceleration when considering gathering effect

In studies of human-induced vibrations, the structural response is affected by factors such as crowd density and pacing frequency. Exploring the impact in these factors on structural vibrations under conditions of pedestrian gathering can provide a basis for vibration control. Therefore, this study further analyzed the specific effects of these factors on bridge vibration acceleration.

4.3.1. Vibration response under different pedestrian densities

To investigate the vibration response of the pedestrian bridge under different pedestrian densities, four different densities of stationary pedestrians were applied at the gathering crowd location, combined with four different densities of moving pedestrians. The pedestrian pacing frequency was set to 2.0 Hz, and the walking speed was set to 0.75 f = 1.5 m/s. The resulting peak acceleration and RMS acceleration responses are shown in Figure 7.

As depicted in Figure 7, both RMS and peak acceleration rise monotonically with increasing densities of moving and stationary pedestrians. A closer look at peak acceleration increments shows that, with a stationary density of 2.0 persons/m², increasing the moving pedestrian density from 0.5 to 1.0 persons/m² raises peak acceleration by 0.137 m/s² (41.3%). Further increasing the moving density from 1.5 to 2.0 persons/m² results in a smaller increment of 0.09 m/s² (15.6%).

Conversely, at a moving pedestrian density of 2.0 persons/m², raising the stationary density from 0.5 to 1.0 persons/m² increases peak acceleration by 0.07 m/s² (18.3%), and from 1.5 to 2.0 persons/m² produces a larger increment of 0.135 m/s² (25.4%). RMS acceleration follows a similar pattern.

Thus, from a pedestrian comfort perspective, controlling stationary pedestrian density is more effective than controlling moving density in limiting acceleration increases on landscape pedestrian bridges with gathering crowd effects.

4.3.2. Acceleration response of the structure at different pacing frequencies

Relevant studies indicate that the typical pacing frequency for pedestrians falls within the range of 1.5 Hz to 2.5 Hz (Leonard, 1966; Živanović et al., 2007). Based on this range, the present study conducts an analysis, assuming a pedestrian mass of 70 kg and a walking speed of 0.75 f (m/s). The analysis considers two scenarios: one with gathering pedestrian effects and one without. The relationship between pacing frequency and peak acceleration is shown in Figure 8.

Within the frequency range from 1.5 to 2.5 Hz, increasing pacing frequency initially raises and then lowers peak acceleration, with localized fluctuations. Specifically, below 1.8 Hz, peak acceleration without pedestrian gathering is generally higher than with it. However, above 1.8 Hz, the opposite occurs, with higher peak acceleration when pedestrians are gathered. Notably, between 2.0 and 2.3 Hz, three pacing frequencies with gathering exceed acceleration limits, compared to only one frequency without gathering.

Between 2.0 and 2.1 Hz, peak acceleration sharply increases in the absence of gathering crowds, with an increase rate of 264%. This suggests that changes in pacing frequency, without considering gathering effects, can significantly elevate acceleration, posing comfort risks and potential panic. Conversely, with pedestrian gathering, acceleration decreases, dropping to 53.1% at 2.1 Hz. It can be observed that at 2.1 Hz, pedestrian gathering significantly reduces the structural vibration response. This phenomenon may be attributed to the fact that the gathering pedestrians alter the dynamic characteristics of the structure, thereby shifting the structure away from the frequency range that could induce severe vibrations. This result further confirms that pedestrian gathering influences the dynamic characteristics of footbridge structures.

According to the comfort criteria outlined in Section 4.3.1, peak acceleration with pedestrian gathering reaches 0.641 m/s² at a pacing frequency of 2 Hz, exceeding the comfort threshold. Without pedestrian gathering, peak acceleration reaches 1.207 m/s² and RMS acceleration hits 0.352 m/s² at 2.1 Hz, both surpassing the comfort threshold and likely causing discomfort. In the frequency range of 1.8 to 2.0 Hz, peak acceleration is higher with pedestrian gathering than without. Most studies have found that typical pedestrian pacing frequencies fall within this range (Ji and Pachi, 2005; Živanović et al., 2007). Furthermore, without gathering, only the 2.1 Hz pacing frequency exceeds the comfort threshold, whereas with an assembled crowd, the thresholds are exceeded at 2 Hz, 2.1 Hz, and 2.3 Hz, increasing the probability of exceeding comfort thresholds. Therefore, targeted control measures should be implemented to manage the increased acceleration response caused by assembled crowds.

This study reveals that for landscape pedestrian bridges with spans like traditional pedestrian bridges, gathering effects significantly influence vibration response to variations in pacing frequency. Specifically, at a pacing frequency of 2.1 Hz, pedestrian gathering effectively reduces resonance. However, with assembled crowd effects considered, an unexpected spike in peak acceleration occurs at 2.3 Hz, despite an overall downward trend. The acceleration time-history curve in Figure 9 shows that peak acceleration begin to rise after 35 seconds, reaching 0.556 m/s² with an RMS acceleration of 0.297 m/s². Both values exceed the comfort thresholds.

On landscape pedestrian bridges, the pedestrians who gather and linger are primarily tourists, who tend to be more sensitive to the bridge’s vibration response. As a result, higher acceleration levels may detract from their overall crossing experience. In contrast, regular commuters, who move at a faster-pace, are less likely to stop on the bridge due to scenic attractions. However, the higher accelerations generated by these faster-moving pedestrians may cause discomfort for the stationary tourists, potentially creating a sense of discord between these two user groups.

4.3.3. Acceleration response under bidirectional pedestrian flow

In daily life, pedestrians typically cross bridges in both directions. Therefore, considering bidirectional pedestrian flow can more accurately simulate real conditions. The above analysis indicates that the acceleration response time history is significantly affected by changes in pacing frequency. Thus, the following section focuses on the impact of asynchronous pacing frequencies on human-induced vibrations under bidirectional pedestrian flow. The loading method for bidirectional pedestrian flow assumes pedestrians enter the bridge simultaneously from both ends, with identical pedestrian loads applied at each end. The load model is shown in equation (5), and the parameter settings are the same as those in Section 4.2.2.

F_{V} (t) = \sqrt{334 / 2} \times [700 + 259 \sin (2 π f t)]

(5)

Figure 10(a) shows that, the trend of peak acceleration with frequency for bidirectional pedestrian flow matches that of unidirectional pedestrian flow, validating the calculation results. Under bidirectional pedestrian flow, the maximum peak acceleration without considering gathering pedestrian effects is 0.456 m/s², but rises to 1.080 m/s² with gathering effects, both at a pacing frequency of 2.1 Hz. This indicates that, pedestrian crowding effects can effectively reduce the maximum acceleration under bidirectional flow condition. Figure 10(b) summarizes the RMS acceleration across different conditions. Under the same conditions, the RMS acceleration with bidirectional pedestrian flow is lower than that with unidirectional flow. This trend is consistent across most pacing frequency conditions. Overall, bidirectional pedestrian flow is more favorable for pedestrian comfort compared to unidirectional flow.

It is noteworthy that at a pacing frequency of 2.1 Hz, the maximum RMS acceleration for bidirectional pedestrian flow without crowding is higher than that for unidirectional flow, reaching 0.414 m/s². This value exceeds the ISO 10137 standard threshold of 0.408 m/s² potentially causing discomfort for both stationary and moving pedestrians on the bridge. As shown in Figure 11, the acceleration time-history curve for bidirectional flow exhibits a “double peaks” phenomenon, leading to an increase in the RMS acceleration value. This behavior can be attributed to the staggered arrival times of pedestrian flows from both ends of the bridge at the reference point. Specifically, as the pedestrian flow from the nearer end reaches the vicinity of the reference point, he acceleration gradually increases, peaks, and then decreases. Similarly, a comparable process occurs when the pedestrian flow from the farther end arrives at the reference point, leading to the emergence of the “double-peak” phenomenon.

5. Conclusion

The present study found in filed investigation that unlike conventional pedestrian bridges, the users of landscape pedestrian bridges can be divided into two categories: daily commuters and tourists. The gathering characteristics of tourists on the bridge deck substantially impact the pedestrian load and traffic flow distribution. Addressing such phenomenon, this paper conducts numerical simulation analysis and systematically explores the influence of assembled crowd effect on the vibration responses of a particular landscape pedestrian bridge. Several conclusions can be drawn as follows:

(1) Tourists on footbridges tend to linger in particular spots, creating an assembled crowd effect and being more sensitive to vibrations. The vibration acceleration they feel may easily exceed the comfort threshold. For landscape pedestrian bridges that serve a substantial number of tourists, suitable vibration control measures should be implemented.

(2) Under different pacing frequencies, the impact of assembled crowd effect on acceleration response varies. Within the pacing frequency range of 1.8 Hz to 2.0 Hz, pedestrian gathering will lead to an increase in structural peak acceleration, which has a negative impact on traffic comfort.

(3) As the density of gathering/moving pedestrians increases, the peak acceleration also increases. At a gathering pedestrian density of 2.0 P/m², the peak acceleration increases as high as 41.3% when the density of moving pedestrians rises from 0.5 P/m² to 1.0 P/m², and by 15.6% when it further increases from 1.5 P/m² to 2.0 P/m². Similarly, at a moving crowd density of 2.0 P/m², the corresponding peak acceleration increases are 18.3% and 25.4%, respectively, with analogous changes in gathering crowd density. The RMS acceleration also exhibits a similar variation pattern. Therefore, for landscape footbridge undergoing assembled crowd effect, controlling the density of gathering pedestrians is a more effective measure to improve traffic comfort than controlling the density of moving pedestrians.

(4) At a pacing frequency of 2.1 Hz, the peak acceleration of the bridge, when accounting for the assembled crowd effect, significantly decreases to 53.1% of the value without such an effect. Hence, at certain pacing frequencies, the gathering of pedestrians can be advantageous for managing the dynamic response of the footbridge.

(5) Due to the time lag as pedestrians from both ends reaching the same reference point, bidirectional traffic flow may cause a “double peaks” phenomenon in the acceleration response, resulting the RMS acceleration to exceed comfort threshold. Nonetheless, the peak acceleration in bidirectional flow is generally much lower than that in unidirectional flow. Thus, maintaining the smooth passage of two-way pedestrian traffic on the bridge is conducive to enhancing traffic comfort.

This study investigated the impact of pedestrian gatherings on bridge dynamic characteristics and the effects of complex crowd-induced excitations on bridge response. It highlights the relationships between crowd density, pacing frequency variations, pedestrian flow patterns (unidirectional/bidirectional), and bridge acceleration response. The findings offer insights for vibration mitigation in landscape pedestrian bridges, suggesting dynamic control measures such as limiting pedestrian numbers or regulating pacing frequencies. In addition, for landscape bridges, landmark bridges, or temporary bridges under event scenarios with high pedestrian flow, the results can help optimize the bridge design and traffic organization strategies and provide a scientific basis for ensuring structural safety and pedestrian comfort.

There are several limitations in the present study. It does not consider the effects of various vibration factors (e.g., pedestrian loads, vehicle loads, and wind loads) on pedestrian bridges or their combined impacts. More complex excitation patterns (e.g., combined wind and pedestrian loads) and experimental verification could be addressed in future research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (Grant No. 52178510), the Natural Science Foundation of Fujian Province (Grant No. 2022J011253, 2022J05283), the Fujian Science and Technology Project (Guiding Project) (Grant No. 2021Y0042) and the Science and Technology Plan Fund of Xiamen Housing and Construction Bureau (Grant No. XJK2021-1-10). The financial support is gratefully acknowledged.

ORCID iD

Longzhen Hua

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