Comfort assessment for a pedestrian passageway suspended under a girder bridge with random traffic flows

Abstract

In recent years, a new type of pedestrian passageway, which is suspended under the main girder bridge, has emerged. Vibration in this type of structure is mainly due to the vehicle traffic loading on the main bridge. However, the current codes are not appropriate for vibration comfort assessments for these pedestrian passageways. In this study, a novel method was proposed for probability-based vibration comfort assessment, which considers the randomness of traffic flows. Using an in-service suspended pedestrian bridge as an example, it was considered to the factors that influence the vibration comfort using field measured traffic loading. The results showed that the peak acceleration response of the pedestrian bridge under a random traffic flow followed a normal distribution and there were some deviations in the vibration comfort assessment results when different codes are used. Factors such as the vehicle type ratio, roughness of the pavement, and vehicle speed had significant effects on the acceleration response of the structure.

Keywords

comfort suspended pedestrian bridge traffic load vehicle–bridge coupling vibration

Introduction

The vibration and vibration comfort properties of pedestrian bridges have been analyzed by many researchers. Živanović et al. (2005) reviewed the current status of vibration comfort research for pedestrian bridges under pedestrian loads, systematically summarized the vibration of pedestrian bridges, described research into step loading, calculated the structure vibration response, comfort index, and pedestrian bridge interaction, and reviewed vibration reduction technologies. Venuti and Bruno (2009) summarized the lateral vibrations between pedestrians and bridges. The vibration comfort properties of pedestrian bridges have attracted much attention since the vibration problem occurred in the London Millennium pedestrian in 2000. Current research interests include step load simulations, pedestrian bridge interactions, vibration comfort evaluation index and checking methods (Butz et al., 2008; Ricciardelli et al., 2007; Sachse et al., 2008; Whittington and Thelen, 2009), which have greatly improved the vibration reduction design of pedestrian bridges. In addition, according to comparisons with independent pedestrian bridges, pedestrian bridges integrated with highway bridges have lower vibration amplitudes under vehicle and pedestrian loadings. Thus, vibration comfort checking is usually unnecessary because of the greater structural stiffness. The form of a suspended pedestrian bridge under a main bridge has been addressed during bridge construction in an effort to reduce the gradients of sidewalks and bicycle lanes, or to increase the spans of pedestrian bridges (Chen, 2007). However, pedestrian bridges suspended under main bridges have low structural stiffness and greater vibration amplitudes, and thus, their vibration comfort properties must be considered. The vibration of a suspended pedestrian bridge is caused by the traffic load on the main bridge. The main factors that control the properties of pedestrian bridge are the traffic situation and pavement roughness, but the current codes are not appropriate for evaluating the load or comfort index for this type of bridge.

To consider specific concerns regarding vibration, some researchers have used probability theory to evaluate the vibration comfort of bridges, which is a common method for addressing random problems. Živanović and Pavia (2009) highlighted the pedestrian bridge vibration amplitude limitation by considering the response of pedestrians using probability theory, thereby determining the relationship between the vibration amplitude of a pedestrian bridge and the vibration comfort of pedestrian subjects. Kajikawa and Kobori (1977) studied the pedestrian bridge vibration comfort index using probability theory and proposed a method based on the effective structure velocity values to check the vibration comfort. Živanović and Pavia (2011) analyzed the probability distribution characteristics of pedestrian bridge vibration responses to a step load using experimental and numerical methods. Based on the psychological annoyance probability model, Fa et al. (2008) constructed an appropriately quantized vibration comfort index to evaluate the annoyance probability due to pedestrian bridge vibration. However, these previous studies focused mainly on pedestrian bridges under pedestrian loads and they did not consider comfort evaluations under random vehicle loadings.

In this study, the vibration comfort of flexible suspended pedestrian bridges was revealed, where a probability evaluation method was employed to assess the vibration comfort by considering the influence of random traffic. Using this method, it was studied for the vibration comfort properties of a flexible suspended pedestrian bridge under construction and considered the factors that affect the vibration comfort of pedestrian bridges. It was considered that research results both vibration comfort evaluation assessment and vibration control for this type of pedestrian bridge.

Computational method

Vibration comfort assessment for suspended pedestrian bridges with random traffic flow can be treated mainly as a vehicle–bridge coupling vibration problem. In this study, a probabilistic method was used to analyze the vibration comfort of a pedestrian bridge vibration by simulating random traffic flow. The traffic load was generated according to a random traffic flow using the following computational procedure.

Generate the random traffic flow samples by the Monte Carlo method;

Compute the structure vibration response of bridges to a random traffic flow according to vehicle–bridge coupling vibration theory;

Extract the structure vibration response under different types of traffic flow. Evaluate the vibration comfort of pedestrian bridges with a statistical probability method according to the vibration comfort evaluation index.

In the following, the traffic flow settings, vehicle–bridge coupling vibration, vibration comfort index, and other parameters were shown.

Simulation of random traffic flow

The actual traffic flow is random; so to obtain a more realistic simulation, we employ cellular automata theory, which (Chen and Wu, 2010) utilized to produce an intelligent simulation where vehicles adjusted their speed according to the road conditions. Moreover, Han and Chen (2008) determined a random distribution law for traffic flow based on traffic load observations and statistical analysis, and simulated the actual traffic flow using the Monte Carlo method according to this law. In this study, based on traffic data measurements (Yi et al., 2013a, 2013b, 2015), the random traffic flow was determined by the Monte Carlo method based on the random characteristics of vehicle distances, vehicle speeds, road roughness, and vehicle locations derived from field measurements on a typical urban road. The actual procedures are described as follows.

Fundamental data: the numbers of lanes on the bridge were specified, vehicle speed limit, vehicle densities, vehicle types, and vehicle proportions in each lane.

Total number of vehicles: the numbers of vehicles of each type was calculated on the bridge at one time according to the vehicle flow densities and proportions of vehicle types, that is, the number of vehicles of all types during the computational period.

Driving behavior (vehicle distances, vehicle speeds, and vehicle locations): according to the previous study (Enright and O’Brien, 2013), vehicle distances follow a gamma distribution, which can be determined after estimating their distribution parameters. According to previous research (Chen et al., 2015; Dey et al., 2006), vehicle speeds follow a normal distribution, where the speed of each vehicle can be estimated from the mean values and variances. The lateral location of each vehicle can be determined according to its position relative to the center line in its own lane, where the distances follow a mean distribution and the vehicle bodies are within the lanes.

Random traffic flow samples were generated based on the random distribution laws mentioned above using the Monte Carlo method.

Vehicle–bridge coupling vibration

The computational theories of vehicle–bridge coupling vibration are comparatively mature, so a brief introduction was given. The vehicle–bridge coupling vibration system comprises a bridge, vehicles, and the road surface contacted between them. The system vibration equation is described by equation (1), which includes the vehicle vibration and bridge vibration systems. For highway bridges, the vibration stimulated by a pedestrian load is sufficiently small to be neglected. Rayleigh damping is adopted for bridge damping according to empirical structural values, where the coefficient of the mass matrix is 0.2 and that of the stiffness matrix is 0.04

\begin{array}{l} [\begin{matrix} M_{V} & 0 \\ 0 & M_{B} \end{matrix}] {\begin{matrix} {\overset{\cdot\cdot}{u}}_{V} \\ {\overset{\cdot\cdot}{u}}_{B} \end{matrix}} + [\begin{matrix} C_{V} & C_{V B} \\ C_{B V} & C_{B} \end{matrix}] {\begin{matrix} {\dot{u}}_{V} \\ {\dot{u}}_{B} \end{matrix}} \\ + [\begin{matrix} K_{V} & K_{V B} \\ K_{B V} & K_{B} \end{matrix}] {\begin{matrix} u_{V} \\ u_{B} \end{matrix}} = {\begin{matrix} F_{R}^{V} \\ F_{R}^{B} + F_{G}^{B} \end{matrix}} \end{array}

(1)

In the equation, M, C, and K denote the mass, damping, and stiffness matrix, respectively, u is the displacement vector, F is the acting force vector within the system, the subscript V denotes vehicles, the subscript B denotes bridge structures, and the subscripts R and G represent the force due to the road surface roughness and the force due to gravity for the vehicles, respectively.

Probability-based vibration comfort assessment

Analyzing the vibration comfort for traffic bridges differs greatly from static structural response computation, where the most unfavorable location of the load cannot be determined by the line of influence and the structural vibration response is related to many factors, such as the road condition, vehicle dynamic properties, and running condition; thus, there is no standard computational method. The vibration comfort limit probability employs the discomfort probability for a pedestrian due to bridge vibration under a traffic load to consider the vibration comfort property as follows

P_{d} = P (S > R) = \int_{0}^{\infty} f_{s} (x) F_{R} (x) d x

(2)

In equation (2), P_d is the probability of discomfort, S is the vibration response of the bridge when stimulated by a traffic load, R is the structural response limit for uncomfortable feelings in pedestrians, f_s(x) is the probability density function of the structure vibration velocity, and F_R(x) is the probability distribution function for the pedestrian’s response.

Different codes and studies provide various methods for the determination of R. In Japan, Kajikawa and Kobori (1977) performed large-scale shaking table experiments to assess people’s feelings and suggested that the effective value is an appropriate index for evaluating the vibration comfort of bridges, with a mean threshold value of 1.7 cm/s, peak value of 2.4 cm/s, and variance of 0.6 cm/s. The codes used in other countries have different limits on the peak acceleration values, as shown in Figure 1 and Table 1. Different acceleration limits were obtained under various frequencies.

Figure 1.

Acceleration limits in the codes employed by different countries.

Table 1.

Acceleration limits in different codes.

Code	Vertical peak acceleration
BS 5400 (British Standards Institution (BSI), 1978)	a _limit = 0.5 f^0.5 m/s²
Euro code 5 (European Committee for Standardization, 1997)	a _limit = 0.7 m/s²
ISO 10137:2007 (2007)	Dashed line in Figure 1
OHBDC standard (Highway Engineering Division, 1983)	a _limit = 0.25 f^0.78 m/s²
French guide (Sétra, 2006)	${\begin{matrix} a_{limit} < 0.5 {m / s}^{2}, Extreme \\ 0.5 < a_{limit} < 1.0 {m / s}^{2}, Medium \\ 1.0 < a_{limit} < 2.5 {m / s}^{2}, Low \end{matrix}$

a _limit is the vertical acceleration limit and f is the fundamental frequency (Hz).

Application examples

Bridge overview

The method was employed to describe the previous section in study the vibration comfort properties of a suspended pedestrian bridge under construction. Figure 2 shows the facade, cross section, and design of the bridge. This bridge comprised a vehicle transit bridge (which we refer to as the main bridge) and a pedestrian bridge suspended beneath. The main bridge (85 + 100 + 100 + 100 + 85 = 416 m) was a pre-stressed concrete continuous box girder bridge with five spans to form a unity, which was divided into two independent decks on the left and right, where each deck was 14 m wide with three lanes and a 10 m-wide pedestrian bridge was suspended between the two parts. Vertical suspenders comprised the main form of connection between the host and the pedestrian bridge, where the suspenders were 4 m from each other, and slant suspenders were deployed at a specific interval to provide lateral stability.

Figure 2.

Diagram illustrating the structure of the bridge: (a) vertical section, (b) cross section, and (c) sketch in cross section showing the design of the bicycle lane.

This was the first pedestrian bridge to be suspended under two different decks. Due to its weak structural stiffness and the use of supports on two independent vehicle transit bridges, the vibration comfort property induced by traffic required special attention.

Vehicle–bridge coupling vibration system

The vehicle–bridge coupling vibration system comprised the bridge, vehicles, and the road surface contacted between them. As shown in Figure 3, space framed rods were used to simulate the bridge structure. The main girder had strong stiffness due to its box cross section and a single beam model was employed in the computation. The suspended structure had low stiffness and its deck was made of orthotropic plates. To consider the local deformation of the orthotropic plates, the grillage method was used. Vertical and slant suspenders were used to connect the main girder and the pedestrian bridge, where the slant suspender elements were allowed to rotate only along the vertical direction, whereas the other directions were restrained.

Figure 3.

Computational model of the bridge: (a) overall model of the structure and (b) cross section of the model.

Table 2 shows the four main natural vibration frequencies and their modes of vibration, which demonstrate that the modes of the low-order frequencies occurred mainly in the form of vertical vibration and torsion, while the pedestrian bridge deformed as well as the main bridge structure. These vibration modes are similar to each other and the structure had a low fundamental frequency of around 1 Hz, which indicates that the structure was flexible, and thus, a vibration comfort evaluation of the structural response was required.

Table 2.

Frequencies of the four main natural vibrations and their corresponding modes of vibration.

Frequency order	Frequency (Hz)	Mode of vibration
1	0.969
2	0.979
3	1.413
4	1.430

The types and models of vehicle in actual vehicle flows are complex, so the vehicle types and parameters based on previous studies were utilized (Chen and Wu, 2010; Han and Chen, 2008), which classified vehicles into three types: heavy lorries, medium-sized vehicles, and ordinary small vehicles. According to the previous study (Zhang et al., 2001), the influence of the dynamic response induced by ordinary small vehicles is very small so in our computation, the former two types of vehicles were used, but the third was neglected. The vehicle modes are shown in Figure 4 and their corresponding parameters are listed in Table 3.

Figure 4.

Different vehicle modes: (a) heavy lorries and (b) medium-sized vehicles.

Table 3.

Parameters for the vehicle modes.

Parameter	Heavy lorry	Medium vehicle
λ ₁ (m)	2.84	1.8
λ ₂ (m)	1.01	1.8
λ ₃ (m)	0.65	–
m ₁ (kg)	21,807	10,000
m ₂ (kg)	719	420
m ₃ (kg)	320	420
m ₅ (kg)	1191	–
m ₇ (kg)	1085	–
J ₁ (kg m²)	62,500	32.4
J ₃ (kg m²)	45.1	–
k ₁ (kN/m)	1200	500
k ₂ (kN/m)	2400	1950
k ₄ (kN/m)	2400	500
k ₅ (kN/m)	4800	1950
k ₆ (kN/m)	2400	–
k ₇ (kN/m)	4800	–
c ₁ (kN s/m)	5	20
c ₂ (kN s/m)	6	20
c ₄ (kN s/m)	10	20
c ₅ (kN s/m)	3.6	20
c ₆ (kN s/m)	10	–
c ₇ (kN s/m)	3.5	–

The road surface contacted between vehicles and a bridge structure is affected by the road condition and the vehicle’s running condition. According to the previous study (Kawatani et al., 2000), a road grade similar to the GOOD grade was adopted in the ISO criteria and we approximated the power spectral density function of the road roughness by the following formula

S_{r} (Ω) = \frac{α}{Ω^{n} + β^{n}}

(3)

In equation (3), S_r is the power spectral density value, cm²/(c/m); Ω is the road frequency; α is the parameter that denotes the road’s planarity; β is the parameter that represents the distribution of shapes, β = 0.08 c/m; and n is an index that denotes the frequency energy distribution according to a normal distribution N (1.92, 0.283).

Simulation of random vehicle flow

The vehicle types and parameters based on previous study were employed (Chen and Wu, 2010; Han and Chen, 2008), which classified vehicles as ordinary small vehicles, medium-sized vehicles, and heavy lorries. According to field measurements obtained on a typical urban road (Chen et al., 2014, 2015), it was found that small vehicles accounted for 80.48% of the total traffic flow, middle-sized vehicles comprised 16.43%, and heavy lorries accounted for 3.09%. The gross vehicle weights (GVW) of the medium-sized vehicles followed a third-order log-normal distribution and those of the heavy lorries followed a third-order normal distribution (Figure 5). The time headway for the following flows obeyed a log-normal distribution and the free flows had a gamma distribution. The vehicle speed followed a normal distribution (Figure 6). The specific parameters for the GVW, time headway, and speed of the traffic flow on the measured road are shown in Tables 4 and 5.

Figure 5.

Probability distributions of the heavy weighted lorries.

Figure 6.

Probability distributions for speed with different flow states: (a) free flow and (b) following flow.

Table 4.

Parameters of the probability distributions for vehicles with different GVWs.

Vehicle type	Distribution type	P	µ	σ
Medium-sized vehicle	Third-order log-normal	0.56	3.41	0.38
Medium-sized vehicle	Third-order log-normal	0.44	10.90	5.99
Heavy lorry	Third-order normal	0.55	14.20	4.05
		0.33	28.90	8.87
		0.12	62.18	6.95

Table 5.

Probability distribution types and parameters for the headway types.

Headway	Distribution type	α	β
Free flow	Gamma	1.38	8.74
Following flow	Log-normal	0.83	0.58

Structural vibration response and comfort assessment

Analysis of the structural vibration response characteristics

To understand the vibration frequency characteristics of the suspended pedestrian bridge when the main bridge had a random vehicle flow, we analyzed the time-history response of the mid-span of the third span to obtain a representative evaluation of the bridge under random flow.

The deflections on the left and right deck of the third span of the pedestrian bridge are compared in Figure 7, which show that when the left and right random flow parameters agreed with each other, the vibration also matched on both sides due to the similar characteristics of the two decks of the main bridge under a random vehicle flow. Under the same random vehicle flow, the influence of the two decks of the main bridge was very small, so we focused on the left side of the pedestrian bridge in the following computations.

Figure 7.

Comparison of the deflection responses of points on the left and right sides of the pedestrian bridge.

Figure 8(a) and (b) compares the time-history responses of the acceleration and their fast Fourier transform (FFT) spectra. According to Figure 8, several conclusions are drawn: (1) As shown by the time-history curves, the responses at the front and back were comparatively smaller at the beginning and the end due to the influence of vehicles entering and leaving the main bridge. (2) The acceleration peaks differ greatly under different random traffic flows. (3) The spectra distributions due to different load samples were quite similar with a first-order frequency of 1 Hz, which was close to the fundamental frequency of the structure. (4) The vibration of the pedestrian bridge contained more frequencies compared with normal pedestrian bridges, which mainly vibrate around their fundamental frequencies. This is because normal pedestrian bridges are structurally independent, thereby resulting in a clear separation of fundamental frequencies, whereas the primary frequencies of the suspended structure were affected by the main bridge and they were densely distributed. Normal pedestrian bridges are affected mainly by the sine wave pedestrian load, which has a frequency similar to the fundamental frequency of the structure. This type of structure is affected mainly by the random vehicle load, which is complex, thereby yielding a complex response.

Figure 8.

Time-history response of the peak acceleration values and FFT spectra of the pedestrian bridge: (a) comparison of the time-history responses in terms of the peak acceleration values and (b) FFT spectral analysis of the responses.

Vibration comfort assessment

The acceleration response of the structure under 100 different random vehicle flows is computed to assess the vibration comfort probability of the pedestrian bridge, where the peak acceleration values are extracted from each sample.

Figure 9 shows the probability histogram for the peak acceleration values of the mid-span on the third span under 100 different random vehicle flows, as well as comparisons with the peak acceleration limits in various countries. According to Figure 9, several conclusions are drawn: (1) The peak acceleration values varied greatly in different random samples ranging from 0.32 to 1.06 m/s², thereby indicating high randomness, which means that it is necessary to perform evaluations based on probabilistic statistics. (2) The peak acceleration values of the samples approximated a normal distribution with a mean value of 0.568 and variance of 0.143, with a confidence coefficient larger than 95%. According to the median value, the vibration comfort probability was 50%. (3) The vibration comfort probabilities and the limits in different codes are shown in Table 6. The European code has a lower requirement and there was a 75% chance of the vibration comfort probability being exceeded under a random vehicle flow, whereas the OHBDC code has a higher demand and the vibration comfort probability is 0%, so the pedestrian bridge did not satisfy this requirement. The French guide offers a range of acceleration values and the pedestrian bridge was in the medium vibration comfort range. Thus, it is useful to study the vibration comfort probability of bridges under random factors as well as eliminating the influence of random factors to obtain an overall assessment.

Figure 9.

Probability histogram showing the peak acceleration values for the pedestrian bridge.

Table 6.

Vibration comfort probability evaluation according to the codes of several countries.

Code	a _limit value (m/s²)	Vibration comfort probability (%)
BS 5400	0.492	28
Euro code 5	0.7	75
OHBDC standard	0.244	0
French guide (Sétra)	[0.0, 0.5] extreme	28
	[0.5, 1.0] medium	100
	[1.0, 2.5] low	100

Parameter analysis

In this study, it was analyzed for the influence of external factors on the vibration comfort, determined the sensitive factors that affect vibration comfort, and provided references to facilitate vibration reduction design for pedestrian bridges. Three factors are studied, that is, the proportions of different vehicle types, road roughness, and vehicle speed, to analyze and compare the characteristic structural responses based on 100 random vehicle flow samples with specific changes on these three parameters.

Figure 10(a) compares the influence of different vehicle types, where the solid line shows the characteristic curve for the pedestrian bridge under a random vehicle flow (initial condition), the dashed line shows the characteristic curve when the relative proportions of heavy lorries, medium-sized vehicles, and small vehicles were 0.1:0.2:0.7, and the other parameters remained the same. It was shown that reducing the proportion of heavy lorries had a significant effect when the peak acceleration was larger than 0.7 m/s², where the majority of the peak acceleration values ranged from 0.45 to 0.65 m/s². The mean value of the normal distribution was determined as 0.56 and the corresponding variance as 0.138, which was close to the distribution under the initial condition. Thus, controlling the number of heavy lorries will not lead to obvious reductions in the vibration response for this type of pedestrian bridge.

Figure 10.

Effects of different parameters on the peak acceleration value for the pedestrian bridge: (a) influence of vehicle types, (b) influence of road roughness, and (c) influence of vehicle speed.

Figure 10(b) shows the influence of the road roughness on vibration. The solid line denotes the initial condition, which is similar to the curve with a Good road roughness in the ISO code. The dashed line represents the curve obtained with the road roughness defined by the following parameters in equation (2): α = 0.001, β = 0.05 c/m, and normal distribution (µ = 2.0, σ = 0.289), according to a previous study (Chen and Wu, 2010) (similar to the extra good road roughness in the ISO code). It was shown that improving the road roughness significantly reduced the vibration response of the structure, thereby resulting in an almost 100% likelihood of satisfying the codes specified in every country. Thus, reducing the road roughness on the main bridges in this type of bridge structure is very important for reducing the peak acceleration value of the suspended pedestrian bridges.

Figure 10(c) shows the influence of vehicle speed, where the vehicle speed increased by 20 km/h for all vehicles but the variance remained the same. The acceleration distribution is similar to a bimodal distribution and not a normal distribution due to the change in the mean speed value, but increasing the speed intensified the peak acceleration value. In the European Code, the vibration comfort probability is reduced to 48%. Thus, speed limits help to reduce the peak acceleration.

Conclusion

In this study, it was evaluated for the vibration comfort probability for a suspended pedestrian bridge under a random traffic flow, by changing the road roughness, vehicle speed, vehicle distance, and locations of the lanes, to establish the probabilistic statistical method for assessing the vibration comfort properties of pedestrian bridges of this type. Furthermore, by analyzing the effects of different parameters, the parameters that significantly influence the vibration comfort for this type of pedestrian bridge are identified. Based on the analysis, the following conclusions were revealed:

Unlike normal independent pedestrian bridges, suspended pedestrian bridges will be influenced by the random vehicle flow load, which is complex, and their acceleration response will not vibrate at the main primary frequency around the fundamental frequencies, but instead it contains multiple primary frequencies.

Considering random factors, the peak acceleration will change greatly, and thus, it is necessary to perform evaluations with probabilistic statistical methods.

The distribution of the peak acceleration values has a normal distribution. According to the different vibration comfort requirements specified in the codes of different countries, the vibration comfort evaluation employed must also vary.

It was found that the vehicle types, road roughness, and vehicle speed affected the peak acceleration response, but the road roughness had the most obvious effect followed by the vehicle speed. Therefore, ensuring the smoothness of the road on the main bridge in this type of structure is very important for reducing the vibration of the suspended pedestrian bridge. In addition, imposing a speed limit can help to reduce vibration by the pedestrian bridge.

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: The authors are grateful for the financial support provided by the Natural Science Foundation of China (Project No. 51508415) and the Natural Science Foundation of Zhejiang Province (Project No. LQ15E080006).

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