2. Brief review of design standards

the last 20 years, after some vibratory events happened in several large-span footbridges [3–5], an intensive research activity has been conducted by the scientific community in order to better characterize the pedestrian-induced vibrations on footbridges. Concretely, these research efforts were mainly focused on two objectives: (i) the accurate definition of the vibration source [6] and (ii) the analysis of a remarkable event, the lateral lock-in phenomenon [7]. On the one hand, the determination of the load induced by pedestrians on footbridges was tackled progressively. Initially, the estimation of the force originated by a single walking or running pedestrian was studied [8, 9]. Subsequently, these results were further extrapolated to the case of a crowd moving on a footbridge [10]. On the other hand, the lateral lock-in instability phenomenon originated by the synchronization of a pedestrian flow walking on a footbridge has been widely studied as well. Based on the outcomes of these researches, different proposals to estimate the number of pedestrians that originates the lateral lock-in phenomenon, as well as limiting values of the modal properties of the structure to avoid the problem, have been

As result of all these studies, several standards [11] and design guidelines [12] were published to facilitate designers the assessment of the vibration serviceability limit state of footbridges under pedestrian action. Although these design codes shed light on this issue, they still present some shortages, so that, the dynamic response of the structure obtained numerically based on these recommendations still differs from the values recorded experi-

In order to overcome these limitations, a new generation of models have been developed and proposed during the last 5 years, giving rise to a new modelling framework. Three key aspects have been additionally taken into account in order to improve the modelling of pedestrian flows and their effect on footbridges [14]: (i) the inter- and intra-subject variability of the pedestrian action, (ii) the pedestrian-structure interaction and (iii) the crowd behaviour. Furthermore, the variability of the pedestrian action is normally simulated via a probabilistic approach, considering that the parameters that characterize the crowd-structure interaction model may be defined as random variables [15]. All these proposed models share a common scheme, and the crowd-structure interaction is simulated via the linking of two sub-models [16, 17]: (i) a pedestrian-structure interaction sub-model and (ii) a crowd sub-model. For the pedestrian-structure sub-model, although different models have been proposed [18], the use of a single-degree-of-freedom (SDOF) system has gained wider popularity in the scientific community. For the crowd sub-model, two approaches have been proposed: either macroscopic or microscopic models [15]. In the first approach, the crowd behaviour is modelled based on fluid mechanics [10], whilse in the second, the position and velocity of each pedestrian follows a multi-agent law [19]. The second approach, which can account explicitly for the inter-subject variability of each pedestrian [20], has been internationally accepted as the best method to simulate numerically the behaviour of pedestrian flows [15]. The linking between the two submodels is achieved by the implementation of several behavioural conditions [20]. In this way, if certain comfort limits are exceed by the pedestrian-structure interaction sub-model, the velocity and step frequency of each pedestrian in the crowd are modified [20, 21]. The new modelling framework, based on these crowd-structure interaction models, has been applied successfully to determine numerically the response of a footbridge under pedestrian action

provided [7].

62 Bridge Engineering

mentally [13].

The international standards for the assessment of the vibration serviceability limit state of footbridges under pedestrian action share two general rules to tackle the pedestrian-induced vibration problem [6]: (i) the establishment of the range of frequencies that characterizes the pedestrian-structure interaction (Table 1) and (ii) the treatment of the problem separately in terms of the direction in which the pedestrian action (longitudinal, lateral or vertical) is applied. However, most of these standards only establish the need to assess the dynamic behaviour of the structure, if some of its natural frequencies are within the interaction range (Table 1), but do not define a methodology to check the required comfort level.

According to the authors' opinion, the Synpex guidelines [12] are currently the most comprehensive standard to assess the vibration serviceability limit state of footbridges under pedestrian action. These guidelines [12] divide the checking of the vibration serviceability limit state in seven steps:



iv. The damping ratio of the affected vibration mode, ζ<sup>f</sup> , is estimated in function of the

Recent Advances in the Serviceability Assessment of Footbridges Under Pedestrian-Induced Vibrations

v. The maximum acceleration has to be evaluated for each design scenario. For this purpose, it is necessary to define a load model which may be characterized by the following

0

where <sup>G</sup> � cos 2 � <sup>π</sup> � <sup>f</sup> <sup>s</sup> � <sup>t</sup> � � is the harmonic load due to a single pedestrian, with <sup>G</sup> being the dynamic load factor (DLF) of the pedestrian step load (280 N for vertical direction, 140 N for longitudinal direction and 35 N for lateral direction); f <sup>s</sup> is the step frequency [Hz], which is assumed equal to the considered natural frequency, f <sup>f</sup> ; ψ is the reduction coefficient that takes into account the probability that the footfall frequency approaches the considered natural frequency and it may be estimated from Figure 1, according to the considered natural frequency; vp is the pedestrian velocity [m/s] which may be assumed around 3 m/s [12] and Lf is the length of the footbridge [m]. In Eq. (1), n<sup>0</sup> is the equivalent number of pedestrians on the foot-

� � <sup>¼</sup> <sup>1250</sup> � cos 2 � <sup>π</sup> � <sup>f</sup> <sup>s</sup> � <sup>t</sup> � � � <sup>ψ</sup> ½ � <sup>N</sup> (2)

� ψ=Lf ½ � N=m (1)

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(3)

65

construction type and the amplitude of the vibrations [12].

Pjog t; vp

bridge, which may be determined from:

n 0

Figure 1. Pedestrian reduction coefficient, ψ, for the equivalent pedestrian load [12].

considered vibration mode, ζ<sup>f</sup> .

<sup>¼</sup> <sup>10</sup>:<sup>8</sup> � ffiffiffiffiffiffiffiffiffiffi <sup>ζ</sup><sup>f</sup> � <sup>n</sup> <sup>p</sup> <sup>1</sup>:<sup>85</sup> � ffiffiffi n p

if

in terms of the number of pedestrians on the deck, n, and the damping ratio of the

d < 1:00 d ≥ 1:00

P=m<sup>2</sup>

P=m<sup>2</sup>

• A pedestrian stream walking is simulated by an equivalent load:

pwalðÞ¼ <sup>t</sup> <sup>G</sup> � cos 2 � <sup>π</sup> � <sup>f</sup> <sup>s</sup> � <sup>t</sup> � � � <sup>n</sup>

• A pedestrian jogging is simulated by a single vertical moving load:

equivalent harmonic loads [12]:

Table 1. Ranges of frequencies of pedestrian-structure interaction according to different international standards [14].

iii. Different design scenarios must be assessed: for each design scenario, the expected traffic class in terms of the pedestrian density, d [P=Person/m<sup>2</sup> ], (Table 2) and its corresponding comfort class in terms of limit acceleration (Table 3) must be determined according to the owner's requirements.


```
Table 2. Traffic classes [12].
```


Table 3. Defined comfort classes with limit acceleration ranges [12].

	- A pedestrian stream walking is simulated by an equivalent load:

$$\boldsymbol{p}\_{\text{wall}}(t) = \mathbf{G} \cdot \cos\left(2 \cdot \boldsymbol{\pi} \cdot \boldsymbol{f}\_s \cdot t\right) \cdot \boldsymbol{n}' \cdot \boldsymbol{\psi}/L\_f \text{ [N/m]} \tag{1}$$

• A pedestrian jogging is simulated by a single vertical moving load:

$$P\_{\text{jeg}}(t, \upsilon\_p) = 1250 \cdot \cos\left(2 \cdot \pi \cdot f\_s \cdot t\right) \cdot \psi \text{ [N]} \tag{2}$$

where <sup>G</sup> � cos 2 � <sup>π</sup> � <sup>f</sup> <sup>s</sup> � <sup>t</sup> � � is the harmonic load due to a single pedestrian, with <sup>G</sup> being the dynamic load factor (DLF) of the pedestrian step load (280 N for vertical direction, 140 N for longitudinal direction and 35 N for lateral direction); f <sup>s</sup> is the step frequency [Hz], which is assumed equal to the considered natural frequency, f <sup>f</sup> ; ψ is the reduction coefficient that takes into account the probability that the footfall frequency approaches the considered natural frequency and it may be estimated from Figure 1, according to the considered natural frequency; vp is the pedestrian velocity [m/s] which may be assumed around 3 m/s [12] and Lf is the length of the footbridge [m]. In Eq. (1), n<sup>0</sup> is the equivalent number of pedestrians on the footbridge, which may be determined from:

$$m' = \begin{array}{c c} 10.8 \cdot \sqrt{\zeta\_f \cdot n} \\ 1.85 \cdot \sqrt{n} \end{array} \text{ if } \begin{array}{c} d < 1.00 \text{ P/}m^2 \\ d \ge 1.00 \text{ P/}m^2 \end{array} \tag{3}$$

in terms of the number of pedestrians on the deck, n, and the damping ratio of the considered vibration mode, ζ<sup>f</sup> .

Figure 1. Pedestrian reduction coefficient, ψ, for the equivalent pedestrian load [12].

iii. Different design scenarios must be assessed: for each design scenario, the expected traffic

Table 1. Ranges of frequencies of pedestrian-structure interaction according to different international standards [14].

comfort class in terms of limit acceleration (Table 3) must be determined according to the

] Characteristics

TC3 <0.50 P/m<sup>2</sup> Unrestricted walking and significantly dense traffic TC4 <1.00 P/m<sup>2</sup> Uncomfortable situation and obstructed walking TC5 <1.50 P/m<sup>2</sup> Unpleasant walking and very dense traffic

Class Degree Vertical acceleration Horizontal acceleration

CL1 Maximum <0.50 m/s<sup>2</sup> <0.10 m/s2 CL2 Medium 0.50–1.00 m/s2 0.10–0.30 m/s2 CL3 Minimum 1.00–2.50 m/s2 0.30–0.80 m/s2 CL4 Discomfort >2.50 m/s<sup>2</sup> >0.80 m/s2

], (Table 2) and its corresponding

class in terms of the pedestrian density, d [P=Person/m<sup>2</sup>

TC1 <15 P (P = Person) Very weak traffic

Table 3. Defined comfort classes with limit acceleration ranges [12].

TC2 <0.20 P/m<sup>2</sup> Comfortable and free walking

Standards Vertical [Hz] Lateral [Hz]

Eurocode 1 (2002) 1.60-2.40 0.80–1.20 Eurocode 5 (2003) <2.50 0.80–1.20

SIA 260 (2003) 1.60–4.50 <1.30 BS 5400 (2006) <5.00 <1.50

Setra (2006) 1.00–2.60/2.60–5.00 0.30–1.30/1.30–2.50

Synpex (2007) 1.25–2.30/2.50–4.60 0.50–1.20

EAE (2011) 1.60–2.40/3.50–4.50 0.60–1.20 IAP-11 (2011) 1.25–4.60 0.50–1.20

owner's requirements.

Class Density d [P/m<sup>2</sup>

Table 2. Traffic classes [12].

LRFD American Guide (2009) <3.00

64 Bridge Engineering

Austroads (2012) 1.50–3.00 Hong Kong Guide (2009) 1.50–2.30 Ontario Guide (1995) <3.00

EHE-08 (2008) <5.00

DIN-Fachbericht 102 (2003) 1.60–2.40/3.50–4.50

To estimate the considered natural frequency for each design scenario, the mass of pedestrians has to be taken into account (with a medium pedestrian weight about 70 kg) when its value is greater than 5% of the modal deck mass.


In spite of the fact that the Synpex design guidelines [12] were an important breakthrough, they still present several limitations, which originate that the numerical prediction of the dynamic response of footbridges, obtained using them, under- or over-estimates the values recorded experimentally. As main limitations, the following ones may be enumerated: (i) the change of the dynamic properties of the structure, due to the presence of pedestrians, is estimated in a simplified form, adding directly the pedestrian mass to the structural mass without considering any additional effect on the remaining modal parameters of the structure, (ii) the proposed methods do not fit well to the case where several vibration modes of the footbridge are affected by the pedestrian-induced excitations, (iii) the effect of the nonsynchronized pedestrians are not taken into account by these recommendations and (iv) the definition of the pedestrian load is performed under a deterministic approach which does not allow considering the inter- and intra-subject variability of the pedestrian action. In order to overcome these limitations, a new generation of models that configure a new modelling framework has been proposed. A brief description of this new modelling framework is included in the next section.

All the pedestrian-structure interaction models based on the use of a SDOF system share a common formulation to solve the pedestrian-structure interaction [22, 26] but, however, they differ in the values adopted to characterize the modal parameters of the SDOF systems. A wide summary of the pedestrian-structure interaction models proposed by different authors can be found in Ref. [18]. The main output obtained from this sub-model is usually the acceleration

Recent Advances in the Serviceability Assessment of Footbridges Under Pedestrian-Induced Vibrations

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67

In the second sub-model, the crowd is usually simulated via a behavioural model [19] that provides a description of the individual pedestrian position, xp, pedestrian velocity, vp, and step pedestrian frequency, f <sup>s</sup>. Additionally, in order to take into account the synchronization among pedestrians, an additional parameter must be included. A common manner to simulate this phenomenon is to add a different phase shift, ϕp, in the definition of the ground reaction

The linking between the two sub-models is usually achieved in the different proposals by taking into account the modification of the pedestrian behaviour in terms of the vibration level that he/she experiences [15, 17, 20–24]. Two additional conditions are commonly included for this purpose: (i) a retardation factor, which reduces the pedestrian velocity in terms of the accelerations experienced by each pedestrian; and (ii) a lateral lock-in threshold, which allows simulating the synchronization among the pedestrians and the structure by the modification of both their step frequencies and the phases [20–23]. This new approach has only been implemented, to the best of the authors' knowledge, in vertical and lateral direction, since there are few reported cases of pedestrian-induced vibration problems in longitudinal direction. In order to illustrate briefly this new modelling framework, one of the most recent crowdstructure interaction models, which has been proposed by the authors, is described in the next sections [23]. Subsequently, the potential of the approach to accurately assess the vibration serviceability limit state of footbridges under pedestrian action is illustrated via its implementation for the analysis of a case study. For clarity, the model is described here only for the lateral

direction, although it may be easily generalized to the vertical direction [14].

experienced by each pedestrian.

Figure 2. Layout of the new modelling framework.

load generated by each pedestrian [14].
