7. Application example: Lateral lock-in phenomenon on a real footbridge

In order to illustrate the potential of this new modelling framework, the analysis of the lateral lock-in phenomenon on a real footbridge, the Pedro e Inês footbridge (Coimbra Portugal) has been performed [23]. The maximum lateral accelerations at the mid-span of the footbridge obtained via three different methods during a lateral lock-in pedestrian test are correlated. The three method used are: (i) the experimental values recorded during a lateral lock-in pedestrian test reported in Ref. [31], (ii) the numerical estimation of the maximum lateral acceleration obtained according to the Synpex guidelines [12] and (iii) the numerical prediction obtained based on the application of the proposed approach [23]. On an updated finite element model of the structure [32].

The footbridge is situated over the Mondego River at Coimbra (Portugal). The structure is configured by five spans (total length of 274.5 m); a central arch of 110 m, two lateral semiarches of 64 m and two transition spans of 30.5 and 6 m, respectively (Figure 5). The deck is configured by a concrete-steel composite box-girder with a variable width between 4 and 8 m. The footbridge presents an anti-symmetrical configuration with respect to the longitudinal axis of the structure. In this way, the intersection of the two parallel decks generates a panoramic square at mid-span of the footbridge (Figure 5). As result of the numerical studies performed during the design phase, it was checked that the structure was prone to pedestrian-induced vibrations in lateral direction. Experimental tests were conducted to assess the dynamic response of the footbridge under pedestrian action in lateral direction. The main outcomes of this experimental work were reported in Ref. [31]. These results have been employed in this chapter to illustrate the potential of the new modelling framework. As the pedestrian is forced to walk in a controlled manner during the lateral lock-in pedestrian test, the crowd-structure model previously described has been applied under the deterministic approach.

graphical representation of these results (Figure 6) allows for identifying the beginning of the instability lateral lock-in phenomenon. As it is illustrated in Figure 6, the number of pedestrians that originates the beginning of the lateral lock-in phenomenon is around 75 [31].

Figure 6. Experimental and numerical variation of the maximum lateral acceleration, ð Þ alat max, during the lateral lock-in

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Subsequently, a numerical lateral lock-in test based on the proposed approach was performed. Each considered group of pedestrians has been simulated considering as initial spatial distribution, a rectangular-shaped grid with an initial distance among pedestrians dp ¼ 0:50 m in longitudinal direction and an equidistant distribution in lateral direction. During the numerical test, according to the assumptions of the experimental test reported in the literature [31], each considered group of pedestrians walks freely along the footbridge, following the curve path illustrated in Figure 5. The number of pedestrians in each group increases gradually between 15 and 85 in increments of 5. The coordinates of the considered lateral vibration

As result of this numerical analysis, the maximum lateral acceleration at mid-span of the structure in terms of the different groups of pedestrians on the footbridge was obtained. The graphical representation of this relationship is shown in Figure 6. A good agreement is achieved between the experimental lateral maximum accelerations and the numerically estimated maximum values, as it is illustrated in Figure 6. Additionally, the estimation of the numerical maximum acceleration obtained, applying the methodology proposed by the Synpex guidelines [12], is also shown in Figure 6. It is clear from Figure 6 that the new modelling framework allows obtaining a more accurate numerical analysis of the lateral lockin phenomenon than these design guidelines. The lateral lock-in criterion established by the

The assessment of the vibration serviceability limit state of footbridges under pedestrianinduced excitation has usually been performed based on the recommendations of the most

modes of the structure follow from the results available in the literature [31].

Synpex guidelines [12] is also illustrated for reference in Figure 6.

8. Conclusions

pedestrian test [23].

The natural frequency (around 0.91 Hz) and associated damping ratio (approximately 0.55%) of the first lateral vibration mode of the footbridge were identified experimentally. As the natural frequency of this vibration mode is within the range that characterizes the pedestrianstructure interaction in lateral direction, a lateral lock-in pedestrian test was conducted to determine experimentally the number of pedestrians that originates the lateral instability phenomenon [31]. The analysis focused on characterizing the beginning of the lateral lock-in phenomenon, since during this part of the phenomenon, the modification of the modal properties of the structure induced by the pedestrian-structure interaction is higher [3]. The lateral acceleration, alat, at mid-span of the structure in terms of the number of pedestrians, which cross along the structure, was recorded in this lateral lock-in pedestrian test. The analysis of the

Figure 5. Scheme of the lateral lock-in pedestrian test on Pedro e Inês footbridge [31].

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Figure 6. Experimental and numerical variation of the maximum lateral acceleration, ð Þ alat max, during the lateral lock-in pedestrian test [23].

graphical representation of these results (Figure 6) allows for identifying the beginning of the instability lateral lock-in phenomenon. As it is illustrated in Figure 6, the number of pedestrians that originates the beginning of the lateral lock-in phenomenon is around 75 [31].

Subsequently, a numerical lateral lock-in test based on the proposed approach was performed. Each considered group of pedestrians has been simulated considering as initial spatial distribution, a rectangular-shaped grid with an initial distance among pedestrians dp ¼ 0:50 m in longitudinal direction and an equidistant distribution in lateral direction. During the numerical test, according to the assumptions of the experimental test reported in the literature [31], each considered group of pedestrians walks freely along the footbridge, following the curve path illustrated in Figure 5. The number of pedestrians in each group increases gradually between 15 and 85 in increments of 5. The coordinates of the considered lateral vibration modes of the structure follow from the results available in the literature [31].

As result of this numerical analysis, the maximum lateral acceleration at mid-span of the structure in terms of the different groups of pedestrians on the footbridge was obtained. The graphical representation of this relationship is shown in Figure 6. A good agreement is achieved between the experimental lateral maximum accelerations and the numerically estimated maximum values, as it is illustrated in Figure 6. Additionally, the estimation of the numerical maximum acceleration obtained, applying the methodology proposed by the Synpex guidelines [12], is also shown in Figure 6. It is clear from Figure 6 that the new modelling framework allows obtaining a more accurate numerical analysis of the lateral lockin phenomenon than these design guidelines. The lateral lock-in criterion established by the Synpex guidelines [12] is also illustrated for reference in Figure 6.

## 8. Conclusions

performed [23]. The maximum lateral accelerations at the mid-span of the footbridge obtained via three different methods during a lateral lock-in pedestrian test are correlated. The three method used are: (i) the experimental values recorded during a lateral lock-in pedestrian test reported in Ref. [31], (ii) the numerical estimation of the maximum lateral acceleration obtained according to the Synpex guidelines [12] and (iii) the numerical prediction obtained based on the application of

The footbridge is situated over the Mondego River at Coimbra (Portugal). The structure is configured by five spans (total length of 274.5 m); a central arch of 110 m, two lateral semiarches of 64 m and two transition spans of 30.5 and 6 m, respectively (Figure 5). The deck is configured by a concrete-steel composite box-girder with a variable width between 4 and 8 m. The footbridge presents an anti-symmetrical configuration with respect to the longitudinal axis of the structure. In this way, the intersection of the two parallel decks generates a panoramic square at mid-span of the footbridge (Figure 5). As result of the numerical studies performed during the design phase, it was checked that the structure was prone to pedestrian-induced vibrations in lateral direction. Experimental tests were conducted to assess the dynamic response of the footbridge under pedestrian action in lateral direction. The main outcomes of this experimental work were reported in Ref. [31]. These results have been employed in this chapter to illustrate the potential of the new modelling framework. As the pedestrian is forced to walk in a controlled manner during the lateral lock-in pedestrian test, the crowd-structure

the proposed approach [23]. On an updated finite element model of the structure [32].

76 Bridge Engineering

model previously described has been applied under the deterministic approach.

Figure 5. Scheme of the lateral lock-in pedestrian test on Pedro e Inês footbridge [31].

The natural frequency (around 0.91 Hz) and associated damping ratio (approximately 0.55%) of the first lateral vibration mode of the footbridge were identified experimentally. As the natural frequency of this vibration mode is within the range that characterizes the pedestrianstructure interaction in lateral direction, a lateral lock-in pedestrian test was conducted to determine experimentally the number of pedestrians that originates the lateral instability phenomenon [31]. The analysis focused on characterizing the beginning of the lateral lock-in phenomenon, since during this part of the phenomenon, the modification of the modal properties of the structure induced by the pedestrian-structure interaction is higher [3]. The lateral acceleration, alat, at mid-span of the structure in terms of the number of pedestrians, which cross along the structure, was recorded in this lateral lock-in pedestrian test. The analysis of the

> The assessment of the vibration serviceability limit state of footbridges under pedestrianinduced excitation has usually been performed based on the recommendations of the most

advanced international standards and design guidelines. However, the numerical estimation of the dynamic response of footbridges obtained according to these codes differs from the value recorded experimentally.

iv. The parameters of the crowd sub-model, normally based on the results of researches of general purpose, should be estimated concretely for the case of pedestrians moving on footbridges, to improve still more the accuracy of the crowd-structure interaction model.

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v. This new modelling framework allows establishing the comfort requirements directly in terms of the maximum accelerations experienced by the pedestrians (instead of the maximum accelerations reached by the deck of the footbridge). A new research line can be opened to establish more accurate thresholds which allow characterizing the vibration

This work was supported by the Ministerio de Economía y Competitividad of Spain and the

\* and Andrés Sáez<sup>2</sup>

1 Department of Building Structures and Geotechnical Engineering, Universidad de Sevilla,

2 Department of Continuum Mechanics and Structural Analysis, Universidad de Sevilla,

[1] Bachmann H, Ammann W. Vibrations in Structures, Induced by Man and Machines.

[2] Zivanovic S, Pavic A, Reynolds P. Vibration serviceability of footbridges under humaninduced excitation: A literature review. Journal of Sound and Vibration. 2005;279(1-2):1-74.

[3] Dallard P, Fitzpatrick AJ, Le Bourva S, Low A, Smith R, Wilford M, Flint A. The London

[4] Dziuba P, Grillaud G, Flamand O, Sanquier S, Tétard Y. La passerelle Solférino comportem-

[5] Jiménez-Alonso JF, Sáez A. A Controlling the Human-Induced Longitudinal Vibrations of a Nielsen-Truss Footbridge Via the Modification of its Natural Frequencies. International

European Regional Development Fund under project DPI2014-53947-R.

Structural Engineering Documents, IABSE, 1987. N 3e

millenium footbridge. The Structural Engineer. 2001;79(22):17-33

ent dynamique. Bulletin Ouvrages Métalliques. 2001;1:34-57 (in French)

DOI: https://doi.org/10.1016/j.jsv.2004.01.019

serviceability limit state better [14].

Acknowledgements

Author details

Spain

Spain

References

Javier Fernando Jiménez-Alonso<sup>1</sup>

\*Address all correspondence to: jfjimenez@us.es

In order to overcome this problem, a new generation of crowd-structure interaction models, that constitute a new modelling framework, has been proposed by the scientific community. All these models share, as common characteristic, that they simulate the crowd-structure interaction phenomenon using two sub-models: (i) a pedestrian-structure interaction submodel and (ii) a crowd sub-model. For the first sub-model, the pedestrian is modelled by a SDOF, MDOF or IP system and the structure via its modal parameters obtained from a finite element model. For the second sub-model, the last tendency is to use a multi-agent method based on the principles of the social force model. The linking between the two sub-models is achieved by the inclusion of several behavioural conditions in the model. Comfort and lateral lock-in threshold are usually considered. Three key aspects are taken into account for this new modelling framework: (i) the inter-and intra-subject variability, (ii) the pedestrian-structure interaction and (iii) the crowd dynamics. The last two aspects are guaranteed by the own formulation of the model, and the first is ensured assuming that the different parameters of the crowd-structure interaction model are random variables.

One of these new crowd-structure interaction models has been described briefly in this chapter, emphasizing the section corresponding to the crowd behaviour.

Finally, the potential of this new modelling framework has been illustrated with a case study, the analysis of the lateral lock-in phenomenon of the Pedro e Inês footbridge (Coimbra, Portugal). As result of this study, a good agreement is achieved between the number of pedestrians which originates the lateral instability phenomenon obtained during the experimental test and the numerical estimation determined via the crowd-structure interaction model.
