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Experimental studies on lateral forces induced by pedestrians
Journal of Constructional Steel Research 64 (2008) 247–252 www.elsevier.com/locate/jcsr
Experimental studies on lateral forces induced by pedestrians Shun-ichi Nakamura a,∗ , Toshitsugu Kawasaki b , Hiroshi Katsuura c , Kaoru Yokoyama d a Department of Civil Engineering, Tokai University, 1117, Kitakaname, Hiratsuka, 259-1259, Japan b Design Section, Fukuken Engineering, 1-11-12 Horidome, Nihonbashi, Tokyo, 103-0012, Japan c Yokohama Laboratory, Mitsubishi Heavy Industry, Nakaku, Yokohama, 231-8715, Japan d Bridge Technical Section, Mitsubishi Heavy Industry, Minatoku, Tokyo, 108-8215, Japan
Received 5 May 2006; accepted 31 May 2007
Abstract It has been observed on a couple of cable supported footbridges that the girder vibrated laterally with a frequency of about 1.0 Hz when a large number of people crossed the bridge. The girder was excited by the lateral dynamic force which was produced by the zigzag movement of pedestrians. Once the bridge started to vibrate, some of the pedestrians synchronized with the girder vibration, which further increased the girder response. Experiments were conducted to find the pedestrians’ dynamic forces on the vibrating deck. Pedestrians walked on the specially tailored deck placed on the shake table, which was vibrated with different frequencies and amplitudes. The experiments showed that the dynamic forces induced by pedestrians increased with the amplitude of the shake table. The synchronous nature of pedestrians was also measured and clarified. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Pedestrian-induced lateral vibration; Footbridge; Cable-supported bridge; Shake table
1. Introduction In recent years a couple of footbridges have suffered lateral vibration induced by pedestrians [1–6]. It was reported that the girder amplitude reached 70 mm on the opening day of the London Millennium Bridge [1], which was closed for 18 months before suppression measures were installed. Fujino and Nakamura et al. experienced this lateral vibration on the Tbridge before this incident and found the mechanism of the lateral vibration [2–5]. The T-bridge is a cable-stayed bridge with a main span of 134 m (Fig. 1). Nakamura and Fujino carried out field measurements on the T-bridge and found that the girder vibrated laterally with a frequency of 0.93 Hz when a large number of people crossed the bridge [2–5]. The girder is excited by the lateral dynamic force which is produced by the zigzag movement of pedestrians (Fig. 2). The gravitational centre of the body moves laterally when a person steps with his right foot and his left foot in turn, which induces this lateral dynamic force (Fig. 2). The frequency of this lateral loading is about 1 Hz (half of the pacing frequency), which is close to the bridge’s lateral natural ∗ Corresponding author. Tel.: +81 463 58 1211; fax: +81 463 50 2045.
E-mail address: [email protected] (S. Nakamura). c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.05.011
Fig. 1. T-bridge (unit: m).
frequency of 0.93 Hz. Therefore, this lateral dynamic force induced by pedestrians can be a resonant or near-resonant force that vibrates excessively the girder and some stay cables whose natural frequencies are close to this frequency (Fig. 3). Once the bridge starts to vibrate, some of the pedestrians synchronize with the girder vibration, which further increases the girder response. The key to predicting the girder response is the lateral dynamic forces of pedestrians when they walk on the vibrating deck; these have so far not been clarified. Bachmann and Ammann found that about 10% of the vertical loading, which is about 4% of the pedestrian’s weight, works laterally when he
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Fig. 2. Movement of gravity center during walking.
Fig. 3. Mechanism of lateral vibration.
Fig. 6. Shake table.
Fig. 4. Shake table (plan view).
Fig. 7. Pedestrian walking on the shake table.
(1), FP , is the modal lateral dynamic force induced by all the pedestrians on the vibrating bridge deck. If FP is clarified, Eq. (1) can be solved easily. In this paper, experiments are conducted to find the pedestrians’ dynamic forces on a vibrating deck. Pedestrians walk on the specially tailored deck placed on the shake table, which is vibrated with different frequencies and amplitudes.
Fig. 5. Shake table (side view).
walks [7]. However, this is the lateral force of pedestrians on the static ground. It was found from field measurements on the T-bridge that the girder vibrated in the first symmetric mode. Therefore, the lateral vibration induced by pedestrians can be modelled as a single-degree-of-freedom dynamic model using the modal analysis of the first lateral mode [4,5]: M B x B00 (t) + C B x B0 (t) + K B x B (t) = FP (t) .
(1)
Eq. (1) is the equation of motion for the first lateral mode; x B is the modal displacement of the girder, x B0 the modal velocity of the girder, x B00 the modal acceleration of the girder, M B the modal mass, C B the modal damping coefficient, and K B the modal stiffness of the bridge. The right-hand side of Eq.
2. Test set-up The testing was conducted using a shake table that was 1500 mm wide and 1000 mm long (Figs. 4–6). A reaction plate 800 mm wide and 500 mm long made of acrylic resin was placed on the shake table. A pedestrian walked on the spot on the reaction plate (Fig. 7). He/she did not move forwards or backwards. The shake table was vibrated by actuators with frequencies from 0.75 to 1.25 Hz, with amplitudes from 10 to 70 mm. The movement of the shake table was measured by the displacement transducers. The reactions between the reaction plate and the shaking table were measured by load-cells. Therefore, when a pedestrian moved and the lateral force was induced, reactions occurred and were measured by load-cells.
S. Nakamura et al. / Journal of Constructional Steel Research 64 (2008) 247–252
249
Fig. 8. Time history of lateral displacements of the shake table and the pedestrian.
Fig. 9. Time history of reactions of the shake table.
Fig. 10. Time history of the pedestrian-induced lateral force.
Fig. 11. Power spectrum of displacements.
Fig. 12. Power spectrum of lateral forces.
Five people were tested (Table 1): two were female and three male. They were not bridge engineers and did not have
any knowledge about this lateral vibration problem. They were not notified of the purpose and procedure of the test. The
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S. Nakamura et al. / Journal of Constructional Steel Research 64 (2008) 247–252
Fig. 13. Non-dimensional lateral forces induced by pedestrians.
accelerometer was attached to the waist belt of the pedestrian. The dominant frequency of the pedestrian movement was compared with that of the shake table to find the rate of synchronized people with the shaking frequency. 3. Test results and discussion Fig. 8 is a typical result of measured displacements of the shake table and the pedestrian ‘a’ for a shaking frequency of 1.0 Hz. An accelerometer was attached to the waist belt of a pedestrian and his/her displacement was obtained by integrating the measured acceleration twice. Fig. 8 shows that the displacements of the shake table and pedestrian ‘a’ have the same frequency and phase. Fig. 9 shows the measured reactions of the reaction plate when nobody was on the table and when pedestrian ‘a’ was on the table for a shaking frequency of 1.0 Hz. The lateral force of the pedestrian is then derived by subtracting the reaction with pedestrian ‘a’ on the table from that without a pedestrian (Fig. 10). Fig. 11 shows the power spectra of the displacements of the shake table and pedestrian ‘a’, which show that both have a peak at a frequency of 1.0 Hz. Fig. 12 is the power spectrum of the lateral force induced by pedestrian ‘a’, which has a sharp peak at 1.0 Hz and a smaller peak at 3.0 Hz. A person first steps
Fig. 14. Rate of synchronized people.
in with his heel and then kicks off with his toe, and this process produces the two peaks in the spectrum. These two peaks were also observed by other researchers [7]. Fig. 13 show the non-dimensional lateral forces induced by pedestrians, the lateral force divided by the pedestrian
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S. Nakamura et al. / Journal of Constructional Steel Research 64 (2008) 247–252 Table 1 Tested pedestrians
Weight (N) Pedestrians
Pedestrian a
Pedestrian b
Pedestrian c
Pedestrian d
Pedestrian e
800 Male
480 Female
500 Male
450 Female
600 Male
Fig. 15. Comparison of summation and superposition of five people.
weight. These figures show the non-dimensional lateral forces of five pedestrians at four shaking frequencies of 0.75 Hz, 0.87 Hz, 1.00 Hz and 1.25 Hz. There is a clear tendency at all these shaking frequencies that the non-dimensional lateral forces gradually increase with shake table amplitude. Taking a figure at the shake table frequency of 1.0 Hz, the nondimensional force can be 10% of the pedestrian’s weight at a shake table amplitude of 10 mm, and 16% at an amplitude of 70 mm. There was no difference in movement between male and female pedestrians. However, as pedestrian forces depended on their weight, the non-dimensional force was found to be a reasonable parameter for describing pedestrian forces. As mentioned before, once the bridge starts to vibrate, some of the pedestrians synchronize with the girder vibration, which further increases the girder response. The rate of pedestrians who synchronize with the girder vibration is an important factor affecting the girder response. It is judged to be synchronized when the frequency of measured pedestrian displacements equals the shaking frequency. Fig. 14 shows the rate of the synchronized people observed in the experiments. At a shaking
frequency of 0.87 Hz, 20% of the tested pedestrians (one out of five people) synchronized with the shaking frequency at all the shake table amplitudes. At a shaking frequency of 1.0 Hz, on average 50% of the tested pedestrians (two or three out of five people) synchronized with the shake table frequency. At shaking frequencies of 0.75 Hz and 1.25 Hz, the synchronization rate was zero. This means that none of them synchronized with the shake table and therefore it is unlikely that the lateral vibration problem occurs with a girder natural frequency lower than 0.75 Hz and higher than 1.25 Hz. Fig. 15 shows a comparison of the two maximum nondimensional lateral forces. One is summation of the maximum lateral forces of five individual people. Another is the maximum lateral force of the superposed time history of five pedestrians. Only one person walked on the shake table, which produced a time history of the pedestrian-induced force, as shown in Fig. 16 (with a shaking frequency of 1.00 Hz and amplitude of 30 mm). Five time histories were then obtained for five pedestrians. Time histories of the three pedestrians were shown in Fig. 16. When the five time histories were superposed, this gave another time history. The maximum value of the superimposed time history is much less than a simple summation of the maximum five individual pedestrian values. This is because there are phase differences between time histories of the five pedestrians (Fig. 16). This reduction in superimposed force may be larger when the deck amplitude becomes higher, as the synchronization rate is lower at an amplitude of 70 mm than those of 30 mm and 50 mm (Fig. 14, shaking frequency of 1.00 Hz). 4. Conclusion The lateral vibration of cable-supported bridge girders induced by pedestrians is a serious problem. To predict the girder response it is essential to evaluate the lateral dynamic forces of pedestrians when they walk on the vibrating deck. Experiments were conducted to find the pedestrians’ dynamic
Fig. 16. Time history of pedestrian-induced lateral forces.
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forces on the vibrating deck. Pedestrians walked on the specially tailored deck placed on the shake table with different frequencies and amplitudes. The experiments showed that the dynamic forces induced by pedestrian increased with the amplitude of the shake table. At a shake table frequency of 1.0 Hz, the non-dimensional force (pedestrian force divided by pedestrian weight) can be 10% of the pedestrian weight at a shake table amplitude of 10 mm, and 16% at an amplitude of 70 mm. The rate of pedestrians synchronized with the girder vibration is an important factor affecting the girder response. At a shake table frequency of 0.87 Hz, 20% of the tested pedestrians synchronized with the shake table frequency at all shake table amplitudes. At a shake table frequency of 1.0 Hz, on average 50% of the tested pedestrians were synchronized. The external force induced by pedestrians is key to predicting the girder response, and the experiments conducted in this research give useful information. However, as the experiments used only five pedestrians, further tests with more
pedestrians would be necessary to understand this problem fully. References [1] Dallard P, Fitzpatrick A, Flint A, Le Bourva S, Low A, Ridsill Smith R, et al. The London Millennium Footbridge. The Structural Engineer 2001; 79(22):17–35. [2] Fujino Y, Pacheco B, Nakamura S, Warnitcahi P. Synchronization of human walking observed during lateral vibration of a congested pedestrian bridge. Earthquake Engineering and Structural Dynamics 1993;22:741–58. [3] Nakamura S, Fujino Y. Lateral vibration on a pedestrian cable-stayed bridge and its suppression by tuned liquid dampers. Structural Engineering International, IABSE 2002;12(4). [4] Nakamura S. Model for lateral excitation of footbridges by synchronous walking. Journal of Structural Engineering, ASCE 2004;130(1):32–7. [5] Nakamura S, Kawasaki T. Lateral vibration of footbridges by synchronous walking. Journal of Constructional Steel Research 2006;62(11):1148–60. [6] Nakamura S. Field measurements of lateral vibration on a pedestrian suspension bridge. The Structural Engineering 2003;81(22):22–6. [7] Bachmann H, Ammann A. Vibrations in structures induced by man and machines. IABSE, Structural Engineering Document. 3e. 1987.
Experimental studies on lateral forces induced by pedestrians Shun-ichi Nakamura a,∗ , Toshitsugu Kawasaki b , Hiroshi Katsuura c , Kaoru Yokoyama d a Department of Civil Engineering, Tokai University, 1117, Kitakaname, Hiratsuka, 259-1259, Japan b Design Section, Fukuken Engineering, 1-11-12 Horidome, Nihonbashi, Tokyo, 103-0012, Japan c Yokohama Laboratory, Mitsubishi Heavy Industry, Nakaku, Yokohama, 231-8715, Japan d Bridge Technical Section, Mitsubishi Heavy Industry, Minatoku, Tokyo, 108-8215, Japan
Received 5 May 2006; accepted 31 May 2007
Abstract It has been observed on a couple of cable supported footbridges that the girder vibrated laterally with a frequency of about 1.0 Hz when a large number of people crossed the bridge. The girder was excited by the lateral dynamic force which was produced by the zigzag movement of pedestrians. Once the bridge started to vibrate, some of the pedestrians synchronized with the girder vibration, which further increased the girder response. Experiments were conducted to find the pedestrians’ dynamic forces on the vibrating deck. Pedestrians walked on the specially tailored deck placed on the shake table, which was vibrated with different frequencies and amplitudes. The experiments showed that the dynamic forces induced by pedestrians increased with the amplitude of the shake table. The synchronous nature of pedestrians was also measured and clarified. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Pedestrian-induced lateral vibration; Footbridge; Cable-supported bridge; Shake table
1. Introduction In recent years a couple of footbridges have suffered lateral vibration induced by pedestrians [1–6]. It was reported that the girder amplitude reached 70 mm on the opening day of the London Millennium Bridge [1], which was closed for 18 months before suppression measures were installed. Fujino and Nakamura et al. experienced this lateral vibration on the Tbridge before this incident and found the mechanism of the lateral vibration [2–5]. The T-bridge is a cable-stayed bridge with a main span of 134 m (Fig. 1). Nakamura and Fujino carried out field measurements on the T-bridge and found that the girder vibrated laterally with a frequency of 0.93 Hz when a large number of people crossed the bridge [2–5]. The girder is excited by the lateral dynamic force which is produced by the zigzag movement of pedestrians (Fig. 2). The gravitational centre of the body moves laterally when a person steps with his right foot and his left foot in turn, which induces this lateral dynamic force (Fig. 2). The frequency of this lateral loading is about 1 Hz (half of the pacing frequency), which is close to the bridge’s lateral natural ∗ Corresponding author. Tel.: +81 463 58 1211; fax: +81 463 50 2045.
E-mail address: [email protected] (S. Nakamura). c 2007 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2007.05.011
Fig. 1. T-bridge (unit: m).
frequency of 0.93 Hz. Therefore, this lateral dynamic force induced by pedestrians can be a resonant or near-resonant force that vibrates excessively the girder and some stay cables whose natural frequencies are close to this frequency (Fig. 3). Once the bridge starts to vibrate, some of the pedestrians synchronize with the girder vibration, which further increases the girder response. The key to predicting the girder response is the lateral dynamic forces of pedestrians when they walk on the vibrating deck; these have so far not been clarified. Bachmann and Ammann found that about 10% of the vertical loading, which is about 4% of the pedestrian’s weight, works laterally when he
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Fig. 2. Movement of gravity center during walking.
Fig. 3. Mechanism of lateral vibration.
Fig. 6. Shake table.
Fig. 4. Shake table (plan view).
Fig. 7. Pedestrian walking on the shake table.
(1), FP , is the modal lateral dynamic force induced by all the pedestrians on the vibrating bridge deck. If FP is clarified, Eq. (1) can be solved easily. In this paper, experiments are conducted to find the pedestrians’ dynamic forces on a vibrating deck. Pedestrians walk on the specially tailored deck placed on the shake table, which is vibrated with different frequencies and amplitudes.
Fig. 5. Shake table (side view).
walks [7]. However, this is the lateral force of pedestrians on the static ground. It was found from field measurements on the T-bridge that the girder vibrated in the first symmetric mode. Therefore, the lateral vibration induced by pedestrians can be modelled as a single-degree-of-freedom dynamic model using the modal analysis of the first lateral mode [4,5]: M B x B00 (t) + C B x B0 (t) + K B x B (t) = FP (t) .
(1)
Eq. (1) is the equation of motion for the first lateral mode; x B is the modal displacement of the girder, x B0 the modal velocity of the girder, x B00 the modal acceleration of the girder, M B the modal mass, C B the modal damping coefficient, and K B the modal stiffness of the bridge. The right-hand side of Eq.
2. Test set-up The testing was conducted using a shake table that was 1500 mm wide and 1000 mm long (Figs. 4–6). A reaction plate 800 mm wide and 500 mm long made of acrylic resin was placed on the shake table. A pedestrian walked on the spot on the reaction plate (Fig. 7). He/she did not move forwards or backwards. The shake table was vibrated by actuators with frequencies from 0.75 to 1.25 Hz, with amplitudes from 10 to 70 mm. The movement of the shake table was measured by the displacement transducers. The reactions between the reaction plate and the shaking table were measured by load-cells. Therefore, when a pedestrian moved and the lateral force was induced, reactions occurred and were measured by load-cells.
S. Nakamura et al. / Journal of Constructional Steel Research 64 (2008) 247–252
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Fig. 8. Time history of lateral displacements of the shake table and the pedestrian.
Fig. 9. Time history of reactions of the shake table.
Fig. 10. Time history of the pedestrian-induced lateral force.
Fig. 11. Power spectrum of displacements.
Fig. 12. Power spectrum of lateral forces.
Five people were tested (Table 1): two were female and three male. They were not bridge engineers and did not have
any knowledge about this lateral vibration problem. They were not notified of the purpose and procedure of the test. The
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S. Nakamura et al. / Journal of Constructional Steel Research 64 (2008) 247–252
Fig. 13. Non-dimensional lateral forces induced by pedestrians.
accelerometer was attached to the waist belt of the pedestrian. The dominant frequency of the pedestrian movement was compared with that of the shake table to find the rate of synchronized people with the shaking frequency. 3. Test results and discussion Fig. 8 is a typical result of measured displacements of the shake table and the pedestrian ‘a’ for a shaking frequency of 1.0 Hz. An accelerometer was attached to the waist belt of a pedestrian and his/her displacement was obtained by integrating the measured acceleration twice. Fig. 8 shows that the displacements of the shake table and pedestrian ‘a’ have the same frequency and phase. Fig. 9 shows the measured reactions of the reaction plate when nobody was on the table and when pedestrian ‘a’ was on the table for a shaking frequency of 1.0 Hz. The lateral force of the pedestrian is then derived by subtracting the reaction with pedestrian ‘a’ on the table from that without a pedestrian (Fig. 10). Fig. 11 shows the power spectra of the displacements of the shake table and pedestrian ‘a’, which show that both have a peak at a frequency of 1.0 Hz. Fig. 12 is the power spectrum of the lateral force induced by pedestrian ‘a’, which has a sharp peak at 1.0 Hz and a smaller peak at 3.0 Hz. A person first steps
Fig. 14. Rate of synchronized people.
in with his heel and then kicks off with his toe, and this process produces the two peaks in the spectrum. These two peaks were also observed by other researchers [7]. Fig. 13 show the non-dimensional lateral forces induced by pedestrians, the lateral force divided by the pedestrian
251
S. Nakamura et al. / Journal of Constructional Steel Research 64 (2008) 247–252 Table 1 Tested pedestrians
Weight (N) Pedestrians
Pedestrian a
Pedestrian b
Pedestrian c
Pedestrian d
Pedestrian e
800 Male
480 Female
500 Male
450 Female
600 Male
Fig. 15. Comparison of summation and superposition of five people.
weight. These figures show the non-dimensional lateral forces of five pedestrians at four shaking frequencies of 0.75 Hz, 0.87 Hz, 1.00 Hz and 1.25 Hz. There is a clear tendency at all these shaking frequencies that the non-dimensional lateral forces gradually increase with shake table amplitude. Taking a figure at the shake table frequency of 1.0 Hz, the nondimensional force can be 10% of the pedestrian’s weight at a shake table amplitude of 10 mm, and 16% at an amplitude of 70 mm. There was no difference in movement between male and female pedestrians. However, as pedestrian forces depended on their weight, the non-dimensional force was found to be a reasonable parameter for describing pedestrian forces. As mentioned before, once the bridge starts to vibrate, some of the pedestrians synchronize with the girder vibration, which further increases the girder response. The rate of pedestrians who synchronize with the girder vibration is an important factor affecting the girder response. It is judged to be synchronized when the frequency of measured pedestrian displacements equals the shaking frequency. Fig. 14 shows the rate of the synchronized people observed in the experiments. At a shaking
frequency of 0.87 Hz, 20% of the tested pedestrians (one out of five people) synchronized with the shaking frequency at all the shake table amplitudes. At a shaking frequency of 1.0 Hz, on average 50% of the tested pedestrians (two or three out of five people) synchronized with the shake table frequency. At shaking frequencies of 0.75 Hz and 1.25 Hz, the synchronization rate was zero. This means that none of them synchronized with the shake table and therefore it is unlikely that the lateral vibration problem occurs with a girder natural frequency lower than 0.75 Hz and higher than 1.25 Hz. Fig. 15 shows a comparison of the two maximum nondimensional lateral forces. One is summation of the maximum lateral forces of five individual people. Another is the maximum lateral force of the superposed time history of five pedestrians. Only one person walked on the shake table, which produced a time history of the pedestrian-induced force, as shown in Fig. 16 (with a shaking frequency of 1.00 Hz and amplitude of 30 mm). Five time histories were then obtained for five pedestrians. Time histories of the three pedestrians were shown in Fig. 16. When the five time histories were superposed, this gave another time history. The maximum value of the superimposed time history is much less than a simple summation of the maximum five individual pedestrian values. This is because there are phase differences between time histories of the five pedestrians (Fig. 16). This reduction in superimposed force may be larger when the deck amplitude becomes higher, as the synchronization rate is lower at an amplitude of 70 mm than those of 30 mm and 50 mm (Fig. 14, shaking frequency of 1.00 Hz). 4. Conclusion The lateral vibration of cable-supported bridge girders induced by pedestrians is a serious problem. To predict the girder response it is essential to evaluate the lateral dynamic forces of pedestrians when they walk on the vibrating deck. Experiments were conducted to find the pedestrians’ dynamic
Fig. 16. Time history of pedestrian-induced lateral forces.
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S. Nakamura et al. / Journal of Constructional Steel Research 64 (2008) 247–252
forces on the vibrating deck. Pedestrians walked on the specially tailored deck placed on the shake table with different frequencies and amplitudes. The experiments showed that the dynamic forces induced by pedestrian increased with the amplitude of the shake table. At a shake table frequency of 1.0 Hz, the non-dimensional force (pedestrian force divided by pedestrian weight) can be 10% of the pedestrian weight at a shake table amplitude of 10 mm, and 16% at an amplitude of 70 mm. The rate of pedestrians synchronized with the girder vibration is an important factor affecting the girder response. At a shake table frequency of 0.87 Hz, 20% of the tested pedestrians synchronized with the shake table frequency at all shake table amplitudes. At a shake table frequency of 1.0 Hz, on average 50% of the tested pedestrians were synchronized. The external force induced by pedestrians is key to predicting the girder response, and the experiments conducted in this research give useful information. However, as the experiments used only five pedestrians, further tests with more
pedestrians would be necessary to understand this problem fully. References [1] Dallard P, Fitzpatrick A, Flint A, Le Bourva S, Low A, Ridsill Smith R, et al. The London Millennium Footbridge. The Structural Engineer 2001; 79(22):17–35. [2] Fujino Y, Pacheco B, Nakamura S, Warnitcahi P. Synchronization of human walking observed during lateral vibration of a congested pedestrian bridge. Earthquake Engineering and Structural Dynamics 1993;22:741–58. [3] Nakamura S, Fujino Y. Lateral vibration on a pedestrian cable-stayed bridge and its suppression by tuned liquid dampers. Structural Engineering International, IABSE 2002;12(4). [4] Nakamura S. Model for lateral excitation of footbridges by synchronous walking. Journal of Structural Engineering, ASCE 2004;130(1):32–7. [5] Nakamura S, Kawasaki T. Lateral vibration of footbridges by synchronous walking. Journal of Constructional Steel Research 2006;62(11):1148–60. [6] Nakamura S. Field measurements of lateral vibration on a pedestrian suspension bridge. The Structural Engineering 2003;81(22):22–6. [7] Bachmann H, Ammann A. Vibrations in structures induced by man and machines. IABSE, Structural Engineering Document. 3e. 1987.