Footbridge 2014 th 5 International Conference Footbridges: Past, present & future
A Study of Vibrations of a Slender Footbridge Due to Human Movements Mehdi SETAREH Professor Virginia Tech Blacksburg, VA, USA setareh@vt.edu
Mico WOOLARD Mechanical Engineering Student Virginia Tech Blacksburg, VA, USA awool012@vt.edu
Amanda SCHLICHTING Architecture Student Virginia Tech Blacksburg, VA, USA amandas8@vt.edu
Summary Slender footbridges can be susceptible to large vibrations due to human movements. The low natural frequencies and damping of these systems can result in excessive or annoying movements. This paper presents details of the vibrations analysis of a two-span steel footbridge, designed and built by a group of architecture students at Virginia Tech, Blacksburg, Virginia, USA. The footbridge is comprised of three main segments that were fabricated in shop and shipped to the site for installation. The main supporting members were made of two 200 mm deep steel I-beams, which resulted in a span/depth ratio of about 66. As the structure was only designed for static loads, design modifications to the boundary conditions and member connections were made to reduce the level of vibrations due to the pedestrians’ movements on the footbridge. Following the completion of the construction, dynamic testing of the structure was conducted. The main objectives of these tests were: (1) to conduct a modal analysis of the footbridge, (2) to study the effects of human-structure interactions (HSI) on the dynamic properties of the footbridge, and (3) to evaluate the vibration of the footbridge due to different numbers of pedestrians walking or running over it. The presence of people resulted in a reduction in the natural frequencies and an increase in damping ratios. The low natural frequency of the footbridge made it susceptible to excessive vibrations when a group of people ran over it. Finally, two observers evaluated the intensity of vibrations as people crossed the bridge. The results were compared with the provisions of some guidelines on human vibration perceptibility. Keywords: steel; footbridge; serviceability; vibration assessment; modal analysis; field testing; human-structure interaction; damping 1. Introduction Light and slender footbridges have become more desirable in the design community due to the savings in material and their aesthetic appeal. High strength of modern construction materials such as steel allows the architects and engineers to achieve their goals. Even though such structures can resist the applied static loads, they might be susceptible to excessive vibrations when people cross them. In extreme cases, this may result in failure of the structure. The first reported case in literature is the collapse of a cast iron bridge in 1831, which was caused by soldiers marching in step [1]. However, in most cases these vibrations may be disturbing to the footbridge users without resulting in structural failures. Reported annoying vibrations are mainly due to the lateral movements of large footbridges. The most publicized case can be found in the London Millennium Bridge over the Thames River in 2000 [2]. Most footbridges with annoying vertical vibrations have had a span of 50 m or less. Matsumoto, et al. [3] studied a footbridge with a span of 48.5 m with large vertical vibrations when pedestrians crossed it at a speed of 120 steps per minute (spm) or 2 Hz. Bachmann and Ammann [4] reported the cases of two small footbridges with very lively vertical vibrations. These included a 40 m span footbridge with a fundamental frequency of 1.92 Hz and a 34 m span footbridge with a fundamental frequency of 2.3 Hz. The presence of humans can change the dynamic properties of structures as they relate to vibration serviceability issues. Some published studies, such as [5], have considered occupants as additional masses on the structure only. However, some researchers have also reported large increases in damping due to human presence [6-8]. There have also been several studies, guidelines, and standards to assess the acceptability of vibrations to humans for different structures. A few of these works are: Lenzen [6]; Bachmann, et al. [9]; Murray, et al. [10]; National Research Council of Canada (NBCC) [11]; Canadian Standards Association [12]; and ISO 10137 [13].
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
This paper presents the results of a series of vibration tests conducted on a footbridge. It details the results of the modal testing and analysis of the structure. The effects of human-structure interactions (HSI) on the dynamic properties of the footbridge were studied by conducting dynamic tests with a number of people on the structure. This paper also discusses the acceptability of measured vibrations from a number of walk tests using different limits found in the literature based on the peak accelerations. Finally, the subjective assessment of the measured vibrations by two bystanders on the footbridge will be discussed and compared to the limits from different published sources. 2. Description of the Footbridge and the Static Design The footbridge is located in Clifton Forge, Virginia. Architectural design and construction of the structure were conducted by a group of architecture students at Virginia Tech as part of a course requirement. The total bridge length, including a main span of 13.6 m and a ramp of 11.2 m, is 28.4 m. Figure 1 shows an elevation and transverse section of the structure. As shown in the figure, the footbridge is made of three main segments: the bridge, the hub, and the ramp, which were prefabricated, shipped to the site, bolted together, and anchored to the end supports. Figure 2 is a photo of the completed structure. Due to the fact that the footbridge was located in the 100-year floodplain, it was designed to resist the flood loads. As shown in Figure 1, the width of the footbridge is about 1.5 m. The deck is made of 19 mm thick wood decking supported by 51 mm x 171 mm wood joists, which were nailed to an end ledger bolted to the main structural steel sections (W200x41.7, W200x35.9, and W150x29.8). For the static design of the structure, a dead load of 960 Pa (including the self-weight of the structure and non-structural elements such as posts and railings) and a live load of 4,070 Pa (based on the recommendations of AASHTO 1997 [14]) were used.
Fig. 1: Elevation and transverse section of the footbridge
The footbridge is a slender structure as the span/depth ratio for the bridge segment is about 66. The architectural design requirement did not allow the use of larger structural sections. Even though the structural elements provided adequate
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
Fig. 2: A photo of the completed footbridge
strength to resist the applied static loads, the deflection under the full design live load (4,070 Pa) was expected to be excessive. One end of the bridge segment is supported by an old existing concrete retaining wall. The compressive strength of the retaining wall using a core sample taken from the wall was found to be 22 MPa. In order to reduce the deflection and provide more stiffness, a 610 mm thick concrete pier was poured behind the existing retaining wall and anchored to the wall at the locations that the bridge segment was to be supported. Four 19 mm epoxied anchor bolts with a minimum embedment length of 108 mm and four 19 mm anchor bolts with hex nuts and a minimum embedment length of 305 mm were used in the newly poured concrete pier. As shown in Figure 1, the hub is supported four 141 mm and three 114 mm steel pipes. The ramp is connected to the hub from one side and supported by a 762 mm x 1,524 mm stem wall on the other. Eight 19 mm anchor bolts with hex nuts and a minimum embedment length of 305 mm were used.
3. Modal Tests and Analysis To estimate the dynamic properties of the footbridge, a series of modal tests and analyses were conducted. For this purpose, an electrodynamic shaker supported by a force plate was placed around the midspan of the bridge segment on one of the support beams. A signal analyzer was used to measure and record the vibration data. Channels 1 and 2 were connected to the force plate and the accelerometer attached to the shaker armature, respectively, to measure the input force. A roving accelerometer technique was used, for which accelerometers connected to channels 3, 5, 6, 8, 9, and 10 were placed on the footbridge main structural steel beams along the edges. Channels 4 and 7 were placed at the center of the deck on the wood flooring. These accelerometers were all oriented vertically. Accelerometers connected to channels 11 through 16 were oriented laterally and clamped to the top of the guardrail. Figure 3 shows the locations of accelerometers on the deck and the guardrail in the bridge segment only. For the sake of clarity of the figure the accelerometer locations for the ramp and hub are not shown. A burst chirp excitation (3-12 Hz range) with 30 seconds on and 15 seconds off (3 averages) was used.
Fig. 3: Locations of accelerometers and human subjects on the bridge portion for modal, HSI, and walk tests
A maximum shift of 7% in the measured resonance frequencies during the tests occurred. This can be attributed to the temperature variations during the tests as it rose from 65º F to 80º F. From the measurements, it is clear that the
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
resonance frequencies decreased as the structure became more flexible as the temperature rose with the modal tests’ progression. Using ME’scopeVES [15], a modal analysis of the measured data was conducted. A local curve-fitting scheme enabled the extraction of modal properties with small variations in the natural frequencies. Only the first two modes of vibration were identified since the higher modes had natural frequencies greater than 8 Hz, which were not susceptible to excitations by human movements on the footbridge. Figure 4 shows the first two mode shapes and Table 1 includes the estimated natural frequencies and damping ratio for the first two modes.
Undeformed shape
Mode 1
Mode 2
Fig. 4: Measured mode shapes of the structure
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
Table 1: Estimated Measured Natural Frequencies and Damping Ratios
Mode (i) 1 2
fi (Hz) 3.61 4.72
i (%) 1.09 0.72
The first identified mode, f1 = 3.61 Hz, is a torsional mode, and the second mode, f2 = 4.72 Hz, is a bending mode. Note that the mode shapes for the ramp segment are not shown due to their insignificant modal amplitudes. Different levels of modal damping ratios for footbridges have been recommended in the available works of literature. Bachmann, et al. [9] and Heinemeyer et al. [16] suggest a minimum damping ratio of 0.2% and an average value of 0.4%. ISO10137 [13] and the Canadian Standards Association [12] recommend a modal damping ratio of 0.6% for composite steel and concrete footbridges. However, Brownjohn [17], AASHTO [14], and Murray, et al. [10] suggest a damping ratio of 1%. Eurocode 8 [18] recommends a range of 1% to 4% for the damping ratios of structures under earthquake loads. Since the first mode is the most susceptible to vibrations due to human movements, it seems that a damping ratio of 1% is more reasonable for such structures (steel footbridges with timber decking). 4. Human-Structure Interactions (HSI) Effects The opportunity for testing the structure was used to check the effects of the presence of people on the dynamic properties of the footbridge. The shaker was left at the midspan of the bridge segment. Fourteen accelerometers were placed on the bridge segment at 1.8 m apart along the main steel W200x41.7 edge beams. No accelerometers were used on the ramp or hub. Accelerometers connected to channels 3 through 16 were oriented vertically as shown in Figure 3. To recheck the dynamic properties of the footbridge prior to the tests, the shaker was run without people with a burst chirp of 2-12 Hz. Table 2 shows the estimated natural frequencies and damping ratios of the first two identified modes. Table 2: Estimated Measured Natural Frequencies and Damping Ratios Inducing HSI Effects
Without People HSI-1 HSI-2
Mode (i) 1 2 1 2 1 2
fi (Hz) 3.49 4.68 3.46 4.47 3.21 4.03
i (%) 0.88 0.71 2.0 1.95 -
Sixteen volunteer subjects with an average age of 20 years old were instructed to stand on the bridge segment while the structure was dynamically excited with a burst chirp forcing function. The same setting as for without people was used. They stood still in two different configurations (see Figures 3 and 5): (1) Every two subjects stood at 1.8 m intervals along each of the steel edge beams of the bridge segment (HSI-1), and (2) Every two subjects stood at 0.9 m intervals along each of the steel edge beams located on the midspan segment of the bridge (HSI-2). A comparison of the results in Tables 1 and 2 shows a very small variation in the modal properties of the footbridge without people. This can be attributed to the temperature effects and variations in the dynamic tests, which are usually expected.
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
Fig. 5: Human subjects on the footbridge during the HSI-1 (left) and HSI-2 (right) Tests
From the results of Table 2, it is clear that the presence of human subjects resulted in a decrease in the measured natural frequencies and an increase in the damping ratios. This is consistent with the results of past studies related to the effects of HSI on low frequency structures. Also, it is concluded that HSI-2 resulted in a larger change in the footbridge natural frequencies than HSI-1. This could have also been expected as HSI-2 configuration results in larger total effective mass of the human subjects and thus, larger change in the dynamic properties of the footbridge. However, the damping ratio for the first mode did not change and a reasonably valid damping ratio for the second modes with people could not be identified due to the high levels of noise in the measured responses due to the presence of human subjects. Figure 6 shows the drive point frequency response functions for the footbridge without people, HSI-1, and HSI2. From this figure, it can be noted that the presence of human subjects resulted in a significant suppression of the second mode of vibration.
Fig. 6: Drive point Frequency Response Functions for Without People, HSI-1, and HSI-2
5. Walk Tests To assess the acceptability of the vibrations when pedestrians cross the footbridge, a number of walking/running tests were conducted with the help of a number of volunteers. Their movements on the structure were synchronized by a metronome. To provide subjective reactions to the footbridge vibrations, two observers were instructed to stand along the two edges of the structure next to the shaker as shown in Figure 3. Prior to the walk tests, the first mode resonance frequency of the footbridge while the observers were standing still on the structure was measured at 3.35 Hz using the same test setup as the HSI tests. The shaker was then removed from the footbridge and two more accelerometers were added on the bridge deck, creating a total of sixteen accelerometers as shown in Figure 3.
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
During the walk tests, two observers stood still while people crossed the entire footbridge at four different speeds: (1) 101 spm (first sub-harmonic of the resonance frequency); (2) 201 spm (resonance frequency); (3) 120 spm (average normal walk speed); and (4) random. For each speed, tests were conducted where 1, 3, and 14 people crossed the entire footbridge with two exceptions: (1) for 101 spm, an additional test was conducted where 7 people crossed the bridge; and (2) after the 7-person test at 201 spm, the observers and the runners indicated great annoyance and it was decided not to allow more than 7 people to cross the bridge at this speed. The observers used a 7-point Likert-Type scale to rate their perceived vibrations as people crossed the footbridge: 1 = not perceptible, 2 = somewhat perceptible, 3 = perceptible, 4 = somewhat uncomfortable, 5 = uncomfortable, 6 = somewhat annoying, 7 = annoying. The available literature has used different parameters of the measured responses of structures to evaluate their vibration serviceability. These include: peak unweighted acceleration (ap), root-mean-squared of unweighted acceleration (arms), maximum running root-mean-squared of frequency-weighted acceleration (aw, rms), and vibration dose value (VDV). Among these, ap is the least accurate parameter used for vibration evaluation [19]. However, it has been chosen here for the vibration evaluations of the footbridge for two reasons: (1) ap is the simplest parameter to use since there is almost no need for post-processing of the measured footbridge response, and (2) several design guides and standards provide acceptable ap limits for vibration serviceability of footbridges. However, these references do not distinguish any difference between the limits for the bystanders (passive users) and the pedestrians (active users). It is assumed that the provisions of these standards and guidelines are applicable to the pedestrians as the main footbridge users. Bachmann, et al. [9], and Murray, et al. [10], recommend a limit of 5% g. The Canadian Standards Association [12] indicates a limit of approximately 6% g for a footbridge with the first natural frequency of about 3.35 Hz. Heinemeyer, et al. [16] suggests various levels of acceptable ap based on the expected levels of comfort: (1) maximum comfort (<5% g), (2) medium comfort (5% g â&#x20AC;&#x201C; 10% g), (3) minimum comfort (10% g â&#x20AC;&#x201C; 25% g), and (4) unacceptable (>25% g). The results of the subjective assessments of the footbridge vibrations based on the average ratings of the two observers (passive users) with respect to ap are as follows: (1) between 2.5% g-5% g (Scale 2); (2) up to 7% g (Scale 3); (3) up to 13% g (Scale 4); (4) up to 15% g (Scale 5); (5) up to 24% g (Scale 6) and (6) up to 55% g (Scale 7). The subjects who crossed the footbridge (active users) indicated that they perceived the vibrations only when 3 people ran at 201 bpm (24% g); however, they had trouble crossing the structure when 7 people ran at 201 bpm (55% g). Comparing the results of the subjective assessment with the provisions of the guidelines and studies mentioned above, it is clear that Bachmann, et al. [9], Murray, et al. [10], and Canadian Standards Association [12] are all very conservative. The recommendations of Heinemeyer, et al. [16] are less conservative as they are closer to the subjective ratings of the observers (passive users), but they are conservative for the active users. However, two points should be noted here: (1) the results presented are from a limited study and a more extensive study is needed for a more definite general conclusion, and (2) as mentioned before, ap is not a reliable parameter for the serviceability evaluation of structures. Figure 7 shows the maximum measured vibrations where the observers were standing while 7 people crossed the footbridge at 201 spm. From the results of the subjective assessment of the footbridge, it was concluded that if a single individual walks or runs at any speed on the structure, the footbridge vibrations will be in the acceptable range for the active and passive users.
Fig. 7: Measured time response of the footbridge where the observers stood
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
6. Summary and Conclusions This paper presented a study of the vibration serviceability of a slender footbridge with natural frequencies that can be excited when pedestrians cross it. Modal test and analysis were conducted and the dynamic properties of the footbridge for the modes that are susceptible to human excitations were identified. Effects of human-structure interactions on the dynamic properties of the structure were measured, which indicated a reduction in the natural frequencies and an increase in modal damping ratios. Subjective assessments of the vibrations when groups of people crossed the footbridge at different speeds were conducted. Comparison of the results with the current guidelines showed that the majority of the provisions on the vibration serviceability limits for footbridges are conservative. However, more studies are required to validate the conclusions made here. 7. Acknowledgement The research presented here was supported by the National Science Foundation under grant number CMMI-1335004. This support is gratefully acknowledged. Any opinions, findings, and conclusions expressed in this paper are those of the writers and do not necessarily reflect the views of the National Science Foundation. The authors would like to acknowledge the assistance of Keith and Marie Zawistowski during various stages of this study. 8. References [1]
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Footbridge 2014 – 5 International Conference - Footbridges: Past, present & future
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