dynamic behavior and analysis of a slender timber footbridge

7/29/2019 Dynamic behavior and analysis of a slender timber footbridge

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Dynamic behavior and analysis of a slender timber footbridge

Anders RnnquistPost doc.Norwegian University of Science and Technology, NTNU

Trondheim, Norway

Lars WollebkConsultant

Scandpower Petroleum Technology ASKjeller, Norway

Kolbein BellProfessor

Norwegian University of Science and Technology, NTNUTrondheim, Norway

SummaryTroublesome pedestrian induced lateral vibrations experienced by a recently built slender timberfootbridge in Norway have led to both experimental and theoretical work at the NorwegianUniversity of Science and Technology. Some of this work, including observations on the bridgeitself, is described briefly, along with some thoughts on possible ways of reducing the problems.

1. Introduction

The Lardal bridge, shown in figure 1, was built during 2001. A creosote impregnated glulaminated(glulam) bridge with a steel cable reinforcement in the mid-section, it has a free span of 92 m and atotal length of about 130 m. The bridge project and its planning is described by Eggen [1].

The bridge is an integral part of a recreation area with two outdoor amphitheaters used for concertsin the vicinity of well known rapids of the Numedalslgen river. Its main purpose is to connect theparking place located on the west side of the river with the amphitheaters on the east side.

On opening day a fair number of people attended which led to a dense flow of people across thebridge. And the London Millennium bridge syndrome repeated itself: very noticeable lateralvibrations were observed and experienced. Some people grabbed the handrails and verbally

Figure 1 The Lardal pedestrian bridge


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expressed concern about the behavior of the bridge. Locals who often attend different arrangementsin the area have later become used to the vibrations, and to some it has even become a source ofamusement. With no major events on, the number of people using the bride on a daily basis is smalland no real problem has been reported. However, it is a matter of some concern that a group ofpeople might find it tempting to excite the bridge and thus perhaps damage vital structural elements.And the present situation, with bridge-keepers limiting the number of people on the bridge at any

one time during public events, can only bee seen as a temporary measure.Being a low budget project it was not unreasonable to seek university advice, and this initiativeeventually led to a shift of focus of an ongoing doctoral study, resulting in a dissertation revolvingaround the problems of this bridge, Rnnquist [2]. This study includes both experimentalinvestigations, in the laboratory as well as on the bridge itself, theoretical work and numericalsimulations. Part of the latter was facilitated by computer program developments carried out as partof another doctoral study, Wollebk [3].In this paper, the how and why of it will be described, followed by a discussion of possible ways ofalleviating the problem. What measures can be applied to the present design and, with hindsightinformation, what modifications could have been made at the design stage? It will be shown thatwhile the problems can be reduced, we do not believe it is possible to eliminate them without makingmajor changes to the design.

2. Structural system

A sketch of the structural system of the bridge is shown infigure 2. It consists of two parallel panels, each of whichis made of three major components: two cantilever trussbeams made of glulam (A-B-E and C-D-F) at each end,and a suspended truss like mid-section consisting of anupper chord made of a curved glulam beam and a lowerchord in the form of a steel cable. Along the top beams(A-B-C-D) the two panels are connected by a steel trusswhose chords are screwed to the glulam beams, at mid-height as indicated in section 1-1 of figure 3. All steelmembers have cross sections of L shape. The bridge deck

(45120 mm planks) is fastened to 4 longitudinal glulambeams (100160 mm) which rest on glulam cross beams(100200 mm) supported by the main beams at every 2,2meters. The deck planks form an angle of 60 degreeswith the bridge axis.It should be noted that the cross beams are fastened to themain beams in such a way that they do not provide muchtransverse stiffness. Hence the bulk of the transverse

stiffness is provided by the horizontal steel truss between the main glulam beams. This truss hasonly diagonals (no verticals), and its effective stiffness is also influenced by the fact that its chordsare fastened to the main beams by screws through oblong holes (to compensate for differenttemperature expansion in timber and steel).

Section 2-2 in figure 3 shows that the lower beams of the side cantilevers, that is A-E (D-F) and E-B(F-C), are strengthened by a horizontal beam, glued and bolted to the main beam, but there is notruss between the two panels along these beams.

65 m13 m 13 m13 m 13 m

2,4 mglulam

steel cablesteel tube

steel (L-section)

AB C

D

E F

Figure 2 Structural system of the Lardal pedestrian bridge

1

12

2

G M H

2400 mm

section 1 - 1

section 2 - 2

Figure 3 Bridge sections

steel

cross-beam

truss


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3. Free vibrations of the bridge

3.1 Measured results

Full-scale measurements were carried out on the Lardal bridge on three occasions during the summermonths of 2002 and 2003. Both horizontal and vertical accelerations were measured with three

different setups of the measuring devices in the mid-span.Three methods of excitation were applied:1. Stamping or jumping at mid-span by one and more persons were used to identify characteristic

eigen-frequencies associated mainly with vertical displacements.2. One and several persons were standing wide footed on the deck at various positions along the

mid-span and rhythmically rocking their bodies sidewise. This method was used to identifycharacteristic eigen-frequencies associated mainly with displacements in the horizontal direction,but also the torsional modes were exited in this manner.

3. A mechanical exciter, consisting of two counter-rotating adjustable masses driven by an electricengine, was placed on the deck at different positions along the mid-span. With this device it waspossible to excite the bridge with a harmonic load with known amplitude and frequency.

In addition to determining the free vibration mode shapes and frequencies, the measurements werealso designed to give information about the corresponding modal mass, stiffness and damping. Themeasurements and the following signal processing are thoroughly described in [2]. In the presentcontext, we are primarily interested in the eigenmodes and frequencies, and the results of theidentification process are summarized in table 1, which also includes the computed results (inparentheses). As can be seen from the table the mode shapes are well separated, and although there

is some coupling, particularly between the horizontal and torsional modes, they are quite distinct andthe main component is dominating in all the nine recorded modes. We shall return to the shape of themodes in the next section.

3.2 Computed results

Both static and dynamic finite element analyses were carried out during the initial design of thebridge. For the static analyses mainly 2D models were used, whereas a fairly crude 3D model wasused for the dynamic (free vibration) analyses. For the latter, ANSYS ED [www.Ansys.com],whichputs severe restrictions on the number of nodes and elements, was used. This model predicted thelowest horizontal mode at 1,29 Hz and the lowest vertical mode at 1,95 Hz.The discrepancies with the measured results led to a much more detailed model being developed foranalysis with a less restrictive version of ANSYS. However, this model, while much better, did notpredict the eigen-frequencies with the desired accuracy. It gave 0,97 Hz for the first horizontal mode,1,24 Hz for the first torsional mode, and 1,59 Hz for the first vertical mode. And while the FEMmodel predicted a shape of the first horizontal mode having a torsional component in the form of ahammock movement, the measurements indicated an inverted pendulum movement.At this stage the problem was considered a suitable test case for the FrameIT program [4] which wasdeveloped by Wollebk in his doctoral research [3]. With versatile modelling and analysis

capabilities and good graphical visualization this program turned out to be well suited for the job.The two properties that need to be adequately modelled in an undamped free vibration analysis is the

Table 1 Recorded (and computed) eigen-frequencies (in Hz) of the Lardal bridge

Mode 1st 2nd 3rd

Horizontal 0,83 (0,83) 2,10 (2,11) 3,30 (3,21)

Torsional 1,12 (1,06) 2,45 (2,61) 4,10 (4,74a)

a. This is a mixed mode with both torsional and vertical movement

Vertical 1,45 (1,45) 2,85 (3,22) 4,85 (4,92)


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mass and the stiffness. The magnitude of the mass of the bridge itself can be determined with fairlygood accuracy, but its distribution is also important. For this reason the railing was included in theFrameIT model, shown in figure 4. The model, which includes all structural elements except the

deck planks, which are not believed to contribute much stiffness to the system (its mass is includedby increased specific weight of the cross beams), consists of about 950 straight beam and barelements, about 815 nodes and 4250 unknown degrees of freedom. It should be noted that the model

used here is slightly different from the model used by Rnnquist [2].It turns out to be very difficult to model the stiffness correctly with no prior knowledge of the realdynamic behavior of the bridge. Assigning normal material stiffness to all elements, and assumingfairly rigid joints, except where hinges are appropriate, overestimate the stiffness quite significantly.However, knowing what we are aiming at, it is possible to tune the stiffness of the model withrespect to the lowest free vibration modes. The crucial structural elements in this tuning process arethe elements of the horizontal steel truss between the main glulam beams, see figure 3. The way inwhich this truss is fastened to the beams makes it far more flexible than if it had been rigidlyconnected to the beams. The cross beams (see figure 3) are also fastened to the main beams in sucha way that they contribute far less stiffness than their axial stiffness suggests.The computed eigen-frequencies with the best tuned model are shown in parentheses in table 1. Forthe three lowest modes the values are very close to the measured ones, particularly for the horizontal

and the vertical modes. We are mainly concerned with the horizontal modes, and we see that thethree lowest modes of this movement are all predicted quite well with the tuned model. Figures 5, 6and 7 show the three lowest modes, respectively. For all three eigenmodes the shapes compare very

well with those observed and recorded on the bridge itself. It is worth noting that in all three cases

the motion is almost limited to the suspended mid-section of the bridge. However, for the lowestmode (figure 5) the tip of the cantilevers also moves sideways, not a great deal, but quite noticeable.

Figure 4 Computational (FrameIT) model of the Lardal bridge

Figure 5 First horizontal mode shape of the bridge - lowest eigen-frequency (0,83 Hz)


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4. Pedestrian induced vibrations

In order to explain the horizontal movements of the bridge caused by and experienced by a group of

pedestrians crossing it, it is vital to understand the forces generated during straightforward walking. Tothis end laboratory experiments were conducted as well as full scale measurements on the bridge itself.

4.1 Human locomotion

The human locomotion is a fairly complex process which is dealt with in some detail in [2]. Here welimit ourselves to some basic definitions, shown in figure 8. The experimental investigations carried

out and described in [2] are aimed at determining the forces exerted by the feet of one or morepedestrians on the pavement during straightforward walking, in particular the horizontal componentof these forces in the lateral direction (i.e. the forces normal to the direction of motion). For thispurpose a test rig consisting of a 6 by 1 m suspended aluminium platform was designed, and a largenumber of tests were carried out in which one person at a time walked in a straight line across theplatform. Several persons, of varying size and mass, took part, and for each person the eigen-

frequencies of the platform were varied (through the stiffness of supporting springs). For ourpurpose, the important results can be summarized as follows:

Figure 6 First torsional mode shape of the bridge - 2ndeigenfrequency (1,06 Hz)

Figure 7 First vertical mode shape of the bridge - 3rdeigenfrequency (1,45 Hz)

gait stride

gait length

stride length

stepwidth

footangle

stance phase (60%) swing phase (40%)

one gait cycle (left or right)

Figure 8 Human locomotion - definition of gait and gait parameters


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1. For each stance phase a horizontal transverse force (pointing outwards from the body) is mea-sured, and the magnitude of the force increases with increasing horizontal displacement ampli-tude (which is a function of the walking pace and the stiffness of the springs and hence the eigen-frequencies of the platform).

2. The ratio of transverse force amplitude to personal mass (the dynamic load factor) is fairly well

independent of the magnitude of the personal mass.3. The effect of the personal (added) mass to the vibrating system seems to be small.

4.2 Observed behavior of the bridge

Important background information for this section is the free vibration characteristics of the bridge,described earlier, and the numbers in table 2, see [2]. We see that the normal pacing frequency for a

pedestrian on a horizontal pavement is about 2 Hz. It should be noted that this is the vertical orgait frequency, i.e. the frequency of each individual step (left and right), whereas the horizontal orstride frequency, i.e. the frequency of each right (or left) step is half the value, that is about 1 Hz.With reference to figure 2 a typical observation when a group of 10 to 15 people cross the bridge,from left to right at a normal pace, is as follows: Little or no movement before the group reach pointG. At this point the bridge starts to move, sometimes with a small initial vertical movement that soonslips into a transverse horizontal movement or it goes directly into transverse horizontal movements.The displacement amplitude increases as the group moves towards the mid point, and soon afterstarting the decent on the right-hand side of the bridge the vibrations decay almost exponentially.The movements of the bridge are very much like those of the lowest horizontal eigenmode, and thereis little doubt that the pedestrians have excited this mode. It seems reasonable to assume that the

groups pace is slightly lower than normal pace as they ascend to the top at mid-point and theirstride frequency must therefore be very close to the measured eigen-frequency of 0,83 Hz. Also thegait frequency is in the vicinity of the eigen-frequency of the first vertical eigenmode (1,45 Hz)which can explain the observed initial vertical movements. The rapid decay of the motion once thegroup starts the descend can be explained by the fact that the pace and hence the excitingfrequency increase on the downhill.Another interesting and consistent observation is that once the movements start, the group memberstend to synchronize their walking and thus increase the amplitude of the resulting exciting force. Asdemonstrated by the lab experiments the transverse horizontal force exerted by an individualpedestrian increases as the horizontal movement increases. Hence, for a group of pedestrians there isa double adverse effect of the movement.The larger the group of pedestrians crossing the bridge is the larger and more noticeable are the

bridge movements. However, the increase is not dramatic, and it does in fact decrease slightly as thethe number of pedestrians reach the bridge capacity. In closing this section it should also bementioned that there has been no reports or observations of noticeable vertical or torsional pedestrianinduced movements of the bridge.

5. Alleviating measures

The current problems can hopefully be reduced, but it is unlikely that they can be eliminated with thepresent design. Three approaches are available: 1) Changing the eigen-frequencies throughstructural measures (mass and/or stiffness), 2) introducing artificial damping into the system, and 3)a combination of 1) and 2), in that order. The latter is probably the one most likely to succeed.The London Millennium bridge solved its problems through massive use of various dampingdevises. For the Lardal bridge the most likely damping devices are tuned liquid dampers

(sloshing). Tests were made with relatively small such devices placed on the deck. However, theeffect was moderate, and it was concluded that it would require a very large liquid mass to provide

Table 2 Characteristic pedestrian pacing frequencies (in Hz)

Walking Slow Normal Fast

Continuous ground contact 1,4 - 1,7 1,7 - 2,2 2,2 - 2,4


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sufficient damping, and that this might jeopardize the static load bearing capacity of the bridge.Changing the dynamic properties seems to be the better option. This can be achieved by changingeither the mass or the stiffness of the system, or both. In practical terms, changing in our case meansincreasing, which in turn means that a simultaneous increase of mass and stiffness may have littleeffect. So it is the one or the other. But first, what is the objective? If we assume that it is thehorizontal lateral eigenmode(s) we need to shift, which way do we want/need to go and how much?

We can perhaps lower the frequency by increasing the mass, but this would also change thefrequencies of the vertical modes, which might bring the second vertical mode into the resonancerange and it will also increase the static loading on the bridge. Shifting the eigen-frequencyupwards, by increasing the horizontal stiffness, seems to be the better way to go, insofar that thismay also be achieved without shifting the vertical eigen-frequencies. How much change do weneed? The reports that came out of the London Millennium bridge investigations seem to indicatethat an eigen-frequency of 1,3 Hz or higher for the first horizontal mode more or less eliminates theproblem of pedestrian induced horizontal vibrations. This limit also seems to agree well with thefindings in [2].Since no problems have been reported or observed regarding the lowest torsional mode or thevertical modes (1st and 2nd) we try to leave these modes unaffected by the changes.

5.1 Stiffening the existing bridgeInspection of the first horizontal eigenmode shows a noticeable displacement of the cantilever tips(points B and C in figure 2). Hence stiffening the cantilevers sidewise may have an effect. Wetherefore place a timber truss, consisting of 100 by 100 mm glulam verticals and diagonals, on thelower beams (E-B and F-C in figure 2), fastened to the horizontal rectangle of the cross section (seesection 2-2 in figure 3). This will be referred to as change I. On the top face of the main beams wealso include timber diagonals (100 by 100 mm glulam) between the two panels, along the entirebridge deck (from A via M to D). Together with the cross beams these diagonals will form ahorizontal truss. It is vital that the cross beams and diagonals are properly fastened to the mainbeams (by screws and bolts and perhaps steel plate mountings). This is change II. Finally a similarhorizontal timber truss (again with 100 by 100 mm glulam rods) is mounted on the bottom face ofthe main beams, but only in the mid-span, from B to C; this is change III. The changes will be

introduced successively into the tuned computational model, so that their individual effect can beassessed.Computed eigen-frequencies (fi ) are (see table 1 for the corresponding values of the tuned model):

Change I: f1 = 0,86 Hz (horizontal), f2 = 1,07 Hz (torsional) and f3 = 1,48 Hz (vertical).Change I+II: f1 = 1,03 Hz (horizontal +), f2 = 1,37 Hz (torsional) and f3 = 1,48 Hz (vertical).Change I+II+III: f1 = 1,18 Hz (horizontal), f2 = 1,46 Hz (vertical) and f3 = 1,54 Hz (torsional).

The first eigenmode of the case (I+II) is predominantly horizontal, but it has a very noticeabletorsional component, and whereas the horizontal mode of the tuned model, and also of the case (I)model, is characterized by a larger amplitude at the deck level than at the steel wire level (invertedpendulum movement), the opposite is observed for the (I+II) and (I+II+III) cases where we see ahammock movement. This type of movement is believed to be less prone to pedestrian inducedvibrations than the observed movement of the first eigenmode.Even with the most extensive strengthening (I+II+III) we are not able to rise the troublesome eigen-frequency more than 0,35 Hz, to 1,18 Hz. And this assumes a rigidity in the connections that will bevery difficult to achieve in practise. An eigen-frequency of about 1,1 Hz is probably about the bestwe can hope to obtain with this modified design. This, combined with the assumption that thehammock movement is less prone to pedestrian induced vibrations than the observed invertedpendulum movement, will clearly reduce the vibration problem, but it will not eliminate it. Also,the increased lateral stiffness will improve the efficiency of artificial damping devices.In order to implement the suggested strengthening it is necessary to remove (and replace) the deckplanks, but otherwise it should be a fairly straightforward operation.

5.2 A stiffer design

What if the dynamic problems had been recognized at the design stage? Without altering the basicidea, we would have replaced the main glulam beams with beams having the same sections as thelower part of the cantilevers, see section 2-2 in figure 3, removed the steel truss altogether and


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replaced it with a glulam truss fastened on the top side of the horizontal part of the new main beamsection, see figure 9. It should be fairly easy to secure the required rigidity in the joints of this truss.

Introducing these changes into our tuned model, we find the following frequencies (in Hz) for the 6lowest eigenmodes: 0,95 (torsional, but with a small horizontal component), 1,10 (torsional with avery small horizontal component), 1,44 (vertical), 2,17 (torsional), 2,83 (torsional) and 3,08(vertical). Although the two lowest eigen-frequencies, shown in figure 9, are still in the danger zone,we believe this design would be less susceptible to pedestrian induced vibrations, due to the shape ofthe modes. In order to excite and sustain the torsional modes, a more deliberate action is required bythe pedestrians. Nevertheless, even this modified design is not a satisfactory one. It is clearlypossible to further increase the lateral stiffness, but we feel inclined to conclude that with the currentratio of width to span length, this bridge design is pushed to its limit, and perhaps beyond.

6. Concluding remarks

The problems experienced by this bridge were not identified in the design process. The structural(finite element) model, while quite adequate for the static analyses failed to predict the dynamicbehavior of the bridge. Perhaps the most important lesson to be learnt here is the significance of thestiffness modelling for a dynamic analysis. The way in which this bridge is built, with its softconnections between both the deck structure and the main glulam beams and between the horizontalsteel truss and the main beams, it is almost impossible to model the stiffness with the accuracyneeded for a reliable prediction of the dynamic behavior of the bridge. The modeling of the joints iscrucial, and more so for a dynamic analysis than a static analysis. Soft connections are verydifficult to model and should be avoided in cases where problems of the nature described here can beexpected.

7. References

[1] Eggen, A. (1998): Wooden footbridge with a single freespan of 90 meters across the riverNumedalslaagen in Norway, Proceedings 5th World Conf. on Timber Eng., Vol. 2, Montreux,Switzerland, pp. 124-128.

[2] Rnnquist, A. (2005) Pedestrian induced lateral vibrations of slender footbridges, DoctoralTheses 2005:102, Norwegian University of Science and Technology, Trondheim. 175 pages.

[3] Wollebk, L. (2005) Analysis of geometrical nonlinearities with application to timberstructures, Doctoral Theses 2005:74, Norwegian University of Science and Technology,Trondheim. 212 pages.

[4] Wollebk, L. and Bell, K.: (2005) FrameIT - A Windows-based program for analysis of 3D

frame type structures - Users Manual, Department of Structural Engineering, NorwegianUniversity of Science and Technology, Trondheim. 48 pages.

modified section

mode # 1 - 0,95 Hz mode # 2 - 1,10 Hz

Figure 9 Modified design and its two lowest free vibration modes

glulam truss

dynamic behavior and analysis of a slender timber footbridge

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