Structural dynamic design of a footbridge under pedestrian loading

Structural dynamic design of a footbridge under pedestrian loading

C. Melchor Blanco1, Ph. Bouillard1, E. Bodarwé2, L. Ney2

1Structural and Material Computational Mechanics Department, Université Libre de Bruxelles,

Av. F. D. Roosevelt 50, C.P. 194/5, 1050 Brussels (Belgium) 2Ney & Partners, Structural Engineering sa/nv, Rue des Hellènes 42, 1050 Brussels (Belgium)

Summary The performed analysis consists of a dynamic analysis of a cable-stayed S-form footbridge of about 202m span over the river Ijzer in the city of Kortrijk (Belgium). The study of the effects caused by human excitation on structures has gained a significant evolution during the last few years. The increased understanding that followed the first studies is difficult to codify and is therefore not yet clearly stated in regulatory guidance on dynamic design of pedestrian bridges. Two of the main issues of this analysis were to decide on the correct approach to model pedestrian-induced vibrations, especially in the case of crowd loading, as well as to define the appropriate verification criteria in order to assess whether the footbridge meets the main safety requirements. Module MECANO in SAMCEF is used to perform the analysis. The objective of the paper is to demonstrate how MECANO can be used to correctly model the pedestrian forcing function as well as the damping of the structure. This latter was subject of specific investigations during the study. Keywords: Non-linear; Dynamic; Vibrations; Steel; Footbridge; Pedestrian

1 Overview of the analysis The analysis performed by the Structural and Material Computational Mechanics Department of the Université Libre de Bruxelles consists of a dynamic study of the above mentioned footbridge using the Finite Element procedure SAMCEF. The analysis carried-out only takes into account pedestrian actions on the structure. The influence of the wind or other type of dynamic excitation on the footbridge is not examined here. The final aim of the analysis is to assess whether the structure satisfies the verification criteria concerning displacement and acceleration which will be described hereafter. The study of the effects caused by human excitation on structures, mainly vibrations, has gained a significant evolution during the last few years. Some shortcomings have been highlighted in existing codes of practice in relation to the dynamic response of footbridges but nevertheless, there is still no clear regulatory guidance on dynamic design of pedestrian bridges.

9th SAMTECH Users Conference 2005 1/19

2 Project situation and characteristics The College bridge project (figure 1) consists of a footbridge located on the Ijzer River between the Ijzerkaai and the Diksmuidekaai in the city of Kortrijk. The S-form bridge is composed of an approach span starting from the Ijzerkaai, one lateral span, the central span and another lateral span leading to the Diksmuidekaai. The main geometrical data are: Total span 202m Central span 86m Lateral spans 42m Approach span 32m Estimated total mass 400 000kg Estimated total vertical inertia (Iz) 13 000 000 cm4

Estimated cross section area 760 cm2

Estimated vertical stiffness at the mid-span 1.7 kN/mm

Figure 1 : The College Brug

3 The vibration issue With current design practice for footbridges, vibrations are becoming an important issue. This is due to several reasons such as high resistant materials, smaller cross sections or larger spans. All this causes a reduction of the stiffness leading to smaller natural frequencies and therefore the structure exhibits a higher risk of resonance with pedestrian excitation. As a consequence, the vibration issue becomes now a main reason for extending the design process to dynamic loads. In order to take the dynamic behaviour into account at the stage of the design study, it is necessary to model pedestrian loading on the footbridge, which result from rhythmical body motions of persons. Vibrations of footbridges lead mainly to serviceability issues such as the comfort of the pedestrians. In order to verify that our structure does not reach such a serviceability limit state, the verification criteria are given at the end of this article highlighting the most appropriate ones for the College footbridge. 3.1 Pacing frequencies When considering the vibration behaviour under pedestrian loading, the major parameter is the stepping rate (frequency) of pedestrians. Typical pacing frequencies for a person walking generally lie between 1,6 and 2,4 Hz with a mean value of 2 Hz. 50% of pedestrians walk at rates between 1,9 and 2,1 Hz and 95% of pedestrians walk at rates between 1,65 and 2,35 Hz.



3.2 Rhythmical human motion – dynamic force Rhythmical human motions during at least 20s tend to lead to almost periodic dynamic forces, which can be described by a Fourier series of the following form:

( ) ( )ip

n

1iip φ- f i 2πsinαGGtF ∑

=

+= (1)

where G is the average weight of a person, fBpB is the pacing frequency and φBi B is the phase lag of the i-th harmonic. Different authors will give or not a special form to this general Fourier series, each one adapting the different intervening parameters to their studies or measurements. See [1], [2], [7] or [10]. Figure 2 represents the time function corresponding to the dynamic load created by a person walking at a rate of 2 Hz. Not only the frequency of the first harmonic of the forcing function (which is equal to the pacing rate) is of relevant importance, but also the frequencies of upper harmonics.

Figure 2 : Time function of the vertical force from walking at a pacing frequency of 2 Hz [1]

3.3 Influence of the number of people Footbridges are commonly excited by several persons. Two different situations may occur: Urandom actionU: the pacing frequency of the pedestrians is distributed according to a

probability curve, while the phase angle of the 1P

stP harmonic is characterized by a

completely random value. Usynchronous actionU: pedestrians walk in uniform step, with same frequency and in

phase. This results in an increase of the force which in the case of the 1P

stP harmonic is

nearly proportional to the number of persons involved. It is very difficult to evaluate at each moment which type of situation is taking place. In general, scientists agree that for small number of persons (not exceeding 15 to 25 individuals), a relatively perfect synchronization can be assumed. Concerning the density of pedestrians, this one is limited by the feasibility of uninhibited walking. A normal realistic value will not exceed 1 person/m².


3.4 Lock-in effect The lock-in effect describes the phenomenon by which, when the structure exceeds a certain threshold value of displacement (depending on direction of vibration, type of activity, etc.), a walking (or jumping, or running) person tends to adapt to and synchronize his/her motions in frequency and phase with the vibrating deck. If the individual limit value of displacement is exceeded, then the user tends to give a certain impulse into every vibration wave of the bridge. As a consequence, the vibration amplitude of the structure increases and due to this, more and more persons are “locked” into synchronization. Values of this threshold have been proposed by several authors and will be exposed further below in section 7. The order of magnitude stands around 10 to 20mm for vibration amplitude. Practice codes contain however very little or none information regarding this phenomenon. 4 Dynamic load model It is possible to find in the literature several dynamic models representing the person’s rhythmical body motion. Among them, Kreuzinger [9] and Stoyanoff [11][15] remain sensitively close concerning the characteristic of a single pedestrian, but differ when describing the effects caused by crowd loading. Bachmann’s [1][2] and Petersen’s studies [10] seem nowadays to be the most complete and convincing among all others. The main difference between both of them relays on the appraisal of the normalised dynamic force, which depends significantly on the pacing frequency in the case of Petersen. The choice of the dynamic model was made taking account of factors such as validity of the proposed model, level of detail, correct and feasible appraisal of all intervening parameters and easy implementation info a finite element procedure. The decision favoured Bachmann’s theory completed by Petersen’s model. 4.1 Pedestrian force due to a single person Petersen gives the following formulae for describing the forcing function created by a single pedestrian:

( ) ( ) ( ) ( )3p32p2p1p tf6πsinGctf4πsinGctf2πsinGcGtF ϕϕ −+−++= (2)

On the contrary to Bachmann, Petersen gives values of the different parameters which vary significantly with respect to pacing frequency. Table 1 shows the Fourier coefficients and phase lags proposed by Petersen for the case of a single pedestrian excitation for 3 different values of excitation frequency. Table 1 Petersen’s parameters and coefficients [10] G weight of pedestrian N f BpB pacing frequency Hz f BpB = 1.5Hz f BpB = 2.0Hz f BpB = 2.5Hz cB1 B coefficient for 1P

stP harmonic 0.073 0.408 0.518

cB2 B coefficient for 2P

ndP harmonic 0.138 0.079 0.058

cB3 B coefficient for 3P

rdP harmonic 0.018 0.018 0.041

φB2 B phase lag of 2P

ndP harmonic π/5 π/5 2π/5 rad

φB3 B phase lag of 3P

rdP harmonic π/5 π/5 2π/5 rad


4.2 Influence of the number of people According to references [1], [2] and [10], for a footbridge with a simple beam behaviour, assumed to be supported by two simple bearings, whose natural frequency remains in the range 1.8 to 2.2Hz, and for a random pedestrian action, the vibration amplitude of the excitation force caused by a single pedestrian can be multiplied by an enhancement coefficient m:

0λTm = (3)

where λ is the mean flow rate of the pedestrians for a certain period of time and TB0 B is the time necessary to cross the bridge. For a synchronized pedestrian action: m = n. In the case where the structure first natural frequency is not close to the mean value 2Hz, or when the synchronization between pedestrians cannot be considered as perfect, the m factor has to be reduced [1] [10]. 4.3 Generalization of Bachmann’s theory Bachmann’s magnification factor m is only valid for a bridge with assumed simple beam behaviour and supported only by two simple bearings. For the College footbridge, this is not the case. Indeed, the structure has an S-form and presents therefore an important torsional component in its behaviour. In addition to this, the footbridge is supported by two simple bearings at both ends, but also by two elastic bearings at 100m approximately and respectively from extremities (suspended cable). A 4 bearing-beam is characterised by a much higher stiffness than a 2 bearing-beam. This suggests that the displacements generated by pedestrians in this first type of structure will be reduced compared to the second type. Therefore, the enhancement coefficient should be reduced with respect to Bachmann’s factor m which was predicted for simple beams. A complete analysis and the description of the method used to obtain the reduction formula is exposed in [16]. The reduction formula, as a function of the ratio d/L (figure 3), obtained for different topologies of hyperstatic bridges is the following:

2

7,8483 3,8255 0, 0114d dreductionL L

π ⎛ ⎞ ⎛ ⎞= − + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(4)

Figure 3: Hyperstatic structure

L

d

5 Dynamic description of the structure 5.1 Basic principles for the dynamic analysis The final aim of this work is to perform a dynamic analysis with the help of SAMCEF (MECANO) in order to evaluate whether the College footbridge satisfies or not verification criteria concerning vibrations for pedestrian excitation. The forcing function (Fourier decomposition) will be applied at mid-span (where displacements and accelerations are maximum) magnified by the enhancement coefficient m given by Bachmann’s generalized theory. The damping characteristics of the structure are of non negligible importance within the dynamic analysis. The parameter that describes at best the dissipating behaviour of a structure is the critical damping coefficient, which will be developed below. The finite element procedure used in the computations is SAMCEF. In order to carry out the natural frequencies computation, with module DYNAM, it is necessary to initially perform a non-linear structural computation. This non-linear computation has for goal to introduce the self-weight stresses into the structure cables and therefore increase their stiffness. As a consequence the natural frequencies slightly increase. It was not possible to perform the dynamic analysis with REPDYN, which would have helped us to carry out a modal analysis, due to the significant non-linearities introduced by the suspension cables in the structure. 5.2 Description of finite element model and meshing Geometry The main structure of the footbridge is modelled by means of shell, beam and rod elements, while the approach span is modelled only with beam elements. The cross-section is actually composed of 5 to 10mm thickness steel plates, which are stiffened in order to absorb local loads. Concerning the global mechanic behaviour of the bridge, the analysis is only concerned by the bending (and torsional) stiffness of the cross-section, and not its local behaviour. This is why the stiffeners are not included in the finite element model. However, this could cause an instability (warping) of the plates, and as a solution, the plate thickness has been artificially increased by a multiplication factor equal to 10, and the elasticity modulus has been decreased by 10 as well. As a consequence, the beam and torsional stiffness of the plates remains unchanged. Concerning the main suspension cables and the lower line of the deck, these have been modelled using beam elements. Loading The only loads applied to the footbridge are the weight load, which is already introduced into the model for the natural frequencies computation, and the pedestrian load given by Petersen’s formulae (see section 6.5). For a correct appraisal of the self-weight, the density of all the materials defined in the finite element model has been modified in order to automatically introduce the load coming from the deck concrete coating, the railings and the stiffeners which do not appear as independent elements in the model.



Meshing The mesh is composed of a total of 16590 nodes, which gives a model characterized by 49770 degrees of freedom. The total number of elements contained in the model is 2446, of which 9 are rod elements, 1776 are 4-node shell elements, 36 are 3-node shell elements and finally 625 are beam elements.

Figure 4: Finite element mesh

5.3 Damping properties Damping in a vibrating structure is associated with a dissipation of mechanical energy. In the presence of a free vibration the influence of damping results in a continuous decay of resulting vibration amplitude. The value of the damping coefficient c affects the decay of the resulting vibration but has weak influence on its frequency. If c > 0, the non-dimensional damping ratio is defined as:

πζncrit mf

cc

c== (fBnB = natural frequency) (5)

By carrying out the analysis with SAMCEF, it is not possible to introduce structural damping by means of the critical damping ratio (ζ ). This is normally possible with the option .AMO but version 9.1 does not allow the use of .AMO unless the option NLIS -1 is included in the data file. The option NLIS suppresses from the analysis all geometrical non-linearities, which is not compatible with our structure due to the suspension cables. For this reason, the only way to take into account natural damping is to define a dissipating visco-elastic behaviour for the materials used in the finite element modelling. The material law of a visco-elastic material depends on the strain velocity. There are several types of visco-elastic models. The model chosen for the current study is described by Kelvin [12]. The equations relating stresses to strains are the following:

v vσ σ ασ= + & and v Hσ ε= where H is the Hooke operator. (6) The visco-elastic coefficient α characterises the behaviour law from the material and it is the only way in which it is possible to introduce damping into the structure. Currently, there are very few references in public or academic literature concerning the value of α and therefore it appeared necessary to establish a certain correspondence between this coefficient and the critical damping ratio (ζ ) which is the typical parameter used in references.


It is possible to establish a relation between parameters ζ and α . Considering that within a visco-elastic material, strain has to be replaced by an “elastic” part plus the viscous contribution ( vσα & ), the system equation can be rewritten, considering no other damping than the one coming from the material:

0=++ kxxkxm &&& α ⇔ αkc = ⇔ kkm

kc

kc crit 2ζζα === (7)

Finally: nfαπζ = (8)

where fBn B is the first natural frequency of the structure. Nevertheless, if we consider that the analysis is carried out in the modal basis, and that material damping is proportional to mass and stiffness ( c k mα β= + ), then the equation for each of the natural modes would be the following [13]:

2 2β αω ζω+ = where 2 fω π= (9)

The critical damping ratio ζ for a certain mode i can be calculated in the following way:

2 2i

ii

β αωζω

= + (10)

In the case we are concerned by, the damping is only proportional to stiffness, which means that β is equal to zero. In that case, the critical damping ratio is a linear function of ω , showed in figure 5. Thus, the material damping is a function of the excitation frequency, and therefore it is impossible to fix the visco-elastic coefficient α to a single value for all frequencies.

Figure 5: Relation between the critical damping ratio and the eigen pulsation 6 Dynamic computations for the College footbridge 6.1 Presentation of the results The results will be presented as a table showing values of displacement and acceleration due to the periodic pedestrian function applied at mid-span (and magnified by m), for 5 different points in the structure. The distribution of these “sensitive” points is given in figure 6 below.

2α

ω

ζ


Figure 6: Sensitive points of the structure These points of the structure are considered to be “sensitive” because they present the higher displacement amplitudes or accelerations among all other significant points of the structure. Points P1 and P2 were kept in order to verify that the supporting cables do not undergo severe movements. 6.2 Enhancement coefficient calculated for the College footbridge according to

Bachmann’s reviewed theory Using the reduction formula (4) obtained in [16], it is possible to calculate several values of the amplification coefficient m. Petersen [10] suggests different values for pedestrian density and velocity with respect to frequency, which were used to calculate the number of persons (n) standing on the bridge for different situations. A final value of 13 was chosen for the enhancement factor m. 6.3 Damping Reference mean values for the critical damping ratio stand around 0.005 for steel and 0.010 for concrete, so that for the College footbridge (whose 22% of the total mass is made of concrete) the appropriate value equals 0.006. Indeed, the main structure of the footbridge is made of steel but the deck supports a 12cm thickness concrete slab along approximately 86m. As it was explained before, the visco-elastic coefficient characterising the material behaviour is a function of the excitation frequency for a given damping ratio. If the value of ζ is fixed to 0.006, then α follows a linear variation depending on excitation frequency. Table 4 contains the values of α for excitation frequencies in the range 1.68 to 2.3Hz.

Point LH0 Nœud 15269

Point P3 Nœud 7372

Point 44 Nœud 2663

Point 73 Nœud 7324

Point P2 Nœud 5388

Point P1 Nœud 15115

Point 15 Nœud 13746


Table 4 Coefficient α for different excitation frequencies Frequency (Hz) α

1.68 0.00113682 1.72 0.00111038 1.76 0.00108515 1.80 0.00106103 1.84 0.00103797 1.88 0.00101588 1.92 0.00099472 1.96 0.00097442 2.00 0.00095493 2.04 0.00093621 2.08 0.00091820 2.12 0.00090088 2.16 0.00088419 2.20 0.00083812 2.24 0.00085262 2.28 0.00083766 2.32 0.00082322

6.4 Natural frequencies A dynamic modal analysis was carried out in order to calculate the natural frequencies from the College footbridge. This analysis is of crucial importance to predict which pacing frequencies will cause high displacement or acceleration amplitudes. The analysis was carried out in SAMCEF with DYNAM. The results are contained in table 6 below. Table 5 College Brug natural frequencies Mode Frequency (Hz)

1 0.79489 2 0.93805 3 1.21240 4 1.53988 5 1.75637 6 1.82897 7 2.21344 8 2.55394 9 2.67660 10 2.82364 11 2.98122 12 3.05543 13 3.06589 14 3.19878 15 3.28652

Description of the natural modes The College footbridge natural modes are complicate to identify. There are no pure bending nor torsional modes and two different types of deformation always appear at the same time, principally due to the length and relative orientation of the lateral spans in relation with the central span. Figure 7 shows the undeformed shape as well as the deformed shapes corresponding to modes 1 up to 5.

UNDEFORMED SHAPE

MODE

MODE 2

MODE 3

MODE 4

MODE 5

Figure 7: Undeformed shape and modes 1 to 5



6.5 Results We decided to carry out dynamic computations for the following the frequency range 1.68 to 2.3Hz, which allows us to consider at least 97% of all probable pacing frequencies. In addition to this, we noticed by analysing the natural frequencies. Table 6 contains the Fourier coefficients (according to Petersen’s formulae) which were used in the computations. Table 6 Petersen’s Fourier coefficients for frequencies 1.68 to 2.3Hz f Bexc B (Hz) cB1 B cB2 B cB3 B φ1 = φ2 1.68 0.194 0.117 0.018 0.226 1.70 0.207 0.114 0.018 0.251 1.71 0.214 0.113 0.018 0.264 1.72 0.220 0.112 0.018 0.276 1.75 0.241 0.109 0.018 0.314 1.78 0.261 0.105 0.018 0.352 1.80 0.274 0.103 0.018 0.377 1.84 0.301 0.098 0.018 0.427 1.90 0.341 0.091 0.018 0.503 1.94 0.386 0.086 0.018 0.553 2.00 0.408 0.079 0.018 0.628 2.30 0.474 0.066 0.032 1.005

Tables 7 and 8 and figures 8 and 9 contain the results (nodal vertical displacement and acceleration) obtained with the finite element software used for the dynamic analysis (SAMCEF) for the range of excitation frequencies in between 1.68 and 2.3Hz. Table 7 Computational results. Nodal displacement for frequencies in the range 1.68 to 2.3Hz

vertical displacement (mm) f Bexc B (Hz)

node 2663 node 15115 node 5388 node 7324 node 15199 1.68 6.600 0.298 0.626 6.670 7.700 1.70 7.050 0.301 0.675 7.220 8.200 1.71 12.400 0.570 1.200 14.000 14.050 1.72 12.300 0.525 1.215 14.450 13.450 1.75 6.480 0.331 0.667 8.550 6.780 1.78 5.040 0.252 0.440 5.410 5.590 1.80 4.240 0.212 0.371 4.030 5.100 1.84 3.180 0.154 0.277 3.100 3.700 1.90 2.530 0.118 0.210 2.570 3.190 1.94 2.485 0.111 0.221 2.280 3.080 2.00 2.290 0.172 0.270 2.600 2.880 2.30 1.650 0.175 0.335 1.355 1.545 Table 8 Computational results. Nodal acceleration for frequencies in the range 1.68 to 2.3Hz

vertical acceleration (m/s²) f Bexc B (Hz)

node 2663 node 15115 node 5388 node 7324 node 15199 1.68 0.395 0.083 0.098 0.400 0.560 1.70 0.425 0.086 0.097 0.420 0.575 1.71 0.780 0.108 0.099 0.850 0.870 1.72 0.760 0.108 0.100 0.890 0.930


1.75 0.425 0.087 0.107 0.540 0.470 1.78 0.330 0.081 0.086 0.340 0.390 1.80 0.292 0.082 0.100 0.263 0.330 1.84 0.240 0.067 0.079 0.190 0.275 1.90 0.193 0.069 0.080 0.163 0.258 1.94 0.180 0.072 0.064 0.150 0.242 2.00 0.205 0.076 0.078 0.188 0.268 2.30 0.300 0.084 0.086 0.124 0.165 All computations were carried out with the following data: - Bachmann’s enhancement coefficient m = 13 - Critical damping ratio ζ = 0.006 - Fourier decomposition given by Petersen The nodes presenting the largest displacement amplitude and acceleration are nodes 2663, 7324 and 15199. Indeed, these nodes are situated at mid-central-span and at mid-lateral-span of the footbridge, respectively, where the structure is the most slender. Displacements for nodes 15115 and 5388, situated at the base of the two cable supports, remain negligible, which is reassuring given that both cables are supposed not to absorb any important motions. The results show a very important peak for both displacement and acceleration for an excitation frequency of 1,72Hz. This is due to resonance with the fifth natural mode of the structure and to damping (causing the natural frequencies to slightly decrease).

0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

16,0

1,65 1,75 1,85 1,95 2,05 2,15 2,25 2,35

excitation frequency (Hz)

vibr

atio

n am

plitu

de (m

m)

node 2663node 15115node 5388node 7324node 15199

2663

7324

538815115

15199

- ζ = 0,0006- Petersen's formula for forcing function- m = 13

84% of pedestrians

Figure 8: Displacement as a function of excitation frequency


0,0

0,2

0,4

0,6

0,8

1,0

1,65 1,75 1,85 1,95 2,05 2,15 2,25 2,35

excitation frequency (Hz)

acce

lera

tion

(m/s

²)

node 2663node 15115node 5388node 7324node 15199

2663

7324

538815115

15199

- ζ = 0,0006- Petersen's formula for forcing function

84% of pedestrians

Figure 9: Acceleration as a function of excitation frequency

7 Verification criteria Vibrations resulting from pedestrian excitation of a structure can lead to several forms of distress:

- Intolerable vibration velocities and accelerations disturbing and discomforting the users

- Overstressing of the structure - Damage to non-structural elements - Excessive noise (due to, for instance, reverberating equipment)

In most cases, only the first factor will have an influence on the design of the strcuture. In order to verify that the structural response of the College Brug stands within the acceptance limit of tolerance, it is important to express the relevant maximum values of vibration amplitude and acceleration for the dynamic problem in order to establish a comparison between these and the results of the study. The determination of vibration tolerable levels is somewhat subjective and based on experience. A review of the different verification criteria proposed in literature and in codes of practice has been made in order to justify our choice. Tables 10 and 11 in Appendix 1 resume the review. 7.1 Chosen verification criteria As it is shown in tables 10 and 11, two types of criteria may be adopted: one related to human comfort and another related to lock-in effect.


Human comfort Practice codes give more accurate data concerning this feature than the criteria found in the literature references. For sake of clarity and simplicity, the British Standards seem to be the most appropriate for the College Brug vertical vibration issue. According to the BS 5400-2, the acceleration resulting from pedestrian excitation has to remain under a certain threshold value given by the following formula:

0max, 5.0 fa vertical ≤

which results in : a Bmax,vertical B ≤ 0.4458m/s² Lock-in effect For this particular issue, Bachmann seems to be a reference. His studies demonstrate that he deeply studied the problem of synchronisation between the pedestrians and the vibrating structure. According to him, two criteria may be applied, one concerning displacement and another concerning acceleration.

max,verticald 10 mm≤

m/s². to . a %g to a ,vertical,vertical 980490105 maxmax ≤⇒≤ 7.3 Remarks concerning the computational results Concerning displacements, it is easy to note, looking at table 7 and figure 8 that the amplitude never exceeds the threshold value given by Bachmann in order to avoid a lock-in effect except for nodes 2663, 7324 and 15199 and for an excitation frequency around 1,71Hz. Table 11 reports all values exceeding the threshold value of 10mm. Table 11 Values of displacement exceeding the threshold value given by Bachmann (in mm) fexc [Hz] node 2663 node 7324 node 15199 1.71 12.400 14.000 14.050 1.72 12.300 14.450 13.450 Concerning accelerations (figure 9), and looking at the criteria for human comfort given in the British Standards BS 5400-2, results related to the same nodes 2663, 7324 and 15199 exceed the threshold value of 0.4458m/s²: These are reported in table 12. Table 12 Values of acceleration exceeding the threshold value given in BS 5400-2 (in m/s²) fexc [Hz] node 2663 node 7324 node 15199 1.68 / / 0.560 1.70 / / 0.575 1.71 0.780 0.850 0.870 1.72 0.760 0.890 0.930 1.75 / 0.540 0.470

As it can be concluded, for excitation frequencies in the range of 1.68 to 1.75Hz, the vibrations caused by the pedestrians could be susceptible to generate a non-negligible degree of discomfort. Nevertheless, it is important to keep in mind that such excitation frequencies are quite rare. Only around 15% of pedestrians walk with pacing frequencies out of the range 1.75 to 2.25Hz so the probability to be confronted to such an excitation is quite low. Concerning the lock-in effect, Bachmann suggests for accelerations not to exceed a threshold value lying between 0.49 to 0.98m/s². As it is possible to conclude from table 12, the values for acceleration concerning nodes 2663, 7324 and 15199 remain in the range 0.56 to 0.93m/s² and never exceed the advised upper limit value of 0.98m/s². However, as shown in table 11, displacements for frequencies 1.71 and 1.72Hz do exceed the value of 10mm predicted by Bachmann as threshold level to avoid lock-in effect. Indeed, the resulting values could generate a vibration that could bring the users to get synchronised with the deck movements. Although concerned frequencies remain quite low and are not susceptible to appear in a large number of cases, some solutions could be brought in order to completely avoid a risk situation. Several authors [1] [2] [3] [4] [5] propose some currently applicable remedial measures in order to either avoid natural frequencies in the range 1.6 to 2.4Hz, increase damping or limit vibration amplitudes by introducing external vibration absorbers. 8 References [1] H. Bachmann, “Lively footbridges – A real challenge”, AFGC and OTUA

Footbridge Conference, Paris, 2002 [2] H. Bachmann, W. Ammann, “Vibration problems in structures“, Birkhäuser,

1995 [3] Bulletin d’information N° 209, “Vibration problems in structures. Practical

guidelines”, CEB, August 1991 [4] David E. Newland, “Vibration of the London Millennium Footbridge. Part 1 –

Cause”, University of Cambridge, February 2003 [5] David E. Newland, “Vibration of the London Millennium Footbridge. Part 2 –

Cure”, University of Cambridge, February 2003 [6] T. Fitzpatrick, “Linking London: The Millennium Bridge”, Royal Academy of

Engineering., 2001. http://www.arup.com/MillenniumBridge/indepth/pdf/linking_london.pdf

[7] M. Willford, “Dynamic actions and reactions of pedestrians”, AFGC and OTUA

Footbridge Conference, Paris, 2002 [8] W. Ammann, H. Bachmann, “Vibrations in structures induced by man and

machines“, A.I.P.C., Zürich, 1987


[9] H. Kreuzinger, “Dynamic design strategies for pedestrian and wind actions”,

AFGC and OTUA Footbridge Conference, Paris, 2002 [10] Ch. Petersen, “Dynamik der Baukonstruktionen”, Vierweg,

Braunschweig/Wiesbaden, 1996 [11] S. Stoyanoff, M. Hunter, D. D. Byers, “Human-induced vibrations on

footbridges”, AFGC and OTUA Footbridge Conference, Paris, 2002 [12] SAMCEF help : http://www.samcef.com [13] O.C. Zienkiewicz, R.L. Taylor, “The finite elements method, Volume 1: The

basis”, Butterworth-Heinemann, 2000 [14] A. McRobie, « Risk management for pedestrian-induced dynamics of

footbridges », AFGC and OTUA Footbridge Conference, Paris, 2002 [15] D. Byers, W. Clawson, S. Stoyanoff, T. Zoli, “Wichita riverfront pedestrian

bridges”, AFGC and OTUA Footbridge Conference, Paris, 2002 [16] C. Melchor, Ph. Bouillard, E. Bodarwé, L. Ney, “Structural dynamic design of

the College footbridge under pedestrian loading”, Finite Elements in Analysis and Design”. Submitted, 2004.



Appendix 1 Table 11 Review of verification criteria appearing in codes of practice

Name Application Frequency range Criteria Comments

ISO 2631

Vertical and/or horizontal vibrations. Random, shock or harmonic vibrations.

1 to 80Hz

Given in graphical form and expressed in relation to effective acceleration:

( )T

2eff max

0

1a = a t dt=0,707 aT ∫ (m/s²)

Vertical (1 to 5Hz): 4,3%effa g≤

Horizontal (1 to 2Hz): 1,7%effa g≤

T = period of time (s) over which the acceleration is measured

DIN 4150/2

Residential buildings.

1 to 80Hz

Perception factor:

2

2

0,8 fKB=d1+0,032 f

(mm/s)

KB is compared to reference values depending on vibration frequency, duration of vibration, etc.

d = displacement amplitude (mm) f = basic vibration frequency (Hz) Maximum values of KB stand between 0,2 and 0,6 for continuous or repeated excitation.

BS 5400-2

Concrete, steel or composite bridges. Only for symmetric structures composed of 1, 2 or 3 spans with constant cross-section.

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2 2

max,vert 0 sa =4π f y Kψ (m/s²)

Used to verify structure the following way : f B0B > 5Hz Verification OK

f B0B < 5Hz max,vertical 0a 0,5 f≤ at any part of

superstructure

f B0B = fundamental natural frequency (Hz) y Bs B = static deflection (m) K = configuration factor Ψ = dynamic response factor

ONT 83

/

/

0,18

max,vertical 0a 0,25 f≤ (m/s²) f B0B = fundamental natural frequency (Hz)

ENV 1995

/

/

For vibration frequency around 2Hz: a Bmax,vertical B = 7%g (equal to 0,7m/sP

2P)

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Table 12 Review of verification criteria appearing in references

Name Criteria Comments Hugo Bachmann (reference [1])

For pedestrian comfort: Vertical: max, 5 10%verticala to g≤

Horizontal: max, 1 2%horizontala to g≤

For avoiding lock-in effect: Vertical: max, 10verticald mm≤

Horizontal: max, 2horizontald mm≤

a = acceleration resulting from the structure’s vibration d = vibration amplitude For Bachmann, acceptance criteria are related in most cases to physiological effects on people representing serviceability problems, rather than safety problems to the structure. For Bachmann, criteria are basically frequency-dependent

Michael Wilford (reference [7])

For pedestrian comfort: Vertical: max, 7%verticala g≤

Horizontal: max, 0,2%horizontala g≤

For avoiding lock-in effect: Vertical: max, 4%verticala g≤

Horizontal: max, 0,25%horizontala g≤

a = acceleration resulting from the structure’s vibration Wilford bases his theory on measurements that took place on the London Millennium Bridge.

Stoyan Stoyannoff (reference [11])

For human comfort: Vertical: max, 0,07verticala g≤

Horizontal: max, 0,02horizontala g≤

a = acceleration resulting from the structure’s vibration According to Stoyannoff, people can tolerate different levels of vibration depending on activity. Acceptable levels for people working in offices will differ significantly from values for people participating in rhythmic activities.

Structural dynamic design of a footbridge under pedestrian loading

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