02_11_ULB_Bouillard

8/3/2019 02_11_ULB_Bouillard

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Structural dynamic design of a footbridge

under pedestrian loading

C. Melchor Blanco1, Ph. Bouillard1, E. Bodarw2, L. Ney2

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

Av. F. D. Roosevelt 50, C.P. 194/5, 1050 Brussels (Belgium)2Ney & Partners, Structural Engineering sa/nv, Rue des Hellnes 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 inthe city of Kortrijk (Belgium).

The study of the effects caused by human excitation on structures hasgained a significant evolution during the last few years. The increasedunderstanding that followed the first studies is difficult to codify and istherefore not yet clearly stated in regulatory guidance on dynamicdesign of pedestrian bridges. Two of the main issues of this analysis

were to decide on the correct approach to model pedestrian-inducedvibrations, especially in the case of crowd loading, as well as to definethe appropriate verification criteria in order to assess whether thefootbridge 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 tocorrectly model the pedestrian forcing function as well as the dampingof the structure. This latter was subject of specific investigationsduring 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 Universit Libre de Bruxelles consists of a dynamic study of the


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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 86mLateral 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 withpedestrian 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


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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:

( ) ( )ipn

1i

ip -fi2sinGGtF =

+= (1)

where G is the average weight of a person, fBpB is the pacing frequency and BiB 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 ormeasurements. 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 1PstP harmonic is characterized by a

completely random value.


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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 persons

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. Bachmanns [1][2] and Petersens

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 Bachmanns theory completed by Petersens model.

4.1 Pedestrian force due to a single person

Petersen gives the following formulae for describing the forcing function created by asingle pedestrian:

( ) ( ) ( ) ( )3p32p2p1p tf6sinGctf4sinGctf2sinGcGtF +++= (2)

On the contrary to Bachmann, Petersen gives values of the different parameters which


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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:

0Tm = (3)

where is the mean flow rate of the pedestrians for a certain period of time and TB0B 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 Bachmanns theory

Bachmanns 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 Bachmanns 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 d

reductionL L

= + +

(4)


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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

Bachmanns 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 hasfor 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 finiteelement 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.


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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:

ncrit mf

c

c

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 ( ). 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 allgeometrical 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


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It is possible to establish a relation between parameters and . Considering that

within a visco-elastic material, strain has to be replaced by an elastic part plus theviscous contribution (

v & ), the system equation can be rewritten, considering no other

damping than the one coming from the material:

0=++ kxxkxm &&& kc = k

km

k

c

k

c crit 2 === (7)

Finally:

nf = (8)where fBnB 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 m = + ), then the equation

for each of the natural modes would be the following [13]:

2 2 + = where 2 f = (9)

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

way:

2 2

ii

i

= + (10)

In the case we are concerned by, the damping is only proportional to stiffness, which

means that is equal to zero. In that case, the critical damping ratio is a linear functionof , 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 to a single

value for all frequencies.

Figure 5: Relation between the critical damping ratio and the eigen pulsation

2


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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

Bachmanns 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 forpedestrian 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 ismade 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

b h i i f i f h i i f f i d i i f h

Point LH0Nud 15269

Point P3Nud 7372

Point 44

Nud 2663

Point 73Nud 7324

Point P2Nud 5388

Point P1Nud 15115

Point 15Nud 13746


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Table 4Coefficient for different excitation frequencies

Frequency (Hz)

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 0.000994721.96 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 5College Brug natural frequenciesMode Frequency (Hz)

1 0.79489

2 0.93805

3 1.21240

4 1.53988

5 1.75637

6 1 82897


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Description of the natural modes

The College footbridge natural modes are complicate to identify. There are no purebending 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 2

MODE 3

MODE 4


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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 Petersens formulae) which were used in

the computations.

Table 6Petersens Fourier coefficients for frequencies 1.68 to 2.3Hz

fBexcB (Hz) cB1B cB2B cB3 B 1 = 2

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 7Computational results. Nodal displacement for frequencies in the range 1.68 to 2.3Hz

vertical displacement (mm)fBexcB (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 14.0501.72 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


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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:- Bachmanns enhancement coefficient m = 13

- Critical damping ratio = 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).

6,0

8,0

10,0

12,0

14,0

16,0

rationamplitude(mm)

node 2663

node 15115

node 5388

node 7324

node 15199

2663

7324

538815115

1519984% of pedestrians


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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)

acceleration

(m/s)

node 2663

node 15115

node 5388

node 7324

node 15199

2663

7324

538815115

15199

- = 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


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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,verticalB 0.4458m/s

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/s.to.a%gtoa ,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 11Values 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 d th th h ld l f 0 4458 / Th t d i t bl 12


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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/s. As it is possible to conclude from table

12, the values for acceleration concerning nodes 2663, 7324 and 15199 remain in therange 0.56 to 0.93m/s and never exceed the advised upper limit value of 0.98m/s.

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, Birkhuser,

1995

[3] Bulletin dinformation 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


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[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. Bodarw, L. Ney, Structural dynamic design of

the College footbridge under pedestrian loading, Finite Elements in Analysis

and Design. Submitted, 2004.


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9th SAMTECH Users Conference 2005 18/19

Appendix 1

Table 11Review of verification criteria appearing in codes of practice

Name ApplicationFrequency

rangeCriteria Comments

ISO 2631 Vertical and/or horizontal

vibrations. Random, shock orharmonic vibrations.

1 to 80Hz Given in graphical form and expressed in relation to

effective acceleration:

( )

T

2

eff max

0

1

a = a t dt=0,707 aT(m/s)

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 f KB=d

1+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 0,6for continuous or repeated excitation.

BS 5400-2 Concrete, steel or composite bridges. Only for symmetric

structures composed of 1, 2 or 3spans with constant cross-section.

/ 2 2max,vert 0 sa =4 f y K (m/s)

Used to verify structure the following way :

fB0B > 5Hz Verification OK

fB0B < 5Hz max,vertical 0a 0,5 f at any part ofsuperstructure

fB0B = fundamental natural frequency (Hz)yBsB = static deflection (m)

K = configuration factor = dynamic response factor

ONT 83 / /0,18

max,vertical 0a 0,25 f (m/s) fB0B = fundamental natural frequency (Hz)

ENV 1995 / / For vibration frequency around 2Hz:aBmax,verticalB = 7%g (equal to 0,7m/s P

2P)

/


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9th SAMTECH Users Conference 2005 19/19

Table 12Review of verification criteria appearing in references

Name Cr it er ia 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 structures vibrationd = vibration amplitude

For Bachmann, acceptance criteria are related in most cases to physiological

effects on people representing serviceability problems, rather than safety problemsto 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 structures vibration

Wilford bases his theory on measurements that took place on the LondonMillennium Bridge.

Stoyan Stoyannoff

(reference [11])

For human comfort:

Vertical: max, 0,07verticala g

Horizontal:max,

0,02horizontala g

a = acceleration resulting from the structures vibration

According to Stoyannoff, people can tolerate different levels of vibration

depending on activity. Acceptable levels for people working in offices will differsignificantly from values for people participating in rhythmic activities.

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02_11_ULB_Bouillard