us 0644a footbridges
TRANSCRIPT
service d'Études
techniques
des routes
et autoroutes
Sétra Tech
nical gu
ide
Footbridges A
ssessmen
t of vib
ration
al beh
aviou
r of fo
otb
ridges
un
der p
edestrian
load
ing
october 2006
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Tech
nical gu
ide
Footbridges A
ssessmen
t of vib
ration
al beh
aviou
r of fo
otb
ridges
un
der p
edestrian
load
ing
Asso
ciation
Fran
çaise de G
énie C
ivil R
égie par la lo
i du
1er ju
illet 1
90
12
8 ru
e des S
aints-P
ères - 75
00
7 P
aris - Fran
cetélép
ho
ne : (3
3) 0
1 4
4 5
8 2
4 7
0 - téléco
pie : (3
3) 0
1 4
4 5
8 2
4 7
9
mél : afgc@
enp
c.fr – in
ternet : h
ttp://w
ww
.afgc.asso.fr
Published by the Sétra, realized within a Sétra/A
FGC (French association of civil engineering) w
orking group
Th
is do
cum
ent is th
e translatio
n o
f the w
ork
"Passerelles p
iéton
nes - E
valuatio
n d
u
com
po
rtemen
t vibrato
ire sou
s l’action
des p
iéton
s", pu
blish
ed in
un
der th
e reference 0
61
1
Fo
otb
ridg
es - Assessm
ent o
f vibra
tiona
l beh
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f footb
ridg
es und
er ped
estrian
loa
din
g –
Pra
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uid
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Contents
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1.1 S
TR
UC
TU
RE
AN
D F
OO
TB
RID
GE
DY
NA
MIC
S...................................................................7
1.2 P
ED
ES
TR
IAN
LO
AD
ING
...............................................................................................10
1.3 P
AR
AM
ET
ER
S
TH
AT
A
FF
EC
T
DIM
EN
SIO
NIN
G:
FR
EQ
UE
NC
Y,
CO
MF
OR
T
TH
RE
SH
OL
D,
CO
MF
OR
T C
RIT
ER
ION
, ET
C.....................................................................................................25
1.4 I M
PR
OV
EM
EN
T O
F D
YN
AM
IC B
EH
AV
IOU
R..................................................................27
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2.1 S
TA
GE
1: DE
TE
RM
INA
TIO
N O
F F
OO
TB
RID
GE
CL
AS
S....................................................31
2.2 S
TA
GE
2: CH
OIC
E O
F C
OM
FO
RT
LE
VE
L B
Y T
HE
OW
NE
R.............................................31
2.3 S
TA
GE
3: DE
TE
RM
INA
TIO
N O
F F
RE
QU
EN
CIE
S A
ND
OF
TH
E N
EE
D T
O P
ER
FO
RM
DY
NA
MIC
LO
AD
CA
SE
CA
LC
UL
AT
ION
S O
R N
OT .......................................................................................32
2.4 S
TA
GE
4 IF N
EC
ES
SA
RY
: CA
LC
UL
AT
ION
WIT
H D
YN
AM
IC L
OA
D C
AS
ES.......................34
2.5 S
TA
GE
5: MO
DIF
ICA
TIO
N O
F T
HE
PR
OJE
CT
OR
OF
TH
E F
OO
TB
RID
GE...........................37
2.6 S
TR
UC
TU
RA
L C
HE
CK
S U
ND
ER
DY
NA
MIC
LO
AD
S........................................................40
T XeMgQ[UThgUMXQNXMdV[RN`OL[QePVRSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSYT
3.1 P
RA
CT
ICA
L D
ES
IGN
ME
TH
OD
S...................................................................................43
3.2 D
YN
AM
IC C
AL
CU
LA
TIO
N A
PP
LIE
D T
O F
OO
TB
RID
GE
S.................................................44
Y XeMgQ[UYhV[RN`OMOV\PUiRRg[XNZNXMQNPOjQ[RQNO`SSSSSSSSSSSSSSSSSSSkY
4.1 E
XA
MP
LE
S O
F IT
EM
S F
OR
A F
OO
TB
RID
GE
DY
NA
MIC
DE
SIG
N S
PE
CIF
ICA
TIO
N..............54
4.2 E
XA
MP
LE
S
OF
IT
EM
S
TO
B
E
INS
ER
TE
D
IN
TH
E
PA
RT
ICU
LA
R
WO
RK
S
TE
CH
NIC
AL
SP
EC
IFIC
AT
ION
FO
R A
FO
OT
BR
IDG
E ........................................................................................54
4.3 D
YN
AM
IC T
ES
TS
OR
TE
ST
S O
N F
OO
TB
RID
GE
S.............................................................54 ^ Mgg[OVNl ^hU[LNOV[URPZQe[VaOMLNXRPZXPORQUWXQNPORSSSSSSkm
1.1 A
SIM
PL
E O
SC
ILL
AT
OR
..............................................................................................58
1.2 L
INE
AR
SY
ST
EM
S A
T N
DO
F......................................................................................67
1.3 C
ON
TIN
UO
US
EL
AS
TIC
SY
ST
EM
S................................................................................76
1.4 D
ISC
RE
TIS
AT
ION
OF
TH
E C
ON
TIN
UO
US
SY
ST
EM
S.......................................................82
c Mgg[OVNl chLPV[ddNO`PZQe[g[V[RQUNMOdPMVSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSm]
2.1 W
AL
KIN
G..................................................................................................................87
2.2 R
UN
NIN
G...................................................................................................................91
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T Mgg[OVNlThVMLgNO`RaRQ[LRSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSnY
3.1 V
ISC
O-E
LA
ST
IC D
AM
PE
RS..........................................................................................94
3.2 V
ISC
OU
S D
AM
PE
RS
....................................................................................................95
3.3 T
UN
ED
MA
SS
DA
MP
ER
S (T
MD
).................................................................................97
3.4 T
UN
ED
LIQ
UID
DA
MP
ER
S.........................................................................................100
3.5 C
OM
PA
RA
TIV
E T
AB
LE
.............................................................................................101 Y Mgg[OVNlYh[lMLgd[RPZZPPQ_UNV`[RSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS ^fT
4.1 W
AR
RE
N-T
YP
E L
AT
ER
AL
BE
AM
S: CA
VA
ILL
ON
FO
OT
BR
IDG
E...................................103
4.2 S
TE
EL
BO
X-G
IRD
ER: S
TA
DE
DE
FR
AN
CE
FO
OT
BR
IDG
E.............................................103
4.3 R
IBB
ED
SL
AB: N
OIS
Y-L
E-GR
AN
D F
OO
TB
RID
GE
.......................................................104
4.4 B
OW
-ST
RIN
G A
RC
H: M
ON
TIG
NY
-LÈ
S-CO
RM
EIL
LE
S F
OO
TB
RID
GE
............................105
4.5 S
US
PE
ND
ED
CO
NS
TR
UC
TIO
N: F
OO
TB
RID
GE
OV
ER
TH
E A
ISN
E A
T S
OIS
SO
NS
.............106
4.6 S
TE
EL
AR
CH
: SO
LF
ER
INO
FO
OT
BR
IDG
E...................................................................106
4.7 C
AB
LE-S
TA
YE
D C
ON
ST
RU
CT
ION
: PA
S-DU
-LA
C F
OO
TB
RID
GE
AT
ST
QU
EN
TIN
..........108
4.8 M
IXE
D C
ON
ST
RU
CT
ION
BE
AM
: MO
NT-S
AIN
T-MA
RT
IN F
OO
TB
RID
GE
.......................109 k Mgg[OVNlkh[lMLgd[RPZXMdXWdMQNPORPZZPPQ_UNV`[RSSSSSSSSSSS ^^f
5.1 E
XA
MP
LE
S O
F C
OM
PL
ET
E C
AL
CU
LA
TIO
NS
OF
FO
OT
BR
IDG
ES....................................110
5.2 S
EN
SIT
IVIT
Y S
TU
DY
OF
TY
PIC
AL
FO
OT
BR
IDG
ES.......................................................119
] MOO[l[]h_N_dNP`UMgeaSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS ^cY
6.1 G
EN
ER
AL
LY
............................................................................................................124
6.2 R
EG
UL
AT
ION
S.........................................................................................................124
6.3 D
AM
PE
RS
................................................................................................................124
6.4 B
EH
AV
IOU
R A
NA
LY
SIS
............................................................................................124
6.5 C
AL
CU
LA
TIO
N M
ET
HO
DS.........................................................................................126
6.6 A
RT
ICL
ES
ON
EIT
HE
R V
IBR
AT
ING
OR
INS
TR
UM
EN
TE
D F
OO
TB
RID
GE
S.......................126
6.7 S
PE
CIF
IC F
OO
TB
RID
GE
S...........................................................................................126
6.8 A
DD
ITIO
NA
L B
IBL
IOG
RA
PH
Y...................................................................................126
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1. Main notations
A(
): dynamic am
plification [C
]: damping m
atrix C
i : damping N
o. i in a system w
ith n degrees of freedom (N
/(m/s))
E: Y
oung's modulus (N
/m2)
F(t): dynam
ic excitation (N)
[F(t)]: dynam
ic load vector F
0 : amplitude of a harm
onic force (N)
[F0 ]: am
plitude vector of a harmonic force (N
) H
r, e (): transfer function input e and the response r
I: inertia of a beam (m
4) J: torsional inertia of a beam
(m4)
[K]: stiffness m
atrix K
i : stiffness No. i in a system
with n degrees of freedom
(N/m
) L
: length of a beam (m
) [M
]: mass m
atrix M
i : mass N
o. i in a system w
ith n degrees of freedom (kg)
S: cross-sectional area of a beam (m
2) [X
]: vector of the degrees of freedom
Ci : generalised dam
ping of mode i
f: natural frequency of a sim
ple oscillator (Hz)
fi : i th natural frequency of an oscillator with n degrees of freedom
(Hz)
f: frequency K
i : generalised stiffness of mode i
Mi : generalised m
ass of mode i
n: number of pedestrians
[p(t)]: modal participation vector
[q(t)]: modal variable vector
t: variable designating time (s)
u(t), v(t), w(t): displacem
ents (m)
x(t): position of a simple oscillator referenced to its balance position (m
) : reduced pulsation
: logarithmic decrem
ent i : i th critical dam
ping ratio of an oscillator with n degrees of freedom
(no unit) : density (kg/m
3) [
i ]: i th eigen vector (
): argument of H
r, e - phase angle between entrance and exit (rad)
0 : natural pulsation of a simple oscillator (rad/s):
=2
f
i : i th natural pulsation of an oscillator with n degrees of freedom
(rad/s): i =
2 fi
: pulsation (rad/s)
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ent o
f vibra
tiona
l beh
avio
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2. Introduction
These guidelines w
ere prepared within the fram
ework of the S
étra/AF
GC
working group on
"Dynam
ic behaviour of footbridges", led by Pascal C
harles (Ile-de-France R
egional Facilities
Directorate and then S
étra) and Wasoodev H
oorpah (OT
UA
and then MIO
). T
hese guidelines were drafted by:
V
alérie BO
NIF
AC
E
(RF
R)
Vu B
UI
(Sétra)
Philippe B
RE
SS
OL
ET
TE
(C
US
T - L
ER
ME
S)
Pascal C
HA
RL
ES
(D
RE
IF and then S
étra) X
avier CE
SP
ED
ES
(S
etec-TP
I) F
rançois CO
NS
IGN
Y
(RF
R and then A
DP
) C
hristian CR
EM
ON
A
(LC
PC
) C
laire DE
LA
VA
UD
(D
RE
IF)
Luc D
IEL
EM
AN
(S
NC
F)
Thierry D
UC
LO
S
(Sodeteg and then A
rcadis-EE
GS
imecsol)
Wasoodev H
OO
RP
AH
(O
TU
A and then M
IO)
Eric JA
CQ
UE
LIN
(U
niversity of Lyon 1 - L
2M)
Pierre M
AIT
RE
(S
ocotec) R
aphael ME
NA
RD
(O
TH
) S
erge MO
NT
EN
S
(Systra)
Philippe V
ION
(S
étphenomenaa)
Actions exerted by pedestrians on footbridges m
ay result in vibrational phenomena. In
general, these phenomena do not have adverse effects on structure, although the user m
ay feel som
e discomfort.
These
guidelines round
up the
state of
current know
ledge on
dynamic
behaviour of
footbridges under pedestrian loading. An analytical m
ethodology and recomm
endations are also proposed to guide the designer of a new
footbridge when considering the resulting
dynamic effects.
The m
ethodology is based on the footbridge classification concept (as a function of traffic level) and on the required com
fort level and relies on interpretation of results obtained from
tests performed on the S
olférino footbridge and on an experimental platform
. These tests w
ere funded by the D
irection des Routes (H
ighway D
irectorate) and managed by S
étra, with the
support of
the scientific
and technical
network
set up
by the
French
"Ministère
de l'E
quipement, des T
ransports, de l'Am
énagement du territoire, du T
ourisme et de la M
er" and specifically by the D
RE
IF (D
ivision des Ouvrages d’A
rt et des Tunnels et L
aboratoire R
égional de l’Est P
arisien). T
his document covers the follow
ing topics: - a description of the dynam
ic phenomena specific to footbridges and identification of the
parameters w
hich have an impact on the dim
ensioning of such structures; - a m
ethodology for the dynamic analysis of footbridges based on a classification according to
the traffic level; - a presentation of the practical m
ethods for calculation of natural frequencies and modes, as
well as structural response to loading;
- recomm
endations for the drafting of design and construction documents.
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Supplem
entary theoretical (reminder of structural dynam
ics, pedestrian load modelling) and
practical (damping system
s, examples of recent footbridges, typical calculations) data are also
provided in the guideline appendices.
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Foreword
This docum
ent is based on current scientific and technical knowledge acquired both in F
rance and
abroad. F
rench regulations
do not
provide any
indication concerning
dynamic
and vibrational phenom
ena of footbridges whereas E
uropean rules reveal shortcomings that have
been highlighted in recent works.
Tw
o footbridges
located in
the centre
of P
aris and
London,
were
closed shortly
after inauguration:
they show
ed ham
pering transverse
oscillations w
hen carrying
a crow
d of
people; this resulted in the need for thorough investigations and studies of their behaviour under pedestrian loading. T
hese studies involved in situ testing and confirmed the existence of
a phenomenon w
hich had been observed before but that remained unfam
iliar to the scientific and technical com
munity. T
his phenomenon referred to as "forced synchronisation" or "lock-
in" causes the high amplitude transverse vibrations experienced by these footbridges.
In order to provide designers with the necessary inform
ation and means to avoid reproducing
such incidents,
it w
as considered
useful to
issue guidelines
summ
arising the
dynamic
problems affecting footbridges.
These guidelines concern the norm
al use of footbridges as concerns the comfort criterion, and
take vandalism into consideration, as concerns structural strength. T
hese guidelines are not m
eant to guarantee comfort w
hen the footbridge is the theatre of exceptional events: marathon
races, demonstrations, balls, parades, inaugurations etc.
Dynam
ic behaviour of footbridges under wind loads is not covered in this docum
ent. D
ynamic analysis m
ethodology is related to footbridge classification based on traffic level. T
his means that footbridges in an urban environm
ent are not treated in the same w
ay as footbridges in open country. T
he part played by footbridge Ow
ners is vital: they select the footbridge comfort criterion and
this directly affects structural design. Maxim
um com
fort does not tolerate any footbridge vibration; thus the structure w
ill either be sturdy and probably unsightly, or slender but fitted w
ith dam
pers thus
making
the structure
more
expensive and
requiring sophisticated
maintenance. M
inimum
comfort allow
s moderate and controlled footbridge vibrations; in this
case the structure will be m
ore slender and slim-line, i.e. m
ore aesthetically designed in general and possibly fitted w
ith dampers.
This docum
ent is a guide: any provisions and arrangements proposed are to be considered as
advisory recomm
endations without any com
pulsory content.
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3. Footbridge dynamics
This
chapter draw
s up
an inventory
of current
knowledge
on the
dynamic
phenomena
affecting footbridges. It also presents the various tests carried out and the studies leading to the recom
mendations described in the next chapter.
3.1 - Structure and footbridge dynamics
3.1.1 - General
By definition static loads are constant or barely tim
e-variant (quasi-static) loads. On the other
hand, dynamic loads are tim
e-related and may be grouped in four categories:
- harm
onic or purely sinusoidal loads; -
periodically recurrent loads integrally repeated at regular time intervals referred to as
periods; -
random loads show
ing arbitrary variations in time, intensity, direction…
; -
pulsing loads corresponding to very brief loads. G
enerally speaking pedestrians loads are time-variant and m
ay be classified in the "periodic load" category. O
ne of the main features of the dynam
ic loading of pedestrians is its low
intensity. Applied to very stiff and m
assive structures this load could hardly make them
vibrate significantly. H
owever, aesthetic, technical and technological developm
ents lead to ever m
ore slender and flexible structures, footbridges follow this general trend and they are
currently designed and built with higher sensitivity to strain. A
s a consequence they more
frequently require a thorough dynamic analysis.
The study of a basic m
odel referred to as simple oscillator illustrates the dynam
ic analysis principles and highlights the part played by the different structural param
eters involved in the process. O
nly the main results, directly useful to designers are m
entioned here. Appendix 1 to
these guidelines gives a more detailed presentation of these results and deals w
ith their general application to com
plex models.
3.1.2 - Simple oscillator
The sim
ple oscillator consists of mass m
, connected to a support by a linear spring of stiffness k and a linear dam
per of viscosity c, impacted by an external force F
(t) (figure 1.1). This
oscillator is supposed to move only by translation in a single direction and therefore has only
one degree of freedom (herein noted "dof") defined by position x(t) of its m
ass. Detailed
calculations are provided in Appendix 1.
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F
igure 1.1: Sim
ple oscillator T
he dynamic param
eters specific to this oscillator are the following:
m k0
= 2
fo : natural pulsation (rad/s), fo being the natural frequency (Hz). S
ince m
is a mass, its S
.I. unit is therefore expressed in kilograms.
mk c
2: critical dam
ping ratio (dimensionless) or critical dam
ping percentage. In
practice, has a value that is alw
ays less than 1. It should be noted that until experim
ental tests have been carried out, the critical damping ratio can only be
assessed. D
amping
has various
origins: it
depends on
materials
(steel, concrete,
timber) w
hether the concrete is cracked (reinforced concrete, prestressed concrete), the steel jointing m
ethod (bolting, welding).
The resonance phenom
enon is particularly clear when the sim
ple oscillator is excited by a harm
onic (or sinusoidal) under the form F
0 sin ( t ).
If, by definition, its static response obtained with a constant force equal to F
0 is:
20
00
/m
F
k Fx
statiq
ue
the dynamic response m
ay be amplified by a factor
)(
Aand is equal to:
)(
max
Ax
xsta
tique
where
0
is the reduced (or relative) pulsation and
22
22
4)
1(
1)
(A
is the dynamic am
plification.
Dynam
ic amplification is obtained as a function of
and . It m
ay be represented by a set of curves param
eterised by . S
ome of these curves are provided in figure 1.2 for a few
specific
values of the critical damping ratio. T
hese curves show a peak for the value of
22
1R
characterising the
resonance and
therefore corresponding
to the
resonance pulsation
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20
21
Rand to the resonance frequency
2R
Rf
. In this case the response is
higher (or even much higher) than the static response.
It should be noted that resonance does not occur for =
0 but for
R. A
dmitting that
structural damping is w
eak in practice, we m
ay consider that resonance occurs for =
0 and
that amplification equals:
2 1)1
(R
A
Figure 1.2: R
esonance curves D
imensioning of the structures based on dynam
ic loading cannot be made using only the
maxim
um intensity of the load im
pact. Thus, for exam
ple, load F(t) =
F0 sin (
1 t) can generate displacem
ents or stresses very much low
er than load F(t) =
(F0 /10) sin (
2 t) which
however has an am
plitude 10 times w
eaker, if this second load has a frequency much closer to
the resonance frequency of the structure. R
esonance am
plification being
directly related
to dam
ping, it
is therefore
necessary to
estimate this param
eter correctly in order to obtain appropriate dynamic dim
ensioning. It should be noted that the sim
ple oscillator study relies on the hypothesis of linear damping
(viscous, with a dam
ping force proportional to speed), which is one dam
ping type among
others. How
ever, this is the assumption selected by m
ost footbridge designers and engineers.
3.1.3 - Complex system
s
The study of real structures, w
hich are generally continuous and complex system
s with an
important num
ber of degrees of freedom, m
ay be considered as the study of a set of n simple
oscillators, each one describing a characteristic vibration of the system. T
he approximation
methods allow
ing such a conversion are detailed in Appendix 1. T
he new item
with regard to
the simple oscillator is the natural vibration m
ode defined by the pair constituted by the frequency and a vibratory shape (
i ,i ) of the system
. Com
putation of natural vibration m
odes is relatively intricate but designers nowadays have excellent softw
are packages to obtain them
, provided that they take, when m
odelling the system, all precautionary m
easures required for m
odel analysis applications. Practical use of natural vibration m
odes is dealt with
in chapter 3.
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It should be emphasized that in som
e cases the problem can even be solved using a single
simple oscillator. In any case, the m
ain conclusions resulting from the sim
ple oscillator study can be generalised to com
plex systems.
3.2 - Pedestrian loading
3.2.1 - Effects of pedestrian walking
Pedestrian loading, w
hether walking or running, has been studied rather thoroughly (see
Appendix 2) and is translated as a point force exerted on the support, as a function of tim
e and pedestrian position. N
oting that x is the pedestrian position in relation to the footbridge
centreline, the load of a pedestrian moving at constant speed v can therefore be represented as
the product of a time com
ponent F(t) by a space com
ponent (x –
vt), being the D
irac operator, that is:
)(
)(
),
(vt
xt
Ft
xP
S
everal parameters m
ay also affect and modify this load (gait, physiological characteristics
and apparel, ground roughness, etc.), but the experimental m
easurements perform
ed show that
it is periodic, characterised by a fundamental param
eter: frequency, that is the number of steps
per second. Table 1.1 provides the estim
ated frequency values.
Designation
Specific features
Frequency range (H
z) W
alking C
ontinuous contact with the ground
1.6 to 2.4 R
unning D
iscontinuous contact 2 to 3.5
Table 1.1
Conventionally, for norm
al walking (unham
pered), frequency may be described by a gaussian
distribution with 2 H
z average and about 0.20 Hz standard deviation (from
0.175 to 0.22, depending on authors). R
ecent studies and conclusions drawn from
recent testing have revealed even low
er mean frequencies, around 1.8 H
z - 1.9 Hz.
The periodic function F
(t), may therefore be resolved into a F
ourier series, that is a constant part increased by an infinite sum
of harmonic forces. T
he sum of all unitary contributions of
the terms of this sum
returns the total effect of the periodic action.
)2
sin(2
sin)
(2
10
im
i
n
i
mt
ifG
tf
GG
tF
with
G0
: static force (pedestrian weight for the vertical com
ponent),
G1
: first harmonic am
plitude,
Gi
: i-th harmonic am
plitude,
fm : w
alking frequency,
i : phase angle of the i-th harm
onic in relation to the first one, n
: number of harm
onics taken into account. T
he mean value of 700 N
may be taken for G
0 , weight of one pedestrian.
At m
ean frequency, around 2 Hz (fm =
2 Hz) for vertical action, the coefficient values of the
Fourier decom
position of F(t) are the follow
ing (limited to the first three term
s, that is n =
3, the coefficients of the higher of the term
s being less than 0.1 G0 ):
G
1 = 0.4 G
0 ; G2 =
G3
0.1 G
0 ;
2 =
3
/2
.
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By resolving the force into three com
ponents, that is, a «vertical» component and tw
o horizontal com
ponents (one in the «longitudinal» direction of the displacement and one
perpendicular to
the transverse
or lateral
displacement),
the follow
ing values
of such
components m
ay be selected for dimensioning (in practice lim
ited to the first harmonic):
Vertical com
ponent of one-pedestrian load: )
2sin(
4,0
)(
00
tf
GG
tF
mv
Transverse horizontal com
ponent of one-pedestrian load:
tf
Gt
Fm
ht
22
sin05,0
)(
0
Longitudinal horizontal com
ponent of one-pedestrian load: )
2sin(
2,0
)(
0t
fG
tF
ml
h
It should be noted that, for one same w
alk, the transverse load frequency is equal to half the frequency of the vertical and longitudinal load. T
his is due to the fact that the load period is equal to the tim
e between the tw
o consecutive steps for vertical and longitudinal load since these tw
o steps exert a force in the same direction w
hereas this duration corresponds to two
straight and consecutive right footsteps or to two consecutive left footsteps in the case of
transverse load since the left and right footsteps exert loads in opposite directions. As a result,
the transverse load period is two tim
es higher than the vertical and longitudinal load and therefore the frequency is tw
o times low
er.
3.2.2 - Effects of pedestrian running
Appendix 2 presents the effects of pedestrian running. T
his load case, which m
ay be very dim
ensioning in nature, should not be systematically retained. V
ery often, the crossing duration of joggers on the footbridge is relatively short and does not leave m
uch time for the
resonance phenomenon to settle, in addition, this annoys the other pedestrians over a very
short period. Moreover, this load case does not cover exceptional events such as a m
arathon race w
hich must be studied separately. A
ccordingly, running effects shall not be considered in these guidelines.
3.2.3 - Random effects of several pedestrians and crowd
In practice, footbridges are submitted to the sim
ultaneous actions of several persons and this m
akes the corresponding dynamic m
uch more com
plicated. In fact, each pedestrian has its ow
n characteristics (weight, frequency, speed) and, according to the num
ber of persons present on the bridge, pedestrians w
ill generate loads which are m
ore or less synchronous w
ith each other, on the one hand, and possibly with the footbridge, on the other. A
dded to these, there are the initial phase shifts betw
een pedestrians due to the different mom
ents when
each individual enters the footbridge.
Moreover, the problem
induced by intelligent human behaviour is such that, am
ong others and facing a situation different to the one he expects, the pedestrian w
ill modify his natural
and norm
al gait
in several
ways;
this behaviour
can hardly
be subm
itted to
software
processing.
It is therefore very difficult to fully simulate the actual action of the crow
d. One can m
erely set out reasonable and sim
plifying hypotheses, based on pedestrian behaviour studies, and then assum
e that the crowd effect is obtained by m
ultiplying the elementary effect of one
pedestrian, possibly weighted by a m
inus factor. Various ideas exist as concern crow
d effects and they antedate the S
olférino and Millenium
incidents. These concepts are presented in the
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following paragraphs together w
ith a more com
prehensive statistical study which w
ill be used as a basis for the loadings recom
mended in these guidelines.
For
a large
number
of independent
pedestrians (that
is, w
ithout any
particular synchronisation) w
hich enter a bridge at a rate of arrival (expressed in persons/second) the
average dynamic response at a given point of the footbridge subm
itted to this pedestrian flow
is obtained by multiplying the effect of one single pedestrian by a factor
Tk
, T being
the time taken by a pedestrian to cross the footbridge (w
hich can also be expressed by T =
L/v
where L
represents the footbridge length and v the pedestrian speed). In fact, this product T
presents the number of N
pedestrians present on the bridge at a given time. P
ractically,
this means that n pedestrians present on a footbridge are equivalent to
N all of them
being synchronised. T
his result can be demonstrated by considering a crow
d with the individuals all
at the same frequency w
ith a random phase distribution.
This result takes into consideration the phase shift betw
een pedestrians, due to their different entrance tim
e, but comprises a deficiency since it w
orks on the assumption that all pedestrians
are moving at the sam
e frequency.
Several researchers have studied the forces and m
oments initiated by a group of persons,
using measurem
ents made on instrum
ented platforms w
here small pedestrian groups m
ove. E
brahimpour &
al. (Ref. [24]) propose a sparse crow
d loading model on the first term
of a F
ourier representation the coefficient 1 of w
hich depends on the number
Np of persons
present on the platform (for a 2 H
z walking frequency):
1 = 0.34 – 0.09 lo
g(n
p ) for N
p <10
1 = 0.25
for Np >
10 U
nfortunately, this model does not cover the cum
ulative random effects.
Until recently dynam
ic dimensioning of footbridges w
as mainly based on the theoretical
model loading case w
ith one single pedestrian completed by rather crude requirem
ents concerning
footbridge stiffness
and natural
frequency floor
values. O
bviously, such
requirements are very insufficient and in particular they do not cover the m
ain problems
raised by the use of footbridges in urban areas which are subject to the action of m
ore or less
dense pedestrian
groups and
crowds.
Even
the above-addressed
N
model
has som
e deficiencies. It
rapidly appears
that know
ledge of
crowd
behaviour is
limited
and this
makes
the availability of practical dim
ensioning means all the m
ore urgent. It is better to suggest simple
elements to be im
proved on subsequently, rather than remaining in the current know
ledge void. T
herefore several crowd load cases have been developed using probability calculations and
statistical processing to deepen the random crow
d issue. The m
odel finally selected consists in handling pedestrians' m
oves at random frequencies and phases, on a footbridge presenting
different modes and in assessing each tim
e the equivalent number of pedestrians w
hich - w
hen evenly distributed on the footbridge, or in phase and at the natural frequency of the footbridge w
ill produce the same effect as random
pedestrians.
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Several digital tests w
ere performed to take into consideration the statistical effect. F
or each test including
N pedestrians and for each pedestrian, a random
phase and a norm
ally
distributed random frequency
2f
centred around the natural frequency of the footbridge
and w
ith 0.175
Hz
standard deviation,
are selected;
the m
aximum
acceleration
over a
sufficiently long period (in this case the time required for a pedestrian to span the footbridge
twice at a 1.5m
/s speed) is noted and the equivalent number of pedestrians w
hich would be
perfectly synchronised is calculated. The m
ethod used is explained in figure 1.3.
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Piétons aléatoires circulant à la vitesse V
= 1,5m
/s sur la passerelle
uences aléatoires (voir loi de distribution ci-contre) et fréqphases com
plètement aléatoires
P =
Néq / L
ongueur totale x 280N x cos( 2
f0 t +
)
+p
+p
-p
ccélération maxim
ale = A
max essai
A
Accélération m
aximale =
Am
ax (Néq )
Am
ax essai = A
max (N
éq )
densité de probabilité pour les fréquences : Loi norm
ale ; moyenne 2H
z ; d'écart-type 0,175 Hz
opoo opoq opro oprq opso opsq opto optq opuo opuqvw xvw yvw zvw {vw |vw }vw ~�w ��w v �w ��w x�w y�w z�w {�w |R
andom pedestrians circulating at speed v =
1.5m/s on the
footbridge random
frequencies (see the distribution law opposite) and
totally random phases
����� ���� ���� ���� ���� � ��� ��� ��� ��� ������ � ��� � ��� � ��� � ��� �� � ���� � �� �� � �� � � ��� �� ��� �� � ���� � ��� � � �� �� � �� �� � �� �� ������� �
���� ��� ��� ��� ��� � �� �� �� �� ���� ¡¢ ¡£¤¢ ¤¡ ¥¢ £¦§¢ ¨§ £¤̈ ¢ ¡ � ¦ ¥ � ¡ ¡ ¦¤ § ¥ ¤ § £� ££ ̈£¥ � £¡ ¦£ ¦© £§ ¡ £¤ ¤¥ � £¥ §¥ ¥ ¥ ¡̈¥̈ �¥ § ¡¥ ¤ ©¡ � ¥¡ ¤¡ ¥ £¡ ¡̈
Piétons répartis, à la m
ême
fréquence (celle de la
passerelle), et en phase, dont le signe du chargem
ent est lié au signe du m
ode.
Déform
ée modale
éq
Mode shape
Maxim
um acceleration =
Am
ax test
Distributed pedestrians, at the
same
frequency (that
of the
footbridge), and in phase, with
the loading sign related to them
ode sign.
P =
Né q / T
otal length x 280N x cos( 2
f0 t +
)
Maxim
um acceleration =
Am
ax (Néq )
donne N
Am
ax test = A
max (N
eq ) yields Neq
Figure 1.3: C
alculation methodology for the equivalent num
ber of pedestrians Neq
These tests are repeated 500 tim
es with a fixed num
ber of pedestrians, fixed damping and a
fixed number of m
ode antinodes; then the characteristic value, such as 95% of the sam
ples
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give a value lower than this characteristic value (95%
characteristic value, 95 percentile or 95%
fractile). This concept is explained in figure 1.4:
Nom
bre de tirages
Num
ber of samples
Nom
de pbre
iétons
95%5%
Num
ber of pedestrians
F
igure 1.4: 95% fractile concept
By varying dam
ping, the number of pedestrians, the num
ber of mode antinodes, it is possible
to infer a law for the equivalent num
ber of pedestrians, this law is the closest to the perform
ed test results. T
he following tw
o laws are retained:
Sparse or dense crow
d: random phases and frequencies w
ith Gaussian law
distribution:
NN
eq8,
10 w
here N is the num
ber of pedestrians present on the footbridge (density
surface area) and the critical dam
ping ratio. V
ery dense crowd: random
phases and all pedestrians at the same frequency:
NN
eq85,1
.
This m
odel is considerably simplified in the calculations. W
e just have to distribute the Neq
pedestrians on the bridge, to apply to these pedestrians a force the amplitude sign is the sam
e as the m
ode shape sign and to consider this force as the natural frequency of the structure and to calculate the m
aximum
acceleration obtained at the corresponding resonance. Chapter 3
explains how this loading is taken into consideration.
3.2.4 - Lock-in of a pedestrian crowd
Lock-in expresses the phenom
enon by which a pedestrian crow
d, with frequencies random
ly distributed around an average value and w
ith random phase shifts, w
ill gradually coordinate at com
mon frequency (that of the footbridge) and enters in phase w
ith the footbridge motion.
So far, know
n cases of crowd lock-in have been lim
ited to transverse footbridge vibrations. T
he most recent tw
o cases, now fam
ous, are the Solférino footbridge and the M
illenium
footbridge which w
ere submitted to thorough in-situ tests. O
nce again, these tests confirm that
the phenomenon is clearly explained by the pedestrian response as he m
odifies his walking
pace when he perceives the transverse m
otion of the footbridge and it begins to disturb him.
To
compensate
his incipient
unbalance, he
instinctively follow
s the
footbridge m
otion frequency. T
hus, he directly provokes the resonance phenomenon and since all pedestrians
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undergo it, the problem is further am
plified and theoretically the whole crow
d may becom
e synchronised. F
ortunately, on the one hand, actual synchronisation is much w
eaker and, on the other, w
hen the footbridge movem
ent is such that the pedestrians can no longer put their best foot forw
ard, they have to stop walking and the phenom
enon can no longer evolve.
Using a large footbridge w
ith a 5.25m x 134m
main span w
hich can be submitted to a very
dense crowd (up to about 2 persons/m
²), Fujino &
al. (Ref. [30]) observed that application of
the above factor gave an under-estimation of about
N 1 to 10 tim
es of the actually observed lateral
vibration am
plitude. T
hey form
ed the hypothesis
of synchronisation
of a
crowd
walking in synchrony w
ith the transverse mode frequency of their footbridge to explain the
phenomenon and w
ere thus able to prove, in this case, the measurem
ent magnitude obtained.
This is the phenom
enon we call "lock-in", and a detailed presentation is provided below
. F
or this structure, by retaining only the first term of the F
ourier decomposition for the
pedestrian-induced load, these authors propose a 0.2N
multiplication factor to represent any
loading which w
ould equate that of a crowd of N
persons, allowing them
to retrieve the m
agnitude of the effectively measured displacem
ents (0.01 m).
Arup's team
issued a very detailed article on the results obtained following this study and tests
performed on the M
illenium footbridge. (R
ef. [38]). Only the m
ain conclusions of this study are m
entioned here. T
he model proposed for the M
illenium footbridge study is as follow
s: the force exerted by a pedestrian (in N
) is assumed to be related to footbridge velocity.
tx
VK
Fped
estrian
,1
where K
is a proportionality factor (in Ns/m
) and V the footbridge velocity
at the point x in question and at time t.
Seen this w
ay, pedestrian load may be understood as a negative dam
ping. Assum
ing a viscous dam
ping of the footbridge, the negative damping force induced by a pedestrian is directly
deducted from it. T
he consequence of lock-in is an increase of this negative damping force,
induced by the participation of a higher number of pedestrians. T
his is how the convenient
notion of critical number appears: this is the num
ber of pedestrians beyond which their
cumulative
negative dam
ping force
becomes
higher than
the inherent
damping
of the
footbridge; the situation would then be sim
ilar to that of an unstable oscillator: a small
disturbance may generate indefinitely-am
plifying movem
ents…
For the particular case of a sinusoidal horizontal vibration m
ode (the maxim
um am
plitude of this m
ode being normalized to 1, f1 representing the first transverse natural frequency and m
1 the generalised m
ass in this mode, considering the m
aximum
unit displacement) and assum
ing an uneven distribution of the pedestrians, the critical num
ber can then be written as:
K
fm
N1
18
K is the proportionality factor, w
ith a value of 300 Ns/m
in the case of the Millenium
footbridge. W
e can then note that a low dam
ping, a low m
ass, or a low frequency is translated by a sm
all critical num
ber and therefore a higher lock-in risk. Consequently, to increase the critical
number it w
ill be necessary to act on these three parameters.
It will be noted that the value of factor K
cannot a p
riori be generalised to any structure and
therefore its use increases the criterion application uncertainty.
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To quantify the horizontal load of a pedestrian and the pedestrian lock-in effects under lateral
motion, som
e tests were carried out on a reduced footbridge m
odel by recreating, using a dim
ensional analysis, the conditions prevailing on a relatively simple design of footbridge
(one single horizontal mode).
The principle consists in placing a 7-m
etre long and 2-metre w
ide slab on 4 flexible blades m
oving laterally and installing access and exit ramps as w
ell as a loop to maintain w
alking continuity (P
hotos 1.1 and 1.2). To m
aintain this continuity, a large number of pedestrians is
of course needed on the loop; this number being clearly higher than the num
ber of pedestrians present on the footbridge at a given tim
e.
CR
OS
S-S
EC
TIO
N V
IEW
(with blade 70 cm
) S
cale 1/25 D
etail A
Detail B
D
etail C
Flexible blade (thickness 8 m
m)
Detail D
A
ttachment axis to the slab
Detail E
H
EB
P
hotographs 1.1 and 1.2: Description of the m
odel B
y recreating the instantaneous force from the displacem
ents measured (previously filtered to
attenuate the effect of high frequencies) (t
kxt
xct
xm
tF
)(
)(
)(
). Figure 1.5 show
s that, in a first step, for an individual pedestrian the am
plitude of the pedestrian force remains
constant, around 50 N, and in any case, low
er than 100N, w
hatever the speed amplitude. In a
second step, we observe that the force am
plitude increases up to 150 N, but these last
oscillations should not be considered as they represent the end of the test.
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C
omparison of F
(t) and v(t) T
est of 01-10-03 – 1 pedestrian at 0.53 Hz – pedestrians follow
ing each other S
peed (m/s)
F pedestrians (N
) T
ime
Exciting force (N
) F
igure 1.5: Force and speed at forced resonant rate
We have a peak value w
hich does not exceed 100 N and is rather around 50N
on average, w
ith the first harmonic of this signal being around 35 N
. T
he following graphs (figures 1.6 and 1.7) represent, on the sam
e figure, the accelerations in tim
e (pink curve and scale on the RH
side expressed in m/s², variation from
0.1 to 0.75 m/s²)
and the «efficient» force (instantaneous force multiplied by the speed sign, averaged on a
period, which is therefore positive w
hen energy is injected into the system and negative in the
opposite case) for a group of pedestrians (blue curve, scale on the LH
side expressed in N).
(N)
(m/s²)
Figure 1.6: A
cceleration (m/s²) and efficient force (N
) with 6 random
pedestrians on the footbridge
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(N)
Beginning of synchronisation: T
here is a majority of
forces the action of which participates in m
ovement
amplification; cum
ulation is achieved in a better way.
Random
rate acceleration: there will be as m
any forcesparticipating in the m
ovement as the forces opposed to it
(m/s²)
Figure 1.7: A
cceleration (m/s²) and efficient force (N
) with 10 random
pedestrians on the footbridge W
e observe that, from a given value, the force exerted by the pedestrians is clearly m
ore efficient and there is som
e incipient synchronisation. This threshold is around 0.15 m
/s² (straight line betw
een the random rate zone and the incipient synchronisation zone). H
owever,
there is only some little synchronisation (m
aximum
value of 100-150 N i.e. 0.2 to 0.3 tim
es the effect of 10 pedestrians), but this is quite sufficient to generate very uncom
fortable vibrations (>
0.6m/s²).
Several test cam
paigns were carried out over several years follow
ing the closing of the S
olferino footbridge to traffic, from the beginning these tests w
ere intended to identify the issues and develop corrective m
easures; then they were needed to check the efficiency of the
adopted m
easures and,
finally, to
draw
lessons useful
for the
scientific and
technical com
munity.
The m
ain conclusions to be drawn from
the Solferino footbridge tests are the follow
ing: -
The lock-in phenom
enon effectively occurred for the first mode of lateral sw
inging for w
hich the double of the frequency is located within the range of norm
al walking
frequency of pedestrians. -
On the other hand, it does not seem
to occur for modes of torsion that sim
ultaneously present vertical and horizontal m
ovements, even w
hen the test crowd w
as made to
walk at a frequency that had given rise to resonance. T
he strong vertical movem
ents disturb and upset the pedestrians' w
alk and do not seem to favour m
aintaining it at the resonance frequency selected for the tests. H
igh horizontal acceleration levels are then noted and it seem
s their effects have been masked by the vertical acceleration.
- T
he concept of a critical number of pedestrians is entirely relative: it is certain that
below a certain threshold lock-in cannot occur, how
ever, on the other hand, beyond a threshold
that has
been proven
various specific
conditions can
prevent it
from
occurring. -
Lock-in appears to initiate and develop m
ore easily from an initial pedestrian w
alking frequency for w
hich half the value is lower than the horizontal sw
inging risk natural frequency of the structure. In the inverse case, that is, w
hen the walking crow
d has a faster initial pace several tests have effectively show
n that it did not occur. This w
ould
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be worth studying in depth but it is already possible to explain that the pedestrian
walking fairly rapidly feels the effects of horizontal acceleration not only differently,
which is certain, but also less noticeably, and this rem
ains to be confirmed.
- C
learly lock-in occurs beyond a particular threshold. This threshold m
ay be explained in term
s of sufficient number of pedestrians on the footbridge (conclusion adopted by
Arup's team
), but it could just as well be explained by an acceleration value felt by the
pedestrian, which is m
ore practical for defining a verification criterion. T
he following graphs (figures 1.8 to 1.13) present a sum
mary of the tests carried out on the
Solferino footbridge. T
he evolution of acceleration over time is show
n (in green), and in parallel the correlation or synchronisation rate, ratio betw
een the equivalent number of
pedestrians and the number of pedestrians present on the footbridge. T
he equivalent number
of pedestrians can be deduced from the instantaneous m
odal force. It is the number of
pedestrians who, regularly distributed on the structure, and both in phase and at the sam
e frequency apparently inject an identical am
ount of energy per period into the system.
S
olferino footbridge: Test 1 A
: Random
crowd/W
alking in circles with increasing num
bers of pedestrians A
cceleration (m/s)
Correlation rate (%
) T
ime (s)
……
Acceleration (m
/s²) ------ Correlation rate (%
) F
igure 1.8: 1A S
olferino footbridge random test: a crow
d is made to circulate endlessly on the footbridge w
ith the num
ber of pedestrians being progressively increased (69 – 138 – 207). In the test show
n in Figure 1.8, it can be seen that below
0.12m/s², behaviour is com
pletely random
, and from 0.15m
/s², it becomes partly synchronised, w
ith synchronisation reaching 30-35%
when the acceleration am
plitudes are already high (0.45m/s²). T
he concept of rate change critical threshold (shift from
a random rate to a partly synchronised rate) becom
es perceptible. T
he various "loops" correspond to the fact that the pedestrians are not regularly distributed on the footbridge, they are concentrated in groups. F
or this reason, it is clear when the largest
group of pedestrians is near the centre of the footbridge (sag summ
it), or rather at the ends of the footbridge (sag trough). It can also be seen that the three acceleration rise sags, that occur at increasing levels of acceleration (0.3m
/s² then 0.4m/s² and finally 0.5m
/s²), occur with the sam
e equivalent
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number of pedestrians each tim
e. This clearly show
s the fact that there is an acceleration rise, halted tw
ice when the group of pedestrians reaches the end of the footbridge.
In the following test, show
n in Figure 1.9, the num
ber of pedestrians has been increased more
progressively.
S
olferino footbridge: Tim
e 1 B: R
andom crow
d/Walking in circles w
ith increasing numbers of pedestrians
Acceleration (m
/s) C
orrelation rate (%)
Tim
e (s) …
… A
cceleration (m/s²) ------ C
orrelation rate (%)
Figure 1.9: 1B
Solferino footbridge random
test: a crowd is m
ade to circulate endlessly on the footbridge with
the number of pedestrians being m
ore progressively increased (115 138 161 92 – 184 – 202). T
his time, the rate change threshold seem
s to be situated around 0.15 – 0.20 m/s². M
aximum
synchronisation rate does not exceed 30%
.
S
olferino footbridge: Test 1 C
: Random
crowd/W
alking in circles with increasing num
bers of pedestrians A
cceleration rate (m/s)
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Correlation rate (%
) T
ime (s)
……
Acceleration (m
/s²) ------ Correlation rate (%
)
Figure 1.10: 1C
Solferino footbridge random
test
Test 1C
, shown in F
igure 1.10, leads us to the same conclusions: change in threshold betw
een 0.10 and 0.15m
/s², then more obvious synchronisation reaching up to 35%
-40%.
S
olferino footbridge: Test 2 A
1: Random
crowd/W
alking grouped together in a straight line 229p A
cceleration (m/s)
Correlation rate (%
) T
ime (s)
……
Acceleration (m
/s²) ------ Correlation rate (%
)
Figure 1.11: 2A
1 Solferino footbridge random
test
In the test shown in F
igure 1.11, the pedestrians are more grouped together and w
alk from one
edge of the footbridge to the other. The rise and subsequent fall in equivalent num
ber of pedestrians better expresses the m
ovement of pedestrians, and their crossing from
an area w
ithout displacement (near the edges) and another w
ith a lot of displacement (around m
id-span). S
ynchronisation rate rises to about 60%. T
his is higher than previously, however, it
should be pointed out, on the one hand, that the level of vibration is higher (0.9m/s² instead of
0.5m/s²) and, on the other, that the crow
d is fairly compact and this favours synchronisation
phenomenon am
ong the pedestrians.
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S
olferino footbridge: Test 2 A
2: Random
crowd/W
alking grouped together in a straight line 160p A
cceleration (m/s)
Correlation rate (%
) T
ime (s)
……
Acceleration (m
/s²) ------ Correlation rate (%
)
Figure 1.12: 2A
2 Solferino footbridge random
test
In the test shown in F
igure 1.12, there are only 160 people still walking slow
ly from one end
of the footbridge to the other. This tim
e, synchronisation rate reaches 50%. A
ccelerations are com
parable with those obtained in tests 1A
, 1B, and 1C
.
S
olferino footbridge: Test 2 B
: Random
crowd/R
apid walking in straight line 160p
Acceleration (m
/s) C
orrelation rate (%)
Tim
e (s) …
… A
cceleration (m/s²) ------ C
orrelation rate (%)
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Figure 1.13: 2B
Solferino footbridge random
test
This
last test
(figure 1.13)
is identical
to the
previous one
except that,
this tim
e, the
pedestrians were w
alking fast. Pedestrian synchronisation phenom
enon was not observed,
although there was a com
pact crowd of 160 people. T
his clearly shows that w
here pedestrian w
alking frequency is too far from the natural frequency, synchronisation does not occur.
These tests show
that there is apparently a rate change threshold in relation to random rate at
around 0.10-0.15 m/s². O
nce this threshold has been exceeded accelerations rise considerably but rem
ain limited. S
ynchronisation rates in the order of 30 to 50% are reached w
hen the test is stopped. T
his value can rise to 60%, or even higher, w
hen the crowd is com
pact. F
or dim
ensioning purposes,
the value
0.10 m
/s² shall
be noted.
Below
this
threshold, behaviour of pedestrians m
ay be qualified as random. It w
ill then be possible to use the equivalent random
pedestrian loadings mentioned above and this w
ill lead to synchronisation rates to the order of 5 to 10%
. Synchronisation rate can rise to m
ore than 60% once this
threshold has been exceeded. Thus, acceleration goes from
0.10 to over 0.60 m/s² relatively
suddenly. Acceleration thus system
atically becomes uncom
fortable. Consequently, the 0.10
m/s² rate change threshold becom
es a threshold not to be exceeded.
The various studies put forw
ard conclusions that appear to differ but that actually concur on several points. A
s soon as the amplitude of the m
ovements becom
es perceptible, crowd behaviour is no
longer random and a type of synchronisation develops. S
everal models are available (force as
a function of speed, high crowd synchronisation rate) but they all lead to accelerations rather
in excess of the generally accepted comfort thresholds.
Passage from
a random rate on a fixed support to a synchronised rate on a m
obile support occurs w
hen a particular threshold is exceeded, characterised by critical acceleration or a critical num
ber of pedestrians. It should be noted that the concept of a critical number of
pedestrians and the concept of critical acceleration may be linked. C
ritical acceleration may
be interpreted as the acceleration produced by the critical number of pedestrians, still random
though after this they are no longer so. A
lthough it is true that the main principles and the variations in behaviour observed on the
two footbridges concur, m
odelling and quantitative differences nevertheless lead to rather different dam
per dimensioning.
The concept of critical acceleration seem
s more relevant that that of a critical num
ber of pedestrians. A
cceleration actually corresponds to what the pedestrians feel w
hereas a critical num
ber of pedestrians depends on the way in w
hich the said pedestrians are organised and positioned on the footbridge. It is therefore this critical acceleration threshold that w
ill be discussed in these guidelines, the w
ay in which the pedestrians are organised depends on the
level of traffic on the footbridge (see below).
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3.3 - Parameters
that affect
dimensioning:
frequency, com
fort threshold,
comfort criterion, etc.
The
problems
encountered on
recent footbridges
echo the
well-know
n phenom
enon of
resonance that ensues from the m
atching of the exciting frequency of pedestrian footsteps and the natural frequency of a footbridge m
ode. Ow
ing to the fact that they are amplified,
noticeable m
ovements
of the
footbridge ensue
and their
consequence is
a feeling
of discom
fort for the pedestrians that upsets their progress. T
hus, it is necessary to review the structural param
eters, vital to the resonance phenomenon,
represented by natural vibration modes (natural m
odes and natural frequencies) and the values of the structural critical dam
ping ratio associated with each m
ode. In reality, even a footbridge of sim
ple design will have an infinity of natural vibration m
odes, frequencies and critical dam
ping ratios associated to it (see Appendix 1). H
owever, in m
ost cases, it is sufficient to study a few
first modes.
Along w
ith this, it is necessary to consider pedestrian walking frequencies, since they differ
from one individual to another, together w
ith walking conditions and num
erous other factors. S
o, it is necessary to bear in mind a range of frequencies rather than a single one.
The first sim
ple method for preventing the risk of resonance could consist in avoiding having
one or
several footbridge
natural frequencies
within
the range
of pedestrian
walking
frequency. This leads to the concept of a range of risk frequencies to be avoided.
3.3.1 - Risk frequencies noted in the literature and in current regulations
Com
pilation of the frequency range values given in various articles and regulations has given rise to the follow
ing table, drawn up for vertical vibrations:
Eurocode 2 ( R
ef. [4] ) 1.6 H
z and 2.4 Hz and, w
here specified, between 2.5 H
z and 5 Hz.
Eurocode 5 ( R
ef. [5] ) B
etween 0 and 5 H
z A
ppendix 2 of Eurocode 0
<5 H
z B
S 5400 ( R
ef. [6] ) <
5 Hz
Regulations in Japan ( R
ef. [30] ) 1.5 H
z – 2.3 Hz
ISO
/DIS
standard 10137 ( Ref. [28] )
1.7 Hz – 2.3 H
z C
EB
209 Bulletin
1.65 – 2.35 Hz
Bachm
ann ( Ref. [59] )
1.6 – 2.4 Hz
T
able 1 .2 A
s concerns lateral vibrations, the ranges described above are to be divided by two ow
ing to the particular nature of w
alking: right and left foot are equivalent in their vertical action, but are opposed in their horizontal action and this m
eans transverse efforts apply at a frequency that is half that of the footsteps. H
owever, on the M
illenium footbridge it w
as noticed that the lock-in phenomenon appeared
even for a horizontal mode w
ith a frequency considerably beneath that of the lower lim
it generally accepted so far for norm
al walking frequency. T
hus, for horizontal vibration modes,
it seems advisable to further low
er the lower boundary of the risk frequency range.
Although risk frequency ranges are fairly w
ell known and clearly defined, in construction
practice, it is not easy to avoid them w
ithout resorting to impractical rigidity or m
ass values. W
here it is impossible to avoid resonance, it is necessary to try to lim
it its adverse effects by
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acting on the remaining param
eter: structural damping; obviously, it w
ill be necessary to have available criteria m
aking it possible to determine the acceptable lim
its of the resonance.
3.3.2 - Comfort thresholds
Before going any further, the concept of com
fort should be specified. Clearly, this concept is
highly subjective. In particular:
from one individual to the next, the sam
e vibrations will not be perceived in the sam
e w
ay. for a particular individual, several thresholds m
ay be defined. The first is a vibration
perception threshold. This is follow
ed by a second that can be related to various degrees
of disturbance
or discom
fort (tolerable
over a
short period,
disturbing, unacceptable).
Finally,
a third
threshold m
ay be
determined
in relation
to the
consequences the vibrations may entail: loss of balance, or even health problem
s. furtherm
ore, depending on whether he is standing, seated, m
oving or stationary, a particular individual m
ay react differently to the vibrations. it is also w
ell known that there is a difference betw
een the vibrations of the structure and the vibrations actually perceived by the pedestrian. F
or instance, the duration for w
hich he is exposed to the vibrations affects what the pedestrian feels. H
owever,
knowledge in this field rem
ains imprecise and insufficient.
Hence,
though highly
desirable, it
is clear
that in
order to
be used
for footbridge
dimensioning, determ
ining thresholds in relation to the comfort perceived by the pedestrian is
a particularly difficult task. Indeed, where a ceiling of 1m
/s² for vertical acceleration has been given by M
atsumoto and al, (R
ef. [16]), others, such as Wheeler, on the one hand, and T
illy et al, on the other, contradict each other w
ith values that are either lower or higher (R
ef. [18] and R
ef. [20]). Furtherm
ore, so far, there have only been a few suggestions relating to transverse
movem
ents. H
ence, where projects are concerned, it is necessary to consider the recom
mended values as
orders of magnitude and it is m
ore convenient to rely on parameters that are easy to calculate
or measure; in this w
ay, the thresholds perceived by the pedestrians are assimilated w
ith the m
easurable quantities generated by the footbridges and it is generally the peak acceleration value, called the critical acceleration a
crit of the structure, that is retained; the comfort criterion
to be respected is represented by this value that should not be exceeded. Obviously, it is
necessary to define appropriate load cases to apply to the structure in order to determine this
acceleration. This verification is considered to be a service lim
it state.
3.3.3 - Acceleration comfort criteria noted in the literature and regulations
The literature and various regulations put forw
ard various values for the critical acceleration, noted a
crit . The values are provided for vertical acceleration and are show
n in Figure 1.14
below:
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ª«ªª ª«¬ª ª«ª ª«®ª ª«¯ª °«ªª °«¬ª °«ª±²³´µ²¶·µ¸¹´¸º»¼
½ ¾ ¿ À ¿ ½ Á  Á ½ ½ à  à ¾ Á À ¿Ä ÅÆÇȪª«ÉʬˬÌÍÎÏÐÑÒÉÊÈˬÓÇ ÌÔÕÓÇ°ª°Ö× ØÙÚÛÜÜÝÞÝß
F
igure 1.14: Vertical critical accelerations (in m
/s²) as a function of the natural frequency for various regulations: som
e depend on the frequency of the structure, others do not .
For vertical vibrations w
ith a frequency of around 2 Hz, standard w
alking frequency, there is apparently a consensus for a range of 0.5 to 0.8 m
/s². It should be remem
bered that these values are m
ainly associated with the theoretical load of a single pedestrian.
For lateral vibrations w
ith a frequency of around 1 Hz, E
urocode 0 Appendix 2 proposes a
horizontal critical acceleration of 0.2 m/s² under norm
al use and 0.4 m/s² for exceptional
conditions (i.e. a crowd). U
nfortunately, the text does not provide crowd loadings.
It is also useful to remem
ber that since accelerations, speeds and displacements are related, an
acceleration threshold may be translated as a displacem
ent threshold (which m
akes better sense for a designer) or even a speed threshold.
movem
ent
Hz
of
frequen
cyon
Accelera
ti2
2
speed
Hz
of
frequ
ency
on
Accelera
ti
2
F
or instance, for a frequency of 2 H
z: acceleration of 0.5m
/s² corresponds to a displacement of 3.2m
m, a speed of 0.04m
/s acceleration of 1m
/s² corresponds to a displacement of 6.3 m
m, a speed of 0.08 m
/s but for a frequency of 1 H
z: acceleration of 0.5m
/s² corresponds to a displacement of 12.7m
m, a speed of 0.08m
/s acceleration of 1m
/s² corresponds to a displacement of 25,3m
m, a speed of 0.16 m
/s
3.4 - Improvem
ent of dynamic behaviour
What is to be done w
hen footbridge acceleration does not respect comfort criteria?
It is necessary to distinguish between a footbridge at the design stage and an existing one.
In the case of a footbridge at the design stage, it is logical to try to modify its natural
frequency vibrations. If it is not possible to modify them
so that they are outside the resonance risk ranges in relation to excitation by the pedestrians, then attem
pts should be made to
increase structural damping.
With
an existing
footbridge, it
is also
possible to
try to
modify
its natural
frequency vibrations. H
owever, experience show
s that it is generally cheaper to increase damping.
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3.4.1 - Modification of vibration natural frequencies
A vibration natural frequency is alw
ays proportional to the square root of the stiffness and inversely proportional to the square root of the m
ass. The general aim
is to try to increase vibration frequency. T
herefore the stiffness of the structure needs to be increased. How
ever, practice indicates that an increase in stiffness is frequently accom
panied by an increase in m
ass, which produces an inverse result, this is a difficult problem
to solve.
3.4.2 - Increasing structural damping
The critical dam
ping ratio is not an inherent fact of a material. M
ost experimental results
suggest that
dissipation forces
are to
all practical
intents and
purposes independent
of frequency
but rather
depend on
movem
ent am
plitude. T
he critical
damping
ratio also
increases when vibration am
plitude increases. It also depends on construction details that may
dissipate energy to a greater or lesser extent (for instance, where steel is concerned, the
difference between bolting and w
elding). It should be noted that although the m
ass and rigidity of the various structure elements m
ay be m
odelled with a reasonable degree of accuracy, dam
ping properties are far more difficult to
characterise. Studies generally use critical dam
ping coefficients ranging between 0.1%
and 2.0%
and
it is
best not
to overestim
ate structural
damping
in order
to avoid
under-dim
ensioning. C
EB
information bulletin N
o. 209 (Ref. [7] ), an im
portant summ
ary document dealing w
ith the general problem
of structure vibrations provides the following values for use in projects:
T
ype of deck C
ritical damping ratio
M
inimum
value A
verage value R
einforced concrete 0.8%
1.3%
P
restressed concrete 0.5%
1.0%
M
etal 0.2%
0.4%
M
ixed 0.3%
0.6%
T
imber
1.5%
3.0%
T
able 1.3 A
s concerns timber, E
urocode 5 recomm
ends values of 1% or 1.5%
depending on the presence, or otherw
ise, of mechanical joints.
Where
vibration am
plitude is
high, as
with
earthquakes, critical
damping
ratios are
considerably higher and need to be SL
S checked. F
or instance, the AF
PS
92 Guide on seism
ic protection of bridges (art. 4.2.3) states the follow
ing:
Material
Critical dam
ping ratio W
elded steel 2%
B
olted steel 4%
P
restressed concrete 2%
N
on-reinforced concrete 3%
R
einforced concrete 5%
R
einforced elastomer
7%
T
able 1.4
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Finally, it should be pointed out that an estim
ate of actual structural damping can only be
achieved through measurem
ents made on the finished structure. T
his said, increased damping
may be obtained from
the design stage, for instance through use of a wire-m
esh structure, or in the case of tension tie footbridges («stress ribbon»), through insertion of elastom
er plates distributed betw
een the prefabricated concrete slabs making up the deck.
The
use of
dampers
is another
effective solution
for reducing
vibrations. A
ppendix 3
describes the different types of dampers that can be used and describes the operating and
dimensioning principle of a selection of dam
pers. The table of exam
ples below show
s this is a tried and tested solution to the problem
:
C
ountry N
ame
Mass
kg T
otal effective m
ass %
Dam
per critical dam
ping ratio
Critical
damping
ratio of
the structure
without
dampers
Critical
damping
ratio of
the structure
with
dampers
Structure
frequency H
z
France
Passerelle
du S
tade de
France
(Football
stadium
footbridge) (S
aint-Denis)
2400 per
span 1.6
0.075 0.2%
to 0.3%
4.3% to 5.3%
1.95 (vertical)
France
Solferino
footbridge (P
aris) 15000
4.7
0.4%
3.5%
0.812 (horizontal)
Ditto
Ditto
10000 2.6
0.5%
3%
1.94 (vertical)
Ditto
Ditto
7600 2.6
0.5%
2%
2.22 (vertical)
England
Millenium
footbridge (L
ondon) 2500
0.6% to 0.8%
2%
0.49 (horizontal)
Ditto
Ditto
1000 to
2500
0.6%
to 0.8%
0.5 (vertical)
Japan
1
0.2%
2.2%
1.8 (vertical)
US
A
Las V
egas (Bellagio-
Bally)
0.5%
8%
South K
orea S
eonyu footbridge
0.6%
3.6%
0.75 (horizontal)D
itto D
itto
0.4%
3.4%
2.03 (vertical)
Table 1.5: E
xamples of the use of tuned dynam
ic dampers
Note: In the case of the M
illenium footbridge, as w
ell as AD
As, viscous dam
pers were also
installed to dampen horizontal m
ovement. ©
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4. Footbridge dynamic analysis m
ethodology
For the purposes of O
wners, prim
e contractors and designers, this chapter puts forward a
methodology and recom
mendations for taking into account the dynam
ic effects caused by pedestrian traffic on footbridges. T
he recomm
endations are the result of the inventory described in Appendices 1 to 4, on the
one hand, and of the work of the S
étra-AF
GC
group responsible for these guidelines, on the other. T
hus, they are a summ
ary of existing knowledge and present the choices m
ade by the group. T
he methodology proposed m
akes it possible to limit risks of resonance of the structure
caused by pedestrian footsteps. Nevertheless, it m
ust be remem
bered that, resonance apart, very light footbridges m
ay undergo vibrational phenomena.
At source, w
hen deciding on his approach, the Ow
ner needs to define the class of the footbridge as a function of the level of traffic it w
ill undergo and determine a com
fort requirem
ent level to fulfil. F
ootbridge class conditions the need, or otherwise, to determ
ine structure natural frequency. W
here they are calculated, these natural frequencies lead to the selection of one or several dynam
ic load cases, as a function of the frequency value ranges. These load cases are defined
to represent the various possible effects of pedestrian traffic. Treatm
ent of the load cases provides the acceleration values undergone by the structure. T
he comfort level obtained can
be qualified by the range comprising the values.
The m
ethodology is summ
arised in the organisation chart below.
Footbridge
class
Calculation of natural frequencies
Classes I to III
No calculation required
Dynam
ic load cases to be studied
Max. accelerations undergone by the structure
Acceleration lim
its
Traffic assessm
en t
Com
fort level
sensitive
negligible
Class IV
Com
fort judged sufficient w
ithout calculating
Com
fort conclusion
Resonance risk
level O
wner
F
igure 2.1: Methodology organisation chart
This chapter also includes the specific verifications to be carried out (S
LS
and UL
S) to take
into account
the dynam
ic behaviour
of footbridges
under pedestrian
loading (see
2.6). O
bviously, the standard verifications (SL
S and U
LS
) are to be carried out in compliance w
ith the texts in force; they are not covered by these guidelines.
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4.1 - Stage 1: determination of footbridge class
Footbridge class m
akes it possible to determine the level of traffic it can bear:
Class
IV:
seldom
used footbridge,
built to
link sparsely
populated areas
or to
ensure continuity of the pedestrian footpath in m
otorway or express lane areas.
Class III: footbridge for standard use, that m
ay occasionally be crossed by large groups of people but that w
ill never be loaded throughout its bearing area. C
lass II: urban footbridge linking up populated areas, subjected to heavy traffic and that may
occasionally be loaded throughout its bearing area. C
lass I: urban footbridge linking up high pedestrian density areas (for instance, nearby presence of a rail or underground station) or that is frequently used by dense crow
ds (dem
onstrations, tourists, etc.), subjected to very heavy traffic. It is for the O
wner to determ
ine the footbridge class as a function of the above information
and taking into account the possible changes in traffic level over time.
His choice m
ay also be influenced by other criteria that he decides to take into account. For
instance, a higher class may be selected to increase the vibration prevention level, in view
of high m
edia expectations. On the other hand, a low
er class may be accepted in order to lim
it construction costs or to ensure greater freedom
of architectural design, bearing in mind that
the risk related to selecting a lower class shall be lim
ited to the possibility that, occasionally, w
hen the structure is subjected to a load where traffic and intensity exceed current values,
some people m
ay feel uncomfortable.
Class
IV
footbridges are
considered not
to require
any calculation
to check
dynamic
behaviour. For very light footbridges, it seem
s advisable to select at least Class III to ensure a
minim
um
amount
of risk
control. Indeed,
a very
light footbridge
may
present high
accelerations without there necessarily being any resonance.
4.2 - Stage 2: Choice of comfort level by the Owner
4.2.1 - Definition of the comfort level
The O
wner determ
ines the comfort level to bestow
on the footbridge. M
axim
um
com
fort: A
ccelerations undergone by the structure are practically imperceptible to
the users. A
verage
com
fort: A
ccelerations undergone by the structure are merely perceptible to the
users. M
inim
um
com
fort: under loading configurations that seldom
occur, accelerations undergone by the structure are perceived by the users, but do not becom
e intolerable. It should be noted that the above inform
ation cannot form absolute criteria: the concept of
comfort is highly subjective and a particular acceleration level w
ill be experienced differently, depending on the individual. F
urthermore, these guidelines do not deal w
ith comfort in
premises either extensively or perm
anently occupied that some footbridges m
ay undergo, over and above their pedestrian function. C
hoice of comfort level is norm
ally influenced by the population using the footbridge and by its level of im
portance. It is possible to be more dem
anding on behalf of particularly sensitive users
(schoolchildren, elderly
or disabled
people), and
more
tolerant in
case of
short footbridges (short transit tim
es). In cases w
here the risk of resonance is considered negligible after calculating structure natural frequencies, com
fort level is automatically considered sufficient.
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4.2.2 - Acceleration ranges associated with comfort levels
The level of com
fort achieved is assessed through reference to the acceleration undergone by the structure, determ
ined through calculation, using different dynamic load cases. T
hus, it is not directly a question of the acceleration perceived by the users of the structure. G
iven the subjective nature of the comfort concept, it has been judged preferable to reason in
terms of ranges rather than thresholds. T
ables 2.1 and 2.2 define 4 value ranges, noted 1, 2, 3 and 4, for vertical and horizontal accelerations respectively. In ascending order, the first 3 correspond to the m
aximum
, mean and m
inimum
comfort levels described in the previous
paragraph. The 4th range corresponds to uncom
fortable acceleration levels that are not acceptable.
A
cceleration ranges 0
0.51
2.5
Range 1
Range 4
Range 3
Range 2
Max
Min
Mean
T
able 2.1: Acceleration ranges (in m
/s²) for vertical vibrations
Acceleration ranges
0 0.15
0.30.8
Range 1
Range 2
Range 3
Range 4
Max
Mean
Min
0.1
T
able 2.2: Acceleration ranges (in m
/s²) for horizontal vibrations
The acceleration is lim
ited in any case to 0.10 m/s² to avoid "lock-in" effect
4.3 - Stage 3: Determination of frequencies and of the need to perform
dynamic
load case calculations or not
For C
lass I to III footbridges, it is necessary to determine the natural vibration frequency of
the structure. These frequencies concern vibrations in each of 3 directions: vertical, transverse
horizontal and longitudinal horizontal. They are determ
ined for 2 mass assum
ptions: empty
footbridge and footbridge loaded throughout its bearing area, to the tune of one 700 N
pedestrian per square metre (70 kg/m
2). T
he ranges in which these frequencies are situated m
ake it possible to assess the risk of resonance entailed by pedestrian traffic and, as a function of this, the dynam
ic load cases to study in order to verify the com
fort criteria. © S
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4.3.1 - Frequency range classification
In both vertical and horizontal directions, there are four frequency ranges, corresponding to a decreasing risk of resonance:
àáâãäå: maxim
um risk of resonance.
àáâãäæ: m
edium risk of resonance.
àáâãäç: low risk of resonance for standard loading situations.
àáâãäè: negligible risk of resonance. T
able 2.3 defines the frequency ranges for vertical vibrations and for longitudinal horizontal vibrations. T
able 2.4 concerns transverse horizontal vibrations.
F
requency 0
1 1.7
2.12.6
5
Range 1
Range 3
Range 2
Range 4
T
able 2.3: Frequency ranges (H
z) of the vertical and longitudinal vibrations
F
requency 0
0.51.1
2.5
Range 1
Range 3
Range 2
Range 4
0.3 1.3
T
able 2.4: Frequency ranges (H
z) of the transverse horizontal vibrations
4.3.2 - Definition of the required dynamic calculations
Depending on footbridge class and on the ranges w
ithin which its natural frequencies are
situated, it is necessary to carry out dynamic structure calculations for all or part of a set of 3
load cases: C
ase 1: sparse and dense crowd
Case 2: very dense crow
d C
ase 3: complem
ent for an evenly distributed crowd (2nd harm
onic effect) T
able 2.5 clearly defines the calculations to be performed in each case.
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Case 1
C
ase 3
Nil
Nil
Case 3
C
ase1
C
ase 2
1 2
3
Natural frequency range
I II III
Load cases to select for acceleration checks
Very
dense
Dense
Sparse
Case N
o. 1: Sparse and dense crow
d Case N
o. 3: Crow
d complem
ent (2nd harmonic)
Case N
o. 2: Very dense crow
d
Case 2
Class
Traffic
T
able 2.5: Verifications – load case under consideration:
4.4 - Stage 4 if necessary: calculation with dynamic load cases.
If the previous stage concludes that dynamic calculations are needed, these calculations shall
enable: checking the comfort level criteria in paragraph II.2 required by the O
wner, under
working conditions, under the dynam
ic load cases as defined hereafter, traditional S
LS
and UL
S type checks, including the dynam
ic load cases.
4.4.1 - Dynamic load cases
The load cases defined hereafter have been set out to represent, in a sim
plified and practicable w
ay, the effects of fewer or m
ore pedestrians on the footbridge. They have been constructed
for each natural vibration mode, the frequency of w
hich has been identified within a range of
risk of resonance. Indications of the way these loads are to be taken into account and to be
incorporated into structural calculation software, and the w
ay the constructions are to be m
odelised are given in the next chapter.
This case is only to be considered for category III (sparse crow
d) and II (dense crowd)
footbridges. The density d of the pedestrian crow
d is to be considered according to the class of the footbridge:
Class
Density d of the crow
d III
0.5 pedestrians/m2
II 0.8 pedestrians/m
2
This crow
d is considered to be uniformly distributed over the total area of the footbridge S.
The num
ber of pedestrians involved is therefore: N =
S d.
The num
ber of equivalent pedestrians, in other words the num
ber of pedestrians who, being
all at the same frequency and in phase, w
ould produce the same effects as random
pedestrians, in frequency and in phase is: 10,8
(N
) 1/2 (see chapter 1). T
he load that is to be taken into account is modified by a m
inus factor w
hich makes
allowance for the fact that the risk of resonance in a footbridge becom
es less likely the further
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away from
the range 1.7 Hz – 2.1 H
z for vertical accelerations, and 0.5 Hz – 1.1 H
z for horizontal accelerations. T
his factor falls to 0 when the footbridge frequency is less than 1 H
z for the vertical action and 0.3 H
z for the horizontal action. In the same w
ay, beyond 2.6 Hz
for the vertical action and 1.3 Hz for the horizontal action, the factor cancels itself out. In this
case, however, the second harm
onic of pedestrian walking m
ust be examined.
0 1
1,7 2,1
1
2,6
0
Structure
freq.
0 0,35
0,51,1
1
1,3 S
tructure freq.
0
F
igure 2.3 : Factor
in the case of walking, for vertical and longitudinal vibrations on the left, and for lateral
vibrations on the right. T
he table below sum
marises the load éêëìíîïðëêð
to be applied for each direction of vibration, for any random
crowd, if one is interested in the vertical and longitudinal m
odes.
represents the critical damping ratio (no unit), and n the num
ber of pedestrians on the footbridge (d x S
).
Direction
Load per m
² V
ertical (v) d
(280N)
cos(fv t)
10.8 (
/n) 1/2
Longitudinal (l)
d (140N
) cos(2
fl t) 10.8
( /n) 1/2
T
ransversal (t) d
(35N)
cos(fv t)
10.8 (
/n) 1/2
T
he loads are to be applied to the whole of footbridge, and the sign of the vibration am
plitude m
ust, at any point, be selected to produce the maxim
um effect: the direction of application of
the load must therefore be the sam
e as the direction of the mode shape, and m
ust be inverted each tim
e the mode shape changes direction, w
hen passing through a node for example (see
chapter III for further details). C
omm
ent 1: in order to obtain these values, the number of equivalent pedestrians is calculated
using the formula 10.8
( n) 1/2, then divided by the loaded area S
, which is replaced by n/d
(reminder n =
S d), w
hich gives d 10.8
( / n) 1/2, to be m
ultiplied by the individual action of these equivalent pedestrians (F
0 cos(t)) and by the m
inus factor C
omm
ent 2: It is very obvious that these load cases are not to be applied simultaneously. T
he vertical load case is applied for each vertical m
ode at risk, and the longitudinal load case for each longitudinal m
ode at risk, adjusting on each occasion the frequency of the load to the natural frequency concerned. C
omm
ent 3: The load cases above do not show
the static part of the action of pedestrians, G0 .
This com
ponent has no influence on acceleration; however, it shall be borne in m
ind that the m
ass of each of the pedestrians must be incorporated w
ithin the mass of the footbridge.
Com
ment 4: T
hese loads are to be applied until the maxim
um acceleration of the resonance is
obtained. Rem
ember that the num
ber of equivalent pedestrians was constructed so as to
compare real pedestrians w
ith fewer fictitious pedestrians having perfect resonance. S
ee chapter 3 for further details.
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This load case is only to be taken into account for class I footbridges.
The pedestrian crow
d density to be considered is set at 1 pedestrian per m2. T
his crowd is
considered to be uniformly distributed over an area S as previously defined.
It is considered that the pedestrians are all at the same frequency and have random
phases. In this case, the num
ber of pedestrians all in phase equivalent to the number of pedestrians in
random phases (n) is 1.85
n (see chapter 1).
The second m
inus factor, , because of the uncertainty of the coincidence betw
een the frequency of stresses created by the crow
d and the natural frequency of the construction, is defined by figure 2.3 according to the natural frequency of the m
ode under consideration, for vertical and longitudinal vibrations on the one hand, and transversal on the other. T
he following table sum
marises the load to be applied per unit of area for each vibration
direction. The sam
e comm
ents apply as those for the previous paragraph:
Direction
Load per m
2
Vertical (v)
1.0 (280N
) cos(
fv t) 1.85 (1/n) 1/2
Longitudinal (l)
1.0 (140N
) cos(
fv t) 1.85 (1/n) 1/2
Transversal (t)
1.0 (35N
) cos(
fv t) 1.85 (1/n) 1/2
This case is sim
ilar to cases 1 and 2, but considers the second harmonic of the stresses caused
by pedestrians walking, located, on average, at double the frequency of the first harm
onic. It is only to be taken into account for footbridges of categories I and II. T
he density of the pedestrian crowd to be considered is 0.8 pedestrians per m
2 for category II, and 1.0 for category I. T
his crowd is considered to be uniform
ly distributed. The individual force exerted by a
pedestrian is reduced to 70N vertically, 7N
transversally and 35N longitudinally.
For category II footbridges, allow
ance is made for the random
character of the frequencies and of the pedestrian phases, as for load case N
o. 1. F
or category I footbridges, allowance is m
ade for the random character of the pedestrian
phases only, as for load case No. 2.
The second m
inus factor, , because of the uncertainty of the coincidence betw
een the frequency of stresses created by the crow
d and the natural frequency of the construction, is given by figure 2.4 according to the natural frequency of the m
ode under consideration, for vertical and longitudinal vibrations on the one hand, and transversal on the other.
0 2,6
3,4 4,2
1
5 S
tructure freq.
0 1,3
1,72,1
1
2,5 S
tructure freq.
F
igure 2.4: Factor
for the vertical vibrations on the left and the lateral vibrations on the right
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4.4.2 - Damping of the construction
The dynam
ic calculations are made, taking into account the follow
ing structural damping:
Type
Critical dam
ping ratio R
einforced concrete 1.3%
P
re-stressed concrete 1%
M
ixed 0.6%
S
teel 0.4%
T
imber
1%
Table 2.6: C
ritical damping ratio to be taken into account
In the case of different constructions combining several m
aterials, the critical damping ratio to
be taken into account may be taken as the average of the ratios of critical dam
ping of the various m
aterials weighted by their respective contribution in the overall rigidity in the m
ode under consideration:
mm
ateria
l
im
mm
ateria
l
im
mk k
,
,
im
ode in w
hich km
,i is the contribution of material m
to the overall rigidity in
mode i.
In practice, the determination of k
m,i rigidity is difficult. F
or traditional footbridges of hardly varying section, the follow
ing formula approach can be used:
mm
ateria
l
m
mm
ateria
l
mmE
I EI
mode
in which E
Im is the contribution of the m
aterial m to the overall rigidity E
I
of the
section, in
comparison
to the
mechanical
centre of
that section.
(such that
) E
IE
Im
ma
terial
m
4.5 - Stage 5: Modification of the project or of the footbridge
If the above calculations do not provide sufficient proof, the project is to be re-started if it concerns
a new
footbridge,
or steps
to be
taken if
it concerns
an existing
footbridge (installation or not of dam
pers). This paragraph gives recom
mendations for reducing the
dynamic effects on footbridges. T
hese recomm
endations are given in decreasing importance.
4.5.1 - Modification of the natural frequencies
The m
odification of the natural frequencies is the most sensible w
ay of resolving vibration problem
s in
a construction.
How
ever, in
order to
modify
the natural
frequencies of
a construction
significantly, it
is very
often necessary
to carry
out extensive
structural m
odifications so as to increase the stiffness of the construction. M
ost of the time, a w
ay of increasing the natural frequencies is sought, so that the first mode,
and thus all the following m
odes, are outside the range of risk. In certain cases, when the
frequency of the first mode is low
, but within the range of risk, and that of the second m
ode is sufficiently high, it m
ay be advantageous to reduce the frequencies so as to bring the first m
ode below the range at risk, provided that the second m
ode remains above that range.
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How
ever, this is not very satisfactory. In addition, by reducing the rigidity of the construction, it becom
es more flexible and static deflection is increased.
Of the w
ays of increasing the natural frequencies of the footbridge, the following can be
quoted:
Let us consider, for exam
ple, the case of the vertical vibrations of a deck formed from
a steel box girder. If the depth of the box girder can be increased, its stiffness can thus be increased w
ithout increasing its mass. It is sufficient to retain the thicknesses of the top and bottom
flanges and to reduce the thicknesses of the w
ebs in proportion to the increase in their depth. B
ut, in a great number of cases, it is not possible to increase the depth of the box girder, or not
by enough, for functional (height of passage under the footbridge) or architectural reasons. If the thicknesses of the flanges and of the w
ebs of the box girder are increased, the inertia increases in proportion to the thickness, but the self-w
eight also increases, which reduces the
overall effect. In this case there is no solution for increasing the natural frequencies, other than by m
odifying the static diagram of the construction: creating recesses in the piers, adding
cable stays, etc. In the case of a deck w
ith a solid-core mixed steel-concrete beam
, an increase in the thickness of the bottom
steel flange is more effective in increasing the frequency of vibration. T
he self-w
eight does not increase as quickly, as it comprises a m
ajor part, due to the mass of the
concrete slab, which does not increase.
In the case of a lattice deck, the inertia varies according to the square of the depth, whereas
the section of the flanges (therefore the mass of the flanges) varies inversely to the depth. It is
therefore advantageous to increase the depth in order to increase the frequency of vibration. In the case of a concrete deck, an increase in the strength of the concrete enables its m
odulus, and thus the stiffness of the deck, to be increased, w
ithout increasing its mass, but in reduced
proportions, as the modulus only increases according to the cube root of the com
pressive strength. A
nother traditional way of increasing stiffness w
ithout increasing the mass consists
of replacing a rectangular section with an I-section. A
deck formed by a box girder w
ill thus have
a higher
frequency of
vibration than
a deck
of the
same
thickness form
ed from
rectangular beam
s. N
ormal concrete can also be replaced by lightw
eight concrete in order to reduce the mass
(with a slight reduction in stiffness) and thus increase the frequency of vibration.
In the case of a cable-stayed deck, an increase in the sections of the stays generally allows the
stiffness to be increased without increasing the m
ass by much. T
his solution is effective, but it is not econom
ical, as the quantity of stays has to be increased, with no offset. A
fan arrangem
ent of stays is stiffer than a harp arrangement. T
aller pylons also lead to an increase in the stiffness w
ithout increasing the mass, and thus to an increase in the frequency of
vibration. In the case of a suspended deck, the frequency of vibration increases according to the square root of the tension of the cables divided by the linear density of the cables and of the deck. T
here is therefore no advantage in simply increasing the section of the cables. T
heir deflection m
ust, in particular, be reduced. T
he vertical stiffness of a deck can also be increased by making the balustrades participate in
the stiffness.
The torsional vibrations of the deck cause it to m
ove vertically, away from
the longitudinal axis of the construction. T
he values of the torsion vibration frequencies of the deck must also
therefore be considered when the vertical vibrations felt by pedestrians are being studied. T
he
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frequency of torsional vibration is proportional to the square root of the stiffness in torsion and inversely proportional to the square root of the polar density of the deck. T
here is therefore every advantage in designing a deck that is rigid in torsion. T
here are several ways of increasing the frequency of the torsion vibrations of a deck. O
ne of these, of course, is to increase the torsional inertia. A
box girder deck thus has greater torsional inertia than a deck form
ed with lateral beam
s. The torsional inertia can be increased
yet more by increasing the cross-sectional area of the box girder. T
he addition, to a deck form
ed from filler joists supported by lateral beam
s, of a bottom horizontal lattice w
ind-brace connecting the bottom
flanges of the two beam
s, will also allow
the torsional stiffness to be increased, but by a sm
aller amount than a box girder.
In the case of a cable-stayed bridge with lateral suspension, the deck of w
hich is formed using
two lateral beam
s, anchoring the cable stays into the axial plane of the construction (on an axial pylon or to the top of an inverted Y
or inverted V pylon), and not into tw
o independent lateral pylons, w
ill allow the torsional frequency to be increased by a factor of close to 1.3
(Ref. [52]). T
his is what w
as done for the footbridge for the Palais de Justice in L
yon.
The frequency of horizontal vibration is proportional to the square root of the horizontal
stiffness and inversely proportional to the square root of the mass of the deck.
One obvious w
ay of increasing horizontal stiffness is to increase the width of the deck. B
ut at a cost that is no less obvious. A
t a given width, one w
ay of increasing horizontal stiffness consists of providing resistant elem
ents on the edges of the deck: for example, tw
o S-section lateral beam
s can be used, rather than four S
/2-section beams equally spaced under the deck.
In the case of cable-stayed or suspended footbridges that are very narrow in relation to their
span, lateral cables can be used to stiffen the construction. This is the case for the suspended
footbridge at Tours, over the C
her.
4.5.2 - Structural reduction of accelerations
If it is not possible to increase the frequencies sufficiently, or if the increase leads to a design that could m
ake the project unviable, or if the bridge is an old one, to which substantial
modification cannot be m
ade, an attempt m
ust be made to reduce accelerations (w
ithout directly affecting the dam
ping). To do this, the m
ass of the construction can be increased by using a "heavy" deck (asphalt, concrete etc.). T
his has a direct effect on accelerations (it should be rem
embered that they are inversely proportional to the m
ass). In addition, if this deck is connected to the structure, the frequencies w
ill not be reduced by too much, and the
damping provided by the deck m
ay make an appreciable contribution to the total dam
ping. A
nother w
ay of
reducing accelerations
is to
use m
aterials that
are naturally
damping.
How
ever, it shall be realised that, for the damping of these m
aterials to be mobilised, they
must play a part in the overall rigidity. Increased dam
ping may be obtained, for exam
ple, by the use of a lattice balustrade. In the case of "stress ribbon" footbridges, elastom
er panels may
be positioned between the pre-cast concrete slabs from
which the deck is form
ed in order to increase the dam
ping.
4.5.3 - Installation of dampers
As a last resort, if the previous solutions do not w
ork, damping system
s can be installed, w
hich will m
ost usually be tuned mass dam
pers (these are the easiest to install: to work
properly, viscous dampers often require the construction of com
plex devices to recreate major
differential m
ovement).
A
tuned m
ass dam
per consists
of a
mass
connected to
the
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construction using a spring, with a dam
per positioned in parallel. This device allow
s the vibrations in a construction to be reduced by a large am
ount in a given vibration mode, under
the action of a periodic excitation of a frequency close to the natural frequency of this vibration m
ode of the construction (see Annexe 3 §3.3).
This shall only be considered as a last resort, as, despite the apparently attractive character of
these solutions (substantial increase in damping at low
cost), there are disadvantages. If tuned m
ass dampers are used, w
hich is the most typical case:
as many dam
pers are needed as there are frequencies of risk. For com
plex footbridges, w
hich have many m
odes (bending, torsion, vertical, transversal, longitudinal modes,
etc.) of risk, it may be very onerous to im
plement;
the damper m
ust be set (within about 2-3%
) at a frequency of the construction that changes over tim
e (deferred phenomena) or according to the num
ber of pedestrians (m
odification of the mass). T
he reduction in effectiveness is appreciable; the addition of a dam
per degenerates, and thus doubles, the natural frequency under consideration:
this com
plicates the
overall dynam
ic behaviour,
and also
the m
easurement of the natural frequencies;
even though manufacturers claim
that dampers have a very long life-span, they do
need a minim
um level of routine m
aintenance: Ow
ners must be m
ade aware of this;
because of the added weight (approxim
ately 3 to 5% of the m
odal mass of the m
ode under consideration), this solution w
ill only work on an existing footbridge if it has
sufficient spare design capacity. On a proposed footbridge, the designer m
ay need to resize the construction; preferably,
3%
of guaranteed
damping
will
be achieved:
on very
lightweight
constructions (for which the ratio of the exciting force divided by the m
ass is high), it m
ay not be sufficient.
4.6 - Structural checks under dynamic loads
4.6.1 - SLS type checks specific to the dynamic behaviour
In addition to the usual checks to be made on service lim
it states, as defined by the relevant regulations, specific service com
binations for the dynamic loads of pedestrians should be built
up, by combining:
the effects (stresses, displacements, etc.) resulting from
one of the dynamic load cases
1 to 4, by applying the same m
ethodology as for the determination of com
fort. Thus,
when accelerations have to be calculated, the corresponding forces m
ust also be calculated to check w
hether they remain perm
issible. If the methodology does not
require the calculation of accelerations, this check need not be made;
the effects of static loading associated with the dynam
ic case under consideration, corresponding on the one hand to the perm
anent loads and on the other hand to the w
eight of the pedestrians set out identically, using a load of 700 N per pedestrian.
It is important to note that static and dynam
ic vibratory shapes are combined, and, therefore,
that precautions must be taken w
hen using calculation software.
4.6.2 - ULS type checks specific to the dynamic behaviour
As these consist of checking the strength of the construction, using levels of stresses that are
assumed to be close to the elastic lim
it of the materials (otherw
ise, obviously, there would be
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no strength problem), the dynam
ic calculations must be m
ade using the structural damping
given in table 2.7 below:
ñòóôõö÷øöùõúûüøúýöù
þÿ�óýù�� ô��ô�õúôô�
��� ö�úô�õúôô�
��ûôýù÷öûøô�øöùøûôúô
�óûô õúûôõõô�øöùøûôúô
��
Table 2.7
In the same w
ay as for service limit states, the ultim
ate limit states set out here should be
carried out in addition to the traditional UL
Ss im
posed by the relevant regulations. An
accidental type combination should be form
ed in order to simulate a case of vandalism
or an exceptionally large public dem
onstration on the footbridge. The check is to be m
ade as soon as a com
fort check for a natural frequency within a range at risk has been m
ade. The load case
to be considered is similar to load case N
o. 1 previously defined (see § 2.3.2), whatever the
category of the footbridge, with the follow
ing modifications:
the density of the crow
d is taken as 1 pedestrian per m²;
apart from the perm
anent loads on the footbridge, allowance m
ust be made for the
static load applied by the crowd, w
hich is taken as 700 N/m
²; the individual effects of pedestrians are com
bined directly, with no m
inus factor (n
equivalent = n and
= 1).
This load case is extrem
ely pessimistic, as it assum
es the perfect coordination of a whole
crowd of pedestrians. H
owever, it could occur under very exceptional conditions (rhythm
ic dem
onstrations, races, processions, etc.) or under lock-in situations. E
xperience has shown that there m
ay be a problem of strength in particular cases only, as it
must not be forgotten that such a load case represents an accidental U
LS
, for which the
permanent loads can be w
eighted differently from the fundam
ental UL
S, or can be m
ore tolerant on the lim
its of the materials.
Whatever, there m
ay, under certain particular circumstances, be a lack of strength if the
construction is optimised and everything precisely dim
ensioned. If such is the case, and if the consequences on sizing are m
ajor (which should not be the case, due to the sm
all difference betw
een sizing with and w
ithout dynamic loads), there m
ay be an advantage in looking to w
hich acceleration
the m
aximum
dynam
ic force
thus calculated
corresponds. If
this acceleration is too great, it could be given a reasonable value less than the acceleration due to gravity (betw
een 0.5 g and 1.0 g), beyond which it can be considered that w
alking becomes
impossible.
In the transverse direction, the strength of the construction is not very likely to be a problem,
as: T
he lateral action of a pedestrian is much w
eaker than the vertical action (35 N instead
of 280 N);
The consequences on sizing m
ay be negligible because of the sizing of the footbridge for w
ind, assuming that extrem
e wind situations and a large crow
d on the footbridge are not com
bined;
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The transverse acceleration beyond w
hich a pedestrian can no longer keep his balance is m
uch smaller than the vertical acceleration. It can, for exam
ple, be limited to a
value of between 0.1 g and 0.3 g.
It is therefore fairly unlikely that an exceptional pedestrian crowd could cause a problem
for the sizing of a pedestrian footbridge, or of its strength, but a check should first be m
ade. F
inally, it should be noted that, in these checks, the individual safety of the pedestrians in the event of accelerations that are too great because of these exceptional public dem
onstrations has not been covered. T
hat is why this guide recom
mends O
wners not to perm
it the holding of exceptional public dem
onstrations on pedestrian footbridges, but to take them into account in
the sizing of the footbridge, as it may be difficult to prevent them
from taking place
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5. Chapter 3: Practical design methods
5.1 - Practical design methods
For the dynam
ic design of continuous constructions and systems, tw
o major m
ethods can be used: direct integration design and m
odal design.
5.1.1 - Direct integration
This m
ethod consists of integrating the equations of the dynamics directly, w
ith an imposed
load. It is used very little in practice for vibrations of footbridges, as the phenomena affecting
them are resonance phenom
ena, which m
eans that, to predict them, the natural frequencies of
the constructions must be know
n. It is used more in earthquake-resistant design, in w
hich the excitation is know
n (imposed earthquake accelerogram
, for example)
The direct integration m
ethod, more expensive in analysis tim
e than modal calculations, m
ay, how
ever, turn out to be necessary in one of the following cases:
when the m
odal superposition method cannot be used w
ith a reduced number of
modes;
when the dam
ping is not proportional or when it is concentrated (viscous dam
pers for exam
ple); w
hen the problem is not linear.
The tw
o latter points may be encountered in certain footbridge vibration problem
s (use of finite elem
ents, dampers, construction w
ith non-linear behaviour). There do exist, how
ever, m
ethods that are close to modal m
ethods, and it is often advantageous, even so, to determine
"apparent frequencies" of normal vibrations in order to be able to set the dynam
ic loads producing the m
aximum
effect.
5.1.2 - Modal calculation
The m
odal method is alw
ays carried out in two stages, nam
ely the determination of the
natural vibration modes initially, and the calculation of the actual response (if necessary) on
the basis formed by these natural vibration m
odes. T
he power of this m
ethod is that the natural vibration modes are the natural vibration m
odes of
the construction.
They
therefore represent
the m
ost likely
vibration m
odes of
the construction. T
he second advantage is that they form a base, and that the actual solution is therefore a linear
combination of these m
odes. The num
ber of degrees of freedom is thus considerably reduced
and non-linked equations are obtained. F
inally, the third advantage is that, when a construction is excited at one of its natural
frequencies, and only in this case, a resonance phenomenon is produced. O
ne of the modes
then has a much larger response than the others. A
problem w
ith several degrees of freedom is
thus restricted to a problem w
ith one degree of freedom, thus easy to resolve. T
here is, therefore,
then only
one unknow
n in
the problem
, the
amplitude
of that
resonance. A
know
ledge of these resonance frequencies is crucial for the analysis of the dynamic problem
.
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5.2 - Dynamic calculation applied to footbridges
5.2.1 - Calculation of natural frequencies and natural vibration modes
In practice, the calculation of the natural vibration modes is m
ade by using software for
complex constructions (w
hich is achieved, in practice, as soon as there are several beams,
etc.). How
ever, a certain number of situations m
ay be determined analytically. It is also often
possible to have orders of magnitude of natural frequencies, even in the case of com
plex constructions.
For a sim
ply-supported beam of constant characteristics, an analytical calculation is possible
(see table 3.1).
Mode
Natural pulsation
Natural frequency
Mode shape
Sim
ple bending to n sags
S
EI
L
nn
2
22
S
EI
L
nf
n2
2
2
L
xn
xv
nsin
)(
T
ension-compression
to n sags S S
E
L nn
S S
E
L nf
n2
L
xn
xu
nsin
)(
T
orsion to n sags
r
nI J
G
L n
r
nI
GJ
L nf
2
L
xn
xn
sin)
(
Table 3.1: analytical m
odes for a simply-supported beam
S
is the linear density of the construction (including the perm
anent loads and the varying
loads), r
I is the m
oment of inertia of the construction, E
S the tension-compression rigidity,
EI the bending rigidity, G
J the torsional rigidity. In practice, for footbridges that are not very w
ide (low w
idth in comparison w
ith span) and rigid in torsion (closed profiles), the torsion m
odes are at high frequencies, as are the tension-com
pression modes. In this case, only an analysis of the bending m
odes is pertinent. If the sections are flexible in torsion (case of open profiles), the torsion m
odes must be taken into
account. If the sections are wide (slabs or double beam
s, for example), allow
ance must also be
made for the differential bending or transverse bending m
odes. (See figure 3.1). In such a
case, "plate and shell" type modelisations m
ay turn out to be necessary.
F
igure 3.1: Torsion m
ode (on the left) and differential bending mode (on the right) to be taken into account w
hen the profiles are open (on the left), or w
hen the sections are wide (on the right), or even in both cases.
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The
following
figures 3.2
illustrate the
bending and
torsion m
odes encountered
on a
footbridge with a w
idth one-fifth of the span.
Mode 1
: Ben
din
g 1
sag
Mode 2
: Ben
din
g 2
sags
M
ode 3
: Torsio
n 1
sag
Mode 4
: Ben
din
g 3
sags
M
ode 5
: Torsio
n 2
sags
Mode 6
: Ben
din
g 4
sags
M
ode 7
: Torsio
n 3
sags
Mode 8
: Ben
din
g 5
sags
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M
ode 9
: Torsio
n 4
sags
Figure 3.2 M
ode shape of a wide sim
ply-supported footbridge
For any beam
, continuous over supports, with different boundary conditions, the natural
pulsations are always in the form
n
n
L
EIS
2w
hich can be broken down into one factor
depending on the shape of the beam
n , one factor depending on the material
Eand one
factor depending on the section IS
, also known as turning radius.
The follow
ing table, taken from the C
EC
M 89 rules (R
ef. [8]), gives the values of the factor
n depending on the support conditions:
T
able 3.2: Influence of the boundary conditions on the natural vibration modes
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Before carrying out any digital calculation of the dynam
ics of a footbridge, whether for
simple cases using frequency and acceleration calculation form
ulae, or for the modelling of
more com
plex constructions on particular calculation programs, a certain num
ber of questions should first be asked on the calculation assum
ptions and the model being created.
The follow
ing list is intended to set out the main points that have to be w
atched carefully to do the m
odelling successfully. These points cover the dynam
ic calculation itself. Obviously,
all the usual static structure modelling assum
ptions and techniques should be carried out. This
concerns, for example, the relevance of the finite elem
ent grid, and thinking about the m
odelling of links in steel constructions, etc. T
he modeliser m
ust take particular care with the follow
ing aspects and questions before starting the digital m
odelling of the construction:
5.2
.1.3
.1
Stru
cture o
f the d
eck
Is the deck topping participating or not? If not, only its mass is relevant, if yes, it w
ill, in addition, provide rigidity w
hich will have a direct effect on frequencies, and dam
ping, which
will have a direct effect on the overall structural dam
ping. T
he links between the various elem
ents must be described and taken into account: sw
ivel joints, recesses, interm
ediate connections will have a direct effect on frequencies.
Welds or bolted connections lead to different dam
ping.
5.2
.1.3
.2
Supports
Particular attention shall be paid to supports. T
he dynamic behaviour of the structure of a
footbridge deck is often taken into account, but not that of its supports. In most typical cases,
this simplification is perm
issible, as the supports are often more rigid than the footbridge deck
and provide more dam
ping. But this is not alw
ays the case. A
t the base of the support, there is the foundation, on piles or a strip footing. This foundation
has certain flexibility linked to that of the ground. In general, it also provides a high level of dam
ping, due to the friction between the ground and the construction. In certain cases, either
its horizontal and/or its vertical components m
ay be taken into account.
F
igure 3.3 On strip footings or piles, a foundation has a certain rigidity k
sol T
hen there is the pier. If this is very high, its flexibility becomes im
portant and it may affect
the natural frequencies of the overall construction.
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F
igure 3.4: Very high pier: T
he rigidity of this type of support, held at its base and free at its top may be
evaluated as kpile =
3E
I/h3 in w
hich h is the height, I the inertia and E the elastic m
odulus.
Finally, the support fittings m
ust not be forgotten, nor their effect on the overall rigidity of the construction and on overall dam
ping. F
or a support fitting in reinforced elastomer, size a
a, the thickness of one layer of w
hich is e, w
ith a shear modulus G
and with n layers, the horizontal stiffness is:
ksupport fitting =
G a
2 / n e A
ll these intermediate elem
ents may m
odify the natural frequencies of the footbridge and their associated dam
ping. The theoretical m
odel and simplified diagram
may becom
e as follow
s:
mapp
Mm
fondatim
pile
kground
kpier
ksupport fitting
Kstructure
Ground
Pier and foundation
Support
fittingS
tructure K
, M
F
igure 3.5: "Sim
plified" diagram of the construction dow
n to the foundations In the case of continuous system
s, there is not necessarily a single stiffness for each of the above elem
ents. A com
plete model, often digital, w
ill then be necessary.
5.2
.1.3
.3
Makin
g a
llow
ance fo
r the m
ass
Certain dynam
ic calculation software rem
ains confused as to making allow
ance for mass. It is
an important factor that m
ust be properly dealt with by the softw
are by distinguishing it from
the static effects of self-weight. C
are must be taken for exam
ple, not to confuse rotation inertia, useful for the calculation of the m
oment of inertia, and torsion inertia, useful for the
calculation of rigidity in torsion. In addition, as the m
ass has a direct influence on the frequencies of the modes, the m
ass of the pedestrians, either standing still or w
alking on the footbridge, shall be taken into account. T
hus, for each mode, the frequency of the m
ode in question is located within a varying range
between tw
o extreme frequencies f1 and f2 , one calculated w
ith the mass of the construction
and of the pedestrians (f1 ) and the other with the m
ass of the construction only (f2 ). For very
lightweight footbridges, these tw
o frequencies may be som
e distance apart.
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They should, therefore, both be calculated. In the sam
e way, w
hen calculating accelerations, there m
ust be coherence between the load case carried out and the m
ass taken into account (and several calculations m
ade if necessary).
5.2
.1.3
.4
Makin
g a
llow
ance fo
r dam
pers
The dynam
ic calculation may be carried out w
ith an allowance for tuned m
ass dampers (in the
case of a recalculation of an existing footbridge that has to be upgraded by the addition of dam
pers, or in the case of a footbridge of a design that would not allow
the saving by using dam
pers to be made).
These dam
pers may be incorporated directly into the digital m
odel if the model allow
s them
to be modelised. A
s, in most cases, it is a question of adding a m
ass, spring, damper system
, design softw
are generally allows this to be done easily.
The analytical calculation m
ay be made m
ode by mode if the frequencies of the footbridge are
known, and also its m
odal masses and m
odal stiffness. In this case, it is possible to calculate the natural frequencies of the follow
ing system for each of the m
odes:
K footbridge
C footbridge
Mfootbridge
Kdam
per
Mdam
per
Cdam
per
F
igure 3.6: Schem
atic representation of a Construction – D
amper system
M
footbridge is the generalised mass of the m
ode of the footbridge under consideration, Kfootbridge
the generalised
stiffness of
the footbridge
mode
under consideration
and C
footbridge its
generalised damping. A
nnexe 3 gives the analytical solution of the dynamic am
plification in accordance w
ith the relationship between the exciting frequency and the initial frequency of
the footbridge. This graph reveals the tw
o natural vibration modes thus degenerated and their
associated amplifications (thus the dam
ping).
5.2
.1.3
.5
Particu
lar fea
tures
The follow
ing points, likely to modify the dynam
ic model of the footbridge, shall, finally, be
taken into account: existence of an initial constraint that influences the natural frequencies (pre-stressing in pre-stressed concrete, pre-tension in cable stays or ties, thrust from
arches, etc.); non-linearity due to the cables or ties, m
aterials, large displacements, pendulum
effects, etc.; distinction betw
een local modes (for exam
ple, vibration of a slab unit on a footbridge, of no danger to a crow
d of pedestrians) and global modes;
�� âá ��characteristics of the m
aterials to be taken into account (concrete, stiffness of the ground, etc.).
5.2.2 - Practical calculation of the loading response
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Rem
ember the equation governing the behaviour of the m
ode i with m
odal damping (see
annexe 1):
i
i
t
ii
ii
ii
im
tF
tp
tq
tq
tq
][]
[)
()
()
(2
)(
2
Each of these responses q
i is calculated separately from the others, then the global response is
deducted by recomposing the m
odes. If the function t
F is harm
onic (t
Ft
Fsin
0),
at the frequency of one of the modes (m
ode j for example), there is then resonance of that
mode. T
he response qj of m
ode j is much greater than the others and the global response, after
a transitory period, is close to:
tq
tq
tX
jj
ni
ii
1
The am
plitude of the dynamic response q
j (t), á��ä���ä��áâ������� ä����� is obtained by m
ultiplying the static response (that obtained as if the load was constant and equal to
, in
other words that:
0F
j
jt
static
jj
m
Fq
02
) by the dynam
ic amplification factor
j2 1
.
The response obtained is therefore:
Displacem
ent: response
y transitor
cos
1
2 10
2t
m
Ft
qj
j
jt
jj
j
Acceleration:
responsey
transitor
cos
2 10
tm
Ft
qj
j
jt
jj
It can be seen that the displacement and the acceleration are out of phase by a quarter of the
natural period in comparison w
ith the excitation. The speed is in phase w
ith the excitation. T
he global displacement response can be w
ritten:
jj
static
tq
Xt
X
and in acceleration:
jj
tq
tX
Rem
ember that the various calculations set out in this guide are based on the notion of
number of equivalent pedestrians, w
hich is the number of fictitious pedestrians that are all in
phase, at the natural frequency of the footbridge, regularly spaced, whose action is in the sam
e direction as the m
ode shape at each point, and, especially, at the maxim
um resonance, in other
words in a steady state. A
s a result, the transitory response is not interesting and only the
amplitude of the perm
anent response j
j
j
t
jm
F0
2 1 is interesting, w
ith j
j
t
jM
m
As specified in the previous chapter, the load m
ust be such that the amplitude of the force is
of the same sign as the m
ode shape.
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In the case of modes w
ith several sags in the longitudinal or transverse direction, this means
that the amplitude of the force m
ust have the shape shown in figure 3.7:
M
ode shape
Figure 3.7: S
ign of the amplitude of the load in the case of a m
ode with several sags
In the case of torsion modes w
ith several sags, the amplitude of the force m
ust be of the shape show
n in figure 3.8
F
igure 3.8: Sign of the am
plitude of the load in the case of a torsion mode w
ith several sags. Noted in red are the
zones in which the am
plitude of the load is positive, and in blue the zones in which the am
plitude of the load is negative.
In the general case, dynamic calculation softw
are must be used to evaluate the natural
frequencies, and also the accelerations obtained with the various loads. S
everal programs are
available that take the dynamic calculation into account. A
lthough most of them
allow a
calculation of the natural vibration modes and natural frequencies, there are few
er that allow
the temporal calculation of the accelerations under varying loads. T
his is because these program
s, that make calculations dynam
ically, are, in practice, intended especially for seismic
calculations, and because, in such calculations, it is, in general, enough to calculate the natural
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vibration modes, and the application of standardised response spectra s that avoids the need
for any real dynamic calculation.
In the
case of
pedestrian footbridges,
these program
s aim
ed at
seismic
design are
not necessarily relevant. H
owever, the results that they provide can be used to calculate the
maxim
um accelerations, in the w
ay explained below:
5.2
.2.3
.1
Pro
gra
ms ca
lcula
ting o
nly th
e natu
ral vib
ratio
n m
od
es an
d m
od
al ch
ara
cteristics.
With this type of program
, the modal inform
ation should be recovered for each natural vibration m
ode within the range at risk: vibratory shape at any point of the m
ode, natural frequency, m
odal damping (if possible), generalised m
ass. Care m
ust be taken with the notion
of generalised mass, w
hich is different from m
odal mass, often given by the program
s. The
generalised mass is the value
jj
t
jM
m w
hich is used in the modal equation of the
dynamics and depends on the standardisation that has been selected (its final result is,
however, independent), w
hile the modal m
ass, the expression of which is
jj
t
j
t
M M2
, is a
value used typically in seismic calculations to evaluate the num
ber of modes to be taken into
account to represent properly the response to a seismic stress. In practice, it enables the
natural vibration modes of low
modal m
ass in relation to the mass of the construction to be
eliminated. B
ut the modal m
ass has no physical reality in relation to the generalised mass of
the m
ode that
is a
proper representation
of the
vibrating m
ass in
the m
ode under
consideration, in the light of the selected standard. In practice, for the m
ode j under consideration, the program m
ust be asked to provide, for each grid i of the m
odel, the values of the displacements of the m
ode shape Vij , together w
ith the
value of
the natural
frequency. T
he generalised
mass
can then
be form
ed by
in which M
i
ji
ij
VM
m2
i is the mass of the grid i.
The m
odal load is then formed, the projection of the load onto the m
ode, by writing
in which F
i
iji
jV
Ff
i represents the value of the force on the grid i. It is written
tF
Fi
icos
. Because of the loading m
ode of the beams,
iF
has the same sign as V
ij . Its
value is given in chapter 2. The second m
ember of the m
odal equation is then written
j j
m f.
The am
plitude of the acceleration response, at the resonance, is then written sim
ply, at the
position of grid k: kj
i
iji
i
iji
j
kj
j j
j
VV
M
VF
Vm f
22 1
2 1 (value independent of the standard selected
for the modes. Indeed, if every V
ij is multiplied by the sam
e constant, the result does not change)
5.2
.2.3
.2
Pro
gra
ms en
ablin
g tem
pora
l dyn
am
ic calcu
latio
ns.
With program
s enabling temporal dynam
ic calculations, it is not necessary to use the tricks set out above. O
ne can either define a load that varies over time in the form
t
qcos
, having previously determ
ined exactly the natural frequencies, band looking at the dynamic response at the end
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of a sufficiently long length of time (in practice, the tim
e at the end of which the am
plitude of the accelerations becom
es just about constant). On the other hand, one m
ust ensure that the frequency of the load is exactly equal to the natural frequency of the construction, w
hich is a risk for the calculation. T
he second method, easier to im
plement and carrying less risk, consists of selecting for the
dynamic calculation only the m
ode under consideration (it must be possible to deactivate the
other modes), and then apply to it the sam
e load as previously, but assumed to be constant
(therefore q
alone, w
ithout the
temporal
section) w
hich is
easier to
define. T
hen the
displacement obtained at the end of a sufficiently long length of tim
e is determined (in other
words w
hen the vibrations have smoothed out) and m
ultiplied by j
j2 1
2 in order to give the
acceleration at the resonance of the point under consideration.
For a footbridge w
ith constant inertia on two supports, the characteristic values of the
footbridge can be easily calculated, as shown in table 3.3.
T
ype of value L
iteral expression
Natural pulsations
S
EI
L
nn
2
22
Natural frequencies
S
EI
L
nf
n2
2
2
Maxim
um deflection
5 4
4m
ax
41
2 1
EI F
L
nv
n
Maxim
um m
oment
3
2
2m
ax
41
2 1F
L
nM
n
Maxim
um shear force
2m
ax
41
2 1F
L
nV
n
Maxim
um acceleration
S Fn
42 1
onA
cceleratim
ax
Table 3.3: m
odal values for a simply-supported beam
subjected to a load on a footbridge depending on the m
ode with n sags. F
represents the amplitude of the force per unit length
.
It can be noted that, if the exciting frequency is not exactly equal to the natural frequency
n , the term
n2 1
is replaced by the dynamic am
plification 2
22
24
)1(
1)
(A
(with
=/
n ).
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6. Chapter 4: Design and works specification, testing
6.1 - Examples of item
s for a footbridge dynamic design specification
The specification shall state that the structural design of the footbridge shall be carried out in
accordance w
ith the
recomm
endations in
the S
ET
RA
/
AF
GC
guide
on the
dynamic
behaviour of pedestrian footbridges. T
he Ow
ner shall, however, specify:
The category of the footbridge (I to IV
) T
he acceptable comfort level (m
aximum
, medium
, minim
um)
6.2 - Examples
of item
s to
be inserted
in the
particular works
technical
s pecification for a footbridge
It is recomm
ended that this guide be annexed to the technical "works" specification of a
footbridge and to state that the working design m
ust follow the recom
mendations in this
guide. The follow
ing shall, therefore, be stated once again: T
he category of the footbridge (I to IV)
The acceptable com
fort level (maxim
um, m
edium, m
inimum
) In addition, if studies have show
n that the dynamic behaviour can only be ensured by the use
of dampers, or that there are probably, but not guaranteed, large dam
ping assumptions, it w
ill be necessary to have dynam
ic tests carried out (see next paragraph) so as to validate these assum
ptions or dampers.
6.3 - Dynamic tests or tests on footbridges
The
carrying out
of dynam
ic tests
on a
new
footbridge, or
the testing
of an
existing footbridge, is a m
ajor, very expensive, operation, which should only be considered in very
particular cases, and which should only be carried out by contractors w
ho are skilled in this field. S
uch an operation should be considered for a new footbridge w
hen the design work has not
succeeded in showing that the construction fully satisfies the previous criteria, w
hen the design has im
posed the use of dampers that have to be validated, or w
hen the dynamic
behaviour is assured using damping assum
ptions that are greater than those recomm
ended in this guide, but probably required. If the dynam
ic behaviour can be assured without dam
pers, and in accordance w
ith the recomm
endations in this guide, tests will not necessarily have to
be carried out. In the case w
here dampers are specified right from
the start, it is strongly recomm
ended to m
easure the frequencies and damping of the m
odes of the completed footbridge before the
final sizing of the dampers, in order to adjust such sizing if applicable. T
he construction program
me w
ill have to make allow
ance for this. O
n an existing footbridge, a series of tests can be programm
ed if the construction has had a num
ber of vibration problems on occasions during its existence, and an im
provement is
needed.
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The follow
ing paragraphs give recomm
endations for the successful completion of such a
series of tests. Depending on the size of the construction, the extent of the phenom
ena encountered and also on w
hat is required to be measured, these recom
mendations can be
adapted. A
s far as equipment is concerned, the follow
ing systems shall be provided:
installation of dynamic sensors, (accelerom
eters or motion m
easurers), together with a
data acquisition unit and a recording system. T
here must be a sufficient num
ber of accelerom
eters, positioned at the sags of the theoretical modes.
installation of a visual monitoring system
of the footbridge so as to be able to connect the dynam
ic measurem
ents to the image of the footbridge. T
his system w
ill have to be synchronised w
ith the data acquisition unit. P
ossib
le use of a mechanical exciter (of the out-of-balance type for exam
ple) so as to be able to characterise precisely the natural vibration m
odes, natural frequencies, m
odal masses and m
odal damping of the footbridge. T
his system m
ust allow reliable
and controlled, horizontal and/or vertical excitation (depending on the footbridge), at program
mable frequencies covering the range 0.5 H
z / 3 Hz m
inimum
. T
he tests shall proceed in the following w
ay:
6.3.1 - Stage 1: Characterisation of the natural vibration modes
This stage shall be carried out using the dynam
ic exciter if high accuracy is required of the values m
easured. In the absence of a dynamic exciter, it is possible to m
easure the vibrations of the footbridge, in norm
al service, and, using suitable computer tools, to infer from
them the
natural frequencies and also the natural vibratory shapes. In this case, certain modes m
ay be forgotten. In addition, it is possible to m
easure the damping using the logarithm
ic decrement
method, but not the generalised m
asses, unless a dynamic exciter is used. D
ynamic analysis
remains possible, but less accurate.
6.3.2 - Stage 2: Crowd tests.
Crow
d tests should be programm
ed, as they enable the dynamic behaviour and the validity of
the dampers to be validated. T
he number of pedestrians is to be determ
ined according to the size of the footbridge, its theoretical characteristics, and also to the com
plexity of the m
anagement of such a crow
d. Several dozen pedestrians w
ill be necessary for a representative test. T
he tests may include the follow
ing elements:
Random
walking, in circles on the footbridge, or continuously, or even from
one end to the other. M
arching in step, at the natural frequencies of the footbridge. M
arching in step, coming to a sudden halt, in order to m
easure the damping of the
footbridge. R
unning, jumping, kneeling, in order to test the footbridge under extrem
e stresses. T
he tests shall be prepared so as to define the apparent alarm levels, beyond w
hich the tests m
ust be stopped. If there are dam
pers, the tests shall be carried out with the dam
pers working and then, as a
second stage, locked into position so as to determine their actual effectiveness.
The tests w
ill have been passed if the vibrations felt during the passage of a desynchronised crow
d result in vibrations acceptable to the Ow
ner, tolerable in the case of a synchronised crow
d, and not too intolerable in the case of exceptional loading tests (jumping, running,
kneeling, vandalism, etc.). In all cases, the decision w
ill lie with the O
wner, w
ho must judge
the comfort level of his footbridge according to his requirem
ents.
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1. Appendix 1: Reminders of the dynam
ics of constructions
1.1 - A simple oscillator
1.1.1 - Introduction
A sim
ple oscillator is the basic element of the m
echanics of vibrations. Its study enables the characteristic phenom
ena of dynamic analysis to be understood. In addition, it can be seen
that the dynamic analysis of constructions reduces them
to simple oscillators.
A sim
ple oscillator comprises a m
ass m, connected to a fixed point via a linear spring of
stiffness k, and a linear viscosity damper c (figure 1.1).
This oscillator is fully determ
ined when its position x(t) is know
n; a single parameter being
enough to describe it, it is said to have 1 degree of freedom(1 D
OF
): it is also called an oscillator at 1 D
OF
. The m
ass may be subjected to a dynam
ic excitation F(t).
We w
ill still assume that the oscillator carries out sm
all movem
ents around its balance configuration: x(t) designates its position around this balance configuration. A
n exhaustive study is carried out on this oscillator: free dam
ped / non-damped oscillation,
forced damped / non-dam
ped oscillation. T
he "non-damped" case is that w
ith a damping value of zero. It is then said that the system
is conservative. T
he "free" case is that of zero excitation. T
he resonance phenomenon can also be show
n: the natural pulsation of a system w
ill be defined. F
inally, a brief description of damping w
ill be given.
1.1.2 - Formation of the equations
Figure 1.1: S
imple oscillator
T
he mass m
(figure 1.1) is subjected to the return force of the spring
k x: if the stiffness k is negative, the spring will travel
in the direction of movem
ent: this will lead to buckling,
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the viscous dissipation force x
c: if the viscosity c is negative, the construction is
unable to dissipate energy: this will lead to dynam
ic instability, the external force F
(t). T
he fundamental relationship of the dynam
ics is therefore written:
xk
xc
Fx
m (E
q. 1.1)
which is rew
ritten in the form:
Fx
kx
cx
m (E
q. 1.2)
In order to solve such an equation, the initial conditions must, obviously, be available:
0 0
)0(
)0(
xx
xx
There is, therefore, a differential equation w
ith constant coefficients and one entry, the force F
(t). From
a practical point of view, the system
being linear, the study of this oscillator can be separated into tw
o rates: the natural or free rate: the system
is excited only by non-zero initial conditions: the free vibrations of the system
are then obtained, the forced rate: only the force F
(t) is non-zero; this force may be sinusoidal, periodic,
random or com
monplace,
It is typical and practical to divide the equation (1.2) by m. W
e thus get:
00
2f
m k: natural pulsation of the oscillator (rad/s);
f0 being the natural
frequency of the oscillator (in Hz).
mk c
2 : critical dam
ping ratio (without dim
ension).
The equation (1.2) becom
es:
m Fx
xx
200
2 (E
q. 1.3)
1.1.3 - Free oscillation
The system
is set to vibrate only by non-zero initial conditions:
0 0
00
)0(
)0(
02
xx
xx
xx
x
The solution of this differential equation w
ith constant coefficients is of the type: x(t) =
A e
(rt)
with r checking the characteristic equation:
r2 +
2
0 r +
0 2 =
0
in which the reduced discrim
inant equals: =
0 2 (
2 1)
It is assumed that
is positive (dissipative system) or zero. T
hus, different rates are obtained depending on w
hether is less than, equal to, or greater than 1 (figure 1.2). W
e will study
here only the case that is typical in practice: is strictly sm
aller than 1.
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F
igure 1.2: Free oscillations
The solution is therefore:
x(t) = A
sin(0 t) +
B cos(
0 t) A
and B are tw
o constants to be determined w
ith the help of the initial conditions:
00 0
)0(
)0(
Ax
x
Bx
x
It should be noted that the solution can also be written in the form
: x(t) =
A co
s(0 t +
)
The integration constants are now
A and the phase
, which are also obtained w
ith the help of the initial conditions.
This is the m
ost interesting case in practice. The solutions of the characteristic equation are:
r1,2 =
0
i 0
21
The general solution of the m
ovement equation is w
ritten according to one of the three follow
ing forms:
)1
cos(
)1
sin()
1cos(
)(
20
1
20
22
01
)1
(
2
)1
(
1
0
00
20
02
00
te
C
te
Bt
eB
eA
eA
tx
t
tt
ti
ti
The pulsation
a is known as the natural pulsation of the dam
ped system or pseudo-pulsation:
a =
0 2
1
The constants are obviously determ
ined thanks to the initial conditions. C
omm
ent: It should be noted that, if the damping
was negative, there w
ould be a solution that w
ould become divergent: the oscillations w
ould have an amplitude that w
ould increase exponentially. T
his problem is encountered in w
ind-excited constructions: the wind can
induce a force proportional to its speed and in the same direction; it is then possible to obtain
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negative damping, as soon as this proportionality factor becom
es greater than the damping of
the construction.
1.1.4 - Forced vibration
The objective is to resolve the equation (1.3). V
arious cases can be examined for F
(t): F m
ay be harm
onic, periodic, random or com
monplace.
From
a mathem
atical point of view, the general solution of (1.3) is the sum
of the general solution of the equation w
ithout a second mem
ber (transitory movem
ent or rate) and of a particular solution of the full equation (perm
anent movem
ent or rate) xS
P (t). It can therefore be w
ritten in the following form
:
tx
te
At
xS
Pa
tcos
0
(E
q. 1.4)
The constants are determ
ined with the initial conditions.
The transitory rate is associated w
ith the free movem
ent: its influence quickly becomes
negligible (in fact, by the end of a few norm
al periods it has already been cancelled out). D
uring the study with harm
onic or periodic excitation, it is not taken into account.
With the help of the harm
onic excitation, the notion of transfer function will be introduced.
This notion is fundam
ental to the mechanics of the vibrations.
A harm
onic excitation is a sinusoidal function: F
(t) = F
0 cos(t) =
(F
0 ei
t)
A particularly interesting solution is in the form
: x(t) =
A co
s(0 t +
)
In order to carry out the calculations more sim
ply and more quickly, com
plex numbers are
used. A solution is therefore sought in the form
: )
()
()
(~
ti
eX
tx
(Eq. 1.5)
with X
()
R+.
The solution sought is therefore the actual part of
tx ~
: t
xt
x~
.
(1.5) is injected into the equation (1.3). After sim
plification by the factor ei
t, we get:
m FX
Xi
Xe
i0
200
2))
()
(2
)(
(
The transfer function is then defined, also know
n as the frequency response, Hx,F (
), between
the excitation and the response (here in movem
ent) by the ratio:
200
20
,2
1
/ )(
)(
im
F
eX
H
i
Fx
(Eq. 1.6)
This function characterises the dynam
ics of the system under study. If
=
/0 is placed,
the dynamic am
plification is defined by:
ix
eX
kF
eX
HA
static
ii
Fx
21
1)
(
/ )(
)(
)(
20
,20
(E
q. 1.7)
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xsta
tic is the value of the displacement that w
ould be obtained if a static force F0 w
as exerted on the oscillator, w
ith a rigidity of k: it is important to note that this value is taken by X
() at
zero frequency. Thus the intercept ordinate of the function H
x,F is determined w
ith a static calculation: this value is often called static flexibility. T
his function A(
) shows that, at a
given F0 , the m
aximum
response of the oscillator will depend effectively on the frequency
(figure 1.3): for certain frequencies, this response is greater (or even much greater) than the
static response.
Figure 1.3: resonance phenom
enon
The com
plex solution is therefore:
02
20
0
2)
(~
i
e
m Ft
x
ti
A study of A
() show
s that, if is betw
een 0 and 1/2
, the resonance phenomenon is
obtained (figure 1.3): A(
) allows a m
aximum
for R =
2
21
which equals:
21
1
2 1)
(R
A
It can be seen that, the lower the dam
ping (i.e. the more
tends towards 0), the higher the
amplification to the level of resonance, w
hich tends towards 2 1
: thus, for =
0.5 %, it can
be seen from figure 1.3 an am
plification in the order of 100.
Com
ment 1: U
sing this oscillator at 1 DO
F, illustrates im
mediately the difference betw
een "static" and "dynam
ic". In static, only the amplitude of the excitation affects the am
plitude of the response; in dynam
ic, frequency also has to be taken into account: exciting a construction to its resonance m
ay produce (especially if the construction does not have much dam
ping) m
ajor displacements, and thus a high level of stresses in the construction. T
his is why it is so
important to determ
ine the resonance frequencies of a construction as soon as there is dynam
ic excitation. C
omm
ent 2: It can be seen that resonance is reached at a pulsation value that is less than the natural
pulsation of
the system
0 .
How
ever, for
systems
without
much
damping,
the displacem
ent resonance pulsation is combined w
ith the natural pulsation of the system. A
s far as the phase is concerned, resonance is reached w
hen it equals /2.
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Com
ment 3: in the above, no allow
ance has been made for initial or transitory conditions, as,
as has
been com
mented
previously, this
component
of the
solution quickly
becomes
negligible, as can be seen in figure 1.4. How
ever, it is clear that, the higher the damping, the
quicker the transitory component becom
es negligible.
response transitory
response transitory
permanent
Figure 1.4: transitory and dam
ping
We return to the previous case. T
he load function can be developed as a Fourier series:
0
)2
exp()
(p
nt
Ti
Ct
F
Thus, by determ
ining the solution xp (t) for each harm
onic p (previous paragraph), we get the
solution by superposing (summ
ing) the xp (t).
Taking the F
ourier transform of the equation (1.3), w
e get the frequential equation:
m
FH
i mF
XF
x
)(
)(
2 /)(
)(
,
02
20
in which X
() and F
() are the F
ourier transforms of x(t) and of F
(t). By inverse F
ourier transform
, we then infer x(t), w
hich is expressed using Duham
el's integral: t
tt
eF
mt
x0
20
)(
20
d))
(1
sin()
(1 1
)(
(E
q. 1.8)
In practice, we carry out a digital integration of the differential equation of the m
ovement
(1.3), using a computer program
.
We assum
e that the excitation F is m
odelised by a stochastic process. The response x of the
system at 1 D
OF
is also a stochastic process: it can, therefore, only be understood by the use of values characterising this process (average, standard deviation, autocorrelation function, spectral pow
er density, etc.). H
owever, the study of random
vibrations exceeds the objective of this annexe, and the reader is referred to the reference w
orks.
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A m
ovement u
(t) is now im
posed on the support: F(t) and the initial conditions are therefore
assumed to be zero.
NB
: x(t) and u(t) designate displacements w
ithin an absolute reference. W
e are only interested in the harmonic m
ovement of the support:
u(t) = U
ei
t
We are looking once again for a solution in the form
: X(
) ei (
t
). By injecting these tw
o expressions into the equation (1.3):
uu
xx
x20
020
02
2
We can infer the transfer function betw
een the base displacement and the m
ass displacement
m:
i
i
U
eX
H
i
ux
21
21
)(
)(
2,
(Eq. 1.9)
The m
odulus of this function is plotted in figure 1.5.
F
igure 1.5: FR
F in m
ovement, com
pared with a base excitation
It is interesting to observe the existence of a fixed point:
, H
x,u (2
) = 1
Because of this, during a base excitation a com
promise m
ust be found in order to optimise the
damping: if high
limit displacem
ents up to 2
0 , after this value low values of
are preferred. T
he base excitation is a much m
ore comm
on excitation than could be expected at first glance: indeed, it is seism
ic excitation (therefore rare) that first comes to m
ind as a priority to illustrate a base excitation. H
owever, all system
s are excited by their support. This is the
reason why w
e try and "isolate" them (elastic bases, S
andow shock cords).
If a system is excited, it transm
its a signal (displacement) to its support: this then produces a
wave that is transm
itted (ground, wall, air, etc.) and causes the displacem
ent of the support of another system
. If the base excitation is periodic or com
monplace, an identical study is carried out to that for
the forced excitation.
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1.1.5 - Damping
We have seen it: the resonance phenom
enon causes displacements in constructions and, thus,
stresses (very) much higher than w
ould be obtained by a static calculation. In addition, the m
aximum
amplification is directly linked to the dam
ping. It is therefore essential for this param
eter to be estimated properly in order to achieve correct dynam
ic sizing. T
he most com
mon sources of dam
ping are: the internal dam
ping linked with the m
aterial itself; its value is linked to temperature
and the frequency of excitation dam
ping by friction (Coulom
b): it is linked to the joints between elem
ents of the construction; it is induced by the relative displacem
ent of two parts in contact.
Experim
entally, it can be seen that the critical damping ratio does not generally depend on
frequency: the damping is then called structural dam
ping: n =
Traditionally, this dam
ping is modelised by viscous dam
ping: this is interpreted as a force that opposes the speed of the construction. It is interesting to know
that, sometim
es, there exist exciting forces that "agree" w
ith the speed (certain modellings of pedestrian behaviour): this
generates "negative damping", w
hich leads to oscillations of the construction that become
greater and greater. Instability then exists (see figure 1.6), which can lead to the collapse of
the construction.
response F
igure 1.6: Divergence induced by "negative dam
ping"
�W
hen an actual system is studied (
is apparently less than 1), modelised by a sim
ple oscillator, it becom
es essential to identify its characteristic mechanical param
eters, 0 and
. T
his can be done by means of a free vibration test: initial conditions are im
posed on the system
and its temporal response m
easured. The follow
ing are then measured (see figure 1.7):
the period Ta betw
een two successive m
axima located at t1 and t2 (T
a = t2
t1 ): we
then get:
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0 2
1=
(2 )/(T
a ),
the value x(t1 ) and x(t2 ) of two successive m
axima: their logarithm
ic decrement
can then be inferred:
)1 2
exp()
exp()
(
)(
)(
exp2
02 1
aT
tx
tx
I.e.: =
Log (x(t1 ) / (x(t2 ))
(2
) /2
1
Therefore, if
<<
1 (which is typical),
/ (2
) T
hus the measurem
ents of T
a and of the logarithmic decrem
ent enable the m
echanical characteristics of a sim
ple oscillator to be determined.
F
igure 1.7: response of the 1 DO
F oscillator
�D
efinition: What is know
n as the bandwidth
, at -3 decib
els, is the difference between
1 and
2 such that:
2 )(
)(
max
2,1
AA
It can be shown that the bandw
idth is linked to the damping by the relationship:
=
2
1 2
21
2 (if the dam
ping is low)
The m
ore this bandwidth tends tow
ards 0, the sharper the resonance and, therefore, the lower
the damping (figure 1.8). A
s a result, to determine
R (and therefore 0 ) and
, A
() is
plotted experimentally, by carrying out a sw
ept sine test. The pulsation corresponding to the
maxim
um of the graph gives
R . In addition, this graph enables the bandwidth to be obtained
and, therefore, .
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F
igure 1.8: Bandw
idth for =
0.2
The dam
ping used in our model is viscous dam
ping: the damping force is proportional to the
speed. This m
odel is very widely used to represent dam
ping. How
ever, in actual fact, there are very few
constructions that are subject to this type of damping: its use is due to its sim
plicity. D
uring harmonic stresses, this dam
ping model indicates that the energy dissipated in each
cycle, at
a given
amplitude
of displacem
ent, depends
on the
exciting frequency:
experimentally, it can be seen that this is not the case. O
ther models m
ust be determined to
overcome this problem
. When the stress is harm
onic, one simple advantageous m
odel is the hysteretic dam
ping model. T
his model consists of using a spring unit, the rigidity k
s of which
is complex:
ks =
k (1 +
i )
in which
is known as the structural dam
ping factor. If the dynamic balance of the m
ass m is
then written:
m
+ k
xs x =
F(t)
Under harm
onic excitation, a equivalent to
can be defined in such a way as to conserve
the energy dissipated per cycle: Wd =
k
x0 2 =
c x0 2
Which leads to:
0
1.2 - Linear systems at n DOF
Having described the sim
ple oscillator, the discrete systems at n D
OF
can be studied directly, as, from
2 DO
F onw
ards, the methods of understanding vibratory phenom
ena are the same,
whatever the D
OF
value n. H
owever, studying system
s at 2 DO
F enables the calculations to be carried out up to the end:
that is why, w
ithout loss of generality when the calculations are m
ade, they are carried out on system
s at 2 DO
F.
The general m
ethod consists of: w
riting the dynamic equations in a m
atrix form: the m
atrices of mass, stiffness and
damping are then defined,
studying the non-damped system
: the natural vibration modes and the m
odal base of the system
are then defined,
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if necessary, determining the tem
poral solution (by going into the modal base, for
example).
Thus, in this chapter w
e will deal w
ith the notions of mass, stiffness and dam
ping matrices,
the notions of natural vibration modes of a construction, and of linking betw
een the DO
F.
1.2.1 - Formation of the equations
System
s at n DO
F are form
ed of n masses connected together by springs and dam
pers (figure 1.9): if this is not the case, n sim
ple oscillators have to be used. Thus, the displacem
ent of a m
ass is dependant on the displacement of another m
ass: it is said that the n masses are
coupled by means of the springs and the dam
pers.
F
igure 1.9: System
at 2 DO
F O
bviously, the n DO
Fs are the positions x
i (t)i =
1,...,n of the n m
asses. F
or the system represented in figure 1.9, the dynam
ic equations are written: )
()
()
()
()
()
()
()
(
)(
)(
)(
)(
)(
)(
)(
)(
21
22
32
12
23
22
2
12
21
21
22
12
11
1
tF
tx
Kt
xK
Kt
xC
tx
CC
tx
M
tF
tx
Kt
xK
Kt
xC
tx
CC
tx
M (E
q. 1.10)
This system
may easily be w
ritten in matrix form
:
)(
2 1
2 1
32
2
22
1
2 1
32
2
22
1
2 1
2
1
)(
)(
)(
)(
)(
)(
)(
)(
0
0
tF
KC
M
tF
tF
tx
tx
KK
K
KK
K
tx
tx
CC
C
CC
C
tx
tx
M
M
(Eq. 1.11)
Which is therefore w
ritten:
)(
)(
)(
)(
tF
tX
Kt
XC
tX
M (E
q. 1.12)
This m
atrix equation recalls the equation governing the oscillator at 1 DO
F. T
he following
names are used:
[X]: vector of the D
OF
s, [M
]: mass m
atrix, [C
]: viscous damping m
atrix, [K
]: stiffness matrix,
[F]: vector of external forces.
The m
ass and stiffness matrices m
ay be found using energy considerations:
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The m
ass matrix is a m
atrix associated with the kinetic energy T
of the system, w
hich is the sum
of the kinetic energies of each mass:
T =
½ M
1 x1 2 +
½ M
2 x2 2 =
½ t[
X] [M
] [X
]
The stiffness m
atrix is a matrix associated w
ith the strain energy J of the system, i.e.
with the elastic potential energy of each spring:
J = ½
K1 x
1 2 + ½
K2 (x
2x
1 ) 2 + ½
K3 x
2 2 = ½
t[X] [K
] [X]
These m
atrices are, therefore, symm
etrical matrices associated w
ith levels of energy.
1.2.2 - Non-damped system
It is assumed here that the dam
ping matrix is zero.
This particular case ([C
] and [F] zero) is fundam
ental to the study of dynamic system
s: it dem
onstrates the notion of natural vibration modes. T
he matrix equation (1.11) thus becom
es:
[ M][
X] +
[ K][X
] = [0]
(Eq. 1.13)
The initial conditions are therefore not zero, otherw
ise the system w
ould obviously remain at
rest. Solutions are sought in the form
: [X
] = [
0 ] er t
By injecting into the above equation, the hom
ogenous system at
0 i is obtained, according to:
[K +
r2 M
] [0 ] =
[0] I.e., for the system
at 2 DO
F in figure 1.9:
0 0
02 01
22
32
2
21
22
1
Mr
KK
K
KM
rK
K
This system
permits the zero solution that cannot be satisfactory as soon as there are non-zero
initial conditions. To obtain an advantageous solution, the determ
inant of the system m
ust be zero. W
e can therefore infer from it the follow
ing equation in r:
Det ([ K
+ r
2 M]) =
0 (Eq. 1.14)
I.e. for the system at 2 D
OF
above, we get:
((K1 +
K2) +
r 2 M
1 ) ((K2 +
K3 )+
r 2 M2)
K2 2 =
0 F
or a system at n D
OF
s this typical equation has 2 n solutions: i
k = 1
,...n . With each index
k, there is an associated natural vector k , solution of the system
: [K
k 2 M] [
k ] = [0]
I.e. for the system at 2 D
OF
s:
0 0
2 1
22
32
2
21
22
1
k k
k
k
MK
KK
KM
KK
The pulsations
k are known as the natural pulsations of the system
. The natural vectors [
k ] are the m
odal vectors of the system. T
he solution is then written in the form
: t
i
k
ti
kk
nk
kk
ee
tX
,...,1
)(
in which the
k and the k are determ
ined by the initial conditions. T
he components of the vector
k are obviously linked. A m
odal vector is defined to within
one multiplicative constant: a standard therefore has to be chosen for these vectors. T
he most
frequent choices are the following:
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one of the components is m
ade equal to 1. each m
ode is unitary: i
= 1,
a standard is set according to the mass m
atrix: t[i ] [M
][i ] =
1
If we select the first norm
alisation:
i
i
2 1]
[
It is now possible to express the general solution of the system
at 2 DO
F, non-dam
ped, under free excitation:
)(
)(
)(
)(
22
11
22
11
43
222
121
2
43
21
1
ti
ti
ti
ti
ti
ti
ti
ti
ee
ee
tx
ee
ee
tx
These expressions are real: x
1 and x2 can be expressed in trigonom
etric form:
))sin(
)cos(
())
sin()
cos((
)(
)sin(
)cos(
)sin(
)cos(
)(
24
23
221
21
121
2
24
23
12
11
1
ta
ta
ta
ta
tx
ta
ta
ta
ta
tx
The constants are determ
ined using the initial conditions. In sum
mary, a m
ode of a linear system is one particular solution of the free, non-dam
ped problem
, in such a way that all the com
ponents of the vector of the DO
Fs are synchronous
(they reach their maxim
a and their minim
a at the same tim
e): t
i
ee
X0
00
mod
The natural pulsations
i are the square roots of the natural positive values of the matrix
[M]
1 [K]. T
he natural forms of a discrete system
are the natural vectors i associated w
ith it. T
he couple (i ,
i) is called a natural vibration mode.
The m
odes are sought directly by writing:
natural pulsation i : this is all of the solutions to the equation:
det([ K
i 2 M
]) = 0
natural vector [i ]:
[ K
i 2 M
] [i ] =
[0]
The natural vibration m
odes are orthogonal in relation to the mass m
atrix and in relation to the stiffness m
atrix: i, j t[
i ] [M] [
j ] = m
i ij
i, j t[i ] [K
] [j ] =
ki
ij
in which
ij is the Kronecker sym
bol and mi and k
i are the mass and the generalised stiffness
associated with the m
ode i: these values will depend on the standard of the natural vectors
selected.
Once again, before passing to a general forced excitation, w
e are interested in the harmonic
excitation: it is thus that the notion of transfer function is generalised. It is considered, therefore, that the construction is excited by a force vector of w
hich each com
ponent is harmonic and in phase w
ith the other components:
[ F(t)] =
ei
t [F0 ]
It is assumed therefore that the solution is in the form
: [X
(t)] = e
it [X
0 ] T
he matrix equation of the dynam
ic therefore becomes:
([K]
2 [M]) [X
0 ] ei
t = e
it [F
0 ]
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From
which:
[ X0 ] =
([K]
2 [M])
1 [F0 ] =
[(
) ] [F0 ]
the matrix [
] is called admittance m
atrix or dynamic influence factor m
atrix: this is a response m
atrix at a frequency between the forced excitation [ F
] and the response [X(t)].
We can dem
onstrate the resonance phenomenon: if it is excited at a natural frequency, the
response becomes infinite.
The general term
of this matrix,
ij , interprets the influence of the component j of the force on
the component
i of the displacement; if all the com
ponents of [F0 ] are zero except the
component j, w
e have: j i
ijF x
0 0)
(
This is not forgetting the transfer function of a system
at 1 DO
F betw
een the excitation F0
j and the response x
0 i . W
e repeat that ij is the transfer function betw
een F0
j and x0
i , or even its frequency response function (F
RF
). C
omm
ent: There is a resonance peak for each natural frequency of the system
. In theory, this peak is infinite. T
his arises from the (unrealistic) assum
ption that the system is not dam
ped.
Rem
ember the equation to be solved:
)](
[)]
([]
[)]
([]
[t
Ft
XK
tX
M
In order to determine the com
ponents of [X(t)], it is alw
ays possible to carry out a temporal
(digital) integration of the equations. This also requires the inverse of the m
ass matrix to be
determined (high digital cost if the num
ber of DO
Fs is high). T
his leads to a differential system
of linked equations. If the equations to be solved were not linked, that w
ould simplify
their solution: there would be n differential equations of second order to solve (see previous
chapter). T
hat is why it is standard practice to act in a different w
ay: a modal m
ethod is used that unlinks the equations. T
he principle is to be positioned in modal base: the solution [ X
(t)] is therefore expressed using a com
bination of the natural vectors, that form a base. T
he change of reference is therefore carried out using the transform
ation:
(Eq. 1.15)
ni
ii
tq
tq
tX
1
)(
][
)](
[][
)](
[
where: [
] = [
1 2
n ]: is a square m
atrix in which the colum
ns are the natural vectors; [ q(t)] is the vector of new
variables, known as m
odal variables. If [ X
(t)] is replaced by its expression (Eq. 1.15), in the m
atrix equation of the dynamic, and if
this equation is pre-multiplied by t[
], we get:
)](
[][
)](
[][]
[][
)](
[][]
[][
tF
tq
Kt
qM
tt
t
In view of the orthogonality of the m
odes in respect of the mass and stiffness m
atrices, we
have:
)(
)(
00
00
00
)(
00
00
00
tF
tq
kt
qm
t
ii
Thus, as the tw
o matrices t[
][M][
] and t[][K
][] are diagonal, w
e get a system of unlinked
differential equations of the type:
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)(
)(
)(
)(
2t
pm
tF
tq
tq
i
i
i
t
ii
i (E
q. 1.16)
The vector [ p(t)] is the vector of the m
odal contribution factors of the force [F(t)]: it is the
component per unit of generalised m
ass of [F(t)] in the m
odal base. W
e thus have to refer to the chapter on the oscillator at 1 DO
F: D
uhamel's integral allow
s each
qi (t) to be obtained. Its vector [X
] can then be inferred by modal superposition, in
accordance with equation (1.15).
Using the principle of m
odal superposition, the study turns from a system
with n D
OF
s to a study of n sim
ple oscillators. P
articular case: For a harm
onic excitation, we have:
qi (t) =
qi0 cos(
t) w
ith:
22
00
i
ii
pq
From
which
nii
ii
tp
X1
22
0)
cos(]
[
This expression clearly dem
onstrates the resonance phenomenon for each natural pulsation.
1.2.3 - Damped system
s
In order to simplify the study, only the viscous dam
ping is considered. The equation of the
problem is thus the equation (1.12) already described:
[M][
X] +
[C][
X] +
[K][X
] = [F
]
Here again, it is possible to integrate the equations directly. W
e will even see that this m
ight be essential to m
ake allowance for dissipation effects.
How
ever, we w
ill apply the modal m
ethod to our system: w
e will then have to discuss its
validity. In the follow
ing, the couple (i , [
i ]) designates the mode i of the equivalent non-dam
ped system
. We w
ill break down the equation (1.12) on the basis of natural vectors, follow
ing the sam
e route as in the previous paragraph: )]
([
][
)](
[][
][
][
)](
[]
[][]
[)]
([
][]
[][
tF
tq
Kt
qC
tq
Mt
tt
t
Although the m
atrices t[
][M][
] and t[
][K][
] are, obviously, still diagonal, the same does
not necessarily apply to the matrix
t[][C
][]; the m
odal vectors would appear not to be
orthogonal in respect of the damping m
atrix: ij =
t[i ][C
][j ]
constant ×
ij
Thus, the projection on the basis of natural vibration m
odes does not unlink the equations from
the
damped
system:
the m
odal m
ethod seem
s to
lose a
lot of
its attraction
for determ
ining a digital solution of [X].
If the linking due to damping w
as zero, the modal m
ethod would becom
e more attractive.
This is only realistic if the distribution of the dam
ping is similar to that of the m
ass and of the rigidity: such an assum
ption does not appear to be justifiable.
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In practice, very little is known about the distribution of the dam
ping. How
ever, if the construction is not very dissipative and the natural frequencies are clearly separated, it can be show
n that the diagonal damping assum
ption is reasonable: this is known as the m
odal dam
ping assumption.
Caughey has show
n that the natural vibration modes are orthogonal at any dam
ping matrix
that is expressed in the general form:
Nk
k
kK
MM
aC
1
11
][
][
][
][
In the particular case where N
equals 2, proportional damping can be found: the dam
ping m
atrix is expressed according to a linear combination of [K
] and [M]; it is then clear that, in
this particular case, the natural vibration modes of the system
are orthogonal in respect of the dam
ping matrix.
Dam
ping is often low if it is w
ell spread out. In this case, it can also be shown that the natural
vibration m
odes of
the non-dam
ped construction
are fairly
close to
the m
odes of
the construction. O
nce again, therefore, the principle of modal superposition is applied, using the
modes of the non-dam
ped construction. T
hus, if we take:
ii i
im
2
we find once m
ore a system of uncoupled equations:
i
i
t
ii
ii
ii
im
Ft
pt
qt
qt
q]
[][
)(
)(
)(
2)
(2
(Eq. 1.17)
Each represents the equation of the m
ovement of a sim
ple damped oscillator under a forced
excitation. In practice, each critical dam
ping ratio i m
ust be recalibrated experimentally. In fact it is
very rare to find a construction with dam
ping that needs simple m
odelling. It is norm
al to consider that the value of the critical damping ratio does not depend on the
mode under consideration:
i =
This constant
is then fixed by experiment or by the
regulations.
The natural m
odes are therefore assumed to be orthogonal in respect of the dam
ping matrix.
A resolution is carried out by m
odal superposition: we are looking for [ X
], the solution to the problem
, using modal variables:
[X] =
[] [q]
Each m
odal variable qi is then subjected to the equation (1.17). T
he solution sought and the excitation are, traditionally, in the form
:
ti
ti
ep
p
eq
q
][
][
][
][
0 0
(Eq. 1.18)
[ p] always being the vector of the factors of m
odal contribution to the force [F]:
ti
i
i
i
t
ii
t
i
t
ie
pm
F
M
Fp
0
][]
[
][]
[][
][]
[
The solution is therefore:
nii
ii
i
t
i
i
nii
ii
ii
ni
ii
Fi
mi
pq
X1
22
12
20
10
0]
[2
][
][
1
2
][
][
][
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It can then be recognised that each term of this sum
is the response of an oscillator at 1 DO
F
under harmonic excitation. T
he response of an oscillator at n DO
F is therefore expressed
according to the response of n oscillators at 1 DO
F �T
hanks to the modal superposition m
ethod, we are led to the study of a system
at a DO
F
under a comm
onplace excitation. The solution is then given by D
uhamel's integral:
t
ii
t
i
ii
id
te
tp
tq
ii
0
2)
(
2))
(1
sin()
(1 1
)(
which leads to the determ
ination of the solution, by modal superposition:
ni
ii
tq
X1
)(
][
][
1.2.4 - Modal truncation
A com
plex construction may be m
odelised by n DO
F, w
here n is very large. Also, it is norm
al to represent the physical D
OF
s by a linear combination of the first N
modes (they are assum
ed to be ranged in increasing order of frequency), w
here N is less, or even m
uch less, than n. S
uch an operation must be carried out w
ith certain precautions: a reliable criterion must be
defined to justify this modal truncation.
This approach w
ill depend on the spectrum of the excitation: if it is restricted to a range of
pulsation less than N , it m
ay be justified. How
ever, for more thoroughness and accuracy, it is
essential to make allow
ance for these neglected modes. If it is thought that these m
odes have an alm
ost-static response, it can be shown that is possible to express the neglected m
odes according to the m
odes retained� ]
[]
[]
[]
[]
[)(
][
][
][
][)
(]
[
12
1
1
12
1
Fm
Kt
q
Fm
tq
X
Nii
i
i
t
iNi
ii
nNi
ii
i
t
iNi
ii
Matrices w
ith residual stiffness are known as K
res and residual flexibility Sres such that:
Nii
i
i
t
i
resres
mK
KS
12
11
][
][
][
][
][
It is very important to note that this m
atrix depends only on static characteristics ([ K]) and on
the first N m
odes: the neglected modes are thus taken into account w
ithout having to calculate them
! C
omm
ent: The above expression com
es from the m
odal decomposition of the inverse of the
stiffness matrix:
nii
i
t
ini
ii
i
t
i
km
K1
12
1]
[]
[]
[]
[]
[ (E
q. 1.19)
If the system is loaded statically, the m
odal equations (1.17) become
i
i
t
ii
m
Fq
][]
[2
and then the modal response is:
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][
][
][
][
][
][]
[]
[]
[]
[1
12
12
1
FK
Fm
m
Fq
Xni
ii
i
t
ini
ii i
t
i
ni
ii
By identification, w
e find once more equation (1.19).
1.2.5 - Rayleigh-Ritz method
We propose an energy approach in order to determ
ine an estimate of the first natural
frequency of a construction: this is the Rayleigh m
ethod. T
his method is based on a principle of approxim
ation of the field of displacement of the
system being studied: w
e suggest the shape of the construction, i.e., here, a displacement
vector]
~[
0X
:
)(
]~
[)]
(~
[0
td
Xt
X
This vector m
ust be kinematically perm
issible.T
he strain and kinematic energies associated w
ith this shape are then calculated. The first
natural pulsation is then estimated by the ratio: ]
~[]
[]~
[
]~
[][]
~[
00
00
2
XM
X
XK
Xt t
i (E
q. 1.20)
If the vector ]
~[
0X
is the first natural vector, then the solution obtained is the first natural
pulsation. The closer one gets to the natural vector, the better the estim
ate.
The w
hole problem is to select a "good" vector
]~
[0
X�in other w
ords to have an idea of the
natural vector. In general, the displacement vector obtained under a static load proportional to
the value of the masses is a vector close to the first natural vector.
This is, to a certain extent, a generalisation of the R
ayleigh method: it enables not only an
estimate of the first N
natural frequencies to be obtained (often N is m
uch lower than n), but
also a close estimate of the natural vectors. T
he principle is the same: w
e propose N vectors
{]
~[
0i
X}
i = 1
N that are independent, kinematically perm
issible: the displacement vector [X
(t)]
is then approached by:
NiN
n
ii
N
Nt
dX
td
X
td
td
XX
tX
tX
10
0
1
001
)](
[]~
[)
(]
~[
)(
)(
]~
~[
)](
~[
)](
[
We then get an estim
ate of the first N natural frequencies by solving the follow
ing problem:
0]
~[]
[]~
[]
~[]
[]~
[0
02
00
NN
t
i
NN
tX
MX
XK
X
If w
e suggest:
]~
[][]
~[
]~
[
]~
[][]
~[
]~
[
00
00
XM
XM
XK
XK
t t
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we therefore com
e to determine the values and natural vectors associated w
ith the problem:
0]
~[
]~
[2
MK
This problem
is more advantageous to solve than the original problem
, to the extent that N is
(much) less than n.
Thus, the
i~
obtained are approximations of the first N
natural pulsations i of the system
.
There is a natural vector that m
atches each of these natural values:
Ni i
i
~ ~
~1
We then get an approxim
ation of the natural vector i by: Nj
ji
ji
iN
iX
XX
X1
00
001
~]
~[
]~[]
~[
]~[]
~~
[]
~[
1.2.6 - Conclusion
To represent a system
at n DO
F by n sim
ple oscillators: this is the fundamental idea of this
chapter and of the mechanics of linear vibrations of discrete system
s in general. This is
interpreted by the diagram in figure 1.10.
F
igure 1.10: Modal approach to a system
at 2 DO
F
1.3 - Continuous elastic systems
1.3.1 - Generally
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Real system
s are rarely discrete. Thus, a beam
is deformable at any point, has a certain inertia
at any point and is capable of dissipating energy at any point. W
e are therefore in the presence of a problem of the dynam
ics of continuous environments: to
the statics equations we m
ust therefore add the term that m
atches the inertia forces. H
owever, w
e must not believe that the study of discrete system
s was useless: w
e will see that,
from a practical point of view
, the study of continuous systems leads finally to the study of
discrete systems. S
uch an approach will necessarily be an approxim
ation, as a discrete system
has a finite number of D
OF
s, whereas a continuous system
has an infinite number of D
OF
s. T
he dynamic study of a continuous elastic system
requires: setting the vibrational problem
(formulating the equations):
�determ
ining the equation for the partial derivatives governing the field of displacem
ent X (x,y,z,t): this is the equation of the m
ovement,
�giving the equations expressing the boundary conditions: these equations m
ay also be equations of partial derivatives,
�giving the initial conditions.
solving the problem:
�calculating the natural m
odes of the continuous elastic system,
�calculating the response of the continuous elastic system
. T
o do this, we assum
e that the systems studied check the follow
ing hypotheses: the continuous elastic system
is only subject to small strains around its natural state,
the continuous elastic system has (by definition) a law
of linear elastic behaviour; its m
echanical characteristics are E, its Y
oung's modulus, and
, its density, the continuous elastic system
vibrates in a vacuum.
1.3.2 - Formation of the equation
In this chapter, we study a beam
(to be understood here in its widest sense): this is a
rectilinear uni-dimensional continuous environm
ent capable of deforming:
by tension-compression (bar),
by pure bending. W
e will study these tw
o cases successively. W
e will only be studying bi-dim
ensional constructions with no real loss of generality: the tri-
dimensional aspect only adds additional variables.
In a general way, the field of displacem
ent X
(x,y,z,t) of a point M
(x,y,z,t) on a beam is
characterised by its DO
Fs:
),
,,
(
),
,,
(
),
,,
(
),
,,
(
tz
yx
tz
yx
w
tz
yx
u
tz
yx
X
in which:
u is the longitudinal displacement, i.e. along the direction of the beam
, w
is the transverse displacement (deflection), i.e. along a direction perpendicular to the
beam,
is the rotation around an orthogonal axis in the plane of the construction: this is the slope adopted by the beam
, due to the bending mom
ent, In addition, w
e are working w
ithin a reference linked to the beam: the axis of the beam
is com
bined with (x
,x) and the system is in the plane (x,z)
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We are only interested therefore in the com
ponent u(x,t) of X(x,t). T
he balance of a length of beam
of thickness (assumed infinitesim
al) dx and of sectional area S, subjected to: the norm
al force at x, the opposite of the norm
al force at x+dx,
an external distributed force (pre-stressing) p(x) d
x. leads to the equation of the m
ovement of a bar of constant sectional area:
),
(2
2
2
2
tx
px u
SE
t uS
(Eq. 1.21)
We study a beam
in pure bending. We use the N
avier hypothesis: straight sections remain
straight. We assum
e, in addition, that transverse shear is negligible (Euler-B
ernoulli beam).
We are only interested therefore in the com
ponent w(x,t) of X
(x,t). The balance of a length of
beam of thickness d
x, of sectional area S and of inertia I, subjected to: the shear force and the bending m
oment at x,
the opposite of the shear force and of the bending mom
ent at x+dx,
an external distributed force F(x,t).
leads to the equation of the movem
ent of a beam in bending of constant sectional area:
),
(4
4
2
2
tx
Fx w
IE
t wS
(Eq. 1.22)
To solve m
ovement equations, there have to be initial conditions (tem
poral) and boundary conditions (spatial). T
he boundary conditions are very important data, as they set the resonance frequencies of the
construction. They are associated w
ith: im
posed displacements (geom
etric condition): housings, supports, hinges, etc. im
posed forces: �
ends free (natural condition): normal force, zero shear, zero bending m
oment,
etc. �
localised masses,
�localised stiffness.
Exam
ple 1: F
ixed beam in bending-tension-com
pression with x =
0 and free at x = L
: x =
0x =
L
u = 0
N =
0 w
= 0
T =
0 =
0 M
= 0
Exam
ple 2: S
upported beam in bending-tension-com
pression with x =
0 and supported at x = L
: x =
0 x =
L
N =
0 N
= 0
w =
0 w
= 0
M =
0
M =
0 E
xample 3:
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L
m
u(L,t)
F
igure 1.11: Fixed bar (E
, S, L
) in tension-compression w
ith x = 0 w
ith a localised mass m
at x = L
. In order to obtain the boundary condition at x =
L, the balance of the m
ass m is w
ritten: )
,(
),
(t
Lu
mt
LN
I.e.:
),
()
,(
tL
um
tL
uS
E
Exam
ple 4:
w(L
,t)L
m
k
F
igure 1.12: Fixed beam
(E, I, L
) in pure bending with x =
0 with a localised m
ass m at x =
L and a transverse
localised stiffness k at x = L
.
In order to obtain the boundary condition at x =
L, the balance of the m
ass M is w
ritten:
0)
,(
),
()
,(
),
(
tL
M
tL
wm
tL
wk
tL
T
I.e.:
0)
,(
),
()
,(
),
()
2(
)3(
tL
w
tL
wm
tL
wk
tL
wI
E
It can be seen that, in the two exam
ples above, there are six boundary conditions: Tw
o associated w
ith the tension-compression (second order equation) and four associated w
ith the bending (fourth order equation).
1.3.3 - Natural modes of a continuous elastic system
In order to determine the natural m
odes of a continuous elastic system, w
e can use: the balance equation, the boundary conditions.
One natural m
ode of a continuous elastic system is a solution u(x,t) to the problem
that is w
ritten in the form:
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u(x,t) =
(x) a(t) (Eq. 1.23)
In addition, it can be shown that the tem
poral function is harmonic:
a(t) = a
0 cos( t +
)
A m
ode consists of the data of a natural shape (x) (shape of the construction) and a natural
pulsation . A
mode is thus a stationary w
ave: all points of the system vibrate in phase.
In order to determine the natural m
odes of the system, the m
ethod is as follows:
the solution (1.23) is injected into the balance equation without a second m
ember and
without dam
ping, the tim
e and space variables are separated: there then appears a constant, the sign of w
hich is clear, in order to have a physically acceptable solution, the equation (probably differential) is then solved, associated w
ith the spatial value: integration constants appear. T
he general solution obtained will then perm
it the natural shapes
(x) of the system to be determ
ined, the integration constants are determ
ined using boundary conditions: a homogenous
system of equations is obtained. A
s its solution cannot be a zero solution (all the constants zero), in order to determ
ine the constants it will be necessary to im
pose the nullity
of the
determinant
of the
system:
this condition
gives the
characteristic equation of the system
, the solution of the characteristic equation then provides the natural pulsations of the system
. W
e apply these principles to the two traditional cases: longitudinal vibration of a bar and
bending vibration of a beam.
The follow
ing solution is injected: u(x,t) =
(x) a(t) in the balance equation of a bar (E, S, L
, ):
02
2
2
2
x uS
Et u
S
After sim
plifications and separation of the variables, we have:
2constant
)(
)(
)(
)(
ta
ta
x xE
The constant is necessarily negative, otherw
ise a solution is obtained that grows exponentially
over tim
e and
is, therefore,
physically unacceptable.
It is
found, therefore,
that a(t) is
sinusoidal: a(t) = a
0 cos( t +
)
In the same w
ay, we get the general form
of the natural shapes of a bar:
xE
Bx
EA
xcos
sin)
(
Thus, the m
odal solution is:
xE
Bx
EA
ta
tx
ucos
sin)
cos()
,(
0
There therefore rem
ains to be determined the integration and pulsation constants
: a
0 and are determ
ined with the initial conditions,
A, B
and are obtained w
ith the boundary conditions. E
xample: M
ixed supported bar: the boundary conditions are written:
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)(
)(
),
(0
),
(
)0(
)(
0)
,0(
Lt
aS
Et
Lx u
SE
tL
N
ta
tu
The follow
ing system can be inferred from
this:
0cos
0
LE
AE
B
As B
is zero, in order to obtain a non-zero solution, it must be:
0cos
LE
We can therefore infer the natural pulsations from
this:
,2,1
2
)12(
nE
L
nn
The natural shapes (defined to the nearest m
ultiplicative constant) are therefore:
xL
nA
xn
2
12
sin)
(
In the same w
ay as the discrete case, a generalised stiffness kn and a generalised m
ass mn can
be defined:
L
nn
L
nn
dx
xS
Ek
dx
xS
m
0
2
0
2
)(
)(
We find the relationship:
n n
nm k
2
The follow
ing solution is injected: w(x,t) =
(x) a(t) into the balance equation:
04
4
2
2
x wI
Et w
S
After sim
plifications and separation of the variables, we have:
2)
4(
constante)
(
)(
)(
)(
ta
ta
x
x
S IE
Here again, the sign of the constant arises from
a physically acceptable solution; we find that
a(t) is sinusoidal. In addition, the differential equation governing is of the fourth order:
0)
()
(2
4
4
xI
E
Sx
xd d
If we take
4
2
IE S
k, the general solution of the above equation (therefore the shape) is:
(x) = A
sin(kx) + B
cos(kx) + C
sinh(kx) + D
cosh(kx) W
e use the boundary conditions to determine the integration and pulsation constants.
Exam
ple: Beam
supported at each end: the boundary conditions are w
ritten:
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),
(0
),
(
0)
,(
),0
(0
),0
(
0)
,0(
tL
wI
Et
LM
tL
w
tw
IE
tM
tw
Whence the system
:
0)
cosh()
sinh()
cos()
sin(
0)
cosh()
sinh()
cos()
sin(
0 0
22
22
kLk
DkL
kC
kLk
BkL
kA
kLD
kLC
kLB
kLA
DB
DB
As this system
is homogenous, there can only be a non-zero solution if the determ
inant of the system
is zero; we can infer from
that that characteristic equation which reduces to:
sin (kL) =
0 T
he resolution of this equation leads to the natural pulsations:
S IE
L
nn
2
22
and, finally, to the mode shapes:
L
xn
An
sin
A generalised m
ass and stiffness can also be defined:
L
nn
L
nn
dx
xI
Ek
dx
xS
m
0
2
0
2
)(
)(
1.4 - Discretisation of the continuous systems
1.4.1 - Introduction
As has been m
entioned, few m
odal analysis problems of continuous elastic system
s have an analytical solution. T
hat is why it is necessary to use approxim
ation methods:
direct mass discretisation m
ethod: the construction is approached by masses and
springs, R
ayleigh-Ritz m
ethod: the spatial variation of functions is apparently postulated, finite elem
ent method: the construction is broken dow
n into sub-sections, to which a
Rayleigh-R
itz method is applied.
These m
ethods are discretisation methods: w
e pass from an infinite num
ber of DO
Fs for
continuous systems to a finite num
ber of DO
Fs.
1.4.2 - Direct discretisation of masses
This
method
may
depend heavily
on the
aptitude of
the "m
odeliser" to
discretise a
construction: it is, however, very im
portant, as it is widely used to carry out civil engineering
modelisations (particularly for earthquake-protection design). It also perm
its the mass and
stiffness matrices to be obtained very sim
ply. T
his method consists of: dividing the construction up geom
etrically,
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concentrating the mass of each divided part on a node (in general its centre of gravity)
in the form of:
�m
ass acting in translation, �
inertia of the mass acting in rotation.
As a consequence, the D
OF
s of the discrete system are the "physical" D
OF
s associated with
the point on which the m
ass is concentrated: it is supposed, therefore, that each section "strictly follow
s" the DO
F of the node, w
hence the importance of the choice of dividing up.
Such a m
ethod leads to a diagonal mass m
atrix. As far as rigidity is concerned, the stiffness is
determined by connecting 2 D
OF
s together: the rigidity matrix is not necessarily diagonal.
Exam
ple: Discretisation of a fixed/free bar (figure 1.13): the first node is located at 1/3 of the
length, the second at 2/3 of the length and the last one at the end. Each node is allocated half
of the mass of each elem
ent to which it is connected; the stiffness of each spring is calculated
from the rigidity of a bar elem
ent. T
he choice of ki and m
i is important and determ
ines the accuracy that there will be in the
natural frequencies. A choice can be m
ade for a division into three sections of the same
length:
F
igure 1.13: Direct discretisation of the m
ass of a bar
1.4.3 - Rayleigh-Ritz method
As the system
to be calculated is continuous, it is determined as soon as the continuous
functions describing the DO
Fs of each point are know
n: translations, rotations. It is proposed therefore to find out about these continuous functions u(M
, t) by the determination of a finite
number
of param
eters {a
i (t)}
i =
1,
,N
(comprising
the vector
[a(t)]) thanks
to the
approximation functions
i (M):
Ni
t
ii
ta
ta
Mt
Mu
tM
u1
)](
[][
)(
)(
),
(~
),
(
with:
ai (t): generalised coordinate. T
hese are our unknowns to be determ
ined: they are grouped together in the vector of generalised coordinates [a(t)]; the dim
ension n of the vector [a(t)] gives the num
ber of DO
Fs of the discretised system
, i (M
): given spatial function. It must be kinem
atically permissible and ��� !��
over the w
hole construction, which occupies the dom
ain V.
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Thus, w
hen the generalised coordinates are known, using the approxim
ation functions i (M
), w
e find u at any point in the system.
The general m
ethod consists of: W
riting the kinetic energy, the system strain energy, and also the w
ork of external forces according to the field approached u ~
. R
e-writing
these energies
in m
atrix form
, by
demonstrating
the vectors
of the
generalised coordinates:
)](
[][
][
)](
[
)](
[][
)](
[2/
1
)](
[][
)](
[2/
1
ta
FF
ta
W
ta
Kt
aJ
ta
Mt
aT
tt
ext
t t
The m
atrices [M] and [K
] which then appear are the m
atrices of the mass and the
stiffness of the discretised system: the governing equation [a(t)] is then:
)](
[)]
([]
[)]
([]
[t
Ft
aK
ta
M
In order to determine the natural m
odes (, [
(x)]), we use the procedure described in
the chapter on discrete systems w
ith n DO
F; w
e resolve: )]
([]
[)]
([]
[2
xM
xK
Let us follow
the procedure described in the introduction to this chapter: C
alculation of the kinetic energy:
)](
[d]
[]
[)]
([
2/1
d)
,(
2/1
][
2t
aV
ta
Vt
Mu
T
M
V
tt
V
We infer from
this the expression of the mass m
atrix of the discretised system:
V
tV
Md]
[]
[]
[
In the simple case of a bar or a beam
, the mass m
atrix therefore becomes:
Lt
xx
xS
M0
d)]
([
)](
[]
[
Calculation of the strain energy:
The general calculation of the strain energy is not m
ade here: depending on the RD
M
hypotheses, we get different "sim
plified" expressions, depending on whether the system
is m
odelised by a beam, a bar, etc. W
e do recall, however, the m
ost typical energies:
Tension-com
pressionL
xx
uS
E0
2d)
(2/
1
Bending
L
xx
wI
E0
2d)
(2/
1
We can then infer from
this the stiffness matrices:
Tension-com
pressionL
tx
xx
SE
K0
d)]
([
)](
[]
[
Bending
Lt
xx
xI
EK
0d
)](
[)]
([
][
C
alculation of the work:
V
tt
t
extt
at
Ft
Ft
aV
tM
ft
Mu
W)]
([
)](
[)]
([
)](
[d
)],
([
)],
([
Know
ing all these matrices, w
e can then infer from them
the balance equation: )]
([
)](
[][
)](
[][
tF
ta
Kt
aM
T
he whole arsenal developed for system
s at n DO
F can then be applied:
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calculation of the natural modes of the discrete system
, response to an external force, etc.
Once the discrete problem
has been solved, we can return im
mediately to the continuous
system. T
hus, if the natural vectors [i ] of the discrete system
associated with the natural
pulsations i have been determ
ined, the first n modes (close) of the continuous system
are determ
ined by the same natural pulsation
i and by the mode shape u
i (x): ]
[)]
([
)(
i
t
ix
xu
Com
ment 1:
As m
any modes are determ
ined as there are DO
Fs. In practice, it can be
estimated that only the first n/2 m
odes are determined w
ith sufficient accuracy. C
omm
ent 2:A
ccuracy is improved w
hen the number of D
OF
s is increased. The estim
ate of the natural pulsations by the R
ayleigh-Ritz m
ethod is always m
ade by excess. C
omm
ent 3:W
hen the system being studied is form
ed from beam
s, it is normal to use
polynomes for the functions
i (x). For exam
ple for a fixed/free beam, w
e choose: i (x) =
x i w
ith i 2. It is easy to check that these functions are all kinem
atically permissible.
Com
ment 4: W
hen n = 1, this m
ethod is known as R
ayleigh's method.
Com
ment 5: T
he static shape of the construction subjected to a load proportional to its self-w
eight gives, in general, a good approximation of the m
ode shape of the first mode.
1.4.4 - Finite element m
ethod
Although it is easy to determ
ine the functions i (x) for a straight fixed/free beam
, the same
does not necessarily apply to systems that are hardly m
ore complex, such as, for exam
ple, two
beams rigidly fixed at one end but non-collinear: in this case, how
do we choose the
interpolation functions? The sim
ple solution is to "break" the system: w
e turn therefore to the finite elem
ent method.
The construction is sub-divided into fields of a sim
ple geometric shape (of w
hich we know
the analytical expression), the contour of w
hich is supported at points (the nodes): these are the elem
ents. T
hese elements have a sufficiently sim
ple shape for it to be possible to determine functions
that are kinematically perm
issible on each of these elements: these are generally polynom
es. W
e then apply the Rayleigh-R
itz method to each of the elem
ents: we calculate the kinetic and
strain energies for each element. W
e can then infer the kinetic and strain energies for the w
hole construction: we use the additivity of the energy. W
e can then infer the mass and
stiffness matrices of the construction: w
e turn once again to the study of a system w
ith n D
OF
s. N
B: T
he approximation functions, called here interpolation functions, are selected in such a
way that the generalised coordinates are the values of the displacem
ents of the nodes: the generalised coordinates thus have a physical m
eaning that enables a continuous displacement
field to be obtained over the construction. W
e can also determine a load vector by w
riting, in matrix form
, the work of the external
forces applied to the system. W
e can therefore study the response of a continuous system
subjected to any load.
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2. Appendix 2: Modelling of the pedestrian load
Pedestrian footbridges are subjected m
ainly to the loads of the pedestrians walking or running
on them. T
hese two types of loading m
ust be treated separately, as there is a difference betw
een them: w
hen walking, there is alw
ays one foot in contact with the ground, but the
same does not apply w
hen a person starts to run (figure 2.1).
F
igure 2.1: changes of the force over time for different types of step (R
ef. [17]) W
hatever the type of load (walking or running), the force created com
prises a vertical com
ponent and a horizontal component, it being possible to break dow
n the horizontal com
ponent into a longitudinal component (along the axis of the footbridge) and a transverse
component (perpendicular to the axis of the footbridge). In addition, apart from
the individual action of a pedestrian, w
e will consider m
aking allowance for a group of people (crow
d) and also the stresses connected w
ith exceptional loadings (inaugurations, deliberate excitation, etc.).
Finally,
when
they exist,
we
will
indicate the
recomm
endations specified
in the
Eurocodes covering dynam
ic models of pedestrian loading.
For activities w
ithout displacement (jum
ping, swaying, etc.), experim
ental measurem
ents show
that it is preferable to make allow
ance for at least the first three harmonics to represent
the stresses correctly, while, for the vertical com
ponent of walking, w
e can restrict ourselves to the first harm
onic, which can be described sufficiently as F
(t). The values of the am
plitudes and of the phase shifts, arising from
experimental m
easurements, are set out below
for the various com
ponents of F in the case of w
alking and of running. It should be noted, however,
that the values one can find in the literature are approximate, in particular as far as the phase
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shifts (i ); the values provided are, therefore, orders of m
agnitude corresponding to an average displacem
ent. A
s can be seen in figure 2.1, the shape of the stress changes between w
alking and running, and also according to the type of w
alking or running (slow, fast, etc.). T
he amplitudes G
i are therefore linked to the frequency
fm and the values indicated subsequently are therefore
average values, corresponding to "normal" w
alking or running.
2.1 - Walking
Walking (as opposed to running) is characterised by continuous contact w
ith the ground surface, as the front foot touches the ground before the back foot leaves it.
2.1.1 - Vertical component
In normal w
alking, the vertical component, for one foot, has a saddle shape, the first
maxim
um corresponding to the im
pact of the heel on the ground and the second to the thrust of the sole of the foot (see figure 2.2). T
his shape tends to disappear when the frequency of
the walking increases, until it is reduced to a sem
i-sinusoid when running (see figure 2.1).
F
igure 2.2: force resulting from w
alking (fm = 2 H
z) (Ref. [3])
F
or norm
al (non-obstructed)
walking,
the frequency
may
be described
as a
Gaussian
distribution of average 2 Hz and standard deviation 0,20 H
z approximately (from
0.175 to 0.22 depending on the authors). A
t the average frequency of 2 Hz (fm =
2 Hz), the values of
the Fourier transform
coefficients of F(t) are as follow
s (as remarked previously, the first
three terms are taken into account, in other w
ords n = 3, the factors of the term
s of greater order being less than 0.1 G
0 ):
G1 =
0,4 G0 ; G
2 = G
3
0,1 G0 ;
2 =
3
/2.
As can be seen above, the values of G
i and of i for the term
s beyond the first are hardly precise, this being due, on the one hand, to uncertainties w
hen taking measurem
ents and, on the other hand, to differences betw
een one person and another. C
om
men
t: for a frequency of walking of 2.4 H
z, the recomm
ended value of G1 is 0.5 G
0 , the other values being unchanged. In the sam
e way, for slow
walking (1 H
z), we have G
1 = 0,1
G0 .
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We have plotted on figure 2.3, for a person of 700 N
walking at a frequency fm of 2 H
z and over 1 second, the change in F
(t), taking into account one, two or three harm
onics in the previous developm
ent. We can see that only w
hen taking the first three harmonics into
account can the saddle shape be found. Apart from
the shape of the signal, the difference lies in the frequential content of the excitation, a fundam
ental aspect when calculating response.
F
igure 2.3: walking – vertical com
ponent (fm = 2 H
z) It should be noted that, in the annex to E
urocode 1 covering dynamic m
odels of pedestrian loads, the recom
mended load w
as (DL
M1):
)2
sin(280
tf
Qv
pv
which corresponds, in fact, to 0
.4 G
0 with G
0 = 700 N
, the weight of the pedestrian. T
his was,
therefore, the first term (dynam
ic) of Fourier's transform
. This m
odel has since been deleted from
the Eurocode, but its physical reality rem
ains no less relevant.
2.1.2 - Horizontal component
The horizontal com
ponent of the load is, admittedly, of less intensity, but cannot, how
ever, be neglected as it can be a source of problem
s. It is apparent that people are very sensitive to being m
oved sideways and w
alking is rapidly disturbed, leading, for example, to keeping
close to the handrail, to a feeling of insecurity or even the closing of the footbridge, etc. A
s announced previously, we w
ill investigate successively being moved transversely and
longitudinally (although longitudinal movem
ent is more rarely a problem
on footbridges). The
transverse component, corresponding to changing from
one foot to the other when w
alking, occurs, therefore, at a frequency of half that of the frequency of w
alking (1 Hz for fm
= 2 H
z). O
n the other hand, the longitudinal component is m
ainly linked to the frequency of walking
(see figure 2.4). In order to present F
ourier's transform of the transverse com
ponent (at the frequencies fm/2
, fm,
3fm
/2) according to the basic frequency of w
alking fm , the solution generally used is to modify
the presentation in the following form
:
tif
Gt
Fm
i
n
i
2sin
)(
2/1
i therefore being able to have (non-whole) values of 1/2, 1, 3/2, 2, etc. In addition, as the
phase shifts are close to 0, they do not therefore appear in the previous expression. Finally, as
opposed to the vertical force, the transverse and longitudinal components do not have, of
course, a static part (no constant term in the expression of F
(t)).
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F
igure 2.4:
changes in
the vertical
force (a),
the transverse (b)
and longitudinal
(c) com
ponents of
the displacem
ent (and of the acceleration) measured – w
alking at 2 Hz R
ef. [22]
As indicated above, tests have show
n that the main am
plitudes of this component are located
at a frequency of about half that of the vertical component, a frequency corresponding to the
lateral oscillations of the centre of gravity of the body when w
alking. The corresponding
values of the Fourier coefficients are thus as follow
s:
G1
/2 = G
3/2
0.05 G0 ; G
1 = G
2
0.01 G0 .
As for the vertical com
ponent (G0 =
700 N, fm =
2 Hz), w
e have marked on figure 2.5 the
transverse component, taking into account from
one to four harmonics. A
s expected, it can be seen that, on the one hand, the frequency is half that of the vertical com
ponent (in other words
the period is double) and, on the other hand, that taking into account the first four terms gives
an appearance close to those measured in figure 2.4 (b).
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F
igure 2.5: walking – transverse com
ponent (fm = 2 H
z)
The m
ain frequency associated with this com
ponent is approximately the sam
e as for the vertical com
ponent (fm = 2 H
z). Its oscillations correspond, for each step, initially to the contact
of the
foot w
ith the
ground, then
to the
thrust exerted
subsequently. F
or this
component, the values of the F
ourier coefficients are:
G1
/2
0.04 G0 ; G
1 0.2
G0 ; G
3/2
0.03 G
0 ; G2
0.1 G0 .
We have show
n in figure 2.6, under the same conditions as for the other com
ponents (G0 =
700 N
, fm = 2 H
z), the change in this component, depending on the num
ber of harmonics
taken into account, the previous comm
ents remaining valid.
F
igure 2.6: walking – longitudinal com
ponent (fm = 2 H
z) It should be noted that, in practice, the longitudinal com
ponent of the walking force of a
pedestrian has, in general, little influence on most footbridges. T
he influence is greater when
the footbridge bears on overhanging, flexible piers, for which the longitudinal com
ponent of the pedestrian leads to the bending of the pier.
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2.2 - Running
As indicated in the introduction, running is characterised by a discontinuous contact w
ith the ground (see figure 2.7 (a)), the frequency fm generally being betw
een 2 and 3.5 Hz.
2.2.1 - Vertical component
In an initial approximation, the vertical com
ponent of the load can be approached by a simple
sequence of semi-sinusoids, represented using the follow
ing expression: pm
mp
pt
Tj
tT
jfo
rt
tG
kt
F)1
()1
( )
sin()
(0
mp
mjT
tt
Tj
for
tF
1
0)
(
with:
kp
: the impact factor (k
p = F
ma
x / G0 ),
j
: the step number (j =
1, 2, etc.),
Fm
ax
: the maxim
um load,
G
0 : the w
eight of the pedestrian,
tp : the period of the contact,
T
m
: the period (Tm =
1
/fm , noted T
p on fig.3.4(b)), fm being the frequency of
running.
F
igure 2.7: running – change in the force according to time (a); im
pact factor according to the relative period of contact (b) R
ef [3] T
he period of contact tp in this m
odel is simply the half-period
Tm , w
hich enables the expression of F
(t) to be written also in the follow
ing form:
mm
mp
Tj
tT
jfo
rt
fG
kt
F2 1
)1(
)2
sin()
(0
mm
jTt
Tj
for
tF
2 1
0)
(
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This approxim
ation of the period of contact tp is an over-estimation of the values m
easured experim
entally, which are represented according to the frequency of running on the graph in
figure 2.8. As for the values of the im
pact factor kp , these are inferred from
figure 2.7 (b), according to the relative period of contact (tp /T
m ). As an exam
ple, running at a frequency of 3 H
z (thus a period Tm =
0.33) gives tp = 0.17 s and k
p = 3 w
hen tp /Tp =
0.5 (instead of kp =
2.4 w
hen tp /Tp =
0.61). As announced, w
e can therefore see from this exam
ple that the semi-
sinusoidal approximation is conservative.
It is also possible to use, for running, a Fourier transform
, which has the advantage of not
revealing explicitly the impact factor k
p , the determination of w
hich is delicate. In order to m
ake allowance for the natural discontinuous contact w
hen running, only the positive part of the transform
is retained, which m
ay be written, w
ith the previous notations:
mm
mi
ni
Tj
tT
jfo
rt
ifG
Gt
F)2 1
()1
( 2
sin)
(1
0
mm
jTt
Tj
for
tF
)2 1(
0)
(
In this case, the phase shifts are assumed to be negligible and the am
plitudes of the first three harm
onics are average values, these coefficients being, in strict logic, as kp , a function of the
frequency of running (figure 2.9) : G
1 = 1.6 G
0 ; G2 =
0.7 G0 ; G
3
0.2 G0 .
Figure 2.8: C
hange in the period of contact F
igure 2.9: Am
plitude of the various according to frequency
harmonics
On figure 2.10 are represented (still over 1 second w
ith G0 =
700 N, but at a frequency fm =
3 H
z), on the one hand, a semi-sinus approxim
ation and, on the other hand, the Fourier series
transform, taking into account 1 to 3 harm
onics. We can see that the various graphs are close.
More precisely, a sem
i-sinus approximation leads to a w
eaker amplitude signal (w
ith the value of k
p used) than taking only one harmonic into account in the F
ourier series.
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F
igure 2.10: running – vertical component (fm =
3 Hz)
This load according to tim
e can be broken down into a F
ourier series, revealing a constant section at about 1250 N
and a varying section of the same frequency as the running for the
first harmonic and of am
plitude 1250 N. It is this value that could be used if a specific
analysis of runners had to be made. T
his guide does not cover specific load cases of runners on a footbridge, as it is considered that the effects of a crow
d of pedestrians are clearly less favourable.
2.2.2 - Horizontal component
To our know
ledge, no measurem
ents have been taken of the horizontal component during a
running race, either of its longitudinal projection, or of its transverse projection. How
ever, it is reasonable to think, on the one hand, that, during a race, the transverse com
ponent (that to w
hich the public is most sensitive) has a low
relative amplitude com
pared with the vertical
component (the race appearing to be a m
ore "directed" progress), whereas the longitudinal
component w
ill be larger (larger motor force). In addition, as for w
alking, it is logical to estim
ate that the frequency of the transverse component w
ill be half the vertical component,
whereas that of the longitudinal com
ponent will be of the sam
e order of magnitude.
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3. Appendix 3: Damping system
s
3.1 - Visco-elastic dampers
The use of visco-elastic m
aterials for controlling vibrations dates back to the 1950s, when
they w
ere used
to lim
it fatigue
damage
caused by
vibrations on
aircraft fram
es. T
he application to civil engineering constructions dates from
the 1960s.
3.1.1 - Principle
The visco-elastic m
aterials used are typically polymers that dissipate energy by w
orking in shear. F
igure 3.1 shows a visco-elastic dam
per formed from
layers of visco-elastic materials
between m
etal plates. When this type of device is installed on a construction, the relative
displacement of the outer plates in respect of the central plate produces shear stresses in the
visco-elastic layer, which dissipates the energy.
F
igure 3.1 – Visco-elastic dam
per U
nder a harmonic load of frequency
, the strain )
(t and the shear stress
)(t
are also at the frequency
, but, in general, out of phase:
(Eq
. 3.1
.))
sin()
(
)sin(
)(
0 0
tt
tt
The param
eters 0
0 , and
depend, in general, on the frequency . T
he shear stress may,
however, be re-w
ritten:
)cos(
)(
)sin(
)(
)cos(
sin)
sin(cos
)(
21
0
0 0
0 00
tG
tG
tt
t
(Eq
. 3.2
.)
By replacing the term
s )
sin(t
and )
cos(t
by 0
/)(t
and )
/()
(0
t, the follow
ing stress-strain ratio is obtained:
)
()
()
()
()
(2
1t
Gt
Gt
(E
q. 3
.3.)
which defines an ellipse (F
igure 3.2), the area of which is the energy dissipated by the
material per unit of volum
e and per cycle of oscillation.
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F
igure 3.2 – Stress-strain diagram
T
his energy is given by:
(Eq
. 3.4
.)sin
)(
)(
00
/20
dt
tt
E
The equation (3
.3) reveals an initial term
in phase with the displacem
ent, representing the elastic m
odulus; the second term describes the dissipation of energy. T
he damping
/)(2
G
leads to expressing the damping factor by:
2
tan
)(
2
)(
1
2
G
G
(Eq
. 3.5
.)
The param
eters )
(2G
and tan
determine com
pletely the behaviour of the visco-elastic dam
per under harmonic excitation. T
he term
tan is called the d
issipatio
n fa
ctor. T
hese factors vary not only according to the frequency, but are also function of the am
bient tem
perature. As the dissipation of energy is in the form
of heat, the level of performance of
the visco-elastic dampers m
ust also be evaluated in relation to the variation in internal tem
perature that appears when they are w
orking.
3.1.2 - Layout and design of dampers
The layout of the dam
pers is an essential parameter in their effectiveness in relation to the
dissipation of energy. How
ever, technical constraints may prevent the m
ost appropriate places from
being selected. The effectiveness of a visco-elastic dam
per is measured by its capacity to
work in shear; it is, therefore, recom
mended that the dam
pers be set out in such a way that
this condition can be checked. T
his type of damper has been little used for footbridges.
3.2 - Viscous dampers
3.2.1 - Principle
Dry or visco-elastic friction dam
pers use the action of solids to dissipate the vibratory energy of a construction (F
igure 3.3). It is, however, also possible to use a fluid.
The first device to com
e to mind is one derived from
a dashpot. In such a device, the dissipation takes place by the conversion of the m
echanical energy into heat, using a piston that deform
s a very viscous substance, such as silicone, and displaces it. Another fam
ily of dam
pers is based on the flow of a fluid in a closed container. T
he piston is not limited to
deforming the viscous substance, but forces the fluid to pass through calibrated orifices, fitted
or not with sim
ple regulation devices. As in the previous case, the dissipation of the energy
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leads to heat being given off. Very high levels of energy dissipation can be reached, but
require adequate technological devices. T
he main difference betw
een these two technologies is as follow
s. In the case of a pot or wall
damper, the dissipative force w
ill depend on the viscosity of the fluid, whereas, in the case of
an orifice damper, this force is due m
ainly to the density of the fluid. Orifice dam
pers will
therefore be more stable in respect of variations in tem
perature than pot dampers or w
all dam
pers. In the case of pedestrian footbridges, the m
ovements are very w
eak and dampers m
ust be selected that w
ill be effective even for small displacem
ents, in the order of one millim
etre. B
ecause of the compressibility of the fluid, friction in the joints, and tolerance in the fixings,
this is not easy to achieve. V
iscous dampers m
ust be located between tw
o points on the construction having a differential displacem
ent in respect of each other. The larger these differential displacem
ents, the more
effective the damper w
ill be. In practice, these dampers w
ill be located either on an element
connecting a pier and the deck a few m
etres away from
the pier (for horizontal or vertical vibrations), or on the horizontal bracing elem
ents of the deck (for horizontal vibrations).
F
igure 3.3 – Exam
ple of viscous damper
3.2.2 - Laws of behaviour of viscous dampers
As part of the study of the behaviour of a construction, it is necessary to have a m
acroscopic behaviour m
odel of the damper. F
or this, it is traditional to use a force-displacement law
as described by a differential equation of order
(Ref. [55]) :
)
()
()
(0
tdt
dx
Ct
dt
fd
tf
(E
q. 3
.6.)
in which
is the force applied to the piston, and the resultant displacem
ent of the piston. T
he parameters
)(t
f)
(tx
,,0
C represent respectively the dam
ping factor at zero frequency, the relaxation tim
e and the order of the damper. T
hese parameters are generally determ
ined experim
entally, although approximations do exist to estim
ate them analytically on the basis of
the characteristics of the materials (R
ef. [56]). N
otes: W
hen 0
, the case for a linear viscous damper is m
ade. It should be pointed out that it is often
the m
ost-used, as,
obviously, it
simplifies
an analysis
of the
behaviour of
the construction. A
nother formulation m
ay be used:
)
()
(t
dt
dx
Ct
f
(Eq
. 3.7
.)
is a coefficient generally between 0.1 and 0.4.
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Note that, for certain fluids, the kinem
atic viscosity also varies according to the rate of shear of the fluid, w
hich is itself proportional to the speed gradient. V
iscous dampers w
ere installed on the Millennium
footbridge in London.
3.3 - Tuned mass dam
pers (TMD)
A tuned m
ass damper (abbreviated to T
MD
) comprises a m
ass connected to the construction by a spring and by a dam
per positioned in parallel. This device enables the vibrations in a
construction in a given vibration mode, under the action of a periodic excitation of a
frequency close to the natural frequency of this mode of vibration of the construction, to be
reduced substantially. T
he first developments of T
MD
s applied essentially to mechanical system
s for which a
frequency of excitation is in resonance with the basic frequency of the m
achine.
3.3.1 - Principle
Consider an oscillator w
ith 1 degree of freedom, subjected to a harm
onic force . T
he response of this oscillator m
ay be reduced in amplitude by the addition of a secondary m
ass (or T
MD
) that has a relative movem
ent in respect of the primary oscillator. T
he equations describing the relative displacem
ent of the primary oscillator in respect of the T
MD
:
)(t
f
(Eq
. 3.8
.))
()
()
()
(
)(
)(
)(
)(
)(
)(
12
22
22
11
1
ty
mt
yk
ty
ct
ym
tf
ty
kt
yc
ty
Kt
yC
ty
M
which is reduced to the single equation:
)
()
()
()
()
(2
11
1t
ft
ym
ty
Kt
yC
ty
mM
(E
q. 3
.9.)
3.3.2 - Den Hartog's solution
In the case where the prim
ary oscillator has zero damping (
0C
), it is possible to determine
the optimum
characteristics of the TM
D for a harm
onic excitation of am
plitude and
of pulsation )
(tf
0 f
. This optim
um solution is the solution deduced by D
en Hartog in his first
studies. For a harm
onic excitation, the impact factor A
, defined as the ratio between the
maxim
um dynam
ic amplitude
and the static displacement
, is expressed by: m
ax1
ystat
1y
2
22
2ada
22
22
22
2ada
22
2
stat1 m
ax1
12
1
2
y yA
(Eq
. 3.1
0.)
where:
osc
;osc
ada;
m k2ada
(Eq
. 3.1
1.)
M K2osc
;ada
ada2
m
c;
M m
(Eq
. 3.1
2)
The dynam
ic amplification factor A
therefore depends on four parameters:
ad
a,
,,
. Figure
3.4 gives the changes of A according to the frequency ratio
, for various damping factors and
for 1
(agreement of frequencies) and
05,0
. This figure show
s that, for the case where
the damping
ad
a is zero, the response amplitude has tw
o resonance peaks. Contrary to w
hen
the damping tends tow
ards infinity, the two m
asses are virtually fused, so as to form a single
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oscillator of mass
with an infinite am
plitude at the resonance frequency. Betw
een these extrem
e cases, there is a damping value for w
hich the resonance peak is minim
al. M
05,1
F
igure 3.4 – Impact factor depending on
(
1,0et
05,0
) T
he objective of a tuned mass dam
per consists, therefore, of bringing the resonance peak dow
n as low as possible. F
igure 3.4 shows the independence in relation to
ad
a of two points
(P and Q
) on the graphs f
A. T
he minim
um am
plitude of the resonance is thus obtained by selecting the ratio
such that these two points are of equal am
plitudes. This optim
um
ratio is thus given by the expression:
M
mopt
/1
1
(Eq
. 3.1
3)
and the amplitudes at points P
and Q are:
2
1R
(E
q. 3
.14
)
The optim
um frequency
and the optimum
damping factor
opt
fopt can be determ
ined for the
tuned mass dam
per (1), in the case w
here the construction does not have its own dam
ping: this is D
en-H
arto
g's so
lutio
n (R
ef. [51]):
s
opt
ff
1
1
(Eq
. 3.1
5)
3
)1(
8 3opt
(E
q. 3
.16
)
3.3.3 - General solution
A m
ore detailed analysis has been carried out by Warburton (R
ef. [54]) in order to determine
the optimum
characteristics of the TM
D for several types of excitations (in the presence of
weak structural dam
ping). The analysis proposed by W
arburton also enables the results to be
(1 ) T
he optimum
frequency is very slightly less than that of the construction, the added mass
always being in the order of a few
hundredths of the mass of the construction.
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used for primary oscillators w
ith one degree of freedom, to cases w
ith several degrees of freedom
. These analogies are based on a breakdow
n on the basis of the standardised natural m
odes of the construction. Optim
ality is not therefore sought on a given mode (m
odal com
ponent), but on a generalised coordinate (displacement or derivatives). T
he reader is therefore referred to reference 54 for further details on these analogies. In addition, this section w
ill be limited to dealing only w
ith the case of oscillators with 1 degree of freedom
. T
he optimum
characteristics of Den H
artog's tuned mass dam
per for a harmonic excitation
have been established, minim
ising the impact factor
A. T
his is the very principle of the determ
ination of an optimum
TM
D: first defining a criterion of optim
ality A, then seeking the
solutions minim
ising this criterion. Apart from
the impact factor, num
erous other criteria can be
selected, such
as the
minim
isation of
the displacem
ent of
the construction,
of the
displacement of the dam
per, of the acceleration of the construction or of forces in the construction, etc., all for various types of excitations. In particular, w
hen the excitation is random
(the primary system
is subjected to a random force – in other w
ords, in our case, the crow
d), the optimum
parameters
and opt
opt are given by the expressions:
1
21
opt
(E
q. 3
.17
)
21
14
4 31
opt
(E
q3
.18
)
3.3.4 - Case of a damped prim
ary oscillator
In the case where the construction has natural dam
ping, the theoretical optimum
frequency is very slightly w
eaker than that given by the above formula (valid for a harm
onic excitation), w
hich is nevertheless sufficient in usual cases (Ref. [53]) :
2
22
22
2
39,
001,0
4,0
12,0
13,0
~9,
10,
16,
27,
1241
,0~
osc
osc
opt
opt
osc
osc
opt
opt
(Eq
. 3.1
9)
in which
osc is the dam
ping factor of the primary m
ass. These expressions show
less than 1%
error for 40
.003.0
and 15.0
0.0
osc
. T
he effectiveness of a tuned mass dam
per is much m
ore sensitive to the natural frequency of the added m
ass, than to the value of the added damping. T
his is why it m
ust be possible to adjust the natural frequency of the m
ass when fixing the device, so as fine-tune it to the actual
natural frequency of the construction. This is generally done by varying the value of the m
ass itself, w
hich is much easier to adjust than the stiffness of the spring.
3.3.5 - Examples
There are various types of tuned m
ass dampers. T
he most typical type consists of a m
ass connected to the construction by m
eans of vertical coil springs and one or more hydraulic or
pneumatic dam
pers. These T
MD
s may be linked together to dam
p torsion vibrations. Figure
3.5 gives an overview of such a device.
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F
igure 3.5 – Tuned m
ass damper for the S
olferino footbridge W
hen the vibration to be damped is horizontal, a device consisting of a m
ass attached to the bottom
of a pendulum, associated w
ith a horizontal hydraulic damper, m
ay be considered.
3.4 - Tuned liquid dampers
In the technology of tuned mass dam
pers, a second mass is attached to the construction using
struts and dampers (figure 3.6a). A
nother category of dampers consists of replacing the m
ass, strut and dam
per by a container filled with a liquid. A
s for a traditional TM
D, the liquid acts
as the secondary mass and the dam
ping is provided by friction with the w
alls of the container. A
s the action of gravity comprises a recall m
echanism, the secondary system
(figure.3.6b) thus form
ed has characteristic frequencies that can be tuned to optimise one perform
ance criterion. T
he first TL
D prototype w
as proposed in the 1900s by Frahm
to control rolling in ships. Since
the 1970s, these dampers have been installed on satellites to reduce long period vibrations.
But it has only been since the 1980s that building applications have been considered.
Fk c
C
M
m
2 K
2 K
(a) T
MD
with m
ass
F
C
M
2 K
2 K
1,
,cm
(b) T
LD
F
igure 3.6 – Com
parison between T
MD
with m
ass and TL
D w
ith fluid T
he principles used for the sizing of TM
Ds also apply to T
LD
s. How
ever, although the param
eters of a TM
D can be optim
ised and analytical formulae provided, the non-linear
response of the fluid in movem
ent in a container makes such optim
isation very difficult. The
response of the "container/construction" system is thus dependant on the am
plitude of the m
ovements.
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h
a2
bh
Boundary layer
z
x
sx
F
igure 3.7 – Description of a T
LD
L
et us consider a rectangular container of length 2a filled w
ith a liquid of viscosity and of
average depth h. The fluid is assum
ed to be incompressible and irrotational. T
he container is subjected to a horizontal displacem
ent x(t). It is assum
ed that the free surface remains
continuous (no waves breaking) and that the pressure
is constant over this free surface (figure 3.7). A
ccording to the linear theory of the boundary layer, the natural vibration frequency of the fluid is:
),
,(
tz
xp
tanh
22
ad
af
gh
aa
; (E
q. 3
.20
)
The dam
ping factor may be approached by the expression (R
ef. [57]) :
b h
hadaf
adaf
12 1
; (E
q. 3
.21
)
in which
is the width of the tank. T
he equations of the coupled system are w
ritten in exactly the sam
e way as equations (3
.8.) and (3
.9.) w
ith b
ab
hm
adaf 2
, adaf
adaf
mc
2,
,
in other words:
2adaf
km
00
02
0
02
1
2 1
2
2
2 1
2 1m
tf
y y
y y
y y
adaf
osc
adaf
adaf
osc
osc
; (Eq
. 3.2
2)
This equation show
s the analogy that can exist between a tuned m
ass damper and a tuned
liquid damper. T
his analogy only makes sense subject to the validity of the expressions (3.20)
and (3.21) selected as equivalent pulsation and damping factor. In reality, the resolution of the
coupled problem is m
ore complex due to the fluid nature of the secondary oscillator.
As opposed to num
erous high-rise towers, to our know
ledge, no footbridge has had tuned fluid dam
pers installed. They w
ere, however, considered for the Ikuchi bridge (Japan), in
order to dampen the horizontal vibrations of the pylons. T
he total fluid mass w
as 4770 kg and the tuning frequency w
as 0.255 Hz.
3.5 - Comparative table
Type of dam
per F
ield of use A
dvantages D
isadvantages
Visco-elastic
Very little used
Dam
ps several m
odes N
eeds to be fixed to work in
shear
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Viscous pot or
wall dam
pers L
ittle used D
amps several
modes
Tem
perature-sensitive, non-linear calculation
Viscous orifice
dampers
Little used
Independent of tem
perature, damps
several modes
Non-linear calculation
Tuned m
ass dam
pers W
idely used E
asy to size A
dditional mass to be
considered, damps a given
mode, needs adjustm
ent in frequency
Tuned liquid dam
pers V
ery little used
"Innovative", additional mass to
be considered, damps a given
mode, needs adjustm
ent in frequency
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4. Appendix 4: Examples of footbridges
We w
ill be reviewing several different designs of footbridges recently constructed, from
the m
ain types of constructions. We w
ill be presenting a construction with lateral beam
s, a steel box-girder construction w
ith orthotropic deck, a construction with a ribbed slab, a bow
-string arch w
ith orthotropic deck, a suspended construction with one steel m
ast, a steel lattice arch, a cable-stayed construction. W
e will indicate the m
ain features of each of the bridges and its m
ethod of construction, and we w
ill give the results of the dynamic studies and, as applicable,
the results of tests.
4.1 - Warren-type lateral beam
s: Cavaillon footbridge
The bridge is a W
arren-type construction with lateral beam
s 3.20 m high; it com
prises two
independent bays of 49.7 m span. T
he deck consists of a reinforced concrete slab bearing on floor beam
s 2.26 m apart. T
he functional width is 3.00 m
. The w
orks were carried out in 2000
- 2001. The steel construction w
as erected with a crane, the m
ass of the framew
ork is 18.8 t per bay. A
modal analysis m
ade using an analytical calculation gave the following results:
Mode 1
Vertical bending
1.95 Hz
This frequency w
as calculated without taking the m
ass of the pedestrians into account.
P
hotograph 4.1 – Cavaillon footbridge
4.2 - Steel box-girder: Stade de France footbridge
The bridge is a box-girder bridge w
ith double intermediate supports, of a total length of
180 m, w
ith a central bay of 64 m span, and end bays of 54 and 50 m
. The structure of the
deck is formed from
a steel box girder with orthotropic topping. T
he functional width is 11.00
m. T
he works w
ere carried out in 1997 - 1998. The construction w
as erected with a crane; the
total mass of the fram
ework is 595.0 t.
The m
odal analysis carried out using finite element design softw
are gave the following
results:
Mode 1
Vertical bending central bay
1.97 Hz
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Mode 2
Vertical bending of end bay 1
2.06 Hz
Mode 3
Vertical bending of end bay 2
2.20 Hz
The dynam
ic study gave the following results:
1 pedestrian deflection 2.7 m
m
acceleration 0.4 m/sec
2
640 pedestrians 25 in phase deflection 70 m
m
acceleration 10 m/sec
2
The dynam
ic study with dam
per of 2.4 t mass:
640 pedestrians 25 in phase deflection 6.5 m
m
acceleration 1 m/sec
2
P
hotograph 4.2 – Stade de F
rance footbridge
4.3 - Ribbed slab: Noisy-le-Grand footbridge
The bridge is a ribbed slab in pre-stressed concrete of varying depth of a total length of 88 m
, w
ith a central bay of 44 m span, and end bays of 22 m
. The depth of the slab varies from
1 m
at the summ
it to 3.05 m at the piers. T
he functional width is 5 m
. The w
orks were carried out
in 1993 - 1994. The tw
o beams com
prising the structure were constructed perpendicularly to
their final position and positioned over the A4 m
otorway by rotation; the total m
ass of the deck is 860.0 t. T
he modal analysis carried out using finite elem
ent design software gave the follow
ing results:
Mode 1
Vertical bending
1.65 H
z
Mode 2
Axi-sym
metrical bending 3.48 H
z
Mode 3
Lateral bending
4.86 H
z
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P
hotograph 4.3 – Noisy-le-G
rand footbridge
4.4 - Bow-string arch: Montigny-lès-Corm
eilles footbridge
The bridge is a bow
-string with polygonal arch of 55 m
span. The structure of the deck
comprises tw
o lateral steel tubes connected by an orthotropic topping. The functional w
idth is 3.50 m
. The w
orks were carried out in 1998 - 1999. T
he construction was erected w
ith a crane; the total m
ass of the framew
ork is 85.0 t. A
modal analysis carried out using finite elem
ent calculation software gave the follow
ing results, w
ith the tubes of the tie filled with cem
ent, and the mass of the bridge being increased
by the mass of the pedestrians corresponding to a(l):
Mode 1
Vertical bending
2.51 H
z
Mode 2
Anti-sym
metrical bending 2.52 H
z
Mode 3
Lateral bending of deck
2.62 Hz
Com
ment: w
ithout the pedestrians, the frequencies increase by 0.5 Hz.
P
hotograph 4.4 – Montigny-lès-C
ormeilles footbridge
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4.5 - Suspended construction: Footbridge over the Aisne at Soissons
The bridge is a suspension bridge of 60 m
span. The structure of the deck is form
ed from a
steel box-girder suspended from a m
ast consisting of four inclined tubes joining at mid-span.
The functional w
idth is 3.00 m. T
he works w
ere carried out in 2000. The structure w
as erected w
ith a crane on temporary legs before the m
ast was erected; the total m
ass of the steel fram
ework is 105.0 t.
The m
odal analysis carried out using finite element design softw
are gave the following
results:
Mode 1
Lateral bending of deck
1.10 H
z
Mode 2
Lateral bending of the arches
2.80 Hz
Mode 3
Vertical bending of the deck
3.10 Hz
P
hotograph 4.5 – Soissons footbridge
4.6 - Steel arch: Solferino footbridge
The bridge is a steel arch of 106 m
span. The structure of the arch is form
ed from tw
o double parabolic arches as a ladder beam
, linked by cross-pieces supporting a lower deck. T
he double upper deck is supported on A
-frames and braces bearing on the tw
o arches. The functional
width varies from
12 m to 14.80. T
he works w
ere carried out in 1998 - 1999. The structure
was erected w
ith a crane on temporary legs; the total m
ass of the framew
ork is 900.0 t. T
he modal analysis carried out using finite elem
ent design software gave the follow
ing results:
Mode 1
Lateral sw
ing 0.71 H
z
Mode 2
Axi-sym
metrical vertical bending 1.03 H
z
Mode 3
Torsion-bending
1.37 Hz
Mode 4
Vertical bending
1.66 Hz
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Mode 5
Torsion-sw
ing 1.66 H
z
The dynam
ic tests and measurem
ents gave the following frequencies em
pty (without dam
per):
Mode 1
Lateral sw
ing 0.81 H
z
Mode 2
Axi-sym
metrical vertical bending 1.22 H
z
Mode 3
Torsion-bending
1.59 Hz
Mode 4
Vertical bending
1.69 Hz
Mode 5
Central torsion-sw
ing 1.94 H
z
Mode 6
Central torsion-sw
ing 2.22 H
z
Mode 7
Bending-torsion
3.09 Hz
The dam
ping for the empty structure varied from
0.3% to 0.5%
. T
he dynamic tests and m
easurements gave the follow
ing accelerations:
16 pedestrians balance mode 1
acceleration 0.5 m/sec
2
16 pedestrians walking m
ode 6 acceleration 2.0 m
/sec2
16 pedestrians running mode 7
acceleration 2.5 m/sec
2
The dam
ping with 16 pedestrians on the footbridge varied from
0.4% to 0.8%
.
106 pedestrians balance mode 1
acceleration 1.5 m/sec
2
106 pedestrians running mode 7
acceleration 5.7 m/sec
2
The dam
ping with m
ore than 100 pedestrians on the footbridge varied from 0.7%
to 1.6%.
Dam
pers
6 pendular systems supporting m
asses of 2.5 t and 1.9 t allow dam
ping of 3.9% in respect of
mode 1 (lateral sw
ing) to be achieved.
8 mass/spring system
s supporting masses of 2.5 t allow
damping of 2.75%
in respect of m
odes 5 and 6 (central torsion-swing) to be achieved.
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Photograph 4.6 – S
olferino footbridge
4.7 - Cable-stayed construction: Pas-du-lac footbridge at St Quentin
The bridge is a dissym
metric cable-stayed bridge of a total length of 188 m
, with, on one side,
two bays w
ith spans of 68 m and 36 m
, and, on the other, two bays, each of 42 m
span. The
structure of the deck is formed from
two steel beam
s, suspended from a single pylon, linked
by floor beams. T
he functional width is 2.50 m
. The w
orks were carried out in 1991 - 1992.
The construction w
as positioned with a crane.
The m
odal analysis carried out using finite element design softw
are gave the following
results:
Mode 1
Lateral bending of deck
1.38 H
z
Mode 2
Displacem
ent-bending of deck 1.85 H
z
Mode 3
Bending of pylon
1.92 Hz
Mode 4
Vertical bending of the deck
1.95 Hz
P
hotograph 4.7 – Pas-du-L
ac footbridge
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4.8 - Mixed construction beam
: Mont-Saint-M
artin footbridge
The bridge has a m
ixed steel-concrete deck with a span of 23 m
. The structure of the deck is
formed from
two slightly cam
bered steel beams linked by floor beam
s. The functional w
idth is 2.50 m
. The w
orks were carried out in 1996. T
he structure was erected w
ith a crane; the m
ass of the framew
ork is 22.0 t. A
modal analysis m
ade using an analytical calculation gave the following results:
Mode 1
Vertical bending
2.15 Hz
Mode 2
Vertical bending
3.99 Hz
Mode 3
Lateral sw
ing 4.50 H
z
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5. Appendix 5: Examples of calculations of footbridges.
This section sets out a full study of tw
o standard footbridges based on actual examples,
together with a sensitivity study of the natural frequencies of typical footbridges.
The
two
complete
examples
of dynam
ic calculation
have been
carried out
using the
methodology of the guide, taking into account the various categories of traffic. If the results
led to unacceptable accelerations, the characteristics of these constructions were m
odified to im
prove their dynamic behaviour and to try and satisfy com
fort conditions.
5.1 - Examples of com
plete calculations of footbridges
5.1.1 - Warren-type lateral beam
footbridge
The first footbridge studied is a construction w
ith a mixed steel-concrete fram
ework form
ing one independent bay w
ith a span of 38.85 m. T
he longitudinal profile is curved to a radius of 450 m
etres.
38,85 m
F
igure 5.1: Warren-type lateral beam
footbridge T
he framew
ork is formed from
two triangulated lateral beam
s. These beam
s, of a constant depth of 1.215 m
, are linked by floor beams located at the level of the bottom
mem
ber. A pre-
cast reinforced concrete slab, 10 cm thick, bears on these floor beam
s.
mem
bers 400
20012
2.90 m
Useful w
idth 2.50 m1.215 m
F
igure 5.2: Cross-section of the lateral beam
footbridge T
he distance between the centre lines of the beam
s is 2.90 m, giving a w
idth of passage for pedestrians of 2.50 m
. C
haracteristics of the deck
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The m
oment of inertia is calculated taking into account the reinforced concrete slab w
ith a hom
ogenisation coefficient of 6, and the mass of the deck is calculated taking into account the
floor beams and the reinforced concrete slab.
Mom
ent of inertia of the deck: 4
030,0
mI
Natural linear density of the deck:
mkg
m/
1456
Young's m
odulus of the steel: 2
9/
10
210m
NE
First of all, w
e will consider class III, in other w
ords a normally-used footbridge that can
sometim
es be crossed by large groups, but never loaded over its whole area.
5.1
.1.1
.1
Ca
lcula
tion
of n
atu
ral m
od
es
The natural frequencies are equal to:
S
EI
L
nf
n2
2
2
S is the linear density of the deck increased by the linear density of the pedestrians, w
hich is calculated for each crow
d density, depending on the class of the footbridge. F
or class III, we are interested in a sparse crow
d, where the density d of the crow
d is equal to 0.5 pedestrians/m
2. T
he number of pedestrians on the footbridge is:
6.48
5.2
85.
385.
0p
n
The total m
ass of the pedestrians is: kg
34026.
4870
The linear density of the pedestrians is:
mkg
mp
/ 6
.87
85.
38/
3402
The linear density is:
mkg
S/
6.
15346.
871456
F
or the first mode, the high-bond and low
bond frequencies are equal to:
Hz
f14.2
1456
030.0
10210
85.
382
19
2
2
1
H
zf
02.2
8.1630
030.0
10210
85.
382
19
2
2
1
For the second m
ode, the high-bond and low bond frequencies are equal to:
Hz
f
54.8
1456
030.0
10210
85.
382
29
2
2
2
H
zf
08.8
8.1630
030.0
10210
85.
382
29
2
2
2
Only the first m
ode is likely to cause uncomfortable vibrations.
5.1
.1.1
.2
Calcu
latio
n o
f the d
ynam
ic load o
f the p
edestria
ns
We w
ill calculate the load for the first mode only, w
ith a critical damping ratio of 0.6%
(m
ixed deck). T
he surface load to be taken into account for the vertical modes is:
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nt
fN
dF
vs
8.10
2cos
280
is equal to 1, as the frequency of the first mode, w
hich is 2.08 Hz, is w
ithin range 1 (1.7 to 2.1 H
z) with a m
aximum
risk of causing resonance.
We have
120.0
6.48 100
/6.0
8.10
8.10
n
The surface load is equal to:
2/
08.2
2cos
8.16
1120.0
08.2
2cos
2805.
0m
Nt
tF
s
The linear load is equal to:
mN
tt
lF
Fp
s/
08.2
2cos
0.42
08,2
2cos
5.2
8.16
T
his load is applied to the whole of the footbridge.
5.1
.1.1
.3
Ca
lcula
tion
of d
yna
mic resp
on
ses
The calculation of the acceleration to w
hich the construction is subjected gives:
2m
ax/
91.2
6.1534 0.42
4100/6
.02
1s
mA
cc
The m
aximum
acceleration calculated is located within range 4 of the accelerations, in other
words at an unacceptable com
fort level (acceleration > 2.5 m
/sec2).
We w
ill next consider class II, in other words an urban footbridge linking populated zones,
subjected to a high level of traffic and likely, on occasions, to be loaded over its whole area.
5.1
.1.2
.1
Ca
lcula
tion
of n
atu
ral m
od
es
For class II, w
e are interested in a dense crowd, w
here the density d of the crowd is equal to
0.8 pedestrians/m2.
The num
ber of pedestrians on the footbridge is: 78
7.77
5.2
85.
388.
0p
n
The total m
ass of the pedestrians is: kg
546078
70
The linear density of the pedestrians is:
mkg
mp
/ 5
.140
85.
38/
5460
The linear density is:
mkg
S/
5.
15965.
1401456
F
or the first mode, the high-bond and low
bond frequencies are equal to:
Hz
f14.2
1456
030.0
10210
85.
382
19
2
2
1
H
zf
02.2
8.1630
030.0
10210
85.
382
19
2
2
1
For the second m
ode, the high-bond and low bond frequencies are equal to:
Hz
f
54.8
1456
030.0
10210
85.
382
29
2
2
2
H
zf
08.8
8.1630
030.0
10210
85.
382
29
2
2
2
Only the first m
ode is likely to cause uncomfortable vibrations.
5.1
.1.2
.2
Calcu
latio
n o
f the d
ynam
ic load o
f the p
edestria
ns
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With 0.6%
of critical damping, as in the previous case, w
e have:
120.0
6.48 100
/6.0
8.10
8.10
n
The surface load is equal to:
2/
04.2
2cos
28.
211
095.0
04.2
2cos
2808.
0m
Nt
tF
s
The linear load is equal to:
mN
tt
lF
Fp
s/
04.2
2cos
2.53
04.2
2cos
5.2
28.
21
5.1
.1.2
.3
Ca
lcula
tion
of d
yna
mic resp
on
ses
2m
ax/
53
.35.
1596 2.53
4100/6
.02
1s
mA
cc
The m
aximum
acceleration calculated is located within range 4 of the accelerations, in other
words at an unacceptable com
fort level (acceleration > 2.5 m
/sec2).
We w
ill finally consider class I, in other words an urban footbridge linking zones w
ith high concentrations of pedestrians (presence of a station, for exam
ple), or frequently used by dense crow
ds (demonstrations, tourists, etc.), subjected to a very high level of traffic.
5.1
.1.3
.1
Ca
lcula
tion
of n
atu
ral m
od
es
For class I, w
e are interested in a very dense crowd, w
here the density d of the crowd is equal
to 1.0 pedestrian/m2.
The num
ber of pedestrians on the footbridge is: 97
5.2
85.
381
pn
T
he total mass of the pedestrians is:
kg
679097
70
The linear density of the pedestrians is:
mkg
mp
/ 8
.174
85.
38/
6790
The linear density is:
mkg
S/
8.
16308.
1741456
F
or the first mode, the high-bond and low
bond frequencies are equal to:
Hz
f14.2
1456
030.0
10210
85.
382
19
2
2
1
H
zf
02.2
8.1630
030.0
10210
85.
382
19
2
2
1
For the second m
ode, the high-bond and low bond frequencies are equal to:
Hz
f
54.8
1456
030.0
10210
85.
382
29
2
2
2
H
zf
08.8
8.1630
030.0
10210
85.
382
29
2
2
2
Only the first m
ode is likely to cause uncomfortable vibrations.
5.1
.1.3
.2
Calcu
latio
n o
f the d
ynam
ic load o
f the p
edestria
ns
We w
ill calculate the load for the first mode only, w
ith 0.6% of critical dam
ping (mixed
deck). T
he surface load to be taken into account for the vertical modes is:
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nt
fN
dF
vs
185.1
2cos
280
is equal to 1, as the frequency of the first mode, w
hich is 2.02 Hz, is w
ithin range 1 (1.7 to 2.1 H
z) with a m
aximum
risk of causing resonance. T
he surface load equals:
²/
02.2
2cos
60.
521
97 185.1
02.2
2cos
2800.
1m
Nt
tF
s
The linear load is equal to:
mN
tt
lF
Fp
s/
02.2
2cos
5.131
02.2
2cos
5.2
60.
52
5.1
.1.3
.3
Ca
lcula
tion
of d
yna
mic resp
on
ses
We have:
2m
ax/
55.8
8.1630 5.131
4100/6
.02
1s
mA
cc
The m
aximum
acceleration calculated is located within range 4 of the accelerations, in other
words at an unacceptable com
fort level (acceleration > 2.5 m
/sec2).
It can be seen that the accelerations are always higher than 2.5 m
/sec², whatever class is
selected. It should be pointed out that this example is a particularly unfavourable case, as the
first natural frequency is in the middle of the range of m
aximum
risk. In order to reduce the accelerations obtained, the stiffness of the construction m
ust be increased. T
o do this, the depth of the triangulated lateral beams can be increased (for
example by 20 cm
) and the thickness of the sheet metal of the m
embers also increased (for
example 14 m
m).
We choose both to increase the thickness of the m
etal in the mem
bers and to increase the depth of the beam
s, and then we recalculate the frequencies, the dynam
ic loads and the corresponding responses.
The sheet m
etal of the mem
bers is increased from 12 to 14 m
m. T
he depth between centre
lines of the mem
bers is increased from 1.215 m
to 1.415 m.
mem
bers 400
20014
2.90 m
1.415 m
F
igure 5.3: Cross-section of the footbridge w
ith mem
bers in 14 mm
sheet metal and depth of 1.415 m
C
haracteristics of the deck T
he mom
ent of inertia of the deck is modified and equals:
4
045.0
mI
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The natural m
ass of the deck is slightly increased, but that is not significant. T
he frequencies of the first modes are m
odified in the following w
ay: C
lass III: 2.57 Hz
Class II: 2.52 H
z C
lass I: 2.50 Hz
The frequency of 2.57 H
z for class III does not lead to any calculation, as this frequency is outside the range 1.7 H
z – 2.1 Hz.
For class I and II, calculations are necessary, but w
ith a coefficient 21.0
for class I and 16.0
for class II. T
his leads to the following accelerations:
Acc =
0.55 m/sec² for class II, w
hich is compatible w
ith a medium
comfort level, and alm
ost m
aximum
(0.50 m/sec²)
Acc =
1.78 m/sec² for class I, w
hich is compatible w
ith a minim
um com
fort level (1 – 2.5 m
/sec²) T
o make this footbridge even m
ore comfortable, it w
ould be possible, for example, to increase
the thickness of the sheet metal to 16 m
m so that the natural frequencies are greater than 2.6
Hz. T
he coefficient is then zero. In this case, the second harm
onic of the pedestrians must
be taken into account, but if it stays around 2.6 Hz, it should not cause any problem
s. 5.1.2 - Box-girder footbridge
The second footbridge studied is a steel box girder w
ith two bays, each of 40 m
, with concrete
deck topping.
40,00m
40.00 m
F
igure 5.4: Steel box girder footbridge
The fram
ework is form
ed from a steel box girder of a constant depth of 1 m
etre. A 10 cm
thick pre-cast reinforced concrete slab bears on this girder.
1.00 m
Concrete slab
4.00 m
0.10 m
sheet 30 mm
Useful w
idth 3.50 m
2.00 m
F
igure 5.5: Cross-section of a m
ixed box girder footbridge T
he width of the slab is 4.00 m
, the width for pedestrian passage is 3.50 m
. C
haracteristics of the deck
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The m
ass of the deck is calculated taking into account the reinforced concrete slab and the m
ass of the balustrades. The m
oment of inertia is calculated taking into account the concrete
slab with a hom
ogenisation coefficient of 6. M
oment of inertia of the deck:
4
057,0
mI
Natural linear density of the deck:
mkg
m/
3055
Young's m
odulus of the steel: 2
9/
10
210m
NE
First of all, w
e will consider class III, in other w
ords a normally-used footbridge that can
sometim
es be crossed by large groups, but never loaded over its whole area.
5.1
.2.1
.1
Ca
lcula
tion
of n
atu
ral m
od
es
The natural m
odes of the deck were calculated using the S
ystus program.
S
is the total linear density, in other words the linear density of the deck (including the
slab), increased by the linear density of the pedestrians, which is calculated for each crow
d density, depending on the class of the footbridge. F
or class III, we are interested in a sparse crow
d, where the density d of the crow
d is equal to 0.5 pedestrians/m
2. T
he number of pedestrians on the tw
o bays is: 140
5.3
402
5.0
pn
T
he total mass of the pedestrians is:
kg
9800140
70
The linear density of the pedestrians is:
mkg
mp
/ 5
.122
80/
9800
The total linear density is:
mkg
S/
5.
31775.
1223055
F
or the first mode, the high-bond and low
bond frequencies are equal to: H
zf
94.1
1
H
zf
86
.11
For the second m
ode, the high-bond and low bond frequencies are equal to:
Hz
f
04.3
2
H
zf
92.2
2
5.1
.2.1
.2
Calcu
latio
n o
f the d
ynam
ic load o
f the p
edestria
ns
First of all, w
e calculate the load for the first mode, and a critical dam
ping ratio of 0.6%
(mixed deck).
The surface load to be taken into account for the vertical m
odes is:
nt
fN
dF
vs
8.10
2cos
280
is equal to 1, as the frequency of the first mode, w
hich is 1.90 Hz, is w
ithin range 1 of the frequencies (1.7 to 2.1 H
z). W
e have: 071
.0140 100
/6.0
8.10
8.10
n
The surface load is equal to:
2/
90.1
2cos
94.9
1071.0
90.1
2cos
2805.
0m
Nt
tF
s
The linear load is equal to:
mN
tt
lF
Fp
s/
90.1
2cos
79.
3490
.12
cos5.
394.9
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5.1
.2.1
.3
Ca
lcula
tion
of d
yna
mic resp
on
ses
The acceleration under the vertical load is equal to:
2m
ax/
16
.15.
317779.
344
100/6
.02
14
2 1s
mSF
Acc
The m
aximum
acceleration calculated is located within range 3 of the accelerations, in other
words at the m
inimum
comfort level (acceleration betw
een 1 and 2.5 m/sec
2). How
ever, the m
edium com
fort level is almost reached.
For the second m
ode, the frequency of 2.96 Hz does not im
pose particular checks.
We w
ill then consider class II.
5.1
.2.2
.1
Ca
lcula
tion
of n
atu
ral m
od
es
For class II, w
e are interested in a dense crowd, w
here the density d of the crowd is equal to
0.8 pedestrians/m2.
The num
ber of pedestrians on the footbridge is: 224
5.3
402
8.0
pn
T
he total mass of the pedestrians is:
kg
15680224
70
The linear density of the pedestrians is:
mkg
mp
/
19680
/15680
The total linear density is:
mkg
S/
3251
1963055
F
or the first mode, the high-bond and low
bond frequencies are equal to: H
zf
94.1
1
H
zf
86
.11
For the second m
ode, the high-bond and low bond frequencies are equal to:
Hz
f
04.3
2
H
zf
92.2
2
5.1
.2.2
.2
Calcu
latio
n o
f the d
ynam
ic load o
f the p
edestria
ns
First of all, w
e calculate the load for the first mode, and a critical dam
ping ratio of 0.6%
(mixed deck).
The surface load to be taken into account for the vertical m
odes is:
nt
fN
dF
vs
8,10
2cos
280
is equal to 1, as the frequency of the first mode, w
hich is 1.879 Hz, is w
ithin range 1 of the frequencies (1.7 to 2.1 H
z).
We have
056.0
224 100/6
.08,
108,
10n
The surface load is equal to:
2/
879.1
2cos
54.
120.
1056
.0879.1
2cos
2808.
0m
Nt
Fs
T
he linear load is equal to: m
Nt
tl
FF
ps
/
879.1
2cos
89.
43879.1
2cos
5.3
54.
12
5.1
.2.2
.3
Ca
lcula
tion
of d
yna
mic resp
on
ses
The acceleration under the vertical load is equal to:
2m
ax/
43.1
325189.
434
100/6
.02
14
2 1s
mSF
Acc
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The m
aximum
acceleration calculated is located within range 3 of the accelerations, in other
words at a m
edium com
fort level (between 1 and 2.5 m
/sec2).
For the second m
ode, the frequency of 2.92 Hz requires us to take the second harm
onic into account. B
ut, with a force of one-quarter of that of the first harm
onic, and taking into account the above acceleration result, the com
fort level obtained is maxim
um.
We w
ill finally consider class I.
5.1
.2.3
.1
Ca
lcula
tion
of n
atu
ral m
od
es
The natural m
odes of the deck were calculated using the S
ystus program.
For class I, w
e are interested in a very dense crowd, w
here the density d of the crowd is equal
to 1.0 pedestrian/m2.
The num
ber of pedestrians on the footbridge is: 280
5.3
402
1p
n
The total m
ass of the pedestrians is: kg
19600280
70
The linear density of the pedestrians is:
mkg
mp
/
24580
/19600
The total linear density is:
mkg
S/
3300
2453055
F
or the first mode, the high-bond and low
bond frequencies are equal to: H
zf
94.1
1
H
zf
86
.11
For the second m
ode, the high-bond and low bond frequencies are equal to:
Hz
f
04.3
2
H
zf
92.2
2
5.1
.2.3
.2
Calcu
latio
n o
f the d
ynam
ic load o
f the p
edestria
ns
First of all, w
e calculate the load for the first mode, and a critical dam
ping ratio of 0.6%
(mixed deck).
The surface load to be taken into account for the vertical m
odes is:
nt
fN
dF
vs
186.1
2cos
280
is equal to 1, as the frequency of the first mode, w
hich is 1.86 Hz, is w
ithin range 1 of the frequencies (1.7 to 2.1 H
z). T
he surface load equals:
²/
86.1
2cos
956.
301
280 185.1
86.1
2cos
2800.
1m
Nt
tF
s
The linear load is equal to:
mN
tt
lF
Fp
s/
86.1
2cos
35.
10886
.12
cos5.
3956.
30
5.1
.2.3
.3
Ca
lcula
tion
of d
yna
mic resp
on
ses
The acceleration under the vertical load is equal to:
2m
ax/
48.3
330035.
1084
100/6
.02
14
2 1s
mSF
Acc
The m
aximum
acceleration calculated is located within range 4 of the accelerations, in other
words at an unacceptable com
fort level (acceleration > 2.5 m
/sec2).
For the second m
ode, the frequency of 2.91 Hz requires us to take the second harm
onic into account. B
ut, with a force of one-quarter of that of the first harm
onic, and taking into account
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the above acceleration result, the comfort level obtained is m
edium (approxim
ately 0.9 m
/sec²).
Class
Frequency in
Hz
Dam
ping ratio A
cceleration in m
/sec2
III 1.90
0.6 1.16
II 1.88
0.6 1.43
I 1.86
0.6 3.48
T
able 5.1 It can be seen that the accelerations are higher than 1 m
/sec² for all categories. If the medium
com
fort level is selected, the project will have to be m
odified. In order to reduce the accelerations obtained, the stiffness of the construction m
ust be increased. T
o do this, the depth of the box girder can be increased (for example by 40 cm
).
The fram
ework is form
ed from a steel box girder of a constant depth of 1.40 m
etres. A 10 cm
thick pre-cast reinforced concrete slab bears on this girder. T
he mom
ent of inertia of the deck is modified and equals:
4
106.0
mI
N
atural linear density of the deck: m
kgm
/
3241
Young's m
odulus of the steel: 2
9/
10
210m
NE
The high bond and low
bond frequencies are modified as follow
s: H
zf
57.2
1
and
Hz
f
47.2
1H
zf
02.4
2H
zf
83.3
2
For class III, the tw
o natural frequencies are outside the range of frequencies that have to be checked. In class III, the com
fort level is therefore automatically m
aximum
. T
he accelerations associated with the first m
ode are, for class I and II:
In class II: 2
max
/
32.0
343736.
104
100/6
.02
14
2 1s
mSF
Acc
In class I: 2
max
/
85.0
343717
.28
4100/6
.02
14
2 1s
mSF
Acc
For class II, the com
fort level is maxim
um. It is m
edium for class I, but acceptable even so.
The conclusions on com
fort levels, taking the second harmonic of pedestrians w
alking into account, for the second natural m
odes, are unchanged.
5.2 - Sensitivity study of typical footbridges.
This section sets out a study of four types of standard footbridges based on actual exam
ples, in w
hich the span of these typical footbridges is varied around the actual span, but remaining
within their field of use.
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The first paragraph sets out the footbridges studied and the second states their natural
frequencies.
5.2.1 - Presentation of the footbridges studied and static pre-sizing
Four m
ain types of footbridges were distinguished: in reinforced concrete, in pre-stressed
concrete, mixed steel-concrete box girder and steel lattice.
For each type, rough pre-sizing w
as carried out, for footbridges of a total length of between
20 and 80 m; the assum
ptions and the procedure used in each case are given in the following
section. In order to allow
a comparison betw
een the various footbridges, the same net w
idth of 3.50 m
was taken for all the footbridges. In the sam
e way, the superstructures are com
parable, even though they have been adapted to each case.
For reinforced concrete, a distinction is m
ade between tw
o fields of span: small spans (20 to
25 m), for w
hich a rectangular reinforced concrete slab is sufficient, and longer spans (25 to 45 m
), for which w
e will use an H
-shaped reinforced footbridge. Only the depth h varies
when the span changes, the fixed dim
ensions are as follows:
h h
3 m 50
3 m 50
0.80 m
0.50 m
0.20
20 m <
L <
25 m
30 m <
L <
45 m
F
igure 5.6: Reinforced concrete footbridges
The depth of the footbridge to be used is that w
hich enables the compression lim
it of the concrete (here 15 M
Pa) to be guaranteed, and to take the steels, either in the 3.50 m
width, for
the slab, or in the two 50 cm
beams, for the H
-shaped footbridge. T
he superstructures of this footbridge, together with their w
eights, are:
- damp-proofing
3.5 x 0.03 x 24 = 2.5 kN
/m
- finish
3.5 x 0.04 x 24 = 4 kN
/m
- sundries
1 kN / m
i.e.
7.5 kN / m
in total. T
he role of the balustrade is played by the H-beam
s themselves. F
or the reinforced slab, two
edge brackets and two balustrades m
ust be added, i.e. 16 kN / m
in total.
In the same w
ay as for reinforced concrete, a distinction is made betw
een two distinct fields:
where L
is between 20 m
and m, a pre-stressed rectangular slab is used. O
n the other hand, as soon as L
becomes longer than 35 m
(up to 50 m), a pre-stressed box girder is preferably to be
used. Only the depth h varies w
hen the span changes, the fixed dimensions are as follow
s:
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h
4 m
4 m
h
2 m
0.2 0.2
0.2
20 m <
L <
30 m
35 m <
L <
50 m
F
igure 5.7: Pre-stressed concrete footbridges
The superstructures of this footbridge, together w
ith their weights, are:
- 2 edge brackets
2 x 25 x (0.25 x 0.34) =
4.5 kN / m
- 2 balustrades
2 x 2 = 4 kN
/ m
- damp-proofing
4 x 0.03 x 24 = 3 kN
/m
- finish
3.5 x 0.04 x 24 =
4 kN /m
- sundries
1 kN
/ m
i.e.
16.5 kN
/ m in total.
This type of footbridge is a steel box girder (w
elded reconstituted beams) in grade S
355 steel. A
concrete slab bears on this box girder; if it is connected to it, it becomes a m
ixed section. If it does not, the load-bearing section is the steel alone, but the thickness of the concrete is still there and adds to the overall w
eight. Only the depth of the footbridge varies w
hen the span changes, the fixed dim
ensions are as follows:
h
0.10 m
0.025 m
0.025 m
4 m
2 m
0.025 m
F
igure 5.8: Mixed footbridge
We w
ill study this type of footbridge for total spans ranging from 40 to 80 m
. For the m
ixed box girder, the sizing is done on the assum
ption that the steel section bears its own w
eight and also that of the w
et concrete (assumed placed in a single pour). O
n the other hand, the mixed
section takes the superstructure (with an equivalence factor n =
18) and the static pedestrian loads (factor n =
6). For the steel box girder, the only load-bearing section is the steel girder.
The superstructures of this footbridge, together w
ith their weights, are:
- 2 edge brackets
2 x 25 x (0.25 x 0.34) = 4.5 kN
/ m
- 2 balustrades
2 x 2 = 4 kN
/ m
- damp-proofing
4 x 0.03 x 24 = 3 kN
/m
- finish
3.5 x 0.04 x 24 = 4 kN
/m
- sundries
1 kN / m
i.e.
16.5 kN
/ m in total.
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The m
ain field of span taken into account for this type of footbridge is 50-80 m. It is assum
ed that the footbridge is properly triangulated. T
he values that remain fixed w
hen the span changes are:
0.23 m
h
3 m 50
0.18 m
0.5 m
0.05 m
F
igure 5.9: Steel lattice footbridge
T
he superstructures of this footbridge, together with their w
eights, are:
- 2 balustrades
2 x 2 =
4 kN / m
- dam
p-proofing
3.5 x 0.03 x 24 =
2.5 kN /m
- finish
3.5 x 0.04 x 24 =
4 kN /m
- sundries
1 kN
/ m
i.e.
11.5 kN
/ m in total.
5.2.2 - Natural frequencies
The four previous types of footbridges are considered to be iso-static bays.
For an iso-static bay, the form
ula giving the natural frequency of the bridge is:
fn
L
EIS
n
12
22
2
fn is the natural frequency of the mode n;
L is the length of the bay in m
; I is the inertia in m
4, vertical or horizontal, depending on what is being sought;
E is the Y
oung's modulus of the m
aterial comprising the structure in N
/ m²;
S is the linear density of the bridge (self-m
ass and mass of the superstructures)
in kg / m, to w
hich is added the mass of the pedestrians on the footbridge. W
e have taken 1 pedestrian / m
², with 70 kg per pedestrian; i.e.
S =
permanent
loads + 3.5 x 70 x 1 in kg / m
. W
e therefore varied the length of the iso-static bay for each type of footbridge. The depth w
as determ
ined according to static sizing. T
he following tables set out the four natural frequencies: the first tw
o vertical and the first tw
o horizontal. "#$%&'($)*#+&%+ ,- %%.'($)*#+&%+ ./-- #0%*#+&%+ L
vert. freq. 1
vert. freq. 2
hor. freq. 1
hor. freq. 2
vert. freq. 1
vert. freq. 2
hor. freq. 1
hor. freq. 2
vert. freq. 1
vert. freq. 2
hor. freq. 1
hor. freq. 2
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40
1.2 4.6
3.6 14
1.6
6.2 3.1
12
50
1.1 4.4
2.3 9.3
1.5
6.0 2.0
8.1
1.4 5.6
2.0 8.3
60
1.1 4.2
1.6 6.5
1.4
5.7 1.5
5.9
1.4 5.7
1.4 5.6
70
1.0 4.1
1.2 4.9
1.4
5.5 1.1
4.5
1.4 5.7
1.0 4.1
80
1.0 3.9
1.0 3.8
1.4
5.3 0.9
3.5
1.5 5.8
0.8 3.1
T
able 5.2
#123
#143
L
vert. freq. 1 vert.
freq. 2
hor. freq. 1
hor. freq. 2
L
vert. freq. 1 vert.
freq. 2
hor. freq. 1
hor. freq. 2
slab 20
2.7
11 19
77 slab
20
2.0 8.0
15 58
RC
25
2.5
10 13
50 P
C
25
1.8 7.1
10 39
H-
sect. 25
3.4
13 15
61
30
1.9 7.7
7.0 28
30
3.0
12 11
44 box girder
30
2.4 9.8
5.2 21
35
2.7
11 8
33
35
2.3 9.1
3.9 15
40
2.5
10 6
26
40
2.1 8.5
3.0 12
45
2.4
10 5
21
45
2.0 8.2
2.4 9.5
50
2.0 7.9
1.9 7.7
T
able 5.3 T
he following com
ments m
ay be inferred from these tables:
O
nly the first mode need be used for these sim
ple types of footbridges, both for vertical and horizontal vibrations;
For vertical vibrations, the first natural frequency of m
etal footbridges is around 1 – 1.5 Hz,
whereas, for concrete footbridges, it is nearer 2 - 3 H
z (more precisely, 2.5 - 3 H
z for reinforced concrete and 2 - 3 H
z for pre-stressed concrete); F
ootbridges that are unable to vibrate under pedestrian excitation are rare; these are H-
shaped reinforced footbridges of 25 m. A
ll others would appear to be classified as "at risk";
Finally, for lateral vibrations, it w
ould be sufficient to study large-span steel footbridges (for all others, the first horizontal frequency is beyond 2.5 H
z).
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6. Annexe 6: Bibliography
6.1 - Generally
1. D
ynamique des constructions. J. A
rmand et a
l.. EN
SM
P courses, 1983.
2. Introduction to structural dynam
ics. J.M. B
iggs. McG
raw-H
ill Book C
ompany, June
1964.
3. V
ibration problems in structures - P
ractical guidelines. H. B
achmann et a
l. Birkhäuser,
1997, 2nd edition.
6.2 - Regulations
4. E
urocode 2 - Design of concrete structures – P
art 2: Concrete bridges . E
NV
1992-2, 1996.
5. E
urocode 5 - Design of tim
ber structures - Part2 : B
ridges. PrE
NV
1995-2, 14 January 1997.
6. P
ermissible vibrations for footbridges and cycle track bridges. B
S 5400 - A
ppendix C.
7. P
ractical guidelines. CE
B, B
. I. 209, August 1991.
8. R
ecomm
endations for the calculation of the effects of wind on constructions. C
EC
M,
1989.
6.3 - Dampers
9. T
wo case studies in the use of tuned vibration absorbers on footbridges. R
.T. Jones;
A.J. P
retlove; R. E
ye. The S
tructural Engineer, June 1981, V
ol. 59B N
o. 2. 10. T
uned vibration absorbers for lively structures. H. B
achmann; B
. Weber. S
tructural E
ngineering International, 1995, No. 1.
11. Tuned m
ass dampers for balcony vibration control. M
. Setareh; R
. Hanson. A
SC
E
Journal of Structural E
ngineering, March 1992, V
ol. 118, No. 3.
12. Tuned m
ass dampers to control floor vibration from
humans. M
. Setareh; R
. Hanson.
AS
CE
Journal of Structural E
ngineering, March 1992, V
ol. 118, No. 3.
13. Manufacturers' docum
entation: Taylor - Jarret.
14. Stade
de F
rance footbridge
– D
esign of
tuned m
ass dam
pers. C
. O
utteryck; S
. M
ontens. French C
ivil Engineering R
eview, O
ctober 1999.
6.4 - Behaviour analysis
15. Hum
an tolerance levels for bridge vibrations. D. R
. Leonard, 1966.
16. Dynam
ic design
of footbridges.
Y.
Matsum
oto; T
. N
ishioka; H
. S
hiojiri; K
. M
atsuzaki. IAB
SE
Proceedings, 1978, P
-17/78, pp. 1-15.
© S
étra - 2
006
Dyn
am
ic beh
avio
ur o
f footb
ridges –
Annexes
1
25
/ 12
7
17. Pedestrian induced vibrations in footbridges. J.E
. Wheeler. P
roceedings AR
RB
, 1980, V
ol. 10, part 3.
18. Prediction and control of pedestrian-induced vibration in footbridges. J. W
heeler. A
SC
E Journal of structural engineering, S
eptember 1982, V
ol. 108, No. S
T9.
19. Structural serviceability - F
loor vibrations. B. E
llingwood; A
. Tallin. A
SC
E Journal of
Structural E
ngineering, February 1984, V
ol. 110, No. 2.
20. Dynam
ic behaviour of footbridges. G.P
. Tilly; D
.W. C
ullington; R. E
yre. IAB
SE
periodical, F
ebruary 1984, S-26/84, page 13 et seq.
21. Vibration of a beam
under a random stream
of moving forces. R
. Iwankiew
icz; J. P
awel S
niady. Structural M
echanics, 1984, Vol. 12, N
o. 1. 22. V
ibrations in structures induced by man and m
achines. H. B
achmann; W
. Am
man.
IAB
SE
, Structural E
ngineering Docum
ents, 1987. 23. O
n minim
um w
eight design of pedestrian bridges taking vibration serviceability into consideration.
H.
Sugim
oto; Y
. K
ajikawa;
G.N
. V
anderplaats. A
SC
E
Journal of
Structural E
ngineering, October 1987.
24. Design live loads for coherent crow
d harmonic m
ovements. A
. Ebrahim
pout; R.L
. S
ack. AS
CE
Journal of Structural E
ngineering, 1990, Vol. 118-4, pp.1121-1136.
25. Durch
Menschen
verursachte dynam
ische L
asten und
deren A
uswirkungen
auf B
alkentragwerke. H
. Bachm
ann. Kurzberichte aus der B
auforschung, Novem
ber 1990, R
eport No. 125.
26. Vibration U
pgrading of Gym
nasia - Dance H
alls and Footbridges. H
. Bachm
ann. S
tructural Engineering International, F
ebruary 1992. 27. C
ase studies of structures with m
an-induced vibrations. H. B
achmann. A
SC
E Journal
of Structural E
ngineering, March 1992, V
ol. 118, No. 3.
28. Bases for design of structures - S
erviceability of buildings against vibration. ISO
10137, 15 A
pril 1992, first edition.
29. Schw
ingungsuntersuchungen für Fu
gängerbrücken. H. G
rundmann; H
. Kreuzinger;
M. S
chneider. Bauingenieur, 1993, N
o. 68.
30. Synchronisation of hum
an walking observed during lateral vibration of a congested
pedestrian bridge.
Y.
Fujino;
B.M
. P
acheco; S
.I. N
akamura;
P.
Warnitchai.
Earthquake engineering and structural dynam
ics, 1993. 31. D
esign criterion for vibrations due to walking. D
.E. A
llen; T.M
. Murray. E
ngineering Journal/ A
merican Institute of S
till Construction, 1993, 4
th Quarter.
32. Guidelines to m
inimise floor vibrations from
building occupants. S. M
ouring; B.
Ellingw
ood. AS
CE
Journal of Structural E
ngineering, February 1994, V
ol. 120, No. 2.
33. Measuring and m
odelling dynamic loads im
posed by moved crow
ds. Various A
uthors. A
SC
E Journal of S
tructural engineering, Decem
ber 1996.
34. Mechanical vibration and shock - E
valuation of human exposure to w
hole-body vibration – P
art 1: General requirem
ents. International Standard IS
O 2631-1, 1997.
35. Serviceability vibration evaluation of long floor slabs. T
.E. P
rice; R.C
. Sm
ith. AS
CE
S
tructures Congress, W
ashington DC
, US
A, 21-23 M
ay 2001. AS
CE
, 1991. 36. D
evelopment of a sim
plified design criterion for walking vibrations. L
. M. H
anagan; T
. Kim
. AS
CE
Structures C
ongress, Washington D
C, U
SA
, 21-23 May 2001. A
SC
E,
2001. 37. A
n investigation
into crow
d-induced vertical
dynamic
loads using
available m
easurements. M
. Willford. T
he Structural E
ngineer, June 2001, Vol. 79, N
o. 12. 38. T
he London M
illennium F
ootbridge. P. D
allard; A.J. F
itzpatrick; A. F
lint; S. L
e B
ourva; A. L
ow; R
.M. R
idsdill Sm
ith; M. W
illford. The S
tructural Engineer, 20
Novem
ber 2001, Vol. 79, N
o. 22.
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/ 12
7
6.5 - Calculation methods
39. Study of vibratory behaviour of footbridges as pedestrians pass. F
. Legeron; M
. L
emoine. R
evue Ouvrages d'A
rt, No. 32. S
ET
RA
, July 1999. 40. D
ynamic study of pedestrian footbridges. G
. Youssouf – course leader F
. Legeron.
MS
OA
thesis of EN
PC
promotion 2000.
41. Cable dynam
ics - A review
. U. S
tarossek. Structural E
ngineering International 3/94, 1994.
42. Using
component
mode
synthesis and
static shapes
for tuning
TM
Ds.
Various
Authors. A
SC
E Journal of S
tructural Engineering, 1992, N
o. 3.
6.6 - Articles on either vibrating or instrumented footbridges
43. Pedestrian-induced vibration of footbridges. P
. Dallard; A
.J. Fitzpatrick; A
. Flint; A
. L
ow; R
. Ridsdill S
mith; M
. Willford. S
tructural Engineer, 5 D
ecember 2000, V
ol. 78, N
o. 23-24.
44. Modal identification of cable-stayed pedestrian bridges. M
. Gardnerm
orse; D. H
uston. A
SC
E Journal of S
tructural Engineering, N
ovember 1993, V
ol. 119, No. 11.
45. A
briefing on
pedestrian-induced lateral
vibration of
footbridges. F
rench C
ivil E
ngineering Review
, October 2000. 4, N
o. 6. 46. D
ynamic
testing of
the S
herbrooke pedestrian
bridge. P
. P
aultre; J.
Proulx;
F.
Légeron; M
. Le M
oine, N. R
oy. 16th IA
BS
E C
ongress, Lucerne, S
witzerland, 2000.
6.7 - Specific footbridges
47. Sacram
ento R
iver pedestrian
bridge U
SA
. C
. R
edfield; J.
Straski.
Structural
Engineering International, 1991, 4/91.
48. Construction of the w
orld’s longest pedestrian stress-ribbon bridge. Various A
uthors. F
IP notes 98-1, 1998.
49. Rebirth of the ribbon. V
arious Authors. B
ridge, 1998, 4th quarter.
6.8 - Additional bibliography
6.8.1 - Damping
51. Mechanical vibrations. J.P
. Den H
artog. Mc G
raw-H
ill, 1940, 4th edition. 52. A
erodynamic behaviour of open-section steel deck cable-stayed bridges. J.C
. Foucriat.
Journées techniques AF
PC
, 1978, part I. 53. O
n the dynamic vibration dam
ped absorber of the vibration system. T
. Ioi; K. Ikeda.
Bulletin of the Japanese S
ociety of Mechanical E
ngineering, 1978, 21, 151. 54. O
ptimal absorber param
eters for various combinations of response and excitation
parameters. G
.B. W
arburton. Earthquake E
ngineering and Structural D
ynamics, 1982,
Vol.10, pp. 381-401.
© S
étra - 2
006
Dyn
am
ic beh
avio
ur o
f footb
ridges –
Annexes
1
27
/ 12
7
55. Fractional-derivative
Maxw
ell m
odel for
viscous dam
pers. N
. M
akris; M
.C.
Constantinou. A
SC
E Journal of S
tructural Engineering, 1991, V
ol. 117, pp. 2708-2724.
56. Dynam
ic analysis of generalised visco-elastic fluids. N. M
akris; G.F
. Dargush; M
.C.
Constantinou. Journal of E
ngineering Mechanics, 1993, V
ol. 119, No. 8.
57. Vibration control by m
ultiple tuned liquid dampers. Y
. Fujino; L
.M. S
un. AS
CE
Journal of S
tructural Engineering, 1993, V
ol. 119, No. 12.
6.8.2 - OTUA Technical Bulletins
58. Steel bridges bulletins.
59. Steel structures bulletins.
6.8.3 - Additional literature on pedestrian behaviour studies
60. Proceedings and technical texts of the 2002 F
ootbridge International Conference, P
aris F
rance, Novem
ber 2002. 61. M
odel for lateral excitation of footbridges by synchronous walking. S
. Nakam
ura. A
SC
E Journal of S
tructural Engineering, January 2004
62. Experim
ental study
on lateral
forces induced
by pedestrians.
S.
Nakam
ura; H
. K
atsuura; K. Y
okoyama. IA
BS
E S
ymposium
, Shanghai, C
hina, Septem
ber 2004.
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Th
ese guid
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un
d u
p th
e state of cu
rrent k
now
ledge o
n d
ynam
ic b
ehavio
ur o
f foo
tbrid
ges un
der p
edestrian
load
ing. A
n an
alytical m
etho
do
logy an
d reco
mm
end
ation
s are also p
rop
osed
to gu
ide th
e d
esigner o
f a new
foo
tbrid
ge wh
en co
nsid
ering th
e resultin
g dyn
amic
effects.
Th
e meth
od
olo
gy is based
on
the fo
otb
ridge classifi
cation
con
cept (as a
fun
ction
of traffi
c level) and
on
the req
uired
com
fort level an
d relies o
n
interp
retation
of resu
lts ob
tained
from
tests perfo
rmed
on
the S
olferin
o fo
otb
ridge an
d o
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latform
.
Th
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• a descrip
tion
of th
e dyn
amic p
hen
om
ena sp
ecific to
foo
tbrid
ges an
d id
entifi
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of th
e param
eters wh
ich h
ave an im
pact o
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structu
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• a meth
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olo
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alysis of fo
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ased o
n a
classificatio
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rdin
g to th
e traffic level;
• a presen
tation
of th
e practical m
etho
ds fo
r calculatio
n o
f natu
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uen
cies and
mo
des, as w
ell as structu
ral respo
nse to
load
ing;
• recom
men
datio
ns fo
r the d
rafting o
f design
and
con
structio
n
do
cum
ents.
Sup
plem
entary th
eoretical (rem
ind
er of stru
ctural d
ynam
ics, ped
estrian
load
mo
dellin
g) and
practical (d
amp
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s, examp
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otb
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ata are also p
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