During the last decades, several studies on suspension bridges under wind actions have been developed in civil engineering and many techniques have been used to approach this structural problem both in time and frequency domain. In this paper, four types of time domain techniques to evaluate the response and the stability of a long span suspension bridge are implemented: nonaeroelastic, steady, quasi steady, modified quasi steady. These techniques are compared considering both nonturbulent and turbulent flow wind modelling. The results show consistent differences both in the amplitude of the response and in the value of critical wind velocity.

1. Computers and Structures 85 (2007) 1032–1048
www.elsevier.com/locate/compstruc
Comparison of time domain techniques for the evaluation of
the response and the stability in long span suspension bridges
Francesco Petrini a, Fabio Giuliano
a
b,*
, Franco Bontempi
a
Department of Civil Engineering, University ‘‘La Sapienza’’, Via Eudossiana 18, 00184 Rome, Italy
b
Department of Structural Mechanics, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy
Received 8 June 2006; accepted 20 November 2006
Available online 16 January 2007
Abstract
During the last decades, several studies on suspension bridges under wind actions have been developed in civil engineering and many
techniques have been used to approach this structural problem both in time and frequency domain. In this paper, four types of time
domain techniques to evaluate the response and the stability of a long span suspension bridge are implemented: nonaeroelastic, steady,
quasi steady, modified quasi steady. These techniques are compared considering both nonturbulent and turbulent flow wind modelling.
The results show consistent differences both in the amplitude of the response and in the value of critical wind velocity.
Ó 2006 Elsevier Ltd. All rights reserved.
Keywords: Aeroelastic forces; Structural dynamics; Suspension bridge; Flutter; Buffeting
1. Introduction
Long span suspension bridges, because of their high
flexibility, low structural damping, and the reduced mass
amount, are very sensitive to wind actions.
The actions of wind on a generic surface, are determined
by the configuration of punctual actions (pressure and tangential stresses), which are caused by the wind impact in
the relative motion.
For a flexible body inside a wind flow, these stresses are
determined by the flow configuration in the surface proximity zone, and so depend on the body motion: from a
mechanic point of view, the couple of the flow and the
body, is an auto-excited dynamic system.
By discretizing the body to a finite number of degrees of
freedom (DOFs), the equation governing the body motion
is the dynamic equilibrium equation:
*
Corresponding author. Tel.: +39 328 5936901.
E-mail address: fabio.giuliano@unipv.it (F. Giuliano).
0045-7949/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruc.2006.11.015
_
_ q
M Á € þ C Á q þ K Á q ¼ F ðbody shape; q; q; €; V ; t; nÞ
q
ð1Þ
where
M
C
K
_ q
q; q; €
V
t
n
mass matrix of the system,
damping matrix of the system,
stiffness matrix of the system,
DOFs of the system and their first an second time
derivates,
incident wind velocity,
time,
oscillation frequencies of the system.
In general, inside the right hand member of Eq. (1) there
is an ‘‘auto-excited’’ component of the aerodynamic forces
_ q
which depends on body motion (q; q; €); in the case that the
inertial terms, if compared with others terms, assume infinitesimal order in Eq. (1), the equation becomes a static
equilibrium statement.
A global picture of all the aeroelastic problems that can
involve a structure is represented by the ‘‘Collar triangle’’,
as shown in the high-left side in Fig. 1: here, focusing on

2. F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
aeroelastic forces dependences, aeroelastic problems are
classified in three categories:
1. Response problems, in which there is a dynamic equilibrium between the body and the wind forces: the expression of the forces contain time-independent terms.
2. Stability problems, in which the interchanging energy
between the body motion and the aeroelastic forces, produces a gradual and unlimited increment of the motion
energy, leading to the dynamic equilibrium instability at
‘‘critical velocities’’. For these problems, the expression
of the forces does not contain time-independent terms.
3. Mixed (stability and response) problems, in which incident wind velocities are close to the critical velocities
and the expression of the forces contains both time-independent and auto-excited terms.
In instability problems, the forces are usually transferred
in the left side hand of the equation in order to obtain a
homogeneous equation in which mass, damping, and stiffness matrices contain auto-excited coefficients.
Considering a generic structure under wind action, two
principal approaches exist to solve the relative aeroelastic
problem:
1. Frequency domain approach, usual for instability problems, based on a modal combinations for the structural
system, is less reliable for structures with strong nonlinear nature.
1033
2. Time domain approach, requiring a time integration of
the equations of the motions of the structure subjected
to wind loads, with large computational costs for structure with many DOFs.
In the present paper, the time domain approach has
been adopted to study the response and the stability problem of a long span suspension bridge under wind actions.
In particular, the incident wind has been modelled as a turbulent flow for the response problem, while for the stability
problem, it has been modelled as nonturbulent.
2. Aeroelastic forces for suspension bridges
In the last decades, referring to aeronautic engineering,
civil engineers have developed many analytical theories to
model wind effects on structures, and simplified approaches
have been adopted to evaluate the aeroelastic terms,
because of the very complex dependence above listed. Both
frequency and time domain techniques to model aeroelastic
forces, derive from wing theory [1–3,5].
In the case of suspension bridges, two peculiar aspects
have to be considered. First, suspension bridges deck sections, usually have so-called ‘‘semi-bluff’’ or ‘‘bluff’’ body
shapes, with (opposite to wing shapes) well predictable
points of flow detachments along the deck surfaces (‘‘live
edges’’), where the turbulence of the flow is very high.
The second particular aspect is related to the intrinsic
turbulent content of an incident wind flow. Furthermore
Collar
Static aeroelastic
stability
Forces: A=Aeroelastic; E=Elastic; I=Inertial; F=Applied
Equilibrium equation:
D
Galloping
G
Vortex Shedding
VS
Flutter
F
Buffeting
B
A− E = 0
For
stability
A+E+I+F=0
Torsional divergence
Dynamic aeroelastic
stability
A+ I − E = 0
Aeroelastic
problems
Static aeroelastic
response
A+ F −E = 0
For
Response
Aeroelastic Phenomena
Tors. Divergence
V. Shedding
Galloping
Classical flutter
Stall flutter
Buffeting
Non Aeroelastic Sciences
D
VS
G
Fc
Fs
B
Mechanics of
vibrations
Mechanics of
rigid body flight
Dynamic aeroelastic
response
MV
MR
A+ I − E + F = 0
Torsional divergence
Galloping
Vortex Shedding
Flutter
Buffeting
Static instability in which
torsional moment due to
aeroelastic forces, overcomes
the elastic resistant moment of
the body
An asymmetry in the flow
produces a vertical oscillation
that generate oscillations in
aerodynamic forces, which
depends on body oscillations
velocity and involves an
aerodynamic damping, which is
opposite to the structural
damping.
If
aerodynamic
damping is greater than the
structural one, the motion may
become unstable.
Dynamic instability very similar
to a resonance. Some vortex
loose themselves in the wake
behind the body. Such physic
configuration generates an
oscillation in aerodynamic
forces,
with
a
definite
frequency. In a certain range of
wind velocity, the oscillation
frequency of the forces “block”
the
structural
oscillations
frequency and it governs the
structural oscillation period.
Dynamic instability in which
two d.o.f. of the forced
structural system are coupled
and,
under
opportune
configurations (defined “critic”)
for frequencies and reciprocal
phase angles, lead the damping
of the system to become
negative, and the oscillations
increase in amplitude
Motion under the action of time
random variable forces both in
intensity and direction. The
buffeting phenomenon assumes
aeroelastic
importance
in
concomitance
of
other
aeroelastic phenomena (like
Flutter or Vortex shedding).
Fig. 1. Aeroelastic problems and instabilities.
A classic buffeting effect, is the
one generated by the intrinsic
turbulence
within
the
atmospheric boundary layer.

3. 1034
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
a larger complexity and rotationally nature of the flow
determines a larger ‘‘aerodynamic delay’’, which is the
transient effect due to the adjustment time of aerodynamic
field, in consequence of a changing in body geometric configuration (i.e. rotation and displacements of the deck). In
order to take into account these effects, the so-called
‘‘memory terms’’, which consider the influence of displacements history in the expression of the aeroelastic forces, are
introduced. These terms are usual implemented by integral
expressions [6].
The uncertainties related to the above phenomena lead
to the use of the experimental approach. In this sense, studies conduced by Scanlan and Tomko [7], Jain et al. [8] and
Scanlan and Simiu [9] had great relevance in wind engineering. Following Scanlan theory, the Self-Excited (SE) components of the aeroelastic forces are determined by a
superposition of the effects, referring to the forces obtained
by wind tunnel tests, acting on a sectional model of the
deck that is moving in simple harmonic oscillations along
three sectional DOFs (the rotational and the two translational ones). Referring to Fig. 2, it results for the Lift force
_
_
_
LSE ðp; p; h; h; #; #; k; xÞ
"
_
_
1
BÁ#
h
2
¼ Á q Á V Á B Á k Á H Ã ðkÞ Á þ k Á H Ã ðkÞ Á
1
2
2
V
V
_
h
p
þ k 2 Á H Ã ðkÞ Á # þ k 2 Á H Ã Á þ k Á H Ã ðkÞ Á
3
4
5
B
V
#
p
2
Ã
þ k Á H 6 ðkÞ Á
B
ð2Þ
where
Analogous expressions of (2) can be written for drag
force and moment.
3. Aeroelastic forces in time domain
Time domain approaches allow to consider directly the
structural nonlinear effects and this is relevant for certain
types of structures, like long span suspension bridges. Furthermore, because the time domain analyses outputs are
merely the time histories of specific variables, it is the most
convenient approach in response problems. Unfortunately,
expressions in time domain which consider aspects
previously mentioned are not trivial to implement. Consequently in the last years simplified formulations for aeroelastic forces have been developed and improved.
The analysis in the time domain consists in a time integration that involves the time step updating of kinematic
parameters and acting forces. Referring to Fig. 3, where
the problem is represented like a two-dimensional problem,
horizontal and vertical components of absolute wind turbulent velocity V a ðtÞ, are considered as composed by mean
components U, W, and fluctuant (or turbulent) components uðtÞ and wðtÞ. The resulting absolute velocity is not
horizontal, and has a time-varying instantaneous angle of
incidence.
Adopting the general notation previously introduced
(Eq. (1)), one has, for the system DOFs, qT ¼ ½ s h # Š.
Moreover, the dependence of the forces from structural
DOFs and their time derivatives can be generally expressed
in matrix form as:
_ q
_
F ðq; q; €; nÞ ¼ P ðt; nÞ Á € þ Qðt; nÞ Á q þ Rðt; nÞ Á q
q
k ¼ xÁB reduced frequency of the system, in simple harV
monic motion,
H Ã ðkÞ functions of the reduced frequency,
i
B
characteristic dimension of the bridge deck,
x
circular frequency of the system, in simple harmonic motion,
V
incident wind velocity.
The functions H Ã ðkÞ are called ‘‘flutter derivatives’’ of
i
the bridge deck, and are determined by wind tunnel tests,
imposing simple harmonic motion to the deck model.
where the time dependence of DOFs has not been noticed:
P ðt; nÞ; Qðt; nÞ and Rðt; nÞ represent in expression (3) coefficient matrices which, in the most general case, depends on
time and on oscillation frequency of the system.
Principal characteristics of the various aeroelastic theories, are summarized in Table 1: it is clear the overall complexity of the classification, while the most used time
domain techniques are presented below.
L(t2)
L(t1)
M(t1)
M(t2)
D(t2)
D(t1)
ϑ
B
y
V
p
x
L(t2)
ϑ
ð3Þ
α
h
M(t2)
D(t2)
Va(t2)
y
L(t1)
x
h
M(t1)
D(t1)
Vy=W+w
Vx=U+u
B
p
Fig. 2. Bridge deck section under wind action.
Fig. 3. Bridge deck section under turbulent wind action.

4. F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
1035
Table 1
Aeroelastic theories
Level
Theory
Hypothesis and approximations
_ q
General form F ðq; q; €; t; nÞ
(Ref. [9])
S.E.P.a
Aerodynamic
coefficients
X
DOFs
dependences
X
X
X
X
Static
Static
Static
Dynamic
X
X
X
Dynamic
X
X
X
Dynamic
Dependent
from n
Nonlinearised
polar lines
qðtÞ qðtÞ €ðtÞ
_
q
0
1
2
3
Nonaeroelastic (NO)
Steady (ST)
Quasi Steady (QS)
Modified Quasi Steady
(QSM)
Extension of Aeroelastic
Derivates in Time Domain
(ADTD)
Scanlan Theory (NSS)
4
4
a
R
R Á qðtÞ
R Á qðtÞ þ QÁqðtÞ
_
_
RðtÞ Á qðtÞ þ QðtÞ Á qðtÞ
X
X
X
X
X
_
q
P ðt; nÞ Á € þ Qðt; nÞ Á q þ Rðt; nÞ Á q X
X
Frequency domain
X
X
X
S.E.P. = Superposition of Effects Principle (concerning the determination of the actions).
3.1. Nonaeroelastic theory (NO)
This is a ‘‘zero level’’ aeroelastic theory: aeroelastic
effects are not considered in the forces formulation but
only the relative angle of incidence between wind and deck,
change with time just in accordance with the turbulence of
the incident wind. Adopting the small displacements
hypothesis, (linearised lift and moment polar line), it
results
1
2
DðtÞ ¼ q Á jV a ðtÞj Á B Á cD ½aðtÞŠ
2
1
2
LðtÞ ¼ q Á jV a ðtÞj Á B Á K L0 Á aðtÞ
2
1
2
MðtÞ ¼ q Á jV a ðtÞj Á B2 Á K M0 Á aðtÞ
2
ð4Þ
where a is the angle of attack, cD is drag coefficient and
KL0, KM0 are the angular coefficients of lift and moment
polar diagrams, respectively.
1
2
DðtÞ ¼ q Á jV a ðtÞj Á B Á cD ½cðtÞŠ
2
1
2
LðtÞ ¼ q Á jV a ðtÞj Á B Á cL ½cðtÞŠ
2
1
2
MðtÞ ¼ q Á jV a ðtÞj Á B2 Á cM ½cðtÞŠ
2
ð6Þ
Adopting the general formulation of Eq. (3), one can write
F ¼ F ðq; tÞ ¼ R Á qðtÞ
ð7Þ
The steady theory has the appeal of simplicity; furthermore, in the case of nonturbulent incident wind, it shows
the fundamental mechanisms of a flutter stability problem
(coalescence of frequency, influence of structural parameters, etc.). Nevertheless it implies many approximations,
such as the neglecting of the dependence of aeroelastic
forces on structural velocities, accelerations and oscillation
frequency, and the linearisation of the relation between
aeroelastic forces and structural DOFs. Furthermore, the
steady theory does not consider the aerodynamic delay,
and utilizes static aerodynamic coefficients.
3.2. Steady theory (ST)
3.3. Quasi Steady theory (QS)
This is a ‘‘first level’’ aeroelastic theory, where the relative angle of incidence between wind and deck, changes
with time due to both the incident wind turbulence and
the rotation (torsion) of the deck. Supposing that the
bridge deck section rotates around a mean equilibrium
position # ¼ #0 , adopting the small displacements hypothesis (both lift and moment polar diagrams are linearised),
the aerodynamic coefficients become
It is a ‘‘second level’’ aeroelastic theory: instantaneous
aeroelastic forces acting on the structure are the same that
act on the structure itself when it moves with constant
translational and rotational velocities, equal to the real
instantaneous ones. The main assumption consists in considering that the body (deck section) is motionless, together
with the wind having velocities and directions equal to the
instantaneous relative (wind-deck) ones: such assumption
is represented in Fig. 4.
_
The coefficients of # (bi, with i ¼ L; M) should be derived
experimentally by wind tunnel tests [10]; it can be derived
also through the use of Computational Fluid Dynamic
(CFD) techniques [11,12].
Adopting the hypothesis of small displacements around
the mean configuration, Eq. (5) are also valid and the
cL ðcÞ ¼ cL ð#0 Þ þ K L0 Á ðc À #0 Þ
cM ðcÞ ¼ cM ð#0 Þ þ K M0 Á ðc À #0 Þ
ð5Þ
in which KL0 and KM0 are the angular coefficients of polar
lines computed in # ¼ #0 . Referring to Fig. 3 and defining
cðtÞ ¼ aðtÞ À #ðtÞ, the aeroelastic forces are expressed as

5. 1036
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
1
DðtÞ ¼ q Á jV aL ðtÞj2 Á B Á cD ½cðtÞŠ
2
1
2
LðtÞ ¼ q Á jV aL ðtÞj Á B Á cà ½cðtÞŠ
L
2
L(t2)
M(t2)
D(t2)
·
·
h
h
− h + bi ⋅ B ⋅ ϑ
Va(t2)
y
ϑ
β
α
x
L(t1)
h
1
2
MðtÞ ¼ q Á jV aM ðtÞj Á B2 Á cà ½cðtÞŠ
M
2
M(t1)
D(t1)
Vy=W+w
ð10Þ
·
h
Vx=U+u − p
2
where ci ðtÞ, jV ai ðtÞj (i ¼ L; MÞ and cD, have the same
meaning as the previous expressions included in QS theory.
In the expressions (10), aerodynamic coefficients cà and cÃ
L
M
are dynamic and they are computed like below
B
p
Fig. 4. Quasi steady theory assumption.
expressions of aeroelastic forces are identical (in the form)
_
_
V ðtÞÀhþbi ÁBÁ#ðtÞ
to the steady theory ones, with bi ðtÞ ¼ arctg y V x Àp
_
(i ¼ L; M) in substitution of aðtÞ:
1
2
DðtÞ ¼ q Á jV aL ðtÞj Á B Á cD ½cðtÞŠ
2
1
2
LðtÞ ¼ q Á jV aL ðtÞj Á B Á cL ½cðtÞŠ
2
1
2
MðtÞ ¼ q Á jV aM ðtÞj Á B2 Á cM ½cðtÞŠ
2
¼ cL ð#0 Þ þ
Z
#
K L d#
#0
cÃ
M
¼ cM ð#0 Þ þ
Z
ð11Þ
#
K M d#
#0
ð8Þ
2
where
ci ðtÞ ¼ bi ðtÞ À #ðtÞ
and
jV ai ðtÞj ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
_
_
_
ðV x À pÞ þ ðV y À h þ bi Á B Á #Þ . Adopting the general
formulation of Eq. (3), one can write
_
_
F ¼ F ðq; q; tÞ ¼ R Á qðtÞ þ Q Á qðtÞ
cÃ
L
ð9Þ
The Quasi Steady theory can consider the dependences of
aeroelastic forces from structural velocities, preserving also
a relatively simple algorithmic implementation. Furthermore, the dependence from oscillation frequency is neglected, and the dependence of aeroelastic forces from the
DOFs of the structure is linearised. The Quasi Steady theory does not consider the aerodynamic delay, utilizing static aerodynamic coefficients, with the possible exception for
the bi coefficients (with i ¼ L; M), whose value are dynamically assessed [10].
Considering the expected low incidence of turbulent
component on the auto-excited forces, and neglecting the
high order infinitesimal terms, it is possible to obtain [10]
more elegant and explicit expressions than (8), in which static, auto-excited and buffeting component are outlined and
expressed separately.
3.4. Modified Quasi Steady theory (QSM)
In this ‘‘third level’’ aeroelastic theory, in respect to the
QS theory, the only changes concern the aerodynamic coefficients for the lift and the moment, which become dynamic
as measured by wind tunnel tests [13]. Referring to Fig. 4,
aeroelastic forces are expressed by the following
expressions:
where cL ð#0 Þ and cM ð#0 Þ are the static aerodynamic coefficients computed in the mean equilibrium configuration
(# ¼ #0 ), and KL, KM are the ‘‘dynamic derivatives’’ computed like below
ocL
K L ¼ h3 Á
o# #¼#
ð12Þ
ocM
K M ¼ a3 Á
o# #¼#
where h3 and a3 are the Zasso’s theory coefficients [15], assessed by dynamic wind tunnel tests. These coefficients are
similar to the Scanlan’s motion derivatives (2), and they depend both from the rotation deck angle and the ‘‘reduced
wind velocity’’ V red ¼ V =ðx Á BÞ (depending from x, which
is the motion frequency). For multi-degree of freedom
structures (MDOFs), the motion frequency is a combination of overall mode shape frequencies, and for nonlinear
structures it varies at every instant, depending on the state
of structure. So the in advance computation of h3 and a3 is
not practicable. To overcome this problem, in the QSM
theory, the fundamental frequency of the structure is used
to compute the reduced velocity V red ¼ V =ðx Á BÞ and the
corresponding h3 and a3 coefficients. Therefore, the dependence of aeroelastic forces from the motion frequency is
not considered.
Adopting the general formulation, one can write
_
_
F ¼ F ðq; q; tÞ ¼ RðtÞ Á qðtÞ þ QðtÞ Á qðtÞ
ð13Þ
The QSM theory has the attractive aspects of the QS theory (together with high analytic difficulty), and implements
dynamic aerodynamic coefficients. Such coefficients take
into account the nonlinearity of the response in respect to
the wind angle of attack, taking also a partial consideration
of the aerodynamic delay. Furthermore they do not consider the dependence of the forces from the oscillation
motion frequency.

6. F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
3.5. Theory of aeroelastic derivates in time domain (ADTD)
In this ‘‘fourth level’’ aeroelastic theory, the basic concept is very similar to the Wagner’s indicial function theory
[6]. The auto-excited component of aeroelastic forces is
computed by a convolution integral:
Z
DSE ðtÞ ¼
functions needs the introduction of further m differential
equations in the /l ðtÞ and h(t) functions [6].
Adopting the general formulation one can write
_ q
_
F ðq; q; €; nÞ ¼ P ðt; nÞ Á € þ Qðt; nÞ Á q þ Rðt; nÞ Á q
q
ð16Þ
From a conceptual point of view, such theory is the most
complete among the time domain formulations: the dependences of aeroelastic forces both on the structural DOFs
and on structural velocities and accelerations are implemented, and that on the motion frequency is also considered. Furthermore the aerodynamic delay is quantified by
Roger’s formulas. Otherwise one can note a great increase
in the analytical difficulties of the method in respect to the
others.
t
ðI DSEh ðt À sÞ Á hðsÞ
À1
þ I DSEp ðt À sÞ Á pðsÞ þ I DSE# ðt À sÞ Á #ðsÞ Á ds
Z t
LSE ðtÞ ¼
ðI LSEh ðt À sÞ Á hðsÞ
À1
þ I LSEp ðt À sÞ Á pðsÞ þ I LSE# ðt À sÞ Á #ðsÞÞ Á ds
Z t
ðI M SEh ðt À sÞ Á hðsÞ
M SE ðtÞ ¼
4. Application on a long span suspension bridge
À1
þ I M SEp ðt À sÞ Á pðsÞ þ I M SE# ðt À sÞ Á #ðsÞÞ Á ds
ð14Þ
where I iSEj (i ¼ D; L; M and j ¼ p; h; #) is the impulsive
function of the auto-excited force i which corresponds
to the generic jth DOF. Such function represents the aeroelastic force component i which acts on a body under a
wind flow which has an impulsive motion along the jth
DOF. By a Fourier transformation of Eq. (14), and supposing that the motions along the three DOFs are sinusoidal with the same oscillation frequency, by comparing Eq.
(14) with Eq. (2), it is possible to obtain the relationships
between the Fourier transform (I iSEj ) of I iSEj (i ¼ D; L; M
and j ¼ p; h; #), and the Scanlan’s flutter derivatives: if
one knows the Scanlan’s flutter derivatives of lift, drag
and moment, one can obtain also the functions I iSEj [6].
Nevertheless the flutter derivatives are known only in
discrete values of reduced frequency (k), and are made
continuous in the frequency domain by the Roger’s
approximating function [6] that can replace the I iSEj . Operating a changing of variable, the Roger’s function are
transposed in Laplace’s domain and, by the Laplace’s inverse transformation, they are transposed finally in the
time domain. Concerning, for example, the part of autoexcited component of the Lift that depends on the sectional vertical DOF (h(t)), using this procedure one can
obtain the following expression:
1
LSEh ðtÞ ¼ q Á jV a ðtÞj2 Á B
2
Á
1037
!
m
B _
B2 € X
Á h þ a3 Á
Áhþ
/l ðtÞ
a1 Á hðtÞ þ a2 Á
jV a j
jV a j2
l¼1
ð15Þ
where ai coefficients and the sum extreme m are those previously defined (during the Roger’s function generation
phase), and the /l ðtÞ are integral terms, which represent
the ‘‘memory terms’’ of the force. The assessment of /l ðtÞ
In this paper, using the above introduced time domain
techniques, the response problem of a long span suspension
bridge under turbulent wind, has been studied. After that,
the stability problem under nonturbulent wind has been
studied, comparing the critical velocities computed by different techniques.
4.1. Descriptions of the bridge and structural performance
aspects
A long span suspension bridge has been examined [16].
The main span of the bridge is 3300 m long, while the total
length of the deck, 60 m wide, is 3666 m (including the side
spans). The deck is formed by three box sections, the outer
ones for the roadways and the central one for the railway
(Fig. 6). The roadway deck has three lanes for each carriageway (two driving lanes and one emergency lane), each
3.75 m wide, while the railway section has two tracks.
The two towers are 383 m high and the bridge suspension system relies on two pairs of steel cables, North and
South, each with a diameter of 1.24 m and a total length,
between the anchor blocks, of approximately 5000 m. Principal characteristics of the structure are summarized in
Figs. 5 and 6. Because of the suspended span size, the sensitivity of the structure at the wind action is foreseeable. In
this sense, the adopted ‘‘multibox’’ section [17] for the deck
section is innovative and it is finalized to optimize the aerodynamic response.
The numerical model was based on the preliminary
design of the Messina Strait Bridge (for a complete description of geometrical and mechanical properties, see [18]) and
it has been developed using 3D beam finite elements, with
each node having six degrees of freedom, as shown in
Fig. 7. The permanent loads and the masses are modelled
as distributed along the elements. For the developed transient step by step analyses, a Newmark time integration
scheme [19,20] has been adopted, in which geometric nonlinearities has been considered.

7. 1038
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
960
777
3300 m
3300
183
+383.00
+54.00
183
810
627
+383.00
+77.00 m
+52.00
+63.00
+118.00
Fig. 5. Bridge profile.
Fig. 6. Bridge deck section.
Fig. 7. 3D FEM model.
In the design stages, numerical analyses are conducted in
order to verify safety and serviceability performance of the
bridge, organized in contingency scenarios and related to
different probabilities of occurrence, i.e., deck accelerations, stresses on substructures, critical wind velocities.
Expected values of each performance are fixed in the basis
of design and verified by structural analyses [21].
ments: complete serviceability (roadway and railway
traffic), partial (only railway traffic) serviceability and
maintaining the structural integrity respectively. Nonaeroelastic (NO), steady (ST), quasi steady (QS) and modified
quasi steady (QSM) theories have been applied in the analyses. Both nonturbulent and turbulent flows have been
considered. The results of the analyses are listed in Tables
2 and 3.
4.2. Analyses developed
4.3. Results
The structural response has been investigated in respect
to three different mean wind velocities at the deck level: 21,
45, 57 m/s, which correspond to a 50, 200 and 2000 years of
return period TR, and to three different structural require-
4.3.1. Response problem
Several geometric configurations of the bridge deck
section (obtained by changing the shape, the traffic and

8. F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Table 2
Response analyses
0.3
0.2
Wind mean
velocity (m/s)
Nonturbulent
flow
Turbulent
flow
Response analyses done by aeroelastic theories: NO, ST, QS, QSM
Time history of midspan 45
Fig. 9
Figs. 12–15
displacements
Statistics
45
Envelopes of deck
displacements
21
45
57
0.1
-10
-8
-6
-4
Fig. 16
Fig. 16
Fig. 16
Fig. 11
45
Fig. 17
0
-2
0
-0.1
2
4
6
8
10
[deg]
-0.2
Figs. 12–15
Envelopes of deck
velocities and
acceleration
1039
L
-0.3
M,
M. a
-0.4
-0.5
Drag
Lift
Moment
D
Fig. 8. Polar lines for response problem.
Table 3
Stability analyses
Formulation
Stability analyses done by aeroelastic theories
Time history of midspan
ALL
displacements
Critical velocities
ALL
Diagram on phases plane
QS
Aerodynamic damping on wind
QS
velocities
Nonturbulent
flow
Fig. 20
Fig. 21
Fig. 23
Fig. 24
wind barriers configurations) have been tested in wind tunnel tests: in Fig. 8, the static polar lines used in this paper
for the response problem are reported.
At first, a preliminary analysis of the responses under
nonturbulent wind has been performed to assess the fundamental characteristics of the responses (oscillation amplitudes, aerodynamic damping, mean values), supplied by
the different theories, regardless the dispersion of results
induced by the turbulence. In Fig. 9, oscillations along
the three sectional DOFs of the railway box mass centre
in the deck midspan, computed by an incident nonturbulent flow having a mean velocity of 45 m/s, are shown. In
Fig. 10 the relative mean values are presented, together
with the experimental ones [23].
The response of the structure is represented by a time
damped oscillation. In QS and QSM results, one can note
_
the presence of an aerodynamic damping (QðtÞ Á qðtÞ, with
reference to the general form), so that the oscillation amplitude decreases more than linearly in time. Concerning the
mean values, they result quite greater than the experimental ones, especially for the deck rotation.
In Fig. 11, time envelopes of transversal and vertical
deck displacements (railway box section mass centre) under
nonturbulent flow having a mean velocity of 45 m/s are
presented, together with the results derived from a static
equivalent formulation, and static analysis. One can note
that the relative differences from a theory to another are
not significant.
After preliminary nonturbulent flow analyses, successive
analyses have been conducted considering a turbulent
wind. Time histories of the wind velocity field have been
generated numerically and obtained by Solari and Carassale [22], and are generated like components of a multivariate, multidimensional Gaussian stationary stochastic
process.
In Figs. 12–14, oscillations along the three sectional
DOFs of the railway box mass centre in the deck midspan,
computed by an incident turbulent flow having a mean
velocity of 45 m/s, are represented. Every displacement
time history has been characterized from a statistic point
of view by the frequency probability density (including
the 5% and 95% fractile values), and by the histogram representing the overcoming frequencies.
In Fig. 15, time histories for the three sectional DOFs of
the railway box mass centre in the deck midspan, computed
by an incident turbulent flow having a mean velocity of
45 m/s, are resumed, and also the computed and experimental mean values are represented.
In general, one can note that by increasing the complexity of the aeroelastic forces representation (following
the succession NO, ST, QS, QSM), both the maximum
amplitude of the oscillations and the variance of computed
time history decrease. Regarding this tendency an exception is represented from the rotation of the deck around
own longitudinal axis, which in QSM results is greater
(both in amplitude and in dispersion) than that obtained
by QS.
One can note that NO results are a similar to those of
ST results, and QS results are similar to the QSM ones.
Concerning the mean values, the similitude of the results
for couples of formulations (NO–ST, QS–QSM) is
confirmed.
Concerning to the mean incident wind velocities of 21
and 57 m/s, analogous analyses have been conducted. In
Fig. 16, regarding the three examined velocities, time envelopes of the transversal and vertical deck displacements
under turbulent flow are represented.

9. 1040
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Transversal
5.60
Vertical
Rotation
-0.0081
-0.275
Uy (m)
5.65
400
900
1400
1 900
2 400
400
2900
900
1400
1 900
2400
2900
time (sec)
-0.0083
-0.279
time (sec)
-0.0085
5.50
-0.287
-0.0087
5.45
-0.291
time (sec)
5.40
400
5.60
1400
1900
2400
-0.295
2900
-0.0089
-0.0091
-0.0088
-0.225
400
Uy (m)
5.65
900
900
1400
1900
2400
-0.229
400
2900
1400
1900
2400
2900
time (sec)
-0.237
-0.0094
-0.241
-0.0096
time (sec)
5.40
5.65
5.60
900
1400
1900
2400
-0.245
2900
-0.0098
-0.0074
-0.338
400
Uy (m )
400
Rot (RAD)
-0.0092
5.50
Uz (m)
900
1400
1900
2400
2900
400
time (sec )
-0.342
900
1400
1900
-0.0078
-0.350
-0.0080
-0.354
2900
-0.0082
5.45
ti me (sec)
5.40
400
5.60
1400
1900
2400
2900
-0.0084
-0.328
400
Uy (m )
5.65
900
-0.358
Rot (RAD)
-0.346
5.50
Uz (m)
5.55
2400
time (sec)
-0.0076
-0.0074
900
1400
1900
2400
2900
400
time (sec)
-0.332
900
1400
1900
-0.0076
-0.336
-0.0078
5.50
-0.340
-0.0080
-0.344
-0.0082
5.45
ti me (sec )
900
1400
1900
2400
-0.0084
-0.348
2900
Fig. 9. Time history of midspan displacements (nonturbulent flow; V ¼ 45 m/s).
Transversal
Vertical
Rotation
0.0
6.0
0.0
-0.1
-0.2
4.0
-0.2
3.0
-0.3
2.0
-0.4
Rotation(DEG)
-0.1
5.0
-0.3
1.0
-0.5
0.0
-0.4
NO
ST
QS
QSM
Experim
-0.6
NO
ST
QS
QSM
Experim
NO
ST
QS
QSM
Fig. 10. Mean values of midspan displacements (nonturbulent flow; V ¼ 45 m/s).
Transversal Max
6.0
5.0
Vertical Max
0.0
ST NO QS QSM
Uz (m)
0
1000
2000
Static
3000
abscissa (m)
4.0
3.0
ST
-0.2
NO
2.0
-0.3
1.0
abscissa (m)
0.0
0
1000
2000
4000
Static
-0.1
3000
4000
QSM
Uz (m)
5.40
400
Uz (m)
5.55
2400
time (sec)
Rot (RAD)
ST
-0.233
5.45
QS
900
-0.0090
ti me (sec)
5.55
QSM
Rot (RAD)
-0.283
Uz (m)
NO
5.55
QS
-0.4
Fig. 11. Envelopes of deck displacements (nonturbulent flow; V ¼ 45 m=s).
Experim
2900

10. F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Time history
10
1000
800
600
6
400
4
200
time (sec)
400
2400
2900
1200
1000
800
ST
8
600
6
400
4
200
2
time (sec)
400
10
1400
1900
2400
2900
1200
Uy (m)
12
900
1000
800
8
QS
Class
0
0
14
Frequency
10
1900
2,
41
3,
91
5,
42
6,
92
8,
43
9,
93
11
,4
3
12
,9
4
12
1400
Uy (m)
14
900
Class
0
0
2,
41
3,
91
5,
42
6,
92
8,
43
9,
93
11
,4
3
12
,9
4
2
600
6
Frequency
NO
8
Frequency
12
Probability density
1200
Uy (m)
14
Frequencies
1041
400
4
200
2
Class
time (sec)
0
900
400
10
2400
2900
1200
1000
800
600
6
400
4
200
2
Class
time (sec)
0
0
400
900
1400
1900
2400
2900
2,
41
3,
91
5,
42
6,
92
8,
43
9,
93
11
,4
3
12
,9
4
QSM
8
Frequency
12
1900
Uy (m)
14
1400
2,
41
3,
91
5,
42
6,
92
8,
43
9,
93
11
,4
3
12
,9
4
0
Fig. 12. Time histories of midspan transversal displacements and their statistic characterization (turbulent flow; V ¼ 45 m=s).
Envelopes confirm the tendency previous evidenced by
the time histories of midspan displacements: increasing
the complexity of the aeroelastic forces representation,
the envelopes decrease. Also in terms of envelopes, the
similitude of the results for couples of formulations (NO–
ST, QS–QSM) is confirmed. Among the examined formulations, the ST is the more sensitive to the increase of mean
wind velocity.
Similar diagrams have been computed concerning velocities and accelerations of the deck: in Fig. 17 the time envelopes of these kinematic entities are represented for the
wind mean velocity of 45 m/s. The deck accelerations, in
particular, have a great relevance in the bridge performance table: they have to maintain themselves under an
imposed limit to ensure the safety during the transit of
the trains.
Also in the velocities and in the accelerations envelopes,
there is a decrease when the complexities of the formulations increase, and the results are similar by couple of
formulations (NO–ST, QS–QSM).
4.3.2. Stability problem
Once evaluated the response under turbulent incident
flow, further analyses have been conducted for the aeroelastic stability problem under nonturbulent wind. Typically, for suspension bridges, the most dangerous
instability phenomenon is flutter (Fig. 1), a dynamic instability in which two DOFs of the forced structural system
are coupled: under opportune configurations (defined
‘‘critical’’) for frequencies and reciprocal phase angles, it
makes the damping of the system become negative, and
the structural oscillations increase in amplitude. For a

11. 1042
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Time history
1000
800
0.5
600
-0.5400
900
1400
1900
2400
2900
400
-1.5
-2.5
0
-1
.7
9
-1
.0
2
-0
.2
5
0.
53
1.
30
2.
07
2.
84
3.
61
time (sec)
-3.5
1000
800
0.5
600
-0.5400
900
1400
1900
2400
2900
0
time (sec)
-1
.7
9
-1
.0
2
-0
.2
5
0.
53
1.
30
2.
07
2.
84
3.
61
-2.5
-3.5
1800
Uz (m)
2.5
1600
1400
1200
1.5
1000
0.5
800
-0.5400
900
1400
1900
2400
2900
-1.5
-1
.7
9
-1
.0
2
-0
.2
5
0.
53
1.
30
2.
07
2.
84
3.
61
Uz (m)
3000
2500
2000
0.5
1500
1400
1900
-3.5
2400
2900
1000
500
time (sec)
Class
0
-1
.7
9
-1
.0
2
-0
.2
5
0.
53
1.
30
2.
07
2.
84
3.
61
900
-1.5
-2.5
Class
3500
1.5
-0.5400
400
0
time (sec)
-3.5
2.5
600
200
-2.5
3.5
Class
200
-1.5
3.5
400
Frequency
ST
1.5
Frequency
2.5
1200
Uz (m)
3.5
QS
Class
200
Frequency
NO
1.5
QSM
Probability density
Frequency
2.5
Frequencies
1200
Uz (m)
3.5
Fig. 13. Time histories of midspan vertical displacements and their statistic characterization (turbulent flow; V ¼ 45 m=s).
suspension bridge deck, the two above sectional DOFs, are
the vertical and the rotational one. Critical configurations
are such as that the oscillation frequency is the same and
the difference between the phase angles is equal to p/2.
Samples of stable, critical and unstable oscillations are
shown in Fig. 18.
The vertical and, in particular, rotational motion frequency, depend on the incident wind velocity. When this
velocity increases, the frequencies come closer to each other
until the ‘‘frequency coalescence’’: during this interval of
time the damping is positive. When the two frequencies
coincide, the damping becomes equal to zero and, if the
wind velocity increases, the damping of the system becomes
negative. The wind velocity which corresponds to zero
damping and incipient flutter is called ‘‘critical wind velocity’’ (Vcrit).
The capability of the examined formulations in computing the flutter phenomenon has been investigated. In
Fig. 19, the polar lines that have been utilized in the stability problem are shown.
Concerning the NO formulation, it is evident from Eq.
(4) that the forces do not depend on the structure motion,
in the case of nonturbulent incident wind, and the forces
are constant in time. An increase of the wind velocity produces an increase of initial amplitudes of the structure oscillation only, so oscillations decrease in time. Consequently,
the NO theory is unable to compute the flutter, while the
others formulations can represent the phenomenon, but
they lead to different values of the critical wind velocity.
In Fig. 20, time histories of unstable oscillations
(V V crit ) are shown. The diagrams refer to the ST, QS
and QSM theories.

12. F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Time history
Frequencies
1043
Probability density
0.015
NO
0.005
-0.005
400
-0.015
1000
800
900
1400
1900
2400
2900
600
Frequency
0.025
Rot (RAD)
1200
400
-0.025
200
Class
-0.035
0
-0.045
-0
.0
47
-0
.0
36
-0
.0
25
-0
.0
14
-0
.0
03
0.
00
8
0.
01
9
0.
03
0
time (sec)
-0.055
ST
0.005
-0.005
400
-0.015
1000
800
900
1400
1900
2400
2900
Frequency
0.015
Rot (RAD)
1200
0.025
600
400
-0.025
Class
200
-0.035
0
-0.045
900
1400
1900
2400
2900
-0.025
-0.035
-0.045
0.015
QSM
0.005
Rot (RAD)
0.025
-0.005
400
-0.015
900
1400
1900
-0.025
-0.035
-0.045
-0.055
time (sec)
Frequency
-0
.0
47
-0
.0
36
-0
.0
25
-0
.0
14
-0
.0
03
0.
00
8
0.
01
9
0.
03
0
time (sec)
-0.055
Class
2400
2900
2000
1800
1600
1400
1200
1000
800
600
400
200
0
Frequency
QS
0.005
-0.005
400
-0.015
2000
1800
1600
1400
1200
1000
800
600
400
200
0
Class
-0
.0
47
-0
.0
36
-0
.0
25
-0
.0
14
-0
.0
03
0.
00
8
0.
01
9
0.
03
0
0.015
Rot (RAD)
0.025
-0
.0
47
-0
.0
36
-0
.0
25
-0
.0
14
-0
.0
03
0.
00
8
0.
01
9
0.
03
0
time (sec)
-0.055
Fig. 14. Time histories of midspan rotational displacements and their statistic characterization (turbulent flow; V ¼ 45 m=s).
It is clear that the three formulations compute the instability, but each one produces different damping for a generic wind velocity. The critical wind velocities computed
by the three formulations are shown in Fig. 21.
To investigate the mechanism of the change in damping
sign with the increase of wind velocity, the damping has
been estimated by identifying it with the exponential
coefficient d of the function qðtÞ ¼ Æ q0 Á eÀdÁt ( identifies
q
q
the static equilibrium position), which envelopes the
generic oscillation (see Fig. 22): in damped oscillations
(V V crit ), critical oscillations (V ¼ V crit ) and amplified
oscillations (V V crit ), it results d 0; d 0; d ¼ 0, respectively. The oscillations in the phase plane (rotation and vertical displacement) and the time projections (3D graphics)
of the planes, are shown in Fig. 23. In such diagrams, the
oscillations become pseudo-circular curves, which implode
in a single point (final configuration) when V V crit , or
they stabilize themselves along a circular curve of constant
amplitude when V ¼ V crit (after a transient initial period
with different amplitude oscillations), or they explode like
a divergent spiral when V V crit .
In Fig. 24, using the QS theory, the amount of damping
d for different velocity of incident flow is shown: here, d
represents the total damping of the structural system,
which is sum of the structural (assumed constant and equal
to 0.5%) and the aerodynamic one (computed as the analytical difference from the total damping and the structural
one). The total damping curve grows when there is an
increasing of the wind velocity; at a certain value it changes
its slope and begins to decrease until the intersection of
x-axis. Such intersection represents the critical flutter
velocity.

13. 1044
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Time history
Probability density
Mean values
7.0
Uy (m)
14
12
6.0
5.0
8
4.0
6
3.0
Transversal
10
4
2.0
2
1.0
time (sec)
0
400
900
NO_V45
1400
ST_V45
1900
QS_V45
2400
QSM_V45
0.0
2900
NO
ST
QS
QSM
Experim
NO
ST
QS
QSM
Experim
NO
ST
QS
QSM
Experim
0.0
Uz (m)
3.5
2.5
-0.1
Vertical
1.5
0.5
-0.2
-0.5400
900
1400
1900
2400
2900
-0.3
-1.5
-2.5
time (sec)
-0.4
-3.5
ST_V45
QS_V45
QSM_V45
Rot (RAD)
NO_V45
0.015
0.0
-0.1
Rotation
0.005
-0.2
-0.005
400
-0.015
900
1400
1900
2400
2900
-0.3
-0.025
-0.4
Rotation(DEG)
0.025
-0.035
-0.5
-0.045
time (sec)
-0.055
NO_V45
ST_V45
-0.6
QS_V45
QSM_V45
Fig. 15. Time histories of midspan displacements, statistic characterization and mean values (turbulent flow; V ¼ 45 m=s).
Vm= 21 m/s
Vm= 45 m/s
Vm= 57 m/s
NO_V21
12
QS_V21
2
QSM_V21
25
ST_V57
NO_V57
NO_V45
QS_V45
20
QSM_V45
8
1.5
QS_V57
15
QSM_57
6
10
1
Static
Analisys_V45
4
2
5
abscissa (m)
abscissa (m)
0
0
500
Uz (m)
0.5
0.4
1000
1500
2000
2500
3000
3500
0
4000
ST_V21
0
4
NO_V21
500
Uz (m)
0
1000
1500
2000
2500
3000
3500
4000
0
500
1000
1500
2000
2500
3000
ST_V45
3
6
ST_V57
4
2
0.2
QS_V21
1
QSM_57
1
0
0
0
500
1000
-0.1
1500
2000
2500
abscissa (m)
3000
3500
0
abscissa (m)
0
4000
500
1000
1500
0
500
1000
1500
2000
abscissa (m)
-0.2
2500
3000
3500
4000
2000
2500
3000
abscissa (m)
0
500
1000
1500
2000
-1
QSM_V21
4000
QS_V45
0
0
500
1000
1500
2000
2500
3000
3500
4000
2500
3000
3500
4000
abscissa (m)
0
2500
3000
QSM_V45
-2
-0.3
3500
abscissa (m)
0
-0.1
QS_V57
2
QSM_V45
QSM_V21
NO_V57
3
QS_V45
0.1
3500
4000
0
500
1000
1500
2000
-1
QSM_57
-2
-3
QS_V57
QS_V21
-0.4
-4
-3
-0.7
NO_V57
-5
ST_V21
NO_V21
-4
-5
ST_V45
-6
Uz (m)
Uz (m)
NO_V45
-0.6
4000
5
NO_V45
0.3
-0.5
3500
7
Uz (m)
abscissa (m)
Vertical MAX
ST_V45
10
0.5
Vertical min
Uy (m)
Uy (m)
ST_V21
Uz (m)
Transversal
2.5
14
Uy (m)
30
3
-7
Fig. 16. Envelopes of deck displacements (turbulent flow; V ¼ 21; 45; 57 m=s).
ST_V57

14. F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Transversal
Velocities
Vertical
2.5
ST_V45
1.5
Vy (m/s)
NO_V45
Vy (m/s)
1.2
1045
ST_V45
NO_V45
QS_V45
QS_V45
0.5
QSM_V45
abscissa (m)
0.2
0
500
1000
1500
QSM_V45
2000
2500
3000
3500
-0.5 0
500
1000
abscissa (m)
1500
2000
2500
3000
3500
-0.8
-1.5
-1.8
-2.5
Accelerations
QS_V45
QSM_V45
NO_V45
ST_V45
ay (m/s^2)
0.7
ST_V45
NO_V45
1.5
ST_V45
QSM_V45
QS_V45
QSM_V45
ST_V45
NO_V45
QS_V45
0.3
-0.1
az (m/s^2)
NO_V45
0.5
abscissa (m)
QS_V45
abscissa (m)
QSM_V45
0
500
1000
1500
2000
2500
3000
3500
0
500
1000
1500
2000
2500
3000
3500
-0.5
-0.5
-1.5
-0.9
NO_V45
ST_V45
QS_V45
NO_V45
QSM_V45
ST_V45
QS_V45
QSM_V45
Fig. 17. Envelopes of deck velocities and accelerations (turbulent flow; V ¼ 45 m=s).
0.520
0.3
0.2
0.515
0.1
0.510
0
-10
-8
-6
-4
-2
0
2
4
-0.1
650
700
750
800
850
900
8
10
-0.2
t (sec)
0.500
600
6
[deg]
0.505
950
-0.3
1000
stable (positive damping)
-0.4
-0.5
0.525
Drag
Lift
Moment
0.520
Fig. 19. Polar lines used for stability problem.
0.515
5. Conclusions
0.510
0.505
t (sec)
0.500
600
650
700
750
800
850
900
950
1000
critical (zero damping)
0.700
0.600
In the present paper the response and the stability problem of a long span suspension bridge have been studied.
Four time domain approaches for aeroelastic forces formulations have been compared. The analyses have been conducted by a three dimensional complete finite element
model of the bridge.
Concerning the response problem one can conclude:
0.500
0.400
t (sec)
0.300
600
650
700
750
800
850
900
950
1000
unstable (negative damping)
Fig. 18. Stable, critical and unstable oscillations.
1. Considering nonturbulent incident wind, the differences
between the formulations on the structural oscillations
damping, the QS and QSM formulations have a damping greater than linear; concerning the time envelopes of
deck displacements, the results obtained from different
formulations are very similar.

15. 1046
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
Rotational
NO
Vertical
NO FLUTTER
NO FLUTTER
0.20
0.0006
0.15
0.0004
ST (70m/s)
0.10
0.0002
0.05
0.0000
0.00
580
-0.05
630
680
730
780
830
880
930
580
980
630
680
730
780
830
880
930
980
830
880
930
980
-0.0002
-0.10
-0.0004
-0.15
-0.0006
-0.20
0.004
0.30
0.003
0.20
QS (75m/s)
0.002
0.10
0.001
0.00
0.000
580
630
680
730
780
830
880
930
980
-0.10
580
-0.001
630
680
730
780
-0.002
-0.20
-0.003
-0.30
-0.004
0.003
0.0005
QSM (90m/s)
0.002
0.0003
0.001
0.0001
0.000
1550
1650
1750
1850
1950
1550
-0.0001
1650
1750
1850
1950
-0.001
-0.0003
-0.002
-0.0005
-0.003
Fig. 20. Midspan unstable oscillations (V V crit ).
0.525
90
q+ q 0
80
Uz; Theta
0.520
V (m/s)
60
50
40
30
NO FLUTTER
70
q = q + q0 ⋅ e −δ ⋅t
0.515
VVcrit
δ 0
q
0.510
66m/s
70m/s
85m/s
0.505
t (sec)
0.500
20
600
650
700
750
800
850
900
950
1000
10
Fig. 22. Envelope of midspan oscillation, to evaluate damping.
0
NO
ST
QS
QSM
Fig. 21. Critical velocities (nonturbulent flow).
2. Considering turbulent incident wind, the differences
between the oscillations amplitude computed by
different formulations become significant. In general,
increasing the complexity of the aeroelastic forces
representation (following the succession NO, ST, QS,
QSM), the maximum response decrease: this is evident
from the time histories of displacements, velocities and
accelerations, from their statistic characterization and
also from time envelopes of the deck motion. These differences increase with the increase of the wind mean
velocity.
Concerning the stability problem:
1. NO formulation cannot compute the flutter phenomenon, while the other formulations can.

16. 1047
Theta
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
final
VVcrit
δ 0
start
Theta
Uz
final
V=Vcrit
δ =0
start
Theta
Uz
final
VVcrit
δ 0
start
Uz
Fig. 23. Damped, critical and amplified oscillations for midspan of the deck (phases plane representation).
Vertical
Rotational
1.0
0.5
0.0
0
10
20
30
40
50
60
70
80
-0.5
Damping (%)
1.5
1.0
Damping (%)
1.5
0.5
0.0
-0.5
0
10
Wind Velocity (m/s)
20
30
40
50
60
70
80
Wind Velocity (m/s)
-1.0
-1.0
-1.5
-1.5
Total
Structural
Aerodynamic
Total
Structural
Aerodynamic
Fig. 24. Damping on incident flow velocity.
2. Increasing the complexity of the aeroelastic forces representation, the value of the critical velocity increases.
3. The variation of aeroelastic damping with the wind incident velocity has been assessed using QS formulation,
where the aerodynamic damping increases its value from
zero velocity to a certain value of the wind velocity;
beyond this value it starts to decrease and finally it
becomes negative.
Acknowledgements
The authors thank Professors R. Calzona and K.J.
Bathe for fundamental supports related to this study. The
financial supports of University of Rome ‘‘La Sapienza’’,
COFIN2004 and Stretto di Messina S.p.A. are acknowledged. Nevertheless, the opinions and the results presented
here are responsibility of the authors and cannot be

17. 1048
F. Petrini et al. / Computers and Structures 85 (2007) 1032–1048
assumed to reflect the ones of University of Rome ‘‘La
Sapienza’’ or of Stretto di Messina S.p.A.
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