cable stayed bridge report

7/21/2019 cable stayed bridge report

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Chapter 1

INTRODUCTION

1.1Preview

Social and economic demand for efficient transportation system has resulted in greater demand forlong span cable-stayed bridges all over the world. Cable-stayed bridges usually span wide rivers,

canals etc., therefore modern long span cable-stayed bridges are very light weight, flexible, and

exhibit low damping. Since cablestayed bridges are frequently constructed along the coastal

areas, they are vulnerable to high wind speed and turbulence. Therefore wind induced vibration is

a very common phenomenon which poses a new challenges to the bridge engineers. The infamous

failure of original Tacoma Narrow Bridge (1940) opens a new chapter in the field of wind induced

vibration of long span cable-supported bridges (Fig. 1.1). Thus to ensure safe and efficient

functionality of bridges and vehicle under wind loading, it is of utmost importance to accurately

determine the behaviors and performances of cable-stayed bridges due to wind induced vibration

and consequently effective measures for mitigating excessive vibration to an acceptable level and

to improve stability to avoid catastrophic collapse.

Figure 1.1 The failure of original Tacoma Narrow Bridge (1940) (Source: Wikipedia

(http://en.wikipedia.org/wiki/File:Image-Tacoma_Narrows_Bridge1.gif)).

1.2 Main features of Cable-stayed Bridges

As shown in Fig. 1.2 a cable-stayed bridge mainly consists of three components- (i) Bridge Deck,

(ii) stay cables and (iii) towers or pylons. Although the concept of cable-stayed bridges actuallyoriginated from suspension bridge but they have very different principles. Cable-stayed bridge is

an optimization between spans longer than cantilever bridges and shorter than suspension bridges.

The deck girder, tower and cables are basic structural features of a cable-stayed bridges. The

components of the bridges are mainly subjected to axial forces. The cables are under tension

whereas the pylon and deck is under compression. Two important advantages are (i) since the
http://en.wikipedia.org/wiki/File:Image-Tacoma_Narrows_Bridge1.gif)http://en.wikipedia.org/wiki/File:Image-Tacoma_Narrows_Bridge1.gif)http://en.wikipedia.org/wiki/File:Image-Tacoma_Narrows_Bridge1.gif)http://en.wikipedia.org/wiki/File:Image-Tacoma_Narrows_Bridge1.gif)


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members are predominantly under axial loading, therefore their performance is better as compared

to any flexural member; (ii) for a symmetric cable-stayed bridge the horizontal force in the deck

Figure 1.2 Schematic diagram showing main components of cable-stayed bridge.

(a) (b) (c)

Figure 1.3 Different types of cable systems (a) harp type, (b) fan type and (c) radial type.

balances, thus eliminate the requirement of large anchorage. There are mainly three types of cable

system as shown in Fig. 1.3, (i) harp or parallel cable system, (ii) fan or intermediate cable systemand (iii) radial or converging cable system. Deck may be either comprising of concrete or steel or

a composite section with concrete slab in a steel frame. Pylons can be of different shapes like H-

shaped pylon, A-shape, the inverted Y and the diamond shape etc.

1.3Organization of the report

Chapter 1 gives introduction about the cable-stayed bridges.

Chapter 2 introduces the concept of mean wind load and aerostatic instability. Also simple finite

element modelling of cable-supported bridges are discussed briefly in this chapter.

Wind induced vibration and aerodynamic instabilities e.g. vortex-induced vibration, gallopinginstability, flutter and buffeting are described in chapter 3.

Chapter 4 presents various control strategies to mitigate wind-induced vibration of cable-supported

bridges.

Finally, chapter 5 gives conclusions of the present study including few possible directions for

future research.


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Chapter 2

AEROSTATIC INSTABILITY

2.1 Wind speed components

The wind speed is usually decomposed into a mean wind speed which acts in the mean winddirection and three mutually perpendicular components. At a given point and time wind velocities

can be written as

In the longitudinal direction: ),,,()( tzyxuzU

In the lateral direction: ),,,( tzyxv

In the vertical direction: ),,,( tzyxw

In which )(zU is the mean wind speed as a function ofz , the height above ground, and wvu ,, are

the fluctuating parts of the wind in the yx, and z direction respectively.

2.2 Mean wind load

The total bridge response is considered as the sum of mean response and random response with

zero mean. Mean response can be calculated from mean wind load which depend on mean wind

speed. As the modern long span cable-stayed bridges are highly flexible in nature, therefore mean

wind loading can cause considerable movement of the bridge components. Furthermore it can

cause aerostatic instability which may lead to collapse of the bridge. Although the mean wind load

is usually expressed with respect to the wind coordinate system, it can also be expressed with

respect to the structural coordinate system as shown in Fig. 2.1.

Figure 2.1 Mean wind load in wind coordinate system and structural coordinate system[Xu 2013].

The drag force, lift force and moment acting on a bridge deck section can be given by the following

expressions

)(2

1)( 2

DD BCUF 2.1(a)


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)(2

1)( 2

LL BCUF 2.1(b)

)(2

1)( 2

MBCUM 2.1(c)

WhereLD

CC , andM

C are the non-dimensional drag, lift and moment coefficients respectively and

all depends upon the wind angle of attack . These coefficients are obtained through wind tunnel

tests of the geometrically scaled model of the prototype bridge. U is the incoming wind velocity

and B is the characteristic dimension of the bridge section, usually taken as the width of the deck.

2.3 Torsional Divergence

It is referred to as the torsional instability which causes continuous increase in the bridge deck

rotation until failure at a critical wind speed. It is a non-oscillatory phenomenon which takes place

abruptly, leading to collapse of the bridge.

1-D Torsional divergence

Figure 2.2 1-D deck model of torsional divergence [Xu 2013].

Fig. 2.2 shows a 1-D model of the bridge deck section. Torsional equilibrium equation is given as

)(2

1 22

MCBUK (2.2)

K torsional stiffness of the bridge girder

Taylor series expansion of )(M

C with respect to zero angle of attack gives (higher order terms

are neglected)

)0()0()(MMM

CCC (2.3)

Substitution of Eq. (2.3) in (2.2) yields

)0(2

1)0(

2

1 2222MM

CBUCBUK

(2.4)

The second term on the left hand side in the parentheses acts as negative torsional stiffness which

increases with wind speed. When the effective torsional stiffness become zero, rotation of the


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bridge section becomes divergent. So the critical wind speed corresponding to the torsional

divergence can be written as

0)0(2

1 22

M

CBUK

)0(

22

M

cr

CB

KU

(2.5)

2.4 3-D Aerostatic instability analysis

Boonyapinyo et al (1994)performed aerostatic instability analysis of cable-stayed bridges. This

is generally associated with lateral-torsional buckling. Boonyapinyo proposed a 3-D nonlinear

analysis which considers geometric nonlinearity and displacement dependent wind forces. In

iterative form the nonlinear equilibrium equation is written as

)](),(),([)](),(),([}{)]([][ 11111 jzjyjxjjzjyjxjjjge MFFFMFFFxxKK (2.6)

][e

K is the elastic linear stiffness matrix.

)]([ 1jg xK is the geometric stiffness matrix at the (j-1)-th step.

jx is the incremental displacement at the j-th step.

jF and 1jF are the structural displacement dependent wind load vectors at j-th and (j-1)-th iteration

respectively.

For a given wind velocity, convergence is reached when Euclidean norm of static aerodynamic

coefficients are less than a prescribed tolerance limit, i.e.

],,[

)]([

)]()([2

1

2

1

2

1

ZYXk

C

CC

kN

jk

N

jkjk

a

a

(2.6)

aN total node number and k prescribed tolerance limit.

2.5 Finite Element Modelling

2.5.1 Spine Beam Model

It is one of the simplest finite element model for a cable-stayed bridge. This simplified model can

effectively demonstrate the dynamic characteristics and overall structural behavior of a bridge

without much computational efforts. Therefore this model is preferred especially for predicting the


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global structural behavior, initial design as well as aerodynamic analysis. However, for local

stress-strain analysis solid and/or shell element must be used. The following line elements are used

in spline beam model: beam elements, truss elements and rigid links.

Pylons and piers are modelled using beam element related to their geometric properties. Truss

elements are used to model the cables. The effects of geometric non-linearity is also taken intoaccount. The spline beam which is referred to as the central beam is used to model the deck. As

the bridge deck usually consists of variety of cross sections, the equivalent cross section of the

beam element is calculated by considering the effective area of all the sections. If different

materials are used in the deck-section, then all should be converted to single material through the

use of modular ratio. Similarly the position of neutral axis and the moment of inertia of the section

about transverse and vertical axis is calculated. Since the bridge section is not is not a circular one,

thus its rotational stiffness must include both pure and warping torsional constants. A typical beam

element is shown in Fig. 2.3

Figure 2.3 A typical beam element

However, this simplified model is unable to capture the local responses of some critical member

like stresses at joints which are prone to local failure. Multi-scale modeling can be used toovercome this problem. In a multi-scale model the components of interest can be modeled with

shell elements or solid elements and other components still with line elements.

2.5.2 Modeling of Cables

The total stiffness matrix of a cable element consists of elastic stiffness matrix and geometricstiffness matrix i.e.

][][][ get KKK (2.7)

The elastic stiffness matrix is affected by the sagging of the cable which is considered by

modifying the modulus of elasticity of the cable material. The total displacement of a cable can be

considered as the sum of elongation due to tension in the cable and negative displacement due to


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self-weight of the cable. This concept is described by Ernsts modulus of elasticity, which relates

the increase in cable length due to increase in cable tension through the stiffness of the cable. The

equivalent modulus of elasticity is given as

3

2

12)(1TAEwL

EEeq (2.8)

Where E modulus of elasticity of cable material; w weight per unit length of the cable; L

horizontal projected length of the cable; A cross-section area of the cable; T mean tension in

cable.

The elastic stiffness matrix for a 3-D cable element is given as

000000

000000

001001

000000

000000

001001

][c

eq

eL

AEK (2.9)

c

L chord length of the cable

In a long-span cable-stayed bridge, the nodal displacements of a cable element is quite large.

Therefore, in addition to the elastic stiffness matrix, the geometric stiffness matrix of the stay cable

should be considered. The geometric stiffness matrix of a 3-D cable element is equal to that of the3-D truss element. It can be obtained by applying the principle of virtual displacements to a straight

element undergoing rigid body rotation and small but finite axial straining. The expression of

geometric stiffness matrix can be written as follows

100100010010

000000

100100

010010

000000

][c

gL

TK (2.10)

2.6 Summary

This chapter first introduces the concept of mean wind load and wind force coefficients. The 1-D

torsional divergence and the critical wind speed are then discussed. Usually cable-stayed bridges


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are stiffer than the suspension bridges, thus less susceptible to torsional divergence. However, it

should be investigated as the modern long-span cable-stayed bridges are becoming more flexible.

The 3-D non-linear aerostatic instability analysis based on the finite element method (FEM) is also

discussed. A brief introduction on finite element modelling which is important for analysis of

various wind load effects on the bridge is also presented.


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Chapter 3

WIND INDUCED VIBRATION AND AERODYNAMIC INSTABILITY

There are four types of wind-induced vibration and aerodynamic instability problems that occur in

long span cable-stayed bridge, i.e. (i) Vortex induced vibration (ii) Galloping instability (iii) Flutter

and (iv) Buffeting.

3.1 Vortex Induced Vibration

Vortex induced vibrations may occur when vortices are shed alternatively from opposite sides of

a structure when it is interacting with an external fluid flow. Vortices are created behind the

structure and detach periodically from either side of the body. Pattern of vortex shedding differ

according to the Reynoldss Number. At low Reynoldss Number the flow around the body is

nearly symmetric but at higher Reynoldss Number (increases with wind speed) the flow becomes

asymmetric with respect to the mid-plane of the body. Therefore different lift forces developed on

each side of the body, resulting in harmonically varying lateral load. The frequency of this load is

same as that of the vortex shedding. This leads to motion transverse to flow.

Lock in Phenomenon

The frequency of vortex shedding ( )stf which depends on flow velocity Uand characteristic

dimension of the structure D (e.g. width of the bridge deck) can be obtained from the expression

D

USf tst (3.1)

tS is a dimensionless number called as the Strouhal number, named after the Czech physicist

Vincenc Strouhal.It is a function of the Reynolds number. However, experimental investigations

show that the Strouhal number is about constant across a wide range of the Reynolds number (102~

107). The Strouhal number is about 0.18 for a cylinder at a Reynolds number range 300 to .107

As the wind speed increases frequency of vortex shedding increases and when stf become equal

to the lowest natural frequency of the structure, first resonance takes place, leading to large

amplitude oscillation. As the vibration of the structure at this stage can be sufficiently large the

vortex shedding frequency is controlled by the structural vibration, this phenomenon is known asLock-in. The resonance can sustain through certain range of wind velocity. When wind velocity

increased to a certain level, again stf is controlled by wind velocity as given by Eq. (3.1). The

lock-in phenomenon is demonstrated in Fig. (3.1)
http://en.wikipedia.org/wiki/Vincenc_Strouhalhttp://en.wikipedia.org/wiki/Vincenc_Strouhal


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Figure 3.1 Vortex shedding frequency vs. wind velocity: Lock-in phenomenon [Simiu and

Scanlan, 1986].

A structure under vortex induced vibration can be expressed by the following governing equation

vsLKXXCXM

(3.2)

Where, KC,M, are structural mass, damping and the stiffness matrix respectively; X,XX, are the

nodal displacement, velocity and acceleration vectors respectively.vs

L is the vortex induced lift

force.

Assuming vortex induced force as simple harmonic the governing equation for SDOF system can

be written as

)sin(

2

1)2( 22 tDCUyyym sLnn (3.3)

m is mass of the structure, y is vertical displacement, is structural damping ratio,n

is natural

frequency of structure,s

is vortex shedding frequency and is phase angle.

At resonance i.e. whenns

, the lock-in response is given as

222

2

max

164 tc

L

n

L

SS

DC

m

UDCy

(3.4)

Where 2D

m

Sc

is the Scruton number. Therefore as the Scruton number increases vortex

induced response decreases. But the above simplified model does not consider the effect of motion

induced force.

Simiu and Scanlan(1986)proposed the following model of vortex induced force


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)sin()()()(

2

1)( 21

2 tKCD

yKY

U

yKYDUtL nLvs

(3.5)

In whichU

BK

is the reduced frequency; )(),(),( 21 KCKYKY L are determined by experiments

and are functions ofKat Lock-in.

In 1990, Ehsan and Scanlan (1990)proposed a revised model which includes the nonlinear aero

elastic damping coefficient. It is given as

)sin()()()1)((

2

1)( 22

2

1

2

tKCD

yKY

U

y

D

yKYDUtL nLvs

(3.6)

is the nonlinear aerodynamic damping coefficients.

3.2 Galloping Instability

Galloping is a typical instability of flexible, lightly damped structures occurring due to the

aerodynamic forces that are induced wholly by the transverse motions of the structure and not

primarily due to vortex shedding. This large amplitude vertical oscillations usually occur at very

low reduced frequency as compared to that of vortex shedding. Although the incoming wind has

a fixed angle of attack, due to the across-wind oscillation of the structure the effective angle of

attack gets modified which leads to the change in aerodynamic forces and thereby introduces self-

excited forces.

Figure 3.2 Flow induced Galloping of a 2-D bridge deck.

As shown in Fig. 3.2, the effective wind velocity U acts on the structure with an angle of attack

with horizontal, although the incoming wind velocity U is horizontal. This is due to the motion

of the deck in y-direction. For the 2-D steady flow the drag and lift forces are expressed as

)(2

1)( 2

DBCUD (3.7a)


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)(2

1)( 2

LBCUL (3.7b)

The force acting in the y-direction is

]sec)(tan)([2

1]cos)(sin)([

2

1)( 22 LDLDy CCBUCCBUF (3.8)

Taylor series expansion of Eq. (3.8) gives (higher order terms neglected)

U

yC

d

dCBUBCU

FFF D

LLyy

0

22

0 2

1)0(

2

1)0()(

(3.9)

Neglecting the static component, the expression for self-excited force is given by the last term of

Eq. (3.9) i.e.

U

yCd

dCBUF DLexcitedselfy

0

2

2

1)(

(3.10)

For a SDOF system undergoing vertical vibration and subjected to aerodynamic force expressed

by Eq. (3.10) can be written as

U

yC

d

dCBUkyycym D

L

0

2

2

1

02

1

0

kyyCd

dCUBcym D

L

(3.11)

Therefore galloping instability will occur when effective damping become negative or at least zero

i.e.

02

1

0

D

LC

d

dCUBc

UB

cC

d

dCD

L

2

0

(3.12)

3.3 Flutter

Flutter is the dynamic instability of the structure due to self-excited aerodynamic forces resulting

from windstructure interaction. Usually the instability occur when the net damping (can be

defined as the inherent structural damping and the negative aerodynamic damping) reduces to zero

and further decrease leads to failure. It can occur both under laminar and turbulent flow of wind

around bridge deck. Classical flutter of a thin airfoil is a coupled vertical and torsional vibration,

also called 2-D flutter. 1-D flutter may also occur in the form of vertical or torsional motion,


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although the torsional motion is more dangerous. The famous failure of original Tacoma Narrow

Bridge (1940) is a consequence of two forms of 1-D flutter-initially at low wind speed vertical

motion takes place and with increase in the wind speed large amplitude catastrophic torsional

flutter.

Figure 3.3 2-D structures for flutter analysis.

3.3.1 Self-excited forces and Aerodynamic Derivatives

Scanlan and Tomko (1971)proposed aerodynamic derivatives to represent the self-excited forces

on a 2-D structure involving vertical and torsional vibration (Fig. 3.3) as

B

hHKHK

U

BKH

U

hKHBUL

se 4

2

3

2

21

2

2

1

(3.13a)

B

hAKAK

U

BKA

U

hKABUM

se 4

2

3

2

21

22

2

1

(3.13b)

Wherese

L andse

M are self-excited lift and moment respectively; h and are the vertical and

torsional displacement of the deck respectively; i

H and iA )41( i are the aerodynamic

derivatives or flutter derivatives which can be obtained either form wind tunnel tests or

computational fluid dynamics.

3.3.2 Theodorsen expression for self-excited forces

For a flat plate airfoil subjected to sinusoidal motion, the self-excited lift and moment are given

by Theodorsen(1934)as

)(2)](1[)(2 2 kCUUbkChkUChbbLse

(3.14a)

)()](2

1[

8)( 2

22

kCUUbkCb

hkUCbMse

(3.14b)


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b half width of the plate; )()()( kiGkFkC is the Theodorsen cyclic function. The variation

of real part )(kF and imaginary part )(kG withk

1are given graphically in Fig. 3.4 (Theodorsen

1934).

Figure 3.4 The functions Fand G againstk

1(Theodorsen1934).

Comparing Equations (3.13) and (3.14) for sinusoidal displacements ),( h the flutter derivatives

can be obtained as

k

kGkF

kKH

k

kFKH

)(2)(1

4)(

)()( 21

(3.15a)

k

kGKH

kkGkF

kKH

)(21

2)(

2

)()(

2)( 423

(3.15b)

k

kGkF

kKA

k

kFKA

)(2)(1

16)(

4

)()( 21

(3.15c)

k

kGKA

kkGkF

kKA

4

)()(

2

)()(

8)( 423

(3.15d)

where 2Kk .

3.3.3 3-D Flutter Analysis in Frequency Domain

Since the self-excited forces are functions of reduced frequency, therefore the flutter instability

analysis is usually performed in frequency domain for computational efficiency. The aim of the

analysis is to obtain the critical flutter wind speed. 3-D Flutter instability problem of cable-

supported bridges has been studied by many researchers (Scanlan 86, Jain et al. 1996, Ding et al.


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2002to name a few). One of the efficient method involve state-space formulation of the governing

equation in modal coordinate and then performing the complex eigenvalue analysis to obtain the

critical wind speed (Ding et al 2002). For 3-D flutter analysis which involve vertical ( h ), lateral (

p ) as well as torsional displacement ( ), the self-excited lift, drag and moment can be expressed

in terms of Scanlans format as follows

B

pHK

U

pKH

B

hHKHK

U

BKH

U

hKHBUtLse 6

2

54

2

3

2

21

2

2

1)(

(3.16a)

B

pPK

U

pKP

B

hPKPK

U

BKP

U

hKPBUtDse 6

2

54

2

3

2

21

2

2

1)(

(3.16b)

B

pAK

U

pKA

B

hAKAK

U

BKA

U

hKABUtMse 6

2

54

2

3

2

21

22

2

1)(

(3.16c)

)61(,, iAPHiii

are the non-dimensional aerodynamic derivatives.

The governing equation can written as

seFKXXCXM (3.17)

seF is the self-excited equivalent nodal force vector.

3.3.4 Flutter in Time Domain

Although the flutter analysis is generally performed in the frequency domain for computational

efficiency, its application is limited only to the linear systems since both the structural and

aerodynamic nonlinearity cannot be taken into consideration in this method. Chen et al. (2000)

proposed a time-domain multimode flutter and buffeting analysis, through the rational function

approximation of the self-excited forces. These functions are obtained from the flutter derivatives

and the span wise coherence of aerodynamic forces. Time domain analysis results of an example

bridge shows good agreement with frequency domain analysis.

3.4 Buffeting

A long span cable-stayed bridge is subjected to both static and dynamic wind forces. Static wind

force is due to mean wind speed whereas the dynamic part comes from the turbulence in the winddue to fluctuating wind speed. Buffeting of a bridge is referred to as the random vibration of the

bridge due to wind turbulence. It occurs under wide ranges of wind speed. However, the model of

buffeting wind load must consists of both the buffeting wind load due to turbulent wind and the

self-excited aerodynamic forces from wind-bridge interaction, since the self-excited forces

increases the magnitude of vibration by providing additional vibration energy to the bridge. As in


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the case of flutter analysis, both the frequency domain (Scanlan 1986, Jain et al. 1996) and time-

domain (Chen et al. 2000) approach can be adopted for buffeting analysis.

Buffeting Forces

The aerodynamic forces in the bridge deck in the transient wind axis system (Fig. 3.5) can be

expressed as

Figure 3.5 Wind and Buffeting forces on bridge deck[Xu 2013].

BCtUtLL

)()(2

1)( 0

2

(3.18a)

BCtUtDD

)()(2

1)( 0

2

(3.18b)

2

0

2)()(

21)( BCtUtM

M (3.18c)

Where0

is the angle of attack of mean wind speed ;U is the change in angle of attack due to

turbulence. From Fig. 3.5, )(2 tU can be expressed as

U

tuUtwtuUtU

)(21)()()( 2

222 (3.19)

Assuming being very small

U

tw

U

tu

U

tw

tuU

tw )()(1

)(

)(

)(tansin

1

(3.20)

The Taylor series expansion (up to first two terms) of Equations (3.18) give

)()()( 000 kkk CCC ; DLk , andM . (3.21)


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Transforming the forces along the mean wind direction, using Eq. (3.18)

)sin()()cos()()(

tDtLtL (3.22a)

)sin()()cos()()(

tDtDtD (3.22b)

)()( tMtM

(3.22c)

Now using equations (3.19) to (3.21), equations (3.22) can be expressed as follows

)()(

)(2

1)()()(

)(2)(

2

1)(

0

0

2

000

2

staticb

LDLL

LtL

BCUU

twCC

U

tuCBUtL

(3.23a)

)()(

)(2

1)()(

)(2)(

2

1)(

0

0

2

00

2

staticb

DDD

DtD

BCUU

twC

U

tuCBUtD

(3.23b)

)()(

)(2

1)()(

)(2)(

2

1)(

0

0

22

00

22

staticb

MMM

MtM

CBUU

twC

U

tuCBUtM

(3.23c)

Where )(),(),( tMtDtLbbb

are the buffeting lift, drag and moment.

Equation of motion of a bridge deck in buffeting can be expressed as

sbse FFFKXXCXM

(3.24)

se

F self-excited force vector as given in Eq. (3.16); bF buffeting forces as given by first part of

Equations (3.23) and s

F mean wind force vector given by second part of Eq. (3.23).


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3.5 Comparison of the four Instabilities

Vortex-Induced

Vibration

Galloping

InstabilityFlutter Buffeting

Occurs at low wind

speed and low

turbulence condition.

Occur at much lower

frequency than vortex

shedding.

Usually occur at very

high wind speed.

Occur over a wide

range of wind speed.

Due to Lock-in,

vortex shedding

frequencynatural

frequency of bridge

components.

Motion of structure in

vertical direction

causes change in

angle of attack of

original flow velocity.

Due to self-excited

aerodynamic forces

resulting from wind

structure interaction.

Due to velocity

fluctuation in the

incoming flow i.e.

turbulence.

Resulting motion

normal to flow, for

bridge deck it is in thevertical direction.

Large amplitude

vibration in normal to

mean wind direction.

Flutter can be 1D

(vertical or torsional),

2D (coupled verticaland torsional motion)

or 3D (coupled

vertical, torsional and

lateral motion).

Random vibration.

Motion can be any

combination oflateral, torsional and

vertical.

Simple harmonic

force due to alternate

vortex shedding as

well as motion

induced force.

Self-excited forces. Self-excited forces. Not self-excited.

Increase in damping

reduces instability.

Increase in damping

reduces instability.

Effect of increase in

damping is very low.

Increase in damping

reduces response.

3.6 Summary

A brief overview of the dynamics of cable-supported bridges under wind loading is given in this

chapter. Four types of wind induced vibration and aerodynamic instabilities are discussed. For

long span cable-stayed bridges flutter instability is the most catastrophic in nature and can lead to

complete collapse of a bridge. It is mainly a coupled torsional and vertical motion of the bridge

deck caused by the self-excited aerodynamic forces due to wind bridge interaction. Determinationof critical flutter wind speed is an important step in flutter instability analysis for which the

Scanlan and Tomkomodel (1971) can be used to represent the self-excited forces on the deck.

The values of aerodynamic derivatives can be taken either form some wind tunnel test data or can

be calculated explicitly from the Theodorsens functions.


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Chapter 4

WIND INDUCED VIBRATION CONTROL

It is well known that the long span cable-supported bridges are very susceptible to wind induced

vibration and aerodynamic instability. Furthermore, multiple loading related fatigue, rain-wind

induced cable vibration, vehicle-bridge interaction effects can also lead to excessive vibration and

collapse of the bridge. In addition, excessive vibration may affect safety and comfort of the vehicle

and the passengers inside it. So in order to ensure proper functioning of the bridge during its service

period as well as to prevent failures, some control measure should be taken.

Broadly vibration control of long span cable-supported bridges can be classified into the following

three categories.

(i) Modification of structural parameters

(ii) Aerodynamic measures and

(iii)

Mechanical measures.

4.1 Structural Modification

This can be done by modifying structural mass, damping and stiffness either at global level or local

level (e.g. applied to bridge components like bridge deck, stay cables and towers). By increasing

the damping of the structure substantial reduction in vortex induced vibration, galloping instability

and buffeting can be achieved. However, the increase in damping has very little effect on flutter

instability.

Torsional stiffness of bridge deck can be increased by selecting proper cross-section of the deck.

To reduce the vibration of the stay cables, cross-ties can be used to increase the in plane stiffness.However, this may hamper cables aesthetic view.

4.2 Aerodynamic Measures

Aerodynamic measures are adopted to modify the wind flow around the bridge components by

changing its configuration through the installation of some aerodynamic devices. Some of the

aerodynamic measures that can be implemented in cable-stayed bridges are as follows

(i) Wedge-shaped fairings

(ii) Longitudinal open slots with or without stabilizer

(iii)

Aerodynamic appendages(iv)

Actively controlled surfaces

(v) deck-flap system

(vi) Guide vanes (e.g. second bay bridge in San Francisco), adjustable wind barrier, grid

plates etc.


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4.3 Mechanical Measures

This includes passive control system, active control system, and semi-active and hybrid control

system. The effects of these measures is to modify the structural characteristics and thereby reduce

the wind induced vibration. In the following section a few active and semi-active control strategies

are discussed briefly.

4.3.1 Active control system

Active control system like active mass damper or active tendon systems are quite effectively used

to reduce the response of the building structures subjected to wind or earthquake. However, active

control system for bridge vibration control is a relatively new and emerging field. Moreover, their

implementation for wind induced vibration mitigation in real long span cable-supported bridges

are very limited. rlinoK and Starossek (2004)proposed two types of active mass damper (AMD)

system for flutter control of bridge deck. These are rotational mass damper (RMD) and movable

eccentric mass damper (MEMD) [Fig. 4.1].

(a) (b)

Figure 4.1 Two types of AMD (a) Rotating Mass Damper; (b) Movable Eccentric Mass Damper [

r l in oK and Starossek 2004]

The bridge is model considered to have two degrees of freedom, vertical displacement )(h and

rotational displacement )( . For the RMD, an additional mass is installed along the center of the

bridge. The control input is the rotational acceleration of the central mass. In case of MEMD, a

mass movable across the deck width is used and the resisting moment is exerted by the gravity of

the additional mass. The control variable in this case is the eccentricity of the movable mass. A

linear optimal static feedback control strategy is used to enhance the flutter stability of the bridge

deck. Hurwitz stability criterion is used to determine the critical flutter wind speed of the

uncontrolled and controlled structure. This criterion is based on the coefficients of the

characteristic polynomial. A time domain analysis is also performed by considering the rational

function approximation of the unsteady aerodynamic forces. Furthermore, a combined numerical-

experimental simulation has been conducted to verify the analytical results. The critical wind speed

of optimally controlled structure increase to 50 m/s from 36 m/s as that of uncontrolled structure.


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The increase in the additional mass results in reduction of control energy requirement. It has been

observed that RMD requires high energy input to produce desired rotational acceleration due to its

low mass moment of inertia as compared to the bridge section. Although the movable eccentric

mass requires lower energy consumption, its movement can induce undesirable horizontal

movement of the real bridge structure. Authors suggested that due to the large energy demand as

well as the effect of saturation of control, bridge deck flutter control cannot be solved only through

these two devices using linear control theories. Therefore, new active control devices and/or more

robust control algorithms may be considered for further future research.

rlinoK and Starossek (2007)performed wind tunnel test on rotational mass damper system to

investigate its suitability in flutter control of bridge deck. Rotational servo actuator is used to

produce stabilizing moment by changing the rotational speed of the control mass. The critical wind

speed obtained through experimentally and numerically are found to be in good agreement for both

uncontrolled and controlled system.

Achkire et al (1998)proposed active tendon control of stay cables to control flutter instability ofthe suspension bridge (Fig. 4.2). An alternative strategy which includes displacement actuator

(active tendon) collocated with a force sensor and the decentralized integral force feedback control

algorithm are implemented. This control law guaranteed the stability of the system. Active tendon

control enhances the flutter stability as it is observed from the shifting of the system poles towards

the left in the complex plane. To confirm the analytical results, a laboratory experiment on a model

using piezoelectric actuator is also performed.

Figure 4.2 Active tendons for flutter control of bridge deck. [Achk ir e et al (1998)]

4.3.2 Semi-active control

A semi-active control strategy using semi-active tuned mass damper (STMD) has been proposed

by Pourzeynali and Datta (2005). The STMD system has two degrees of freedom, vertical

displacement and rotational displacement. It is installed at the middle of the center span. A semi-

active hydraulic damper (SHD) is incorporated between the TMD mass and the bridge deck


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(Fig.4.3). The variable damping provided by SHD is controlled using fuzzy logic controller (FLC).

The displacement and velocity at center of bridge where the STMD is installed, are taken as the

input to the controller. The output from the FLC system is the variable damping ratio of the semi-

active damper.

Figure 4.3 Semi-active tuned mass damper (STMD) system: (a) location of the STMD in the bridge;

(b) cross-section showing bridge deck and STMD. [Pourzeynali and Datta (2005)]

The Vincent-Thomas suspension bridge is considered for numerical studies. The self-excited

forces per unit length of the bridge span is taken as that given by Jain et al (1996). Comparison

of the effectiveness of passive TMD and STMD for bridge deck flutter control is reproduced here

as given by the authors.

Table 4.1 Comparison of the efficiencies of passive TMD and STMD control the bridge.

[Pourzeynali and Datta (2005)]

CaseWind Speed

(m/s)

Max. Torsional

amp. (rad.)

Uncontrolled55.52 (flutter, sustained

oscillation)0.02

Controlled with tuned mass damper (20%

damping)

98 (flutter, sustained

oscillation)0.02

Controlled with semi-active tuned mass

damper (max. damping 21.6%)110 (decaying oscillation) 0.0063

The table clearly shows the effectiveness of STMD over passive TMD not only in the increase of

flutter wind speed but also in the reduction of maximum torsional amplitude. Various parametric

studies with different initial conditions and different fuzzy rule bases are conducted to investigate

the performance of the system. It is observed that STMD can bring the whole system in stable

condition within few seconds (50-60 sec.). Also being a semi-active system the power

consumption is very low as compared to purely active control system. The maximum control force

requirement which in turn depends on the maximum damping to be provided by the SHD is


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dependent on the initial condition and fuzzy rule base, as the parametric study show. The time

delay effect is ignored in this study and therefore, it can form the scope of future studies as

suggested by authors. Nevertheless, STMD system with variable damping is a very effective

method to mitigate flutter instability of cable-supported bridges.

4.4 Summary

This chapter gives an overview of the different wind induced vibration control strategies like

structural modification, aerodynamics measures and mechanical measures. Giving the priority to

the mechanical measures, few active and semi-active control devices and their effectiveness are

discussed in detail. From these literatures review it can be concluded that there exist ample scope

of future research on new active, semi-active and hybrid control systems with robust control

algorithms to mitigate aerodynamic instability e.g. application of hybrid mass damper using direct

feedback control algorithms, consideration of time delay effect etc.


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Chapter 5

CONCLUSIONS

With the ever increasing prospects of developing super long span cable-stayed bridges to cross

straits around the world, the studies on advanced bridge wind engineering is becoming more

important day by day. These long span bridges are highly susceptible and vulnerable to wind

induced vibration and aerodynamic instability due to their high flexibility and inherently low

damping. Therefore, the effective control measures to mitigate the wind induced vibration and

their implementation to the real bridge structures are highly motivating and challenging task.

The following aspects regarding the wind effects on cable-stayed bridges may be considered as

the topic of future studies.

Non-linear flutter and buffeting analysis.

Vibration due to stay cables and bridge deck interactions.

Wind effects on coupled vehicle-bridge systems. Study on new active, semi-active and hybrid control systems with robust control algorithms

to mitigate aerodynamic instability.

In the next eight months a FEM model for flutter analysis of a cable stayed bridge will be

developed. The free vibration analysis will be performed to determine the natural frequencies and

mode shapes. To determine the critical flutter wind speed, the self-excited aerodynamic forces will

be considered either using the Scanlan and Tomko (1971) model in frequency domain or the

model based on rational function approximation of self-excited forces (Chen et al 2000) for time

domain analysis.


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REFERENCES

1. Achkire, Y., Bossens, F., and Preumont, A. (1998), Active damping and flutter control of

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2. Boonyapinyo, V., Yamada, H., and Miyata, T. (1994), Wind-induced non-linear lateral-

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9. Pourzeynali, S., and Datta, T.K. (2005), Semi active fuzzy logic control of suspension bridge

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Documents

cable stayed bridge report