Lateral Response Analysis of Suspension Bridge Under Wind Loads

COMMUNICATIONS IN NUMERICAL METHODS IN ENGINEERINGCommun. Numer. Meth. Engng 2006; 22:861–874Published online 9 February 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/cnm.854

A simpli�ed method for lateral response analysis ofsuspension bridges under wind loads

Jin Cheng∗;†;‡ and Ru-Cheng Xiao§

Department of Bridge Engineering; Tongji University; Shanghai 200092; China

SUMMARY

A simpli�ed method for analysing lateral response of suspension bridges under wind loads is pro-posed in this paper. The geometric non-linearity in the de�ection theory and the three componentsof displacement-dependent wind loads are taken into account in the method. The analytical formulasfor calculating the torsional, vertical, and lateral responses of suspension bridges under wind loads arederived. An iterative procedure, which has a high convergence rate for solving the problem, is devel-oped. The proposed method is su�cient and simple to use. Wind-induced lateral response analysis ofa long-span suspension bridge demonstrates the proposed method’s e�ciency and accuracy. Copyright? 2006 John Wiley & Sons, Ltd.

KEY WORDS: suspension bridges; series; lateral response; wind loads

1. INTRODUCTION

With the increasing central span length of suspension bridges, in addition to the typical �utterand bu�eting issues [1, 2], it becomes especially important to accurately estimate wind-inducedlateral response and take this into account in the design of suspension bridges, because bridgegirders become more �exible as spans become longer. On the other hand, long-span suspensionbridges are usually wind sensitive, and large lateral de�ection occurs under wind loads. Foraccurate lateral response, it is necessary that the analysis technique incorporate the non-lineare�ects arising from bridge structure and the three components of wind loads. This problemcan be solved by the non-linear �nite element method (NFEM). However, the use of NFEMis usually computationally too intensive with the increase in structural size, and for long-spansuspension bridges, input preparation is time consuming.To reduce the computation e�ect, researchers have proposed simpli�ed methods for calcu-

lation of lateral response of suspension bridges under wind loads. Two simpli�ed methods,

∗Correspondence to: Jin Cheng, Department of Bridge Engineering; Tongji University; Shanghai 200092; China.†E-mail: [email protected]‡Research Associate Professor.§Professor.

Received 1 June 2005Revised 5 December 2005

Copyright ? 2006 John Wiley & Sons, Ltd. Accepted 14 December 2005

862 J. CHENG AND R.-C. XIAO

uniform distribution method and elastic distribution method, were �rst introduced by Moissei�and Lienhard [3]. In the uniform distribution method, the load transferred to the cables fromthe sti�ening truss (restitution force) is assumed to be uniform [4]. In the elastic distributionmethod, the distribution of restitution between the cable and the truss is taken into account.However, this method is a trial-and-error process and requires a considerable amount of workfor accuracy [4]. Gursoy [4] applied a similar method to study the e�ect of lateral wind loadson the response of side spans of suspension bridges. However, all these simpli�ed methodsneglect the non-linear e�ects arising from bridge structure and the three components of windloads. Thus, they are inadequate in capturing the true lateral behaviour of suspension bridgesunder wind loads.The purpose of this paper is to propose a new, simpli�ed method for the accurate prediction

of the lateral response of suspension bridges under wind loads. The proposed method takesinto account the geometric non-linearity in the de�ection theory and the three components ofwind load. The proposed method is simple to use and is suitable for engineering application.The simplicity and accuracy of the proposed method is demonstrated through the wind-inducedlateral response analysis of a long-span suspension bridge with an 888m central span length.The results due to the proposed method are compared with those produced by the uniformdistribution method and NFEM.

2. DESCRIPTION OF THREE COMPONENTS OF WIND LOADS

The three components of wind load are drag force, lift force, and pitch moment. Consider asection of bridge deck in a smooth �ow, as shown in Figure 1. Assuming that under the

Figure 1. Motion of bridge deck and three components of wind loads in di�erent axes.

Copyright ? 2006 John Wiley & Sons, Ltd. Commun. Numer. Meth. Engng 2006; 22:861–874

LATERAL RESPONSE ANALYSIS OF SUSPENSION BRIDGES 863

e�ect of the mean wind velocity V with the angle of incidence �0, the torsional displacementof deck is �. Then the e�ective wind angle of attack is � = �0 + �. The components of windforces per unit span acting on the deformed deck can be written in wind axes as

drag force: Fy(�) = 12�V

2Cy(�)D (1a)

lift force: Fz(�)= 12�V

2Cz(�)B (1b)

pitch moment: M (�)= 12�V

2CM (�)B2 (1c)

where Cy(�), Cz(�), and CM (�) are the coe�cients of drag force, lift force, and pitch momentin local bridge axes, respectively; B is bridge width; D is vertical projected area.The wind forces in (1) are a function of the torsional displacement of the structure. They

vary as the girder displaces. Therefore, the three components of wind load are displacementdependent. Theoretical coe�cient curves of three components of wind loads from the experi-mental points are generally non-linear. In this paper, the non-linear curves are approximatelyrepresented by linear curves between two experimental points.

3. ASSUMPTIONS

The following assumptions are made in this paper:

(1) Hanger is densely distributed along the bridge length direction.(2) The stretching of hangers under wind loads is ignored.(3) Con�guration of cable is a quadratic parabola; the stress of a sti�ened girder due to

dead load is ignored.(4) Under the action of drag force of displacement-dependent wind loads, the load trans-

ferred to the sti�ened girder from the cables is assumed to be an isosceles triangledistribution (see Figure 2).

Figure 2. Lateral wind load distribution on sti�ened girder.



4. FORMULATION

The formulation of the proposed method involves the derivation of structural responses underwind loads. The derivation of formula is based on the de�ection theory. This is mainlybecause: (1) this theory can, to a certain extent, account for the geometric non-linearity ofstructure (the sti�ening e�ect of the tension force in the cable); (2) this theory is simple andaccurate. This conclusion has been demonstrated by the fact that the majority of the existinglong-span suspension bridges were correctly designed using the de�ection theory before thecomputer era [5]. A detailed review of the de�ection theory can be found in Reference [6].

4.1. Derivation of torsional displacements of sti�ened girder under pitch moment M (�)

The di�erential equation for the twisted sti�ened girder of suspension bridges subjected todistribution torque moment without considering warping e�ect can be written as [7]

−(GId +

H · B2c4

)d2�dx2

+8fl2BcHp1 −M (�)=0 (2)

where H is the horizontal component of the cable tension owing to dead and wind loads(two cables); Hp1 is increment in the horizontal component of the cable tension owing to thepitching moment of displacement-dependent wind loads (per cable); Bc is width between thecentre lines of the cables; f is sag of the cable; l is length of the main span and GId istorsional sti�ness of the deck cross-section.According to the consistency condition of the cables, Hp1 can be expressed as [6]

Hp1 =AcEcLp

4fl2Bc∫ l

0�(x) dx

Lp = l

(1 +

83

(fl

)2) (3)

where Ac is the cross-section area of the cables and Ec is modulus of elasticity of the cables.A solution of Equation (2) may be expressed in terms of a series function that satis�es all

the boundary conditions of the torsional sti�ened girder

�(x)=∞∑r=1brfr(x) in which fr(x)= sin

r�xl

(4)

where br is the unknown coe�cients to be determined.The coe�cient curves of pitch moment of the wind loads are approximately represented by

linear curves between two experimental points. CM (�) may be expressed as

CM (�)= e1 + e2� (5)

where e1, e2 are the coe�cients.Substituting (4) and (5) into (1c), M (�) can be rewritten as

M (�)=∞∑r=1

(2�e1V 2B2

r�+12�e2V 2B2br

)sin(r�xl

)(6)



Substituting (3), (4), and (6) into (2), Equation (2) can be rewritten as

((GId +

HB2c4

)(r�l

)2+128f2B2cACEC(�rl)2Lpl

− 12�e2V 2B2

)br

+∑m �=r

128f2B2cACEcm�2l3Lpr

bm=2V 2�e1B2

r�(7)

where r, m are the odd and even integer values, respectively.

4.2. Derivation of vertical displacements of sti�ened girder under lift force Fz(�)

The vertical equilibrium equation for suspension bridges can be expressed as [8, 9]

EId4�d4x

−H d2�d2x

= Fz(�)−Hp d2yd2x

d2yd2x

=−8fl2

(8)

where � is the vertical displacements of sti�ened girder; Hp is increment in horizontal com-ponent of the cable tension owing to the lift force of displacement-dependent wind loads (twocables), EI is vertical sti�ness of the deck cross-section and y is vertical co-ordinate of thecable (the top of the tower being the reference point).According to the consistency condition of cables, Hp can be expressed as [6]

Hp=AcEcLP

8fl2

∫ l

0−� dx (9)

A solution of Equation (8) may be expressed in terms of a series function that satis�es allthe boundary conditions of the deformed sti�ened girder

�(x)=∞∑r=1arfr(x) (10)

where ar is the unknown coe�cients to be determined.Coe�cient curves of lift force of wind loads are approximately represented by linear curves

between two experimental points. Cz(�) may be expressed as

Cz(�)= c1 + c2� (11)

where c1, c2 = coe�cients.Substituting (4) and (11) into (1b), Fz(�) can be rewritten as

Fz(�)=∞∑r=1

(2�c1V 2Br�

+12�c2V 2Bbr

)sin(r�xl

)(12)



Substituting (9), (10), and (12) into (8), Equation (8) can be rewritten as

ar =

2�V 2c1Br�

+12�V 2c2Bbr +

32fHpr�l2

EI(r�l

)4+H

(r�l

)2 (13)

4.3. Derivation of lateral displacements of sti�ened girder and main cable under dragforce Fy(�)

Because the torsional displacement at each node of sti�ened girder is di�erent under anygiven wind velocity, the drag force acting at each node of sti�ened girder is also di�erent. Forconvenience, the drag force, Fy(�), acting on the sti�ened girder is approximately representedby equivalent drag force, �Fy(�) · �Fy(�) may be calculated as

�Fy(�)=

∫ l0 Fy(�)l

(14)

Coe�cient curves of the drag force of wind loads are approximately represented by linearcurves between the two experimental points. Cy(�) may be expressed as

Cy(�)=d1 + d2� (15)

where d1, d2 are the coe�cients.Substituting (1a), (4), and (15) into (14), �Fy(�) can be obtained by

�Fy(�)=12�V 2D

(d1 + d2

∑r=1;3;5:::

2brr�

)(16)

According to the isosceles triangle distribution assumption, the maximum value of the loadtransferred to sti�ened girder from cable, xy, can be calculated as [10]

xy =

5Fy(�)l4

384EIy− Fyc(�)l2

8Hhtp+

l4

120EIy+

l2

12H

Fyc(�) = 12�V

2CycDc

(17)

where Fyc(�) is the drag force of the displacement-dependent wind loads acting on the cables,Dc is diameter of the main cable, Cyc is drag coe�cient of the main cable, EIy is lateralsti�ness of the deck cross-section, ht is length of the hanger at the midpoint of the centralspan and p is lift force of displacement-dependent wind loads and dead load.The lateral displacement of sti�ened girder, u(x), is computed from the method of equiva-

lently simpli�ed beam under drag force and the load transferred to sti�ened girder from cable,



and is given by

u(x)=−

(Fy(�)lx3

12−Fy(�)x

4

24−Fy(�)l

3x24

)

EIy+

(xylx3

24− xyx5

60l− 5xyl3x

192

)EIy

(06 x6

l2

)(18)

u(x) =−

(Fy(�)lx3

12− Fy(�)x4

24− Fy(�)l3x

24

)

EIy

+

⎛⎜⎜⎝xy

(x5

20− 14lx4)

3l+3xylx3

24− xyl2x2

24− xyl3x

64− xyl4

960

⎞⎟⎟⎠

EIy

(l26 x6 l

)(19)

where x is the distance measured from the end of the main span.The geometrical continuity between the de�ected sti�ened girder and the cable can be

expressed as [4]

u(x)− w(x)h(x)

=

2xyxlp

(06 x6

l2

)(20)

u(x)− w(x)h(x)

=

2xy(l− x)lp

(l26 x6 l

)(21)

Substituting (18) and (19) into (20) and (21), the lateral displacement of the maincable, w(x), can be obtained.

5. METHOD OF SOLUTION

An iterative method is employed for the solution of structural lateral response under windloads. The solution procedure can be summarized as follows:

(1) Calculate the initial horizontal component of the cable tension owing to dead loads ofcables, hangers and sti�ened girder from

H0 =ql2

8f(22)

where q is the dead load of the two cables, hangers and sti�ened girder.



(2) Give an initial wind velocity, V = 40 m=s.(3) Initialize the iteration counter i = 1.(4) If i = 1, the coe�cient b(i)r = 0, H

(i)p = 0, and H (i) = H0 in Equation (13).

(5) Determine the coe�cient a(i)r by substituting b(i)r , H(i)p and H (i) into Equation (13).

(6) Calculate the vertical displacement of sti�ened girder, �(x), from (10).(7) Set i = i + 1.(8) Substitute �(x) into (9), and determine the new value of H (i)

p .(9) Let H (i) = H0 +H

(i)p .

(10) Substitute H (i) into (7), and determine the coe�cient b(i)r .(11) Check the convergence value � using

�=∣∣∣∣H (i) −H (i−1)

H (i−1)

∣∣∣∣ (23)

If �¿�max, and go to Step 5. If �6 �max, continue.(12) Determine the values of �Fy(�) and xy by substituting b

(i)r and H (i) into Equations (16)

and (17).(13) Calculate the lateral displacement of sti�ened girder, u(x) and the lateral displacement

of the main cable, w(x), from (18), (19), (20) and (21), respectively.(14) Check if the prescribed wind velocity, Vt is attained

If V¡Vt , update the wind velocity V using

V =V +�V (24)

where �V is the incremental wind velocity (in this paper, �V = 1 m=s), and go toStep 2; otherwise, STOP (end of calculation).

It has been found that only a few iterations are needed and a small number of seriesterms are su�cient for achieving the solutions of the problem. A �ow chart for the solutionprocedure is given in Figure 3.

6. EXAMPLE

An example of a long-span suspension bridge with a main span of 888m (Hu Men Bridge inChina) shown in Figure 4 was analysed using the proposed method. The deck cross-sectionis an aerodynamically shaped closed box steel girder 35.6m wide and 3.012m high. Thedistance between the two cables is 33m; the hanger spacing is 12.0m; and the height oftwo towers is 150m. The section material and geometrical features of the main member areindicated in Table I. The static aerodynamic coe�cients for the bridge are shown in Figure 5.The three components of the displacement-dependent wind loads were only considered for thebridge deck while only the initial drag force was considered for the cables. The drag forceacting on the towers was not considered. For more details of the bridge, the reader is referredto [11–13]. In the analysis, the following parameters are used: (1) the angle of incidence�0 = 0; (2) Dc=0:6 mCyc = 0:7; and (3) from Figure 5, it can be seen that the coe�cient



Yes

Yes

Calculate H0 (Eq.22)

Set V

i=1,nstep (nstep: total numbers of iteration)

Determine (i)ar (Eq.13)

Calculate (i)Hp (Eq.9)

Determine (i)br(Eq.7)

Set i =i+1

Convergence? (Step 11)

Stop

V=

V+

∆V

No

No

Calculate u(x), w(x) (Eqs.18,19,20 and 21)

V < Vt ? (Step 14)

i=1, (i)br = 0,(i)HpH (i) =H0= 0,

Figure 3. Flow chart for solution procedure.

curves of the lift force and the pitch moment of wind loads are almost linear. Therefore, linearcurve �tting is used. The coe�cients (e1; e2; c1; c2) are given by e1 = 0:00877, e2 = 0:01838,c1 = −0:02462, and c2 = 0:0789. However, because the coe�cient curve of the drag force ofwind loads is non-linear, di�erent linear curve �ttings are used. The coe�cients (d1; d2) are



Figure 4. General con�guration of Hu Men Bridge.

Table I. Section geometrical and material feature of the main member.

Substructures Jd (m4) I2 (m4) I3 (m4) m (t=m) E (MPa) �

Steel box girder 5.1 124.39 1.98 18.33 210000.0 0.3Cable — — — 2.397 200000.0 —Hanger — — — 0.172 160000.0 —

m = mass per unit length; E = modulus of elasticity; Jd = St: Venant constant; I2 =Out-of-planemoments of inertia; I3 = In-plane moments of inertia; � =Poisson ratio.

-6 -4 -2 0 4-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Cz

Cy

Cm

Angle of Attack (deg)

Stat

ic A

erod

ynam

ic C

oeff

icie

nts

2 6

Figure 5. Static aerodynamic coe�cients as function of angle of attack.

given by

when 06 �6 1 (degree); d1 = 0:81993; d2 = 0:02075

when 1¡�6 2 (degree); d1 = 0:83759; d2 = 0:00309

when 2¡�6 3 (degree); d1 = 0:87531; d2 = − 0:01577when 3¡�6 4 (degree); d1 = 0:9354; d2 = − 0:0358when 4¡�6∞ (degree); d1 = 1:0202; d2 = − 0:057

r = 1; 3; 5; 7; 9.



Table II. Comparison of characteristics of di�erent methods.

Characteristics of Method IIdi�erent methods Method I (Proposed method) Method III

Wind loads Only drag forces ofwind loads acting onsti�ened girder andcables

Three components ofwind loads acting onsti�ened girder and dragforce acting on cables

Three components ofwind loads acting onsti�ened girder anddrag forces acting ontowers and cables

Consideration of geo-metric non-linearity ofstructure

No Yes Yes

(Only consideration ofthe sti�ening e�ect ofthe tension force in thecable)

Consideration of e�ectof changes in the cabletensions due to di�er-ent wind velocities

No Yes Yes

E�ciency of calcula-tion

High High Low

Rate of convergence — Good Rather poor

Accuracy of results Rather poor Good Accurate

Three methods are used to analyse the lateral response of the bridge under displacement-dependent wind loads. Method I is the uniform distribution method [3]. Method II is theproposed method. Method III is the NFEM [14, 15]. The capabilities of the three methods arecompared in Table II.Figures 6 and 7, respectively, illustrate the lateral displacements of the sti�ened girder and

the main cable at the midpoint of centre span for di�erent methods. Figure 8 shows the re-lationship between drag force and wind velocity. Figure 9 shows the curve of the tensionsin the cables versus the wind velocity. From these �gures and Table II, it can be seen that:(1) the drag forces of wind loads obtained from the proposed method (Method II) and theNFEM (Method III) are almost identical, as shown in Figure 8. It may be further observedfrom Figure 8 that the proposed method provides more accurate results than the uniformdistribution method; (2) the tensions in cables decrease as the wind velocity increases. Thee�ects of changes in the tensions in cables due to di�erent wind velocities are not consideredin the uniform distribution method (Method I). This problem can be solved by the proposedmethod (Method II). The proposed method does o�er a signi�cant improvement over the uni-form distribution method (Method I); (3) compared to the results obtained from the NFEM(Method III), the proposed method (Method II) yields good correlations in lateral displace-ments of suspension bridges. The uniform distribution method (Method I) gives rather poorresults; (4) only the results obtained from the proposed method and the NFEM (Method III)show the non-linear characteristics of lateral displacement response of suspension bridges



40 60 80 100 1200

1

2

3

4

5

Lat

eral

dis

plac

emen

t(m

)

Wind velocity(m/s)

Method I

Method II

Method III

Figure 6. Comparison of lateral displacement of the sti�ened girder at midpointof centre span for di�erent methods.

40 60 80 100 1200

1

2

3

4

5

Lat

eral

dis

plac

emen

t(m

)

Wind velocity(m/s)

Method III

Method II

Method I

Figure 7. Comparison of lateral displacement of the main cable at midpointof centre span for di�erent methods.

under displacement-dependent wind loads, and (5) the proposed method requires minimaldata preparation e�ort. Also, the personal computer running time is absolutely negligiblewhen compared with the NFEM (Method III).



40 60 80 100 1200

5

10

15

20

25

Dra

g fo

rce

(KN

/m)

Wind velocity(m/s)

Method IMethod II(proposed method)

Method III

Figure 8. Drag force–wind velocity relationship.

40 60 80 100 120

20000

40000

60000

80000

100000

120000

140000

160000

Method III(leeward)

Cab

le te

nsio

n(K

N)

Wind velocity(m/s)

Method I

Method IIMethod III(windward)

Figure 9. Cable tension–wind velocity relationship.

7. CONCLUDING REMARKS

A new, simpli�ed method is proposed for the lateral response analysis of suspension bridgesunder wind loads. In the proposed method, de�ection theory is used to account for the sti�en-ing e�ect of the tension force in the cable, thereby partly taking the geometric non-linearity ofstructure into consideration. On the other hand, the use of di�erent linear curve �tting makesit possible to consider additional displacement-dependent wind loads (drag force, lift force,and pitch moment). These are two main advancements over the existing simpli�ed meth-ods (e.g. the uniform distribution method). By introducing the series method, the problemof lateral response of suspension bridges under wind loads is greatly simpli�ed, thus savinga signi�cant amount of computation time. This is an advancement over the non-linear �niteelement method. The proposed method is particularly useful for the preliminary design stageof suspension bridges where numerous analysis iterations need to be carried out. The accuracy,



simplicity, and reliability of the proposed method are veri�ed through a comparative studyon the wind-induced lateral response analysis of a long-span suspension bridge with a centrespan length of 888 m.The proposed method is simple, accurate, economical and reliable. However, the practical

application of the proposed method is limited to the lateral response analysis of single-spansuspension bridges under wind loads. It should be noted out that the proposed method couldalso be extended to include the lateral response prediction of multi-span suspension bridgesunder wind loads.

ACKNOWLEDGEMENTS

The authors wish to thank the referees for their suggestions that helped to improve the presentation ofour work.

REFERENCES

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Lateral Response Analysis of Suspension Bridge Under Wind Loads