preliminary design of very long-span suspension bridge

Engineering Structures 22 (2000) 1699–1706www.elsevier.com/locate/engstruct

Preliminary design of very long-span suspension bridges

Paolo Clementea,*, Giulio Nicolosi b, Aldo Raithelb

a ENEA—Centro Ricerche Casaccia, via Anguillarese 301, 00060 S.M. Galeria, Roma, Italyb Dipartimento di Analisi e Progettazione Strutturale, Universita` di Napoli Federico II, via Claudio 21, 80125 Napoli, Italy

Received 27 January 1999; received in revised form 25 November 1999; accepted 26 November 1999

Abstract

The feasibility of very long-span suspension bridges is analysed. The relation between the geometrical parameters, the load andthe material characteristics is first discussed in order to find out the limit value of the span length. An iteration procedure is thenproposed to study the main aspects of the structural behaviour in the deflection theory. The results of a numerical investigationallow to state that the behaviour of a very long-span suspension bridge, both in terms of stresses and deflection, is similar to thebehaviour of an unstiffened cable, the contribution of the girder being negligible. 2000 Elsevier Science Ltd. All rights reserved.

Keywords:Suspension bridges; Long-span bridges; Limit span; Deflection theory

1. Introduction

Suspension bridges were already built by primitiveman, by using ropes of creepers, anchored to towers ofnatural rock. The first iron chain bridge appeared per-haps in China [1]. In the eighteenth century, chains ofwrought iron were introduced in England and the USA.Most of the chain bridges collapsed because of oscil-lations due to wind actions. The first wire cable suspen-sion bridges were built at the beginning of the nine-teenth century.

In 1883 the 580 m span Brooklyn Bridge was success-fully completed. It was considered as the eighth wonderin the world and its success encouraged the engineers touse the suspension bridge for all the new long-spanbridges.

Between the last decades of the nineteenth century andthe first ones of this century, first the elastic theory andthen the deflection theory developed [2]. These advancesin the understanding of suspension bridge behaviouraffected a great expansion of their use, especially in theUSA. The George Washington Bridge of about 1000 mspan, opened in 1931, represented a great step forwardthe construction of long-span bridges. A few years later,

* Corresponding author. Tel.:+ 39-06-30486297; fax + 39-06-30484872.

E-mail address:[email protected] (P. Clemente).

0141-0296/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (99)00112-1

in 1937, the Golden Gate Bridge, with a main span on1240 m, was completed.

The glamorous failure of the Tacoma Narrows Bridgein 1940 caused a stop in the span increasing and majorefforts were made in order to understand the behaviourof suspension bridges under wind loads.

In recent years very long-span suspension bridgeswere designed. The Great Belt East Bridge in Denmark,characterised by a wind transparent deck, shows a 1624m main span. The Akashi Kaikyo Bridge in Japan hasa main span of 1991 m. A suspension bridge of 3300 mhas been proposed to span the Messina Strait and finally,a new type with three spans of 5000 m has been analysedfor the Gibraltar Strait crossing [3].

The reasons for increased spans are known. First ofall the increase of the horizontal navigation clearances,in order to accommodate the increasing size and volumeof marine traffic. Then the economic trade off of spanlength cost of deep water foundations, as opposed toshallow water foundations, and the risk of ship collisionwith piers.

The feasibility of longer spans is related to theimplementation of new high-strength lightweightmaterials. As a matter of fact, as spans become longer,cables become heavier. Therefore, a high percentage ofthe cable stress is related to its own self-weight. Further-more the stiffening contribution of the deck on the struc-tural behaviour becomes negligible.

In the present paper the analysis of the structural

1700 P. Clemente et al. / Engineering Structures 22 (2000) 1699–1706

behaviour of very long-span suspension bridges is car-ried out. A very simple relationship between the span,the load and the cable cross-sectional area is discussed,in order to find out the maximum value of the spanlength. A very rapid iteration procedure is proposed thatallows to analyse the behaviour of very long-span sus-pension bridges in the deflection theory. The results ofa large numerical investigation are shown, that point outthe influence of the span on stresses and deflection.

2. Limit span

Consider first the case of an unstiffened cable, i.e. asuspension bridge without stiffening girders. This model,which is suitable (a) to perform the preliminary designof the structure and (b) to have a first glance at its dis-placements, is shown in Fig. 1. We assume the sag ratiof/, in the middle of the main span, and the maximumvalueam of the slope anglea, with respect to the hori-zontal, to be fixed. So the following non-dimensionalgeometrical parameter is defined:

k 5 f/,·cosam (1)

For the usual values off/, (>0.10) we can supposethe total self-weight of the cable to be uniformly distrib-uted on the main span. IfAc is the cable cross-sectionalarea andgc the cable weight per unit volume, then theuniformly distributed self-weight of the cable is

wc 5 gcAc·Lc/, 5 gAc

Lc being the cable length. The ratioLc/, depends on thecable shape only. So, for a fixed shape, i.e. for a givenk, g is proportional togc. If wp is the permanent load onthe main span, which is also supposed to be uniform,the dead load per unit length can be written:

w 5 wp 1 gAc (2)

Two kinds of travelling loads are considered: a uni-form load acting on the whole main span

Fig. 1. Unstiffened cable model.

p1 5 β1·w (3)

and a uniform load coming to the main span

p2 5 β2·w (4)

whose maximum length isc2, (c2 , 1).The horizontal component of tension in the cable is

Hw under dead loads only andH under combined deadand live loads. We conventionally put:

H 5 Hw 1 Hp (5)

Hp as being the increment of tension caused by live loadsacting on the structure.

Considerpeq to be the uniform load acting on thewhole main span, that causes the same horizontal tensionin the cable of the loadsp1 and p2. If p2 is symmetricaround to the mid-span and starts at a distance ofc1,from the left pier, then:

peq 5 p1 1 p2·(1 2 4c21) (6)

If we put

β 5 (wp 1 peq)/gAc (7)

then the value of the horizontal tension and the corre-sponding maximum value of the axial force in the cableare, respectively:

H 5 gAc·(1 1 β),/(8f/,) Nmax 5 gAc·(1 (8)

1 β),/8k

The non-dimensional parameterβ is the ratio betweenthe additional load, which the cable supports, and itsself-weight. If β = 0, then the cable bears its self-weight only.

By settingNmax to the allowable value of the tensionin the cable,sAc , we deduce, for a givenβ, the limitvalue of the span, i.e. the maximum value of the spanlength for a cable of cross-sectional areaAc, compatiblewith the assigned loads and material strength:

,lim 5 8k·(s/g)/(1 1 β) (9)

1701P. Clemente et al. / Engineering Structures 22 (2000) 1699–1706

Consider first the limit case of the cable subject toself-weight only (β = 0). The limit value of the spanbecomes independent of the cable section and dependsonly on the characteristics of the material and on thedefined geometry of the structure

,0,lim 5 8k·s/g (10)

If live loads are acting on the structure (β . 0), thenthe limit value of the span depends also on the value ofthe live loads and on the cable cross-sectional area, thatdetermine the value ofβ. Eq. (9) can be used for thepreliminary design of the cable, by solving it forAc:

Ac 5(wp + peq)8ks/, − g (11)

Eq. (11) is meaningful only for, , ,0,lim. In fact,Ac→` when ,→,0,lim.

In Fig. 2 the diagram of the non-dimensional para-meter,lim/(ks/g) versusβ is plotted. As one can see,lim

decreases rapidly whenβ increases. For the usual caseof k = 0.1, and considering a steel cable for whichs/g>104 m, the diagram in Fig. 2 gives the limit spanvalue in km. High values of the sag ratio determinelower stresses in the cable and so allow to span largerdistances, but the cost of the piers becomes prohibitive.In the case of materials characterised by higher valuesof the ratios/g, the theoretical limit span grows remark-ably.

Fig. 2 can also used in a different way. It produces,for a fixed,, the maximum value ofβ, i.e. the maximumload increment, as a percentage of the self-weight, whichthe cable can support.

Fig. 2. ,lim/(ks/g) againstβ.

3. Iteration procedure for the non-linear analysisof suspension bridges

Under dead loads only, the co-ordinates of the cablearez andy (z1 andy1 in the side span—see Fig. 1). Whena generic live loadp(z) is acting, the system shows thevertical displacementv(z), which must satisfy the equi-librium equation (see Appendix A)

EI·vIV(z) 2 H·v0(z) 5 p(z) 2 wHp/Hw (12)

and the compatibility condition relative to the cablelength. The solution can be found by using an iterativeprocedure [4].

Alternatively, in order to determine the horizontalcomponent of cable tensionH, the procedure proposedby Franciosi [5] can be used, that rapidly leads to thesame results. It is based on the consideration that, in thesame approximation of the deflection theory, in a struc-ture subject to a fixed axial force system, the principleof the superposition of the effects, and so the theoremsof the theory of elasticity based on it, are still valid refer-ring to the transversal loads. In a suspension bridge theaxial force system is made up byw andHw, whereas thelive loads represent the transversal ones.

In order to apply the Betti theorem, consider the sys-tems of Fig. 3(a) and (b). The model of Fig. 3(a) is theactual structure, subject to dead loadsw (w1 on the sidespan), to live loadp(z) and to temperature variation±DT, in which the horizontal constraints at the ends havebeen substituted by the horizontal component of thereaction Hw + Hp. The model of Fig. 3(b) (auxiliarysystem) is the same as the structure of Fig. 3(a) subjectto load w and to the horizontal forcesHw + Hp at theends. It shows the vertical displacementsh(z) and therelative horizontal displacementd between the two ends.The Betti theorem enables us to write

E ph(z)·dz 1 Hpd 5 2 E Hp

cosawDTds (13)

Where w is the thermal expansion coefficient of thecable material. It is:

Hp 5 2

E ph·dz

d + EwDTcosa

ds

(14)

From Eq. (14) we can determine the increment of thehorizontal component of the cable tension due to liveload, whenh(z )andd are established. IfX is the bendingmoment in the stiffening girder at sections C and D ofthe system in Fig. 3(b), and we puta2 = H/EI (a1

2 =H/EI1), h(z) and d are produced by the followingrelations:

h(z) 5 Ceaz 1 De− az 2Hp

Hy 1

XH,

z 2Hp

Ha2y0


Fig. 3. Actual (a) and auxiliary (b) systems.

h1(z1) 5 C1ea1z1 1 D1e− a1z1 2Hp

Hy1 1

XH,1

z1 (15)

2Hp

Ha21y0

1

in the main and side spans respectively, and

d 5Ha

EALs 1 2

HEI1

E,1

0

y1h1dz 1HEIE

,

0

yhdz (16)

11615Sf2

1,1

EI11

f2,

2EIDHp 223Ff1,1

EI11

f,EIGX

in which h is the height of the pylon (see Fig. 1) and

y1 5 y1 2 (,1 2 z1)·h/,1 (17)

In Appendix A, how to achieve at Eqs. (15) and (16) andalso how to determineX is explained. The expressions ofconstantsC, D, C1, D1 andLs are also given in Appen-dix A.

Eqs. (14)–(16) can be solved using an iterative pro-cedure. This can be started by assigning a value ofH,calculatingX (see Appendix A) and thenh(z) andd fromEqs. (15) and (16). Then Eq. (14) produces a new valueof H and so on. The solution is found with a few iter-ations.

If H is known, the equilibrium configuration can bedetermined from Eq. (12), whose solution has beenfound, in the following numerical investigation, by usinga finite differences procedure.

4. Stresses and deflections in very long-spansuspension bridges

A numerical investigation has been carried out byusing the previously described procedure. The analysedmodel is shown in Fig. 4. The main span has been sup-posed to be simply supported at C and D (X = 0). Theinfluence of , on the structural behaviour has beenpointed out, in particular. This study being referred tofuture bridges, values of, from 1000 to 3500 m havebeen considered, withf/, = 0.10.

The analysis has been carried out by referring torealistic values of the loads. The permanent loadw1, i.e.,the summation of the weight of the deck and the weightof the suspension elements, has been assumed equal to250 kN/m. The uniform loadp1 acting on the wholemiddle span is equal to 20 kN/m. A uniform loadp2 =300 kN/m, coming to the bridge and whose maximumlength is 750 m, has also been considered. Loadp1 rep-resents a slight vehicular load,p2 both heavy vehicularand railway loads. The cable cross-sectional area hasbeen calculated by using Eq. (11), in whichs = 850MPa andg = 0.078 MN/m3 have been assumed, thesebeing the typical values for steel cables.

In Fig. 5 the diagram of the ratiop/w against the span, is plotted,p being the total live load distributed on thewhole main span:p = (p1 + p2 c2). It is apparent that,because of the increase ofw, this ratio becomes verylow as , gets higher. It isp/w = 0 for , = ,0,lim.

The maximum valueHmax of the horizontal compo-nent of the cable tension occurs whenp2 is placed sym-metrically around mid-span. For this load condition the


Fig. 4. Investigated model.

Fig. 5. p/w against,.

diagram of the non-dimensional ratioHmax/w, against,is shown in Fig. 6. It diminishes as, gets higher andtends to a limit value related to the ratiof/,.

In Fig. 7 the diagrams of the non-dimensionalmaximum displacementv/, against ,, for differentvalues of the girder bending stiffnessEI [MN.m2], areplotted. The Young’s modulusEc = 180 000 MPa hasbeen assumed for the cable. IfEI = 0, thenv/, gets loweras , increases. This behaviour is due to the increase ofw. In fact, w→` when ,→,0,lim. A slope variation isapparent in the curve; the value of, for which thisoccurs is obviously related to the length of loadp2 (750m). The curves relative to different values ofEI showthat the influence of the girder bending stiffness is negli-gible for high values of,. For , = 3500 m, it isv/, =0.005. As, gets shorter the contribution ofEI becomesstronger so that a reduction ofv/, can be observed when, decreases. The value of, for which v/, is maximumis related to the length of the loadp2. This behaviour

Fig. 6. Hmax/w, against,.

confirms that for a normal span a noticeable reductionof the displacement is obtained if the girder bendingstiffness is high.

The section in which the maximum displacementoccurs and the relative position of the moving loadp2

are shown in Fig. 8. In more detail, the non-dimensionalabscissazv/, of the section in which the maximum dis-placementv occurs and the relative abscissa of the headof the coming loadzp/, are plotted. The first is practi-cally independent of the span, and equal to 0.27. Thesecond varies in a wider range. In both the diagrams avariation in the slope is evident at,>1500 m, related tothe length of the loadp2 (750 m). The diagram of theratio H/w, for the same load condition is plotted in Fig.9, in which the previously pointed out singularity canalso be observed.


Fig. 7. Non-dimensional maximum displacementv/, against,.

Fig. 8. zv/, andzp/, against,.

5. Conclusions

The structural analysis has been performed using aniteration procedure in the same approximation of thedeflection theory. The numerical investigation, carriedout by means of a specific computer code, allowed tocheck the fast convergence of the procedure. The mainaspects of the structural behaviour have been analysedboth in terms of stresses and deflections considering realloading. The result can be summarised as follows:

O The behaviour of a suspension bridge when the spanincreases, tends to the behaviour of an unstiffenedcable and the contribution of the girder becomes neg-ligible;

Fig. 9. H/w, against,.

O The deformability, in terms of non-dimensionalmaximum displacementv/,, becomes independent ofthe girder bending stiffness and decreases when,increases;

O The load condition, which gives the maximum dis-placement, individualised by the head of the loadp2,zv/,, and the section in which the maximum displace-ment occurs, whose non-dimensional abscissa iszL/,,are independent of the span,.

A very simple relationship between the span, the loadand the cable cross-sectional area has also been dis-cussed. It has been deduced considering vertical staticloads only and referring to a linear analysis. The non-linear analysis under static load gives longer values ofthe limit span, but the analysis in presence of wind load-ing determines a reduction of the real limit span.

Obviously, the limit span of a suspension bridge isalso related to the material characteristics. Newmaterials, characterised by high values of the ratios/gseem to allow to span very long distances in the future,but in this case the effects of wind loading are muchhigher. The use of new structural types is advisable inorder to span longer distances in the future [6].

Appendix A

A.1. Equilibrium equation

The bending momentM in the generic section is

M(z) 5 M0(z) 2 (Hw 1 Hp)·(y(z) 1 v(z))

whereM0 is the bending moment in a simply supportedbeam of the same span and subject to the same load.The second derivative of the previous expression gives:


2 EI·vIV(z) 5 2 (w 1 p(z)) 2 (Hw 1 Hp)·y0(z)

2 (Hw 1 Hp)·v0(z)

And keeping into account thatHw·y0(z) = 2 w, weobtain finally:

EI·vIV(z) 2 H·v0(z) 5 p(z) 2 wHp/Hw

A.2. Determination ofh and d

In order to calculate the displacementsh(z) [Fig.3(b)], consider the equilibrium equation in the mainspan:

2 Hh 2 Hpy 1 X 5 2 EIh0

whereX is the bending moment at sections C and D,Eand I the Young’s modulus and the moment of inertiaof the girder respectively. By putting

a2 5 H/EI

the previous equation can be written in the form

h0 2 a2h 5Hp

EIy 2

XEI

The complete solution, sum of the complementary andparticular solutions, is

h(z) 5 Ceaz 1 De− az 2Hp

Hy 1

XH

2Hp

Ha2y0

where:

y0 5 2 8f/,2

With the boundary conditions

h(0) 5 0 h(,) 5 0

the constantsC andD are:

C 5 2Hp

H·

8fa2,2·

1 − e− a,

ea, − e− a, 21 − e− a,

ea, − e− a,·XH

D 5 2Hp

H·

8fa2,2·

ea, − 1ea, − e− a, 2

ea, − 1ea, − e− a,·

XH

For the side span we obtain, analogously:

h01 2 a2

1h1 5Hp

EI1y1 2

XEI1

z1

,1

in which

y1 5 y1 2 (,1 2 z1)·h/,1

The complete solution is:

h1(z1) 5 C1ea1z1 1 D1e− a1z1 2Hp

Hy1 1

XH,1

z1 2Hp

Ha21

y01

in which

C1 5 2Hp

H·8f1a2

1,21·

1 − e− a1,1

ea1,1 − e− a1,12

1ea1,1 − e− a1,1

·XH

D1 5 2Hp

H·8f1

a21,

21

·ea1,1 − 1

ea1,1 − e− a1,11

1ea1,1 − e− a1,1

·XH

From the compatibility condition

h91(,1) 5 h9(0)

we deduce the value ofX

X 5

A1a1ea1,1 − A3a1e− a1,1 − B1a + B3a −Hp

H S4f,

+4f1,1D

1H,1

− A2a1(ea1,1 + e− a1,1) + B2a − B4a

the constantsAi andBi having the following expressions:

A1 5Hp

H·8f1a2

1,21

·1 − e− a1,1

ea1,1 − e− a1,1A2 5

1H

1ea1,1 − e− a1,1

A3 5Hp

H·8f1a2

1,21

·e− a1,1 − 1

ea1,1 − e− a1,1

B1 5Hp

H·

8fa2,2·

1 − e− a,

ea, − e− a, B2 51H

1 − e− a,

ea, − e− a,

B3 5Hp

H·

8fa2,2·

ea, − 1ea, − e− a, B4 5

1H

e− a, − 1ea, − e− a,

The displacementd can be evaluated by means of thePrinciple of Virtual Work. Assume the system in Fig.3(b) as displacement system and that of Fig. 10 as forcesystem. For the force system it is:

N 5 1/cosa M 5 2 y

N1 5 1/cosa1 M1 5 2 y1

in main and side span respectively. In the displacementsystem the cable tension due to the transversal loads is:

N 5 Hp/cosa

while the bending moment is, respectively in the mainand side spans:

M(z) 5 2 Hh 2 Hpy 1 X

M1(z1) 5 2 Hh1 2 Hpy1 1X,1

z1

The Principle of Virtual Work produces in this case:

d 5Ha

EALs 1 2

HEI1

E,1

0

y1h1dz 1HEIE

,

0

yhdz


Fig. 10. Force system to calculate displacementd by means of the Betti theorem.

11615Sf2

1,1

EI11

f2,

2EIDHp 223Ff1,1

EI11

f,EIGX

where:

Ls 5 EB

A

(1 1 y92)3/2dz>2·,1

1cos3 a1

S1 1 8f2

1

,21D

1 ,S1 1 8f2

,2DReferences

[1] Pugsley A. The theory of suspension bridges, 2nd ed. London:Arnold Ltd, 1968.

[2] Buonopane SG, Billington DP. Theory and history of suspensionbridge design from 1923 to 1940. J Struct Engng ASCE1993;119(3):954–77.

[3] Lin TY, Chow P. Gibraltar Strait crossing—a challenge to bridgeand structural engineers. Struct Engng Int J IABSE1991;1(2):53–8.

[4] Ulstrup CC. Rating and preliminary analysis of suspension bridges.J Struct Engng ASCE 1993;119(9):2653–79.

[5] Franciosi V. Lezioni di ponti. Napoli: Pellerano Del Gaudio, 1956.[6] Nicolosi G, Raithel A, Clemente P. Static issues in very long-span

suspension bridge design. In: Proceedings of the IABSE Sym-posium on Long-span and High-rise Structures. Zurich: IABSE,1998:545–6.

preliminary design of very long-span suspension bridge

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