research on limit span of self-anchored

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Research on Limit Span of Self-Anchored

Suspension Bridge

Wenliang Qiu1 , Meng Jiang

1and Zhe Zhang

1

1School of Civil & Hydraulic Engineering, Dalian University of Technology, Dalian 116024,

P.R. China

Abstract. Self-anchored suspension bridges are increasingly appreciated by engi-

neers for their aesthetic look, low cost and more adaptive for geological environ-

ment than earth-anchored suspension bridges. Many self-anchored suspension bridges have been completed or are in construction in the world. For huge hori-

zontal component of main cable in the stiffening girder and the different construc-

tion method, self-anchored suspension bridge has much less limit span than earth-

anchored suspension bridges. In this paper, the limit span of three-span self-

anchored suspension bridge with two towers is deduced. Some factors, such as ra-

tio of rise to span of main cable, ratio of side-span to mid-span, second dead load

and live load, are analyzed in this paper. Based on the common used material and

considering the main factors, the limit spans of self-anchored suspension bridges

with concrete stiffening girder and with steel stiffening girder, are discussed in de-tail, and the corresponding limit spans are given.

Keywords: suspension bridge, self-anchored, limit span

1 Introduction

The horizontal component of the main cable force is carried by stiffening girder in

a self-anchored suspension bridge (John and David, 1999). The value of the hori-

zontal component force vary with the deadweight of the girder, main cables, deck

system and the live load, and increase with increase of the length of span. Assum-

ing the strength of the main cable is certain, the main cable will reach its ultimate

strength when the span increases to a certain limit. Under such circumstances, it is

apparent that the deadweight can’t not be afforded needless to say load by the stif-

fening girder and the bridge beck system and so on. Besides, the axial compres-

sion force can be carried by the girder is limited when its own material strengthand cross section are given. The limit span of self-anchored suspension bridges

∗ Corresponding author, e-mail: [email protected]

© Springer Science+Business Media B.V. 2009

Y. Yuan, J.Z. Cui and H. Mang (eds.), Computational Structural Engineering, 1173–1180.



differs from that of the earth-anchored suspension bridges as both the main cable

strength and the girder strength are in charge.

The limit span of bridge is influenced by factors such as material strength, con-

struction technique and economic rationality (Li et al., 1999; Wu, 1996a; Wu,1996b). To make it simple, the limit spans of self-anchored suspension bridges are

studied only considering the material strength in this paper.

2 Deduction of Limit Span of Self-Anchored Suspension Bridge

Figure 1 shows a symmetric three-span suspension bridge with two towers. The

mid-span length of the main cable is L with the rise of f , and the side-span lengthof main cable is L1. To simplify the problem, we assume: the configuration of the

main cable is second order parabola (Cobo and Aparicio, 2001), stiffening girder

is horizontal and deadweight of the deck system is distributed uniformly along the

whole bridge, live load is also distributed uniformly along the whole girder and

the structural deformation caused by live load is neglected. So the sum of dead-

weight and live load distributed horizontally is q.

l mbbcc qq A Aq +++= γ γ (1)

wherec A is cross section of the main cable (m2),

b A is cross section of the stif-

fening girder (m2),cγ is density of main cable (kN/m3),

bγ is density of the girder

(kN/m3),mq is load of bridge deck system (kN/m) including deck pavement,

bump wall , handrail ect.l q is live load (kN/m) including motor vehicles, human

crowds or non-motor vehicles ect.

1

1 1

1

Figure 1. Three-span self-anchored suspension bridge with two towers

As the configuration of main cable is second order parabola, and the horizontalcomponent of the mid-span main cable force is H.

1174 Research on Limit Span of Self-Anchored Suspension Bridge



2

( )8 8

c c b b m

qL L H A A q q

f γ γ

λ = = + + +

(2)

where λ is ratio of rise to span and / Lλ = . The horizontal component of the

side-span main cable force is H 1.

1

1

1

2

188

1

λ

qL

f

qL H ==

(3)

where1λ is ratio of rise to side span and

1 1 1/ f Lλ = .

Since the unbalanced horizontal component born by cable tower is negligible,the following expression can be obtained when the horizontal components of both

sides of cable tower are equal.

f f L

L f 2

2

2

11 β ==

(4)

where1

/ L L β = is ratio of side-span to mid-span.

So the stress in girder is /b b

H Aσ = and the maximum tensile stress of the

main cable is)cos( maxθ

σ c

c A

H = , where

maxθ is the bigger one of the two angles

θ and θ 1. θ and θ 1 are the inclination angles of mid-span cable and side-span cable

at the upper supporting point respectively, and they can be deduced from the equa-

tion of main cable configuration. For the mid-span, we have

22 )4(1

1

))(tg(1

1)cos(

λ θ θ

+

=

+

=

(5)

and for the side-span,

)4(1

)(tg 2

1 L

d ++= λ β λ

β θ

(6)

where d is the vertical distance between the lowest point of the mid-span main ca-

ble and the anchored end of the side-span main cable. From the above expression,

we know that if d is increased, the tension force in the main cable will increase

correspondingly due to the accretion of )tg( 1θ . In order to lessen the tension force

T in the main cable at a given horizontal component force H , d should be de-

Wenliang Qiu et al. 1175



creased. In this paper, the minimum value d =0 is discussed to obtain the limit

span, which yields,

222

1

1

))4(1

(1

1))(tg(1

1)cos(

λ β λ β

θ θ

++

=

+

=

(7)

hence, comparing )cos(θ with )cos( 1θ ,maxcos( )θ can be achieved from the fol-

lowing expression:

)}(cos),(cosmin{)(cos 1max θ θ θ = (8)

The stresses in the stiffening girder and the main cable depend directly on the

horizontal component H , which is directly affected byc A and

b A . Therefore,

when the shape of the main cable and cross-sections of the main cable and the stif-

fening girder are given, that isc A ,

b A , λ , β are determined, span L will reach its

maximum value L j under the condition that the compressive stress in the stiffening

girder and the tensile stress in the main cable reach their allowable stress ][ cσ

and ][c

σ respectively, that is,

][ b

b A

H σ =

,

][)cos( max

c

c A

H σ

θ =

(9)

Substitute the above expressions into equation (2), we can get the following

expression,

b

l mbc

c

b

b j

Aqq

L+

++

=

γ γ θ σ

σ

σ λ

)cos(][][

][8

max (10)

The above expression gives the limit span of three-span self-anchored suspen-

sion bridge with two towers.

3 Upper Limit of the Limit Span

If the suspension bridge only carry its self-weight and doesn’t carry loadl m qq + ,

the limit span L j will get to its upper limit L j,max when the materials of the structure

reach their allowable stress, that is,




][)cos(][

8

max

max,

b

b

c

c j L

σ

γ

θ σ

γ

λ

+

=

(11)

It can be seen that the upper limit span L j,max depends on the parameters of the

suspension bridge, such as ratio of rise to span of main cable λ , ratio of side-span

to mid-span β , the weight density of stiffening girder and main cable materials

(bγ ,

cγ ) and their allowable stress ( ][ bσ , ][ cσ ) as well. Hence, to gain higher up-

per limit of limit span, the following measures can be taken:

1. To enhance ratio of rise to span of main cable of mid-span λ .

2. To build the main cable using high strength and lightweight material.

3. To build the stiffening girder using high strength and lightweight material.

Further analysis of commonly used materials in current suspension bridges can

be seen as follow.

For self-anchored suspension bridge with reinforced concrete stiffening girder,

C50 concrete is commonly used whose allowable stress is 3105.17][ ×=bσ kPa.

Since the girder is subjected to axial compression force, only very few reinforce-

ments are required and the density of girder is 25kN/m3. Considering that the anc-

hor blocks of the hanger and the transverse structures don’t carry axial compres-

sion force, the effective area b A of the stiffening girder doesn’t include these

parts of concrete. Correspondingly, a factor should be added to the original weight

density of concrete (25kN/m3), and 25.3125.125 =×=bγ kN/m3.

For self-anchored suspension bridge with steel stiffening girder, Steel Q345 is

commonly used in china currently, whose allowable axial compressive stress is310200][ ×=bσ kPa. Same as the concrete suspension bridge, the density of steel

girder is 125.9825.15.78 =×=bγ kN/m3.

Parallel steel wires of ultimate strength of 1670Mpa are used to constitute a

main cable. With a safety factor of 3.0, the allowable stress3

c 107.556][ ×=σ kPa. The weight density of main cable is 5.78=cγ kN/m3.

Figure 2 shows the relationships of the upper limit span L j,max and the ratio of

rise to span of main cable λ when ratio of side-span to mid-span β equals 0.4, 0.6,

and 0.8 respectively.

For concrete self-anchored suspension bridge, the L j,max～λ curves shown in

Figure 2(a) denote that the relationships between L j,max and λ are nearly linear.

The curves of different β maintain nearly the same line. Thus, ratio of side-span to

mid-span β has little influence on the limit span. Therefore, regardless of the in-

fluence of β , limit spans corresponding to different ratios of rise to span of main

cable can be obtained by taking the average result of the 6 curves, which are given

in table 1.




For steel self-anchored suspension bridge, To some extent, the L j,max～λ curves

shown in Figure 2(a) denote that the relationships between L j,max and λ are non-

linear, and the ratio of side-span to mid-span β has little influence on the limit

span. Therefore, regardless of the influence of β , limit spans corresponding to dif-ferent ratios of rise to span of main cable can be obtained by taking the average re-

sult of the 6 curves, which are given in table 2.

(a) Concrete suspension bridge (b) Steel suspension bridge

Figure 2. L j,max～λ curve of self-anchored suspension bridge

Table 1. Value of L j,max corresponding with different for concrete suspension bridge

λ 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12

L j,max(m) 1006.2 813.0 681.5 586.3 514.4 458.0 412.8 375.6 344.6

Table 2. Value of L j,max corresponding with different for steel suspension bridge

λ 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12

L j,max(m) 2887.3 2377.4 2015.9 1747.4 1540.6 1376.9 1244.1 1134.4 1042.2

4 Study of the Influence of ql + qm on Limit Span

Limit span, regardless of second dead load and live load, has been studied pre-

viously under deadweight of the bridge. Such a result is the upper limit span with

the consideration of ql +qm. However, the result can not be applied practically. Thefollowing study on limit span is related to the influence of ql +qm. It has been stu-

died that the influence of ratio of side-span to mid-span β can be neglected, so

β =0.4 will be applied in the following calculation.




First of all,

bb

l m

A

qq

γ α

+＝ , α denotes the ratio of sum of second dead load and

live load to deadweight of stiffening girder. Then

bbc

c

b

b j L

αγ γ γ θ σ

σ

σ λ

++

=

)cos(][

][

][8

max (12)

Main cable is constituted by parallel steel wires. Stiffening girder is constituted

by concrete and steel respectively. Allowable stress and specific weight are taken

as before. By the above expression, the relationship curves of the limit span L j and

ratio of load α , corresponding to different ratios of rise to span of main cable λ ,are shown in Figure 3.

(a) Concrete suspension bridge (b) Steel suspension bridge

Figure 3. L j～α curve of self-anchored suspension bridge

As shown in the figures, limit span decreases rapidly with the increase of ratio

of load α . In practical design, live load ql is given by the design requirements of

bridge. Therefore, in order to get larger span, second dead load qm should be re-

duced probably by taking some measures, that is, the stiffening girder surface is

paved by materials with light weight and small thickness, and the steel guardrails

with light weight are applied to avoid the use of concrete bumper with heavy

weight.




5 Conclusions

In this paper, the limit span of three-span self-anchored suspension bridge with

two towers is deduced. Some factors, such as ratio of rise to span of main cable,ratio of side-span to mid-span, second dead load and live load, are analyzed in this

paper. Based on the common used material and considering the main factors, the

limit spans of self-anchored suspension bridges with concrete stiffening girder and

with steel stiffening girder, are discussed in detail, and the corresponding limit

spans are given. The study shows that, we can meet the aim of enlarging the limit

spans of self-anchored suspension bridges though the following methods:

(1) To enhance ratio of rise to span of mid-span main cable.

(2) To build the main cable and stiffening girder with high strength andlightweight materials.

(3) To reduce second dead load as much as possible by paving the bridge surface

light weight materials and small thickness, and by using steel guardrails with

light weight.

References

Cobo D.A. and Aparicio A.C. (2001). Preliminary static analysis of suspension bridges. Engi-

neering Structure, 1096-1103.

John A.O. and David P.B. (1999), Self-anchored suspension bridges. Journal of Bridge Engi-

neering , 4, 151-156.

Li J.SH., He D.R. and Liu Y. (1999). Analysis of the limit span of cable-stayed bridge. East

China Highway, 1, 7-8 [in Chinese].

Wu X.G. (1996a). Study on limit span of cable-stayed bridge. Journal of Chongqing Jiaotong

University, 3, 36-38 [in Chinese].

Wu X.G. (1996b). Study on limit span of arch bridge. North and East Highway, 3, 54-56 [in

Chinese].


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