Research ArticleAnalytical Calculation Method for the Preliminary Analysis ofSelf-Anchored Suspension Bridges

Shaorui Wang1 Zhixiang Zhou12 Yanmei Gao1 and Yayi Huang3

1School of Civil Engineering Architecture and Construction Chongqing Jiaotong University Chongqing 400074 China2State Key Laboratory Breeding Base of Mountain Bridge and Tunnel Engineering Chongqing 400074 China3Editorial Department of Applied Mathematics and Mechanics Chongqing Jiaotong University Chongqing 400074 China

Correspondence should be addressed to Shaorui Wang ruiruiplace163com

Received 20 October 2014 Revised 31 January 2015 Accepted 27 February 2015

Academic Editor Daniela Boso

Copyright copy 2015 Shaorui Wang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The stiffening girder of self-anchored suspension bridge (SSB) is subjected to huge axial force because the main cable is directlyanchored on the end of the stiffening girder To obtain a simple model and accurately understand the mechanical behavior of thewhole structure in preliminary design this paper proposed an analytical calculationmethod considering the combined effects of themain cable-suspender-stiffening girder On the basis of the deflection theory of the stiffening girder the relation between the girdershape and the suspender force was explored The relation between the main cable end force (MCEF) and the suspender force wasderived through segmental catenary theory and iteration method was further improved to avoid the divergence condition Finallythe solution was obtained through satisfying the compatibility condition The proposed method does not need to iterate manuallyand can save calculation time Examples are introduced to verify the applicability of this method with the result that this methodconsiders the combined effects of the main cable-suspender-stiffening girder and the finished bridge state satisfies the minimumstrain energy of the stiffening girder Results also indicate that this method has fast convergence speed and high precision

1 Introduction

In the early years (1801ndash1870) the spans of suspension bridge(SB) were relatively small and the main cables were relativelylight so linear-elastic theories are usually used to study themechanical behavior of the whole structure Because thenonlinearity of the main cable and suspender is growingwith the increase of the span length the deflection theory isput forward and rsearchers have done so afterwards Stein-man [1] introduced the traditional analytical method of SBrepresented by the so-called deflection theory Jennings [2]modified the deflection theory allowing it for the structuralbehavior of suspension bridges inwhich the deck is supportedby two or more cables with different profiles Wollmann [3]presented a practical method for the preliminary analysis ofsuspension bridges based on the deflection theory Ohshimaet al [4] presented a practical analysis of a suspension bridgeunder vertical loads by means of the stiffness matrix methodWith the development of computer technology modernsuspension bridges are typically analyzed by using computer

programs with nonlinear analysis capabilities based on finite-element formulations Jung et al [5] presented a nonlinearanalysis method based on the unstrained element length fordetermining initial shape of suspension bridges under deadloads Montoya et al [6] presented a new methodology todetermine the safety of suspension bridge main cables Theapproach is the first one incorporating a finite-element (FE)model to predict the cablersquos failure loadwhich can account forthe load recovery due to friction in brokenwires and simulatethe reduced cablersquos strength as a three-dimensional randomfield

The finite-element theory was also applied in self-anchored suspension bridges (SSB) Kim et al [7 8] pro-posed a nonlinear shape-finding analysis for a self-anchoredsuspension bridge The procedure consists of two successivesteps of nonlinear analysis The first step is for the cable-onlysystem and the second for the total bridge system In thesecond step the fixed boundary of the anchor points of themain cables and hangers treated in first step must be changedmanually which is burdensome

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 918649 12 pageshttpdxdoiorg1011552015918649

2 Mathematical Problems in Engineering

Such models based on finite-element theory may havethousands of degrees of freedom So there is a need forsimpler models to better understand the structural behaviorin a way not offered by finite-element analysis Such modelsare useful for the preliminary design and independent checksof more complex models And many scholars have beenexploring such models

Tan [9] based on the analysis of finite displacementtheory and analytical iteration method proposed a determi-nation method of the reasonable finished bridge state of SSBconsidering themain cable alignment the force of suspenderand the stiffening girderHan et al [10] studied themain cableshape-finding method of SSB with spatial cables Li et al [11]based on the reasonable internal force requirements of SSBmain cable and the suspender put forward a determinationmethod of the reasonable suspender force of SSB and thismethod considered the effect of stiffening girder alignmentthrough finite-element method but it is relatively compli-cated because the finite-element analysis procedure must beiterated manually

In summary so far the analysis method of SSB mainlyincludes (1) linear-elastic theory and deflection theory (2)finite displacement theory (3) finite-element theory (4)combined method based on finite-element theory and thenumerical analytical method Methods (1) and (2) are effec-tive methods in the early stage but they have many assump-tions and are only suitable for some small span bridgesThesetwo methods are barely used now Method (3) reduces theassumptions and the calculation result meets the actual circsbetter However the method needs a model with thousandsof degrees of freedom and the model is complex The fixedboundary also needs to be changed manually in the secondstep of Method (3) and the calculation time is very long InMethod (4) themain cable is considered through the numer-ical analytical method and the stiffening girder is consideredthrough finite-element method The method is a more accu-rate calculation for main cable but the analysis proceduremust be iterated manually which is time-consuming

So this paper proposes an analytical calculation methodconsidering the combined effect of themain cable-suspender-stiffening girder Compared to the above mentioned meth-ods in the method the main cable and the stiffeninggirder are all considered through the numerical analyticalmethod then the calculation procedure has no need to iteratemanually which can save calculation time

The method also can quickly and effectively find thereasonable finished bridge state of SSB and help designers tobetter understand the structural behavior in amanner offeredby a simpler and accurate model

The paper is organized as follows Section 2 brieflypresents the principle of the proposed method Sections 3ndash5 are dedicated to the main cable analysis the stiffeninggirder analysis and the deformation compatibility conditionof the calculation model respectively Section 6 introducesthe numerical calculation process of the proposed methodExamples are provided in Section 7 to illustrate the appli-cation of the method Finally Section 8 summarizes thefindings of this paper and presents some conclusions

2 The Principle of the Proposed Method

Self-anchored suspension bridge is a high-order staticallyindeterminate structure thus the direct calculation hascertain difficulties In the proposed method one model isestablished which includes three parts namely the maincable the stiffening girder and the deformation compatibilitycondition The main cable and the stiffening girder areanalyzed independently and then they are coupled by thedeformation compatibility condition The discrete graph ofSSB is shown in Figure 1

As to Part I of the calculation model the stiffening girderis assumed as a simply supported girder along the wholebridge The shape of the stiffening girder (SSG) needs to begiven firstly and then calculate the suspender force (SF) Thescheme of calculation principle is shown in Figure 2

The support of pylon to the stiffening girder is replaced asconcentrate forcesThe counterweights set at the pylon-girderjoint and girder end can also be assumed as concentrate forcesat supports and they have slight influence upon the internalforce of the stiffening girder The MCEF (the main cableend force) eccentrically acts upon the stiffening girder andthrough axis shift formula it can be divided into a horizontalforce acting on the neutral axis of stiffening girder and anadditional bending moment1198721 as shown in Figure 1

As to Part II of the calculation model the main cable SFcan be given from Part I and calculate the MCEF satisfyingthe rise to span ratio and other requirements

On the basis of deflection theory and the principle ofminimum strain energy of the stiffening girder the SF andMCEF given from Part I and II are substituted into PartIII of the calculation model the deformation compatibilitycondition and finally checkwhether the SF andMCEF satisfythe criterion (the displacements at hanging point equal zero)if they do the calculation stops otherwise modify the SF andrepeat the iteration

When the error of SSG meets the convergence conditionof precision requirement the iteration stops and the lastcalculation is the expected result

For SSB with overhanging span the bearing reaction 119877

and the self-weight of anchor span 119866 can be equivalent toadditionalmoment1198722 and concentrated force119866-119877 acting onthe end of anchor section as shown in Figure 1The influenceof shrinkage and creep of concrete pylon to the alignment canbe considered by a reduction of the pylon height And moreinformation could be found in relative references

3 Part I of Calculation ModelThe Stiffening Girder Analysis

Under the condition of the known SSG (the shape of thestiffening girder) Part I is used to calculate SF (the suspenderforce) The calculation procedure needs to iterate and theinfluential matrix 119861 of SSG to SF should be obtained

31 The Control Principle of SSG The reasonable finishedbridge state of SSB is measured by the stress state of thestiffening girders under dead load Furthermore reasonable

Mathematical Problems in Engineering 3

Center line ofmain cable

Neutral axis ofstiffening girder

Center lineof support

Center line ofmain cable

T1 T3

Ti

Tnminus3 Tnminus1T2 Tnminus2

V

H

V

H

V

H

V

H

F0 Fn

M1 + M2 M1 + M2T1T2

T3 Ti Tnminus3 Tnminus2 Tnminus1Mj1 Mj1

Mj2 Mj2Mj3 Mj3 Mji Mji Mjiminus3 Mjiminus3 Mjiminus2 Mjiminus2 Mjiminus1

Mjiminus1

L1L2

L3 LiLnminus3 Lnminus2 Lnminus1 Ln

Main cableMain tower Main tower

Stiffening girder

H

Anchorage beam segment Standard beam segment

Neutral axis ofstiffening girder

Center lineof support

R

Overhanging span

Center lineof support

G

M2 G minus R

Figure 1 The discrete graph of SSB

T1T2 T3 T4 T5V1

H1

V9984001

H9984001

Suspender force

Shape of the stiffening girder

Part I the stiffening girder

T1 T2 T3T4 T5

V2

H2

V9984002

H9984002

Suspender force

Part II the main cable

T1T2 T3 T4 T5V2

H2

V9984002

H9984002match shape of the stiffening girder

Part III the deformation compatibility condition

Use the shape errorto correct suspender

force Ti

Meet the shape requirements ofthe stiffening girder calculate

the suspender force Ti

Meet the shape requirements ofthe main cable calculate the

Reaction forces H2 V2 and suspender force Ti

reaction forces H2 and V2

Figure 2 The scheme of the calculation principle

stiffening stress state can be ensured through the reasonablesuspender force which can be solved by rigid supportedcontinuous beam method zero displacement method orminimumbending energymethodThe solving process usingminimum bending energy method is described as follows

The structural cost can be measured by bending strainenergy thus the smaller the bending strain energy is the lessthe materials the structure costs The bending strain energyof the stiffening girder can be obtained through

119880 =1

2int119904

1198722

119864119868119889119904 (1)

According to the principle of minimum bending energy119880 (bending energy) should meet the following relationship

120597119880

120597119883119894

= 0 (2)

where119883119894means the suspender force

Actually the physical meanings of rigid supported con-tinuous beam method zero displacement method and min-imum bending energy method are the same as describedas (2) that is the vertical displacement of the suspendingpoint is zero under the joint action of suspender force thehorizontal component of the main cable and the dead load

32 Impact Analysis of SSG to SF To complete nonlinearanalysis of the stiffening girder the following two assump-tions are made

(i) Neglect the influences of the shear deformation on thealignment of the stiffening girder

(ii) Neglect the influences of the axial deformation on thealignment of the stiffening girder

Hence the alignment of the stiffening girder is onlyrelated to the bending moment

As shown in Figure 3 119902means the uniformly distributedload of stiffening girder self-weight 119871

119894means the length of

4 Mathematical Problems in Engineering

y0i

yi

BH

Mjiminus1A

Y

X

H

Fiminus1

Mji

Fi

Δi

Li

Ti

q

Figure 3 The calculation diagram of the girder segment

girder segment 119867 means the horizontal end reaction of themain cable 119872

119895119894means the bending moment at the hanging

point 119865119894means the shearing force of girder 119879

119894means the

suspender force and 119910119894means the alignment of the girder

segment under the joint action of the suspender tension forcethe horizontal component of themain cable and the constantload119910

0119894is the initial camber of the stiffening girderwhich can

meet the precision requirements using designed alignmentin calculation The starting point of the girder segment isassumed as the origin point

According to (2) the stress state of the girder end underuniformly distributed load 119902 meets equilibrium equationwhen ensuring that no vertical displacement occurs in points119860 and 119861

119865119894= 119865119894minus1

+ 119879119894minus 119902119871119894

119872119895119894=

119902119871119894

2

2minus 119865119894minus1

119871119894+ 119872119895119894minus1

+ 119867Δ119894+ 1198721015840

119872119894=

1199021199092

2minus 119865119894minus1

119909 + 119872119895119894minus1

+ 119867119910

Δ119894= 119910119894minus 119910119894minus1

(3)

Neglecting the shearing deformation the equation can bedescribed as follows

119872119894

119864119868= minus(

11991010158401015840

119894

(1 + 11991010158402

119894)32

minus11991010158401015840

0119894

(1 + 11991010158402

0119894)32

) (4)

From (3) deflection differential equilibrium equation canbe obtained

11991010158401015840

119894+

119867

119864119868119910119894=

minus119902

21198641198681199092+

119865119894minus1

119864119868119909 minus

119872119895119894minus1

119864119868+ 11991010158401015840

0119894 (5)

In the equation assume

1198602=

119867

119864119868 119861 =

minus119902

2119864119868

119862119894=

119865119894minus1

119864119868 119863

119894= minus

119872119895119894minus1

119864119868

(6)

Because 1199100119894is a continuous function we can spread out

power series at 119909 = 0 to conveniently solve the differentialequation

1199100119894=

infin

sum

119898=0

119866119894

119898119909119898 (7)

where 119866119894119898

= 119910(119898)

0119894(0)119898

In practical calculation when 119898 = 4 the result can meetthe precision requirement

1199100119894=

4

sum

119898=0

119866119894

119898119909119898

= 119866119894

0+ 119866119894

1119909 + 119866

119894

21199092+ 119866119894

31199093+ 119866119894

41199094

(8)

11991010158401015840

0119894= 2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092 (9)

Substitute (9) into the differential equation (4)

119910119894= 1198641119894sin (119860119909) + 119864

2119894cos (119860119909)

+

(1198611199092+ 119862119894119909 + 119863

119894)1198602minus 2119861

1198604

+

(2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092)1198602minus 24119866

119894

4

1198604

(10)

Equation (10) is the deflection equation of the girder seg-ment considering geometrical nonlinearity With the knownquantities 119860 119861 119862

119894 119863119894 1198661198942 1198661198943 and 119866

119894

4and combining with

the controlling condition of the minimum strain energy ofthe stiffening girder 119910

119894(0) = 0 119910

119894(119871119894) = Δ

119894 the expressions

of 1198641119894 1198642119894are given by

1198641119894=

Δ119894

sin (119860119871119894)minus 1198642119894ctan (119860119871

119894)

minus

(119861119871119894

2+ 119862119894119871119894+ 119863119894)1198602minus 2119861

1198604 sin (119860119871119894)

minus

(2119866119894

2+ 6119866119894

3119871119894+ 12119866

119894

4119871119894

2)1198602minus 24119866

119894

4

1198604 sin (119860119871119894)

1198642119894=

2119861

1198604minus

119863119894

1198602+

24119866119894

4

1198604minus

2119866119894

2

1198602

(11)

For each girder segment its deflection equation thatmeets theminimum stain energy theory can be obtainedwithknown119879

119894and119867 Besides the deflection equation should also

satisfy the compatibility condition of the stiffening girderthus

120575119894= 1199101015840

119894+1(0) minus 119910

1015840

119894(119871119894) = 0 (12)

Then differentiate (12)

119889120575119894= 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894) = 0 (13)

Mathematical Problems in Engineering 5

In (13) 119879119894is unknown and differentiate 1199101015840

119894

1198891199101015840

119894+1(0) =

sin (119860119871119894+1

) minus 119860119871119894+1

1198602 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

119889119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

119889119879119895

1198891199101015840

119894(119871119894) = (

1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

119889119879119895

+1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

119889120575 = 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894)

=cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

+ (sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)lowast

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

minus(1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

)119889119879119895

(14)

Transform (14) into matrix form

119889120575 = 119870119889119879

119889119879 = 119870minus1119889120575

(15)

where

119861 = 119870minus1

119889119879 =

[[[[

[

1198891198791

119889119879119899minus1

]]]]

]

119870 =

[[[[

[

11987011

sdot sdot sdot 1198701119899minus1

d

119870119899minus11

sdot sdot sdot 119870119899minus1119899minus1

]]]]

]

119889120575 =

[[[[

[

1205751

120575119899minus1

]]]]

]

119870119894119895=sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)

120597119862119894+1

120597119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

120597119863119894+1

120597119879119895

minus (1

1198602minusctan (119860119871

119894)

119860119871119894)

120597119862119894

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

120597119863119894

120597119879119895

(16)

H

VT

l

O X

h

Y

H998400

V998400T998400

Figure 4 The calculation scheme of the main cable segment

Matrix 119861 is the influential matrix of SSG to SF whichrepresents the relationship between 119889120575 (the first derivateincrement of the stiffening girder shape) and 119889119879 (the changeof suspender force) 119889119879 can be solved out through knowngirder shape and then modify 119879(119894) to adjust the suspenderforce

4 Part II of Calculation ModelThe Main Cable Analysis

Under the condition of the known SF calculated from PartI Part II is used to calculate MCEF which satisfies the shaperequirements of main cable by shape-finding

41 The Segmental Catenary Theory of Main Cable For theSSB the segmental catenary method assumes that the align-ment of the main cable is a catenary which is more approxi-mate with the actual situation In the main cable calculationthe segmental catenary method is adopted [12ndash19]

According to the differential equilibrium equations geo-metrical equations and the physical equations of cablesegment the relation between the shape and the internal forceof cable segment is obtainedThe calculation scheme is shownin Figure 4

119897 = minus1198671198780

119864119860minus

119867

119902ln (119881 + radic1198672 + 1198812)

minus ln(119881 minus 1199021198780+ radic1198672 + (119881 minus 119902119878

0)2

)

ℎ =1199021198780

2minus 2119881119878

0

2119864119860

minus1

119902[radic1198672 + 1198812 minusradic1198672 + (119881 minus 119902119878

0)2

]

(17)

In formulas (17) 119902 represents the self-weight of theunstressed main cable 119864 represents the elastic modulus119860 represents the cross-sectional area 119897 represents the spanlength of the cable segment ℎ represents the elevationdifference of two ends and 119878

0represents the unstressed

length of cable segment 119867 and 119881 represent the horizontaland vertical component of cable segment force in the left end

6 Mathematical Problems in Engineering

The calculation process of the main cable alignment is asfollows assume the value of119867 and119881 and calculate 119878

0 119897 and ℎ

of each cable segment according to the calculated suspenderforce and then check whether the result meets the alignmentrequirements or not If it does calculation stops otherwisemodify the value of 119867 and 119881 and repeat the above steps tillthe result meets the shape requirements

42 Improved E-M IterationMethod for theMain Cable Shape-Finding in the Middle Span Formulas (17) are nonlinearequations to solve the equations the value of119867 and119881 shouldbe assumed to get 119878

0 and then check whether ℎ of the target

point meets the requirements or not The solving process isan iterative process

The numerical iterative methods of the main cable in themiddle span of SBmay not converge in some casesThemainreason is that in the iteration process the elevation require-ments of end points and intermediate points are consideredindependently so that the mutual influences between themare ignored The proposed E-M iteration method consideredthe elevation requirements of both the ending points andintermediate points besides when modifying the horizontalforce 119867 and vertical force 119881 the influences of ending pointselevation error 119889119910

119890and intermediate points elevation error

119889119910119898are consideredAs shown in Figure 5 for any Point 119860 on the cable

segment according to the equilibrium equation of moment

119881119898 minus

119899

sum

119894=1

(119879119894119863119894+ 119878119894119908119862119894) minus 119867119910 = 0 (18)

where119867 and 119881 represent the horizontal and vertical compo-nents of the cable segment force in the left end 119879

119894represents

the suspender force 119878119894represents the unstressed length of

cable segment 119862119894represents the distance from the gravity

center of cable segment to Point119860119863119894represents the distance

from hanging point to Point 119860 119910 and 119898 represent thehorizontal and vertical distances from the left ending point ofthe main cable to Point 119860 respectively 119910

119890and 119910

119898represent

the elevation of the ending point and the middle pointrespectively and 119908 represents the self-weight of the maincable

Formula (18) can be transformed into

119910 =(119881119898 minus sum

119899

119894=1(119879119894119863119894+ 119878119894119908119862119894))

119867 (19)

where 119910 is a function of 119867 and 119881 and the differential of 119910 isgiven by

119889119910 =sum119899

119894=1(119879119894119863119894+ 119878119894119908119862119894) minus 119881119898

1198672119889119867 +

119898

119867119889119881 (20)

Substitute formula (19) into (20)

119889119910 = minus119910

119867119889119867 +

119898

119867119889119881 (21)

For the ending point 119910 = 119910119890119898 = 119871 then

119889119910119890= minus

119910119890

119867119889119867 +

119871

119867119889119881 (22)

where 119871 is the span length of the cable segment

S1S2

A

T1 T2T3

T4

T5H

VH998400

V998400

L

S3

m

C1

C2

C3

D2D1

Figure 5 The calculation scheme of the E-M method

For the middle point 119910119898

= 119891 then

119889119910119898

= minus119891

119867119889119867 +

119898

119867119889119881 (23)

where 119891 is the sag of the main cable at the middle pointBased on formulas (22)-(23) considering the elevation

error of the ending andmiddle points119867 and119881 are modified

[

119889119910119890

119889119910119898

] =

[[[

[

minus119910119890

119867

119871

119867

minus119891

119867

119898

119867

]]]

]

[

119889119867

119889119881] (24)

Then

[

119889119867

119889119881] =

1198672

119891119871 minus 119910119890119898

[[[

[

119898

119867minus119871

119867

119891

119867minus119910119890

119867

]]]

]

[

119889119910119890

119889119910119898

] (25)

For a three-span SB elevations of two pylons are equaland the middle point is at the mid-span of the intermediatespan where 119910

119890= 0 119910

119898= 119891119898 = 1198712 and then formula (25)

can be simplified as

[

119889119867

119889119881] =

[[[

[

119867

2119891minus119867

119891

119867

1198710

]]]

]

[

119889119910119890

119889119910119898

] (26)

where 119889119910119890= 0 minus 119910

119890 119889119910119898

= 119891 minus 119910119898

119891 (1198670 1198810 1198781) = 0

119891 (119867119894minus1

119881119894minus1

119878119894)

= minus119867119894minus1

119878119894

119864119860minus

119867119894minus1

119902

sdot ln(119881119894minus1

+ radic119867119894minus1

2+ 119881119894minus1

2)

minus ln(119881119894minus1

minus 119902119878119894+ radic119867

119894minus1

2+ (119881119894minus1

minus 119902119878119894)2

) minus 119897119894= 0

119891 (119867119899minus1

119881119899minus1

119878119899) = 0

(27)

Mathematical Problems in Engineering 7

Assuming the initial valuesof H and V

Substitute them into the maincable shape equation and

calculate the shape of each cablesegment

Check whether the elevation errorof ending point and middle point

elevation meets the precisionrequirements

Calculation ends

ModifyH V by

E-Mmethod

Yes

No

Figure 6 The calculation step of the E-M method

119881119897 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119898= 0

119881119871 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119890= 0

(28)

E-M iteration method is to solve the nonlinear equations(27) and (28) to obtain the numerical solution For formula(27) Newton method is used to calculate the elevation of thetarget point then substitute the elevation error into (28) andmodify119867 and119881 finally substitute themodified119867 and119881 into(27) to do the new round of iterationThe steps are as follows

(i) Assume the values of 119867 and 119881 and substitute theminto the main cable shape equation and calculatethe shape of each cable segment through Newtoniteration method

(ii) Check whether the elevation error of ending pointand middle point elevation meet the precisionrequirements (|119889119910

119890| lt 119862 and |119889119910

119898| lt 119863) if so the

calculation ends otherwise execute step (3)

(iii) Modify 119867 119881 through E-M method and then substi-tute themodified119867 and119881 into step (1)The flow chartis shown in Figure 6

43 The Main Cable Shape-Finding in the Side Span For themain cable in the side span of SB the shape-finding can alsouse formulas (17) Unlike in the middle span 119867 is knownand equals the horizontal component of the main reactionin the middle span and in the iteration process 119867 remainsconstant and only modifies 119881 as shown in Figure 7 Manyscholars have done research on this method then there is nomore discussion here

V998400

H998400

H

V

T1

T2

T3

Anchor point

Main cable

Tower support

H998400 = H = horizontal force of themain cable in the middle span

Figure 7 The calculation scheme of the main cable shape-findingin the side span

5 Part III of Calculation ModelThe Deformation Compatibility Condition

In Part III SF and MCEF are substituted into the stiffeningequation and check whether the SF and MCEF satisfies theSSG if it does the calculation stops and the last calculationis the expected result otherwise modify the SF by matrix 119861

(Section 32) and repeat the iteration process

6 The Numerical Calculation Process ofthe Proposed Method

According to the theories introduced before an analyticalcalculation method considering the combined effect of themain cable-suspender-stiffening girder is programmedbyVBprograming language The main steps are as follows

(i) Under the condition of the known SSG (the shapeof the stiffening girder) calculate the suspender force1198790(119894) according to the known girder segments119873 seg-

ment length 119871(119894) elastic modulus 119864 cross-sectionalmoment of inertia 119868 uniform load 119902 design shapeequation of the stiffening girder 119910

0119894 and the error

precision requirement 119890(ii) With the suspender force119879

0(119894) solve out the horizon-

tal force1198670of the stiffening girder end

(iii) Substitute 1198670and 119879

0(119894) into the equation of the

stiffening girder then the influence matrix 119861 119889120575 and119889119879 can be obtained

(iv) Check whether |119889120575|meets the precision requirementsor not if |119889120575| lt 119890 the calculation ends otherwisemodify 119879

0(119894) (1198790(119894) = 119879

0(119894) + 119889119879) until |119889120575| meet the

precision requirements(v) Substitute the new 119879

0(119894) into step (ii) repeat steps (ii)

to (iv)

The flow chart is shown in Figure 8

7 Example Analysis

71 Example of the Proposed Method Figure 9 is the generallayout of a self-anchored suspension bridge with the span

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Such models based on finite-element theory may havethousands of degrees of freedom So there is a need forsimpler models to better understand the structural behaviorin a way not offered by finite-element analysis Such modelsare useful for the preliminary design and independent checksof more complex models And many scholars have beenexploring such models

Tan [9] based on the analysis of finite displacementtheory and analytical iteration method proposed a determi-nation method of the reasonable finished bridge state of SSBconsidering themain cable alignment the force of suspenderand the stiffening girderHan et al [10] studied themain cableshape-finding method of SSB with spatial cables Li et al [11]based on the reasonable internal force requirements of SSBmain cable and the suspender put forward a determinationmethod of the reasonable suspender force of SSB and thismethod considered the effect of stiffening girder alignmentthrough finite-element method but it is relatively compli-cated because the finite-element analysis procedure must beiterated manually

In summary so far the analysis method of SSB mainlyincludes (1) linear-elastic theory and deflection theory (2)finite displacement theory (3) finite-element theory (4)combined method based on finite-element theory and thenumerical analytical method Methods (1) and (2) are effec-tive methods in the early stage but they have many assump-tions and are only suitable for some small span bridgesThesetwo methods are barely used now Method (3) reduces theassumptions and the calculation result meets the actual circsbetter However the method needs a model with thousandsof degrees of freedom and the model is complex The fixedboundary also needs to be changed manually in the secondstep of Method (3) and the calculation time is very long InMethod (4) themain cable is considered through the numer-ical analytical method and the stiffening girder is consideredthrough finite-element method The method is a more accu-rate calculation for main cable but the analysis proceduremust be iterated manually which is time-consuming

So this paper proposes an analytical calculation methodconsidering the combined effect of themain cable-suspender-stiffening girder Compared to the above mentioned meth-ods in the method the main cable and the stiffeninggirder are all considered through the numerical analyticalmethod then the calculation procedure has no need to iteratemanually which can save calculation time

The method also can quickly and effectively find thereasonable finished bridge state of SSB and help designers tobetter understand the structural behavior in amanner offeredby a simpler and accurate model

The paper is organized as follows Section 2 brieflypresents the principle of the proposed method Sections 3ndash5 are dedicated to the main cable analysis the stiffeninggirder analysis and the deformation compatibility conditionof the calculation model respectively Section 6 introducesthe numerical calculation process of the proposed methodExamples are provided in Section 7 to illustrate the appli-cation of the method Finally Section 8 summarizes thefindings of this paper and presents some conclusions

2 The Principle of the Proposed Method

Self-anchored suspension bridge is a high-order staticallyindeterminate structure thus the direct calculation hascertain difficulties In the proposed method one model isestablished which includes three parts namely the maincable the stiffening girder and the deformation compatibilitycondition The main cable and the stiffening girder areanalyzed independently and then they are coupled by thedeformation compatibility condition The discrete graph ofSSB is shown in Figure 1

As to Part I of the calculation model the stiffening girderis assumed as a simply supported girder along the wholebridge The shape of the stiffening girder (SSG) needs to begiven firstly and then calculate the suspender force (SF) Thescheme of calculation principle is shown in Figure 2

The support of pylon to the stiffening girder is replaced asconcentrate forcesThe counterweights set at the pylon-girderjoint and girder end can also be assumed as concentrate forcesat supports and they have slight influence upon the internalforce of the stiffening girder The MCEF (the main cableend force) eccentrically acts upon the stiffening girder andthrough axis shift formula it can be divided into a horizontalforce acting on the neutral axis of stiffening girder and anadditional bending moment1198721 as shown in Figure 1

As to Part II of the calculation model the main cable SFcan be given from Part I and calculate the MCEF satisfyingthe rise to span ratio and other requirements

On the basis of deflection theory and the principle ofminimum strain energy of the stiffening girder the SF andMCEF given from Part I and II are substituted into PartIII of the calculation model the deformation compatibilitycondition and finally checkwhether the SF andMCEF satisfythe criterion (the displacements at hanging point equal zero)if they do the calculation stops otherwise modify the SF andrepeat the iteration

When the error of SSG meets the convergence conditionof precision requirement the iteration stops and the lastcalculation is the expected result

For SSB with overhanging span the bearing reaction 119877

and the self-weight of anchor span 119866 can be equivalent toadditionalmoment1198722 and concentrated force119866-119877 acting onthe end of anchor section as shown in Figure 1The influenceof shrinkage and creep of concrete pylon to the alignment canbe considered by a reduction of the pylon height And moreinformation could be found in relative references

3 Part I of Calculation ModelThe Stiffening Girder Analysis

Under the condition of the known SSG (the shape of thestiffening girder) Part I is used to calculate SF (the suspenderforce) The calculation procedure needs to iterate and theinfluential matrix 119861 of SSG to SF should be obtained

31 The Control Principle of SSG The reasonable finishedbridge state of SSB is measured by the stress state of thestiffening girders under dead load Furthermore reasonable

Mathematical Problems in Engineering 3

Center line ofmain cable

Neutral axis ofstiffening girder

Center lineof support

Center line ofmain cable

T1 T3

Ti

Tnminus3 Tnminus1T2 Tnminus2

V

H

V

H

V

H

V

H

F0 Fn

M1 + M2 M1 + M2T1T2

T3 Ti Tnminus3 Tnminus2 Tnminus1Mj1 Mj1

Mj2 Mj2Mj3 Mj3 Mji Mji Mjiminus3 Mjiminus3 Mjiminus2 Mjiminus2 Mjiminus1

Mjiminus1

L1L2

L3 LiLnminus3 Lnminus2 Lnminus1 Ln

Main cableMain tower Main tower

Stiffening girder

H

Anchorage beam segment Standard beam segment

Neutral axis ofstiffening girder

Center lineof support

R

Overhanging span

Center lineof support

G

M2 G minus R

Figure 1 The discrete graph of SSB

T1T2 T3 T4 T5V1

H1

V9984001

H9984001

Suspender force

Shape of the stiffening girder

Part I the stiffening girder

T1 T2 T3T4 T5

V2

H2

V9984002

H9984002

Suspender force

Part II the main cable

T1T2 T3 T4 T5V2

H2

V9984002

H9984002match shape of the stiffening girder

Part III the deformation compatibility condition

Use the shape errorto correct suspender

force Ti

Meet the shape requirements ofthe stiffening girder calculate

the suspender force Ti

Meet the shape requirements ofthe main cable calculate the

Reaction forces H2 V2 and suspender force Ti

reaction forces H2 and V2

Figure 2 The scheme of the calculation principle

stiffening stress state can be ensured through the reasonablesuspender force which can be solved by rigid supportedcontinuous beam method zero displacement method orminimumbending energymethodThe solving process usingminimum bending energy method is described as follows

The structural cost can be measured by bending strainenergy thus the smaller the bending strain energy is the lessthe materials the structure costs The bending strain energyof the stiffening girder can be obtained through

119880 =1

2int119904

1198722

119864119868119889119904 (1)

According to the principle of minimum bending energy119880 (bending energy) should meet the following relationship

120597119880

120597119883119894

= 0 (2)

where119883119894means the suspender force

Actually the physical meanings of rigid supported con-tinuous beam method zero displacement method and min-imum bending energy method are the same as describedas (2) that is the vertical displacement of the suspendingpoint is zero under the joint action of suspender force thehorizontal component of the main cable and the dead load

32 Impact Analysis of SSG to SF To complete nonlinearanalysis of the stiffening girder the following two assump-tions are made

(i) Neglect the influences of the shear deformation on thealignment of the stiffening girder

(ii) Neglect the influences of the axial deformation on thealignment of the stiffening girder

Hence the alignment of the stiffening girder is onlyrelated to the bending moment

As shown in Figure 3 119902means the uniformly distributedload of stiffening girder self-weight 119871

119894means the length of

4 Mathematical Problems in Engineering

y0i

yi

BH

Mjiminus1A

Y

X

H

Fiminus1

Mji

Fi

Δi

Li

Ti

q

Figure 3 The calculation diagram of the girder segment

girder segment 119867 means the horizontal end reaction of themain cable 119872

119895119894means the bending moment at the hanging

point 119865119894means the shearing force of girder 119879

119894means the

suspender force and 119910119894means the alignment of the girder

segment under the joint action of the suspender tension forcethe horizontal component of themain cable and the constantload119910

0119894is the initial camber of the stiffening girderwhich can

meet the precision requirements using designed alignmentin calculation The starting point of the girder segment isassumed as the origin point

According to (2) the stress state of the girder end underuniformly distributed load 119902 meets equilibrium equationwhen ensuring that no vertical displacement occurs in points119860 and 119861

119865119894= 119865119894minus1

+ 119879119894minus 119902119871119894

119872119895119894=

119902119871119894

2

2minus 119865119894minus1

119871119894+ 119872119895119894minus1

+ 119867Δ119894+ 1198721015840

119872119894=

1199021199092

2minus 119865119894minus1

119909 + 119872119895119894minus1

+ 119867119910

Δ119894= 119910119894minus 119910119894minus1

(3)

Neglecting the shearing deformation the equation can bedescribed as follows

119872119894

119864119868= minus(

11991010158401015840

119894

(1 + 11991010158402

119894)32

minus11991010158401015840

0119894

(1 + 11991010158402

0119894)32

) (4)

From (3) deflection differential equilibrium equation canbe obtained

11991010158401015840

119894+

119867

119864119868119910119894=

minus119902

21198641198681199092+

119865119894minus1

119864119868119909 minus

119872119895119894minus1

119864119868+ 11991010158401015840

0119894 (5)

In the equation assume

1198602=

119867

119864119868 119861 =

minus119902

2119864119868

119862119894=

119865119894minus1

119864119868 119863

119894= minus

119872119895119894minus1

119864119868

(6)

Because 1199100119894is a continuous function we can spread out

power series at 119909 = 0 to conveniently solve the differentialequation

1199100119894=

infin

sum

119898=0

119866119894

119898119909119898 (7)

where 119866119894119898

= 119910(119898)

0119894(0)119898

In practical calculation when 119898 = 4 the result can meetthe precision requirement

1199100119894=

4

sum

119898=0

119866119894

119898119909119898

= 119866119894

0+ 119866119894

1119909 + 119866

119894

21199092+ 119866119894

31199093+ 119866119894

41199094

(8)

11991010158401015840

0119894= 2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092 (9)

Substitute (9) into the differential equation (4)

119910119894= 1198641119894sin (119860119909) + 119864

2119894cos (119860119909)

+

(1198611199092+ 119862119894119909 + 119863

119894)1198602minus 2119861

1198604

+

(2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092)1198602minus 24119866

119894

4

1198604

(10)

Equation (10) is the deflection equation of the girder seg-ment considering geometrical nonlinearity With the knownquantities 119860 119861 119862

119894 119863119894 1198661198942 1198661198943 and 119866

119894

4and combining with

the controlling condition of the minimum strain energy ofthe stiffening girder 119910

119894(0) = 0 119910

119894(119871119894) = Δ

119894 the expressions

of 1198641119894 1198642119894are given by

1198641119894=

Δ119894

sin (119860119871119894)minus 1198642119894ctan (119860119871

119894)

minus

(119861119871119894

2+ 119862119894119871119894+ 119863119894)1198602minus 2119861

1198604 sin (119860119871119894)

minus

(2119866119894

2+ 6119866119894

3119871119894+ 12119866

119894

4119871119894

2)1198602minus 24119866

119894

4

1198604 sin (119860119871119894)

1198642119894=

2119861

1198604minus

119863119894

1198602+

24119866119894

4

1198604minus

2119866119894

2

1198602

(11)

For each girder segment its deflection equation thatmeets theminimum stain energy theory can be obtainedwithknown119879

119894and119867 Besides the deflection equation should also

satisfy the compatibility condition of the stiffening girderthus

120575119894= 1199101015840

119894+1(0) minus 119910

1015840

119894(119871119894) = 0 (12)

Then differentiate (12)

119889120575119894= 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894) = 0 (13)

Mathematical Problems in Engineering 5

In (13) 119879119894is unknown and differentiate 1199101015840

119894

1198891199101015840

119894+1(0) =

sin (119860119871119894+1

) minus 119860119871119894+1

1198602 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

119889119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

119889119879119895

1198891199101015840

119894(119871119894) = (

1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

119889119879119895

+1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

119889120575 = 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894)

=cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

+ (sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)lowast

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

minus(1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

)119889119879119895

(14)

Transform (14) into matrix form

119889120575 = 119870119889119879

119889119879 = 119870minus1119889120575

(15)

where

119861 = 119870minus1

119889119879 =

[[[[

[

1198891198791

119889119879119899minus1

]]]]

]

119870 =

[[[[

[

11987011

sdot sdot sdot 1198701119899minus1

d

119870119899minus11

sdot sdot sdot 119870119899minus1119899minus1

]]]]

]

119889120575 =

[[[[

[

1205751

120575119899minus1

]]]]

]

119870119894119895=sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)

120597119862119894+1

120597119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

120597119863119894+1

120597119879119895

minus (1

1198602minusctan (119860119871

119894)

119860119871119894)

120597119862119894

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

120597119863119894

120597119879119895

(16)

H

VT

l

O X

h

Y

H998400

V998400T998400

Figure 4 The calculation scheme of the main cable segment

Matrix 119861 is the influential matrix of SSG to SF whichrepresents the relationship between 119889120575 (the first derivateincrement of the stiffening girder shape) and 119889119879 (the changeof suspender force) 119889119879 can be solved out through knowngirder shape and then modify 119879(119894) to adjust the suspenderforce

4 Part II of Calculation ModelThe Main Cable Analysis

Under the condition of the known SF calculated from PartI Part II is used to calculate MCEF which satisfies the shaperequirements of main cable by shape-finding

41 The Segmental Catenary Theory of Main Cable For theSSB the segmental catenary method assumes that the align-ment of the main cable is a catenary which is more approxi-mate with the actual situation In the main cable calculationthe segmental catenary method is adopted [12ndash19]

According to the differential equilibrium equations geo-metrical equations and the physical equations of cablesegment the relation between the shape and the internal forceof cable segment is obtainedThe calculation scheme is shownin Figure 4

119897 = minus1198671198780

119864119860minus

119867

119902ln (119881 + radic1198672 + 1198812)

minus ln(119881 minus 1199021198780+ radic1198672 + (119881 minus 119902119878

0)2

)

ℎ =1199021198780

2minus 2119881119878

0

2119864119860

minus1

119902[radic1198672 + 1198812 minusradic1198672 + (119881 minus 119902119878

0)2

]

(17)

In formulas (17) 119902 represents the self-weight of theunstressed main cable 119864 represents the elastic modulus119860 represents the cross-sectional area 119897 represents the spanlength of the cable segment ℎ represents the elevationdifference of two ends and 119878

0represents the unstressed

length of cable segment 119867 and 119881 represent the horizontaland vertical component of cable segment force in the left end

6 Mathematical Problems in Engineering

The calculation process of the main cable alignment is asfollows assume the value of119867 and119881 and calculate 119878

0 119897 and ℎ

of each cable segment according to the calculated suspenderforce and then check whether the result meets the alignmentrequirements or not If it does calculation stops otherwisemodify the value of 119867 and 119881 and repeat the above steps tillthe result meets the shape requirements

42 Improved E-M IterationMethod for theMain Cable Shape-Finding in the Middle Span Formulas (17) are nonlinearequations to solve the equations the value of119867 and119881 shouldbe assumed to get 119878

0 and then check whether ℎ of the target

point meets the requirements or not The solving process isan iterative process

The numerical iterative methods of the main cable in themiddle span of SBmay not converge in some casesThemainreason is that in the iteration process the elevation require-ments of end points and intermediate points are consideredindependently so that the mutual influences between themare ignored The proposed E-M iteration method consideredthe elevation requirements of both the ending points andintermediate points besides when modifying the horizontalforce 119867 and vertical force 119881 the influences of ending pointselevation error 119889119910

119890and intermediate points elevation error

119889119910119898are consideredAs shown in Figure 5 for any Point 119860 on the cable

segment according to the equilibrium equation of moment

119881119898 minus

119899

sum

119894=1

(119879119894119863119894+ 119878119894119908119862119894) minus 119867119910 = 0 (18)

where119867 and 119881 represent the horizontal and vertical compo-nents of the cable segment force in the left end 119879

119894represents

the suspender force 119878119894represents the unstressed length of

cable segment 119862119894represents the distance from the gravity

center of cable segment to Point119860119863119894represents the distance

from hanging point to Point 119860 119910 and 119898 represent thehorizontal and vertical distances from the left ending point ofthe main cable to Point 119860 respectively 119910

119890and 119910

119898represent

the elevation of the ending point and the middle pointrespectively and 119908 represents the self-weight of the maincable

Formula (18) can be transformed into

119910 =(119881119898 minus sum

119899

119894=1(119879119894119863119894+ 119878119894119908119862119894))

119867 (19)

where 119910 is a function of 119867 and 119881 and the differential of 119910 isgiven by

119889119910 =sum119899

119894=1(119879119894119863119894+ 119878119894119908119862119894) minus 119881119898

1198672119889119867 +

119898

119867119889119881 (20)

Substitute formula (19) into (20)

119889119910 = minus119910

119867119889119867 +

119898

119867119889119881 (21)

For the ending point 119910 = 119910119890119898 = 119871 then

119889119910119890= minus

119910119890

119867119889119867 +

119871

119867119889119881 (22)

where 119871 is the span length of the cable segment

S1S2

A

T1 T2T3

T4

T5H

VH998400

V998400

L

S3

m

C1

C2

C3

D2D1

Figure 5 The calculation scheme of the E-M method

For the middle point 119910119898

= 119891 then

119889119910119898

= minus119891

119867119889119867 +

119898

119867119889119881 (23)

where 119891 is the sag of the main cable at the middle pointBased on formulas (22)-(23) considering the elevation

error of the ending andmiddle points119867 and119881 are modified

[

119889119910119890

119889119910119898

] =

[[[

[

minus119910119890

119867

119871

119867

minus119891

119867

119898

119867

]]]

]

[

119889119867

119889119881] (24)

Then

[

119889119867

119889119881] =

1198672

119891119871 minus 119910119890119898

[[[

[

119898

119867minus119871

119867

119891

119867minus119910119890

119867

]]]

]

[

119889119910119890

119889119910119898

] (25)

For a three-span SB elevations of two pylons are equaland the middle point is at the mid-span of the intermediatespan where 119910

119890= 0 119910

119898= 119891119898 = 1198712 and then formula (25)

can be simplified as

[

119889119867

119889119881] =

[[[

[

119867

2119891minus119867

119891

119867

1198710

]]]

]

[

119889119910119890

119889119910119898

] (26)

where 119889119910119890= 0 minus 119910

119890 119889119910119898

= 119891 minus 119910119898

119891 (1198670 1198810 1198781) = 0

119891 (119867119894minus1

119881119894minus1

119878119894)

= minus119867119894minus1

119878119894

119864119860minus

119867119894minus1

119902

sdot ln(119881119894minus1

+ radic119867119894minus1

2+ 119881119894minus1

2)

minus ln(119881119894minus1

minus 119902119878119894+ radic119867

119894minus1

2+ (119881119894minus1

minus 119902119878119894)2

) minus 119897119894= 0

119891 (119867119899minus1

119881119899minus1

119878119899) = 0

(27)

Mathematical Problems in Engineering 7

Assuming the initial valuesof H and V

Substitute them into the maincable shape equation and

calculate the shape of each cablesegment

Check whether the elevation errorof ending point and middle point

elevation meets the precisionrequirements

Calculation ends

ModifyH V by

E-Mmethod

Yes

No

Figure 6 The calculation step of the E-M method

119881119897 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119898= 0

119881119871 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119890= 0

(28)

E-M iteration method is to solve the nonlinear equations(27) and (28) to obtain the numerical solution For formula(27) Newton method is used to calculate the elevation of thetarget point then substitute the elevation error into (28) andmodify119867 and119881 finally substitute themodified119867 and119881 into(27) to do the new round of iterationThe steps are as follows

(i) Assume the values of 119867 and 119881 and substitute theminto the main cable shape equation and calculatethe shape of each cable segment through Newtoniteration method

(ii) Check whether the elevation error of ending pointand middle point elevation meet the precisionrequirements (|119889119910

119890| lt 119862 and |119889119910

119898| lt 119863) if so the

calculation ends otherwise execute step (3)

(iii) Modify 119867 119881 through E-M method and then substi-tute themodified119867 and119881 into step (1)The flow chartis shown in Figure 6

43 The Main Cable Shape-Finding in the Side Span For themain cable in the side span of SB the shape-finding can alsouse formulas (17) Unlike in the middle span 119867 is knownand equals the horizontal component of the main reactionin the middle span and in the iteration process 119867 remainsconstant and only modifies 119881 as shown in Figure 7 Manyscholars have done research on this method then there is nomore discussion here

V998400

H998400

H

V

T1

T2

T3

Anchor point

Main cable

Tower support

H998400 = H = horizontal force of themain cable in the middle span

Figure 7 The calculation scheme of the main cable shape-findingin the side span

5 Part III of Calculation ModelThe Deformation Compatibility Condition

In Part III SF and MCEF are substituted into the stiffeningequation and check whether the SF and MCEF satisfies theSSG if it does the calculation stops and the last calculationis the expected result otherwise modify the SF by matrix 119861

(Section 32) and repeat the iteration process

6 The Numerical Calculation Process ofthe Proposed Method

According to the theories introduced before an analyticalcalculation method considering the combined effect of themain cable-suspender-stiffening girder is programmedbyVBprograming language The main steps are as follows

(i) Under the condition of the known SSG (the shapeof the stiffening girder) calculate the suspender force1198790(119894) according to the known girder segments119873 seg-

ment length 119871(119894) elastic modulus 119864 cross-sectionalmoment of inertia 119868 uniform load 119902 design shapeequation of the stiffening girder 119910

0119894 and the error

precision requirement 119890(ii) With the suspender force119879

0(119894) solve out the horizon-

tal force1198670of the stiffening girder end

(iii) Substitute 1198670and 119879

0(119894) into the equation of the

stiffening girder then the influence matrix 119861 119889120575 and119889119879 can be obtained

(iv) Check whether |119889120575|meets the precision requirementsor not if |119889120575| lt 119890 the calculation ends otherwisemodify 119879

0(119894) (1198790(119894) = 119879

0(119894) + 119889119879) until |119889120575| meet the

precision requirements(v) Substitute the new 119879

0(119894) into step (ii) repeat steps (ii)

to (iv)

The flow chart is shown in Figure 8

7 Example Analysis

71 Example of the Proposed Method Figure 9 is the generallayout of a self-anchored suspension bridge with the span

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Center line ofmain cable

Neutral axis ofstiffening girder

Center lineof support

Center line ofmain cable

T1 T3

Ti

Tnminus3 Tnminus1T2 Tnminus2

V

H

V

H

V

H

V

H

F0 Fn

M1 + M2 M1 + M2T1T2

T3 Ti Tnminus3 Tnminus2 Tnminus1Mj1 Mj1

Mj2 Mj2Mj3 Mj3 Mji Mji Mjiminus3 Mjiminus3 Mjiminus2 Mjiminus2 Mjiminus1

Mjiminus1

L1L2

L3 LiLnminus3 Lnminus2 Lnminus1 Ln

Main cableMain tower Main tower

Stiffening girder

H

Anchorage beam segment Standard beam segment

Neutral axis ofstiffening girder

Center lineof support

R

Overhanging span

Center lineof support

G

M2 G minus R

Figure 1 The discrete graph of SSB

T1T2 T3 T4 T5V1

H1

V9984001

H9984001

Suspender force

Shape of the stiffening girder

Part I the stiffening girder

T1 T2 T3T4 T5

V2

H2

V9984002

H9984002

Suspender force

Part II the main cable

T1T2 T3 T4 T5V2

H2

V9984002

H9984002match shape of the stiffening girder

Part III the deformation compatibility condition

Use the shape errorto correct suspender

force Ti

Meet the shape requirements ofthe stiffening girder calculate

the suspender force Ti

Meet the shape requirements ofthe main cable calculate the

Reaction forces H2 V2 and suspender force Ti

reaction forces H2 and V2

Figure 2 The scheme of the calculation principle

stiffening stress state can be ensured through the reasonablesuspender force which can be solved by rigid supportedcontinuous beam method zero displacement method orminimumbending energymethodThe solving process usingminimum bending energy method is described as follows

The structural cost can be measured by bending strainenergy thus the smaller the bending strain energy is the lessthe materials the structure costs The bending strain energyof the stiffening girder can be obtained through

119880 =1

2int119904

1198722

119864119868119889119904 (1)

According to the principle of minimum bending energy119880 (bending energy) should meet the following relationship

120597119880

120597119883119894

= 0 (2)

where119883119894means the suspender force

Actually the physical meanings of rigid supported con-tinuous beam method zero displacement method and min-imum bending energy method are the same as describedas (2) that is the vertical displacement of the suspendingpoint is zero under the joint action of suspender force thehorizontal component of the main cable and the dead load

32 Impact Analysis of SSG to SF To complete nonlinearanalysis of the stiffening girder the following two assump-tions are made

(i) Neglect the influences of the shear deformation on thealignment of the stiffening girder

(ii) Neglect the influences of the axial deformation on thealignment of the stiffening girder

Hence the alignment of the stiffening girder is onlyrelated to the bending moment

As shown in Figure 3 119902means the uniformly distributedload of stiffening girder self-weight 119871

119894means the length of

4 Mathematical Problems in Engineering

y0i

yi

BH

Mjiminus1A

Y

X

H

Fiminus1

Mji

Fi

Δi

Li

Ti

q

Figure 3 The calculation diagram of the girder segment

girder segment 119867 means the horizontal end reaction of themain cable 119872

119895119894means the bending moment at the hanging

point 119865119894means the shearing force of girder 119879

119894means the

suspender force and 119910119894means the alignment of the girder

segment under the joint action of the suspender tension forcethe horizontal component of themain cable and the constantload119910

0119894is the initial camber of the stiffening girderwhich can

meet the precision requirements using designed alignmentin calculation The starting point of the girder segment isassumed as the origin point

According to (2) the stress state of the girder end underuniformly distributed load 119902 meets equilibrium equationwhen ensuring that no vertical displacement occurs in points119860 and 119861

119865119894= 119865119894minus1

+ 119879119894minus 119902119871119894

119872119895119894=

119902119871119894

2

2minus 119865119894minus1

119871119894+ 119872119895119894minus1

+ 119867Δ119894+ 1198721015840

119872119894=

1199021199092

2minus 119865119894minus1

119909 + 119872119895119894minus1

+ 119867119910

Δ119894= 119910119894minus 119910119894minus1

(3)

Neglecting the shearing deformation the equation can bedescribed as follows

119872119894

119864119868= minus(

11991010158401015840

119894

(1 + 11991010158402

119894)32

minus11991010158401015840

0119894

(1 + 11991010158402

0119894)32

) (4)

From (3) deflection differential equilibrium equation canbe obtained

11991010158401015840

119894+

119867

119864119868119910119894=

minus119902

21198641198681199092+

119865119894minus1

119864119868119909 minus

119872119895119894minus1

119864119868+ 11991010158401015840

0119894 (5)

In the equation assume

1198602=

119867

119864119868 119861 =

minus119902

2119864119868

119862119894=

119865119894minus1

119864119868 119863

119894= minus

119872119895119894minus1

119864119868

(6)

Because 1199100119894is a continuous function we can spread out

power series at 119909 = 0 to conveniently solve the differentialequation

1199100119894=

infin

sum

119898=0

119866119894

119898119909119898 (7)

where 119866119894119898

= 119910(119898)

0119894(0)119898

In practical calculation when 119898 = 4 the result can meetthe precision requirement

1199100119894=

4

sum

119898=0

119866119894

119898119909119898

= 119866119894

0+ 119866119894

1119909 + 119866

119894

21199092+ 119866119894

31199093+ 119866119894

41199094

(8)

11991010158401015840

0119894= 2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092 (9)

Substitute (9) into the differential equation (4)

119910119894= 1198641119894sin (119860119909) + 119864

2119894cos (119860119909)

+

(1198611199092+ 119862119894119909 + 119863

119894)1198602minus 2119861

1198604

+

(2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092)1198602minus 24119866

119894

4

1198604

(10)

Equation (10) is the deflection equation of the girder seg-ment considering geometrical nonlinearity With the knownquantities 119860 119861 119862

119894 119863119894 1198661198942 1198661198943 and 119866

119894

4and combining with

the controlling condition of the minimum strain energy ofthe stiffening girder 119910

119894(0) = 0 119910

119894(119871119894) = Δ

119894 the expressions

of 1198641119894 1198642119894are given by

1198641119894=

Δ119894

sin (119860119871119894)minus 1198642119894ctan (119860119871

119894)

minus

(119861119871119894

2+ 119862119894119871119894+ 119863119894)1198602minus 2119861

1198604 sin (119860119871119894)

minus

(2119866119894

2+ 6119866119894

3119871119894+ 12119866

119894

4119871119894

2)1198602minus 24119866

119894

4

1198604 sin (119860119871119894)

1198642119894=

2119861

1198604minus

119863119894

1198602+

24119866119894

4

1198604minus

2119866119894

2

1198602

(11)

For each girder segment its deflection equation thatmeets theminimum stain energy theory can be obtainedwithknown119879

119894and119867 Besides the deflection equation should also

satisfy the compatibility condition of the stiffening girderthus

120575119894= 1199101015840

119894+1(0) minus 119910

1015840

119894(119871119894) = 0 (12)

Then differentiate (12)

119889120575119894= 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894) = 0 (13)

Mathematical Problems in Engineering 5

In (13) 119879119894is unknown and differentiate 1199101015840

119894

1198891199101015840

119894+1(0) =

sin (119860119871119894+1

) minus 119860119871119894+1

1198602 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

119889119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

119889119879119895

1198891199101015840

119894(119871119894) = (

1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

119889119879119895

+1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

119889120575 = 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894)

=cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

+ (sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)lowast

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

minus(1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

)119889119879119895

(14)

Transform (14) into matrix form

119889120575 = 119870119889119879

119889119879 = 119870minus1119889120575

(15)

where

119861 = 119870minus1

119889119879 =

[[[[

[

1198891198791

119889119879119899minus1

]]]]

]

119870 =

[[[[

[

11987011

sdot sdot sdot 1198701119899minus1

d

119870119899minus11

sdot sdot sdot 119870119899minus1119899minus1

]]]]

]

119889120575 =

[[[[

[

1205751

120575119899minus1

]]]]

]

119870119894119895=sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)

120597119862119894+1

120597119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

120597119863119894+1

120597119879119895

minus (1

1198602minusctan (119860119871

119894)

119860119871119894)

120597119862119894

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

120597119863119894

120597119879119895

(16)

H

VT

l

O X

h

Y

H998400

V998400T998400

Figure 4 The calculation scheme of the main cable segment

Matrix 119861 is the influential matrix of SSG to SF whichrepresents the relationship between 119889120575 (the first derivateincrement of the stiffening girder shape) and 119889119879 (the changeof suspender force) 119889119879 can be solved out through knowngirder shape and then modify 119879(119894) to adjust the suspenderforce

4 Part II of Calculation ModelThe Main Cable Analysis

Under the condition of the known SF calculated from PartI Part II is used to calculate MCEF which satisfies the shaperequirements of main cable by shape-finding

41 The Segmental Catenary Theory of Main Cable For theSSB the segmental catenary method assumes that the align-ment of the main cable is a catenary which is more approxi-mate with the actual situation In the main cable calculationthe segmental catenary method is adopted [12ndash19]

According to the differential equilibrium equations geo-metrical equations and the physical equations of cablesegment the relation between the shape and the internal forceof cable segment is obtainedThe calculation scheme is shownin Figure 4

119897 = minus1198671198780

119864119860minus

119867

119902ln (119881 + radic1198672 + 1198812)

minus ln(119881 minus 1199021198780+ radic1198672 + (119881 minus 119902119878

0)2

)

ℎ =1199021198780

2minus 2119881119878

0

2119864119860

minus1

119902[radic1198672 + 1198812 minusradic1198672 + (119881 minus 119902119878

0)2

]

(17)

In formulas (17) 119902 represents the self-weight of theunstressed main cable 119864 represents the elastic modulus119860 represents the cross-sectional area 119897 represents the spanlength of the cable segment ℎ represents the elevationdifference of two ends and 119878

0represents the unstressed

length of cable segment 119867 and 119881 represent the horizontaland vertical component of cable segment force in the left end

6 Mathematical Problems in Engineering

The calculation process of the main cable alignment is asfollows assume the value of119867 and119881 and calculate 119878

0 119897 and ℎ

of each cable segment according to the calculated suspenderforce and then check whether the result meets the alignmentrequirements or not If it does calculation stops otherwisemodify the value of 119867 and 119881 and repeat the above steps tillthe result meets the shape requirements

42 Improved E-M IterationMethod for theMain Cable Shape-Finding in the Middle Span Formulas (17) are nonlinearequations to solve the equations the value of119867 and119881 shouldbe assumed to get 119878

0 and then check whether ℎ of the target

point meets the requirements or not The solving process isan iterative process

The numerical iterative methods of the main cable in themiddle span of SBmay not converge in some casesThemainreason is that in the iteration process the elevation require-ments of end points and intermediate points are consideredindependently so that the mutual influences between themare ignored The proposed E-M iteration method consideredthe elevation requirements of both the ending points andintermediate points besides when modifying the horizontalforce 119867 and vertical force 119881 the influences of ending pointselevation error 119889119910

119890and intermediate points elevation error

119889119910119898are consideredAs shown in Figure 5 for any Point 119860 on the cable

segment according to the equilibrium equation of moment

119881119898 minus

119899

sum

119894=1

(119879119894119863119894+ 119878119894119908119862119894) minus 119867119910 = 0 (18)

where119867 and 119881 represent the horizontal and vertical compo-nents of the cable segment force in the left end 119879

119894represents

the suspender force 119878119894represents the unstressed length of

cable segment 119862119894represents the distance from the gravity

center of cable segment to Point119860119863119894represents the distance

from hanging point to Point 119860 119910 and 119898 represent thehorizontal and vertical distances from the left ending point ofthe main cable to Point 119860 respectively 119910

119890and 119910

119898represent

the elevation of the ending point and the middle pointrespectively and 119908 represents the self-weight of the maincable

Formula (18) can be transformed into

119910 =(119881119898 minus sum

119899

119894=1(119879119894119863119894+ 119878119894119908119862119894))

119867 (19)

where 119910 is a function of 119867 and 119881 and the differential of 119910 isgiven by

119889119910 =sum119899

119894=1(119879119894119863119894+ 119878119894119908119862119894) minus 119881119898

1198672119889119867 +

119898

119867119889119881 (20)

Substitute formula (19) into (20)

119889119910 = minus119910

119867119889119867 +

119898

119867119889119881 (21)

For the ending point 119910 = 119910119890119898 = 119871 then

119889119910119890= minus

119910119890

119867119889119867 +

119871

119867119889119881 (22)

where 119871 is the span length of the cable segment

S1S2

A

T1 T2T3

T4

T5H

VH998400

V998400

L

S3

m

C1

C2

C3

D2D1

Figure 5 The calculation scheme of the E-M method

For the middle point 119910119898

= 119891 then

119889119910119898

= minus119891

119867119889119867 +

119898

119867119889119881 (23)

where 119891 is the sag of the main cable at the middle pointBased on formulas (22)-(23) considering the elevation

error of the ending andmiddle points119867 and119881 are modified

[

119889119910119890

119889119910119898

] =

[[[

[

minus119910119890

119867

119871

119867

minus119891

119867

119898

119867

]]]

]

[

119889119867

119889119881] (24)

Then

[

119889119867

119889119881] =

1198672

119891119871 minus 119910119890119898

[[[

[

119898

119867minus119871

119867

119891

119867minus119910119890

119867

]]]

]

[

119889119910119890

119889119910119898

] (25)

For a three-span SB elevations of two pylons are equaland the middle point is at the mid-span of the intermediatespan where 119910

119890= 0 119910

119898= 119891119898 = 1198712 and then formula (25)

can be simplified as

[

119889119867

119889119881] =

[[[

[

119867

2119891minus119867

119891

119867

1198710

]]]

]

[

119889119910119890

119889119910119898

] (26)

where 119889119910119890= 0 minus 119910

119890 119889119910119898

= 119891 minus 119910119898

119891 (1198670 1198810 1198781) = 0

119891 (119867119894minus1

119881119894minus1

119878119894)

= minus119867119894minus1

119878119894

119864119860minus

119867119894minus1

119902

sdot ln(119881119894minus1

+ radic119867119894minus1

2+ 119881119894minus1

2)

minus ln(119881119894minus1

minus 119902119878119894+ radic119867

119894minus1

2+ (119881119894minus1

minus 119902119878119894)2

) minus 119897119894= 0

119891 (119867119899minus1

119881119899minus1

119878119899) = 0

(27)

Mathematical Problems in Engineering 7

Assuming the initial valuesof H and V

Substitute them into the maincable shape equation and

calculate the shape of each cablesegment

Check whether the elevation errorof ending point and middle point

elevation meets the precisionrequirements

Calculation ends

ModifyH V by

E-Mmethod

Yes

No

Figure 6 The calculation step of the E-M method

119881119897 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119898= 0

119881119871 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119890= 0

(28)

E-M iteration method is to solve the nonlinear equations(27) and (28) to obtain the numerical solution For formula(27) Newton method is used to calculate the elevation of thetarget point then substitute the elevation error into (28) andmodify119867 and119881 finally substitute themodified119867 and119881 into(27) to do the new round of iterationThe steps are as follows

(i) Assume the values of 119867 and 119881 and substitute theminto the main cable shape equation and calculatethe shape of each cable segment through Newtoniteration method

(ii) Check whether the elevation error of ending pointand middle point elevation meet the precisionrequirements (|119889119910

119890| lt 119862 and |119889119910

119898| lt 119863) if so the

calculation ends otherwise execute step (3)

(iii) Modify 119867 119881 through E-M method and then substi-tute themodified119867 and119881 into step (1)The flow chartis shown in Figure 6

43 The Main Cable Shape-Finding in the Side Span For themain cable in the side span of SB the shape-finding can alsouse formulas (17) Unlike in the middle span 119867 is knownand equals the horizontal component of the main reactionin the middle span and in the iteration process 119867 remainsconstant and only modifies 119881 as shown in Figure 7 Manyscholars have done research on this method then there is nomore discussion here

V998400

H998400

H

V

T1

T2

T3

Anchor point

Main cable

Tower support

H998400 = H = horizontal force of themain cable in the middle span

Figure 7 The calculation scheme of the main cable shape-findingin the side span

5 Part III of Calculation ModelThe Deformation Compatibility Condition

In Part III SF and MCEF are substituted into the stiffeningequation and check whether the SF and MCEF satisfies theSSG if it does the calculation stops and the last calculationis the expected result otherwise modify the SF by matrix 119861

(Section 32) and repeat the iteration process

6 The Numerical Calculation Process ofthe Proposed Method

According to the theories introduced before an analyticalcalculation method considering the combined effect of themain cable-suspender-stiffening girder is programmedbyVBprograming language The main steps are as follows

(i) Under the condition of the known SSG (the shapeof the stiffening girder) calculate the suspender force1198790(119894) according to the known girder segments119873 seg-

ment length 119871(119894) elastic modulus 119864 cross-sectionalmoment of inertia 119868 uniform load 119902 design shapeequation of the stiffening girder 119910

0119894 and the error

precision requirement 119890(ii) With the suspender force119879

0(119894) solve out the horizon-

tal force1198670of the stiffening girder end

(iii) Substitute 1198670and 119879

0(119894) into the equation of the

stiffening girder then the influence matrix 119861 119889120575 and119889119879 can be obtained

(iv) Check whether |119889120575|meets the precision requirementsor not if |119889120575| lt 119890 the calculation ends otherwisemodify 119879

0(119894) (1198790(119894) = 119879

0(119894) + 119889119879) until |119889120575| meet the

precision requirements(v) Substitute the new 119879

0(119894) into step (ii) repeat steps (ii)

to (iv)

The flow chart is shown in Figure 8

7 Example Analysis

71 Example of the Proposed Method Figure 9 is the generallayout of a self-anchored suspension bridge with the span

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

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Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

y0i

yi

BH

Mjiminus1A

Y

X

H

Fiminus1

Mji

Fi

Δi

Li

Ti

q

Figure 3 The calculation diagram of the girder segment

girder segment 119867 means the horizontal end reaction of themain cable 119872

119895119894means the bending moment at the hanging

point 119865119894means the shearing force of girder 119879

119894means the

suspender force and 119910119894means the alignment of the girder

segment under the joint action of the suspender tension forcethe horizontal component of themain cable and the constantload119910

0119894is the initial camber of the stiffening girderwhich can

meet the precision requirements using designed alignmentin calculation The starting point of the girder segment isassumed as the origin point

According to (2) the stress state of the girder end underuniformly distributed load 119902 meets equilibrium equationwhen ensuring that no vertical displacement occurs in points119860 and 119861

119865119894= 119865119894minus1

+ 119879119894minus 119902119871119894

119872119895119894=

119902119871119894

2

2minus 119865119894minus1

119871119894+ 119872119895119894minus1

+ 119867Δ119894+ 1198721015840

119872119894=

1199021199092

2minus 119865119894minus1

119909 + 119872119895119894minus1

+ 119867119910

Δ119894= 119910119894minus 119910119894minus1

(3)

Neglecting the shearing deformation the equation can bedescribed as follows

119872119894

119864119868= minus(

11991010158401015840

119894

(1 + 11991010158402

119894)32

minus11991010158401015840

0119894

(1 + 11991010158402

0119894)32

) (4)

From (3) deflection differential equilibrium equation canbe obtained

11991010158401015840

119894+

119867

119864119868119910119894=

minus119902

21198641198681199092+

119865119894minus1

119864119868119909 minus

119872119895119894minus1

119864119868+ 11991010158401015840

0119894 (5)

In the equation assume

1198602=

119867

119864119868 119861 =

minus119902

2119864119868

119862119894=

119865119894minus1

119864119868 119863

119894= minus

119872119895119894minus1

119864119868

(6)

Because 1199100119894is a continuous function we can spread out

power series at 119909 = 0 to conveniently solve the differentialequation

1199100119894=

infin

sum

119898=0

119866119894

119898119909119898 (7)

where 119866119894119898

= 119910(119898)

0119894(0)119898

In practical calculation when 119898 = 4 the result can meetthe precision requirement

1199100119894=

4

sum

119898=0

119866119894

119898119909119898

= 119866119894

0+ 119866119894

1119909 + 119866

119894

21199092+ 119866119894

31199093+ 119866119894

41199094

(8)

11991010158401015840

0119894= 2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092 (9)

Substitute (9) into the differential equation (4)

119910119894= 1198641119894sin (119860119909) + 119864

2119894cos (119860119909)

+

(1198611199092+ 119862119894119909 + 119863

119894)1198602minus 2119861

1198604

+

(2119866119894

2+ 6119866119894

3119909 + 12119866

119894

41199092)1198602minus 24119866

119894

4

1198604

(10)

Equation (10) is the deflection equation of the girder seg-ment considering geometrical nonlinearity With the knownquantities 119860 119861 119862

119894 119863119894 1198661198942 1198661198943 and 119866

119894

4and combining with

the controlling condition of the minimum strain energy ofthe stiffening girder 119910

119894(0) = 0 119910

119894(119871119894) = Δ

119894 the expressions

of 1198641119894 1198642119894are given by

1198641119894=

Δ119894

sin (119860119871119894)minus 1198642119894ctan (119860119871

119894)

minus

(119861119871119894

2+ 119862119894119871119894+ 119863119894)1198602minus 2119861

1198604 sin (119860119871119894)

minus

(2119866119894

2+ 6119866119894

3119871119894+ 12119866

119894

4119871119894

2)1198602minus 24119866

119894

4

1198604 sin (119860119871119894)

1198642119894=

2119861

1198604minus

119863119894

1198602+

24119866119894

4

1198604minus

2119866119894

2

1198602

(11)

For each girder segment its deflection equation thatmeets theminimum stain energy theory can be obtainedwithknown119879

119894and119867 Besides the deflection equation should also

satisfy the compatibility condition of the stiffening girderthus

120575119894= 1199101015840

119894+1(0) minus 119910

1015840

119894(119871119894) = 0 (12)

Then differentiate (12)

119889120575119894= 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894) = 0 (13)

Mathematical Problems in Engineering 5

In (13) 119879119894is unknown and differentiate 1199101015840

119894

1198891199101015840

119894+1(0) =

sin (119860119871119894+1

) minus 119860119871119894+1

1198602 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

119889119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

119889119879119895

1198891199101015840

119894(119871119894) = (

1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

119889119879119895

+1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

119889120575 = 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894)

=cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

+ (sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)lowast

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

minus(1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

)119889119879119895

(14)

Transform (14) into matrix form

119889120575 = 119870119889119879

119889119879 = 119870minus1119889120575

(15)

where

119861 = 119870minus1

119889119879 =

[[[[

[

1198891198791

119889119879119899minus1

]]]]

]

119870 =

[[[[

[

11987011

sdot sdot sdot 1198701119899minus1

d

119870119899minus11

sdot sdot sdot 119870119899minus1119899minus1

]]]]

]

119889120575 =

[[[[

[

1205751

120575119899minus1

]]]]

]

119870119894119895=sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)

120597119862119894+1

120597119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

120597119863119894+1

120597119879119895

minus (1

1198602minusctan (119860119871

119894)

119860119871119894)

120597119862119894

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

120597119863119894

120597119879119895

(16)

H

VT

l

O X

h

Y

H998400

V998400T998400

Figure 4 The calculation scheme of the main cable segment

Matrix 119861 is the influential matrix of SSG to SF whichrepresents the relationship between 119889120575 (the first derivateincrement of the stiffening girder shape) and 119889119879 (the changeof suspender force) 119889119879 can be solved out through knowngirder shape and then modify 119879(119894) to adjust the suspenderforce

4 Part II of Calculation ModelThe Main Cable Analysis

Under the condition of the known SF calculated from PartI Part II is used to calculate MCEF which satisfies the shaperequirements of main cable by shape-finding

41 The Segmental Catenary Theory of Main Cable For theSSB the segmental catenary method assumes that the align-ment of the main cable is a catenary which is more approxi-mate with the actual situation In the main cable calculationthe segmental catenary method is adopted [12ndash19]

According to the differential equilibrium equations geo-metrical equations and the physical equations of cablesegment the relation between the shape and the internal forceof cable segment is obtainedThe calculation scheme is shownin Figure 4

119897 = minus1198671198780

119864119860minus

119867

119902ln (119881 + radic1198672 + 1198812)

minus ln(119881 minus 1199021198780+ radic1198672 + (119881 minus 119902119878

0)2

)

ℎ =1199021198780

2minus 2119881119878

0

2119864119860

minus1

119902[radic1198672 + 1198812 minusradic1198672 + (119881 minus 119902119878

0)2

]

(17)

In formulas (17) 119902 represents the self-weight of theunstressed main cable 119864 represents the elastic modulus119860 represents the cross-sectional area 119897 represents the spanlength of the cable segment ℎ represents the elevationdifference of two ends and 119878

0represents the unstressed

length of cable segment 119867 and 119881 represent the horizontaland vertical component of cable segment force in the left end

6 Mathematical Problems in Engineering

The calculation process of the main cable alignment is asfollows assume the value of119867 and119881 and calculate 119878

0 119897 and ℎ

of each cable segment according to the calculated suspenderforce and then check whether the result meets the alignmentrequirements or not If it does calculation stops otherwisemodify the value of 119867 and 119881 and repeat the above steps tillthe result meets the shape requirements

42 Improved E-M IterationMethod for theMain Cable Shape-Finding in the Middle Span Formulas (17) are nonlinearequations to solve the equations the value of119867 and119881 shouldbe assumed to get 119878

0 and then check whether ℎ of the target

point meets the requirements or not The solving process isan iterative process

The numerical iterative methods of the main cable in themiddle span of SBmay not converge in some casesThemainreason is that in the iteration process the elevation require-ments of end points and intermediate points are consideredindependently so that the mutual influences between themare ignored The proposed E-M iteration method consideredthe elevation requirements of both the ending points andintermediate points besides when modifying the horizontalforce 119867 and vertical force 119881 the influences of ending pointselevation error 119889119910

119890and intermediate points elevation error

119889119910119898are consideredAs shown in Figure 5 for any Point 119860 on the cable

segment according to the equilibrium equation of moment

119881119898 minus

119899

sum

119894=1

(119879119894119863119894+ 119878119894119908119862119894) minus 119867119910 = 0 (18)

where119867 and 119881 represent the horizontal and vertical compo-nents of the cable segment force in the left end 119879

119894represents

the suspender force 119878119894represents the unstressed length of

cable segment 119862119894represents the distance from the gravity

center of cable segment to Point119860119863119894represents the distance

from hanging point to Point 119860 119910 and 119898 represent thehorizontal and vertical distances from the left ending point ofthe main cable to Point 119860 respectively 119910

119890and 119910

119898represent

the elevation of the ending point and the middle pointrespectively and 119908 represents the self-weight of the maincable

Formula (18) can be transformed into

119910 =(119881119898 minus sum

119899

119894=1(119879119894119863119894+ 119878119894119908119862119894))

119867 (19)

where 119910 is a function of 119867 and 119881 and the differential of 119910 isgiven by

119889119910 =sum119899

119894=1(119879119894119863119894+ 119878119894119908119862119894) minus 119881119898

1198672119889119867 +

119898

119867119889119881 (20)

Substitute formula (19) into (20)

119889119910 = minus119910

119867119889119867 +

119898

119867119889119881 (21)

For the ending point 119910 = 119910119890119898 = 119871 then

119889119910119890= minus

119910119890

119867119889119867 +

119871

119867119889119881 (22)

where 119871 is the span length of the cable segment

S1S2

A

T1 T2T3

T4

T5H

VH998400

V998400

L

S3

m

C1

C2

C3

D2D1

Figure 5 The calculation scheme of the E-M method

For the middle point 119910119898

= 119891 then

119889119910119898

= minus119891

119867119889119867 +

119898

119867119889119881 (23)

where 119891 is the sag of the main cable at the middle pointBased on formulas (22)-(23) considering the elevation

error of the ending andmiddle points119867 and119881 are modified

[

119889119910119890

119889119910119898

] =

[[[

[

minus119910119890

119867

119871

119867

minus119891

119867

119898

119867

]]]

]

[

119889119867

119889119881] (24)

Then

[

119889119867

119889119881] =

1198672

119891119871 minus 119910119890119898

[[[

[

119898

119867minus119871

119867

119891

119867minus119910119890

119867

]]]

]

[

119889119910119890

119889119910119898

] (25)

For a three-span SB elevations of two pylons are equaland the middle point is at the mid-span of the intermediatespan where 119910

119890= 0 119910

119898= 119891119898 = 1198712 and then formula (25)

can be simplified as

[

119889119867

119889119881] =

[[[

[

119867

2119891minus119867

119891

119867

1198710

]]]

]

[

119889119910119890

119889119910119898

] (26)

where 119889119910119890= 0 minus 119910

119890 119889119910119898

= 119891 minus 119910119898

119891 (1198670 1198810 1198781) = 0

119891 (119867119894minus1

119881119894minus1

119878119894)

= minus119867119894minus1

119878119894

119864119860minus

119867119894minus1

119902

sdot ln(119881119894minus1

+ radic119867119894minus1

2+ 119881119894minus1

2)

minus ln(119881119894minus1

minus 119902119878119894+ radic119867

119894minus1

2+ (119881119894minus1

minus 119902119878119894)2

) minus 119897119894= 0

119891 (119867119899minus1

119881119899minus1

119878119899) = 0

(27)

Mathematical Problems in Engineering 7

Assuming the initial valuesof H and V

Substitute them into the maincable shape equation and

calculate the shape of each cablesegment

Check whether the elevation errorof ending point and middle point

elevation meets the precisionrequirements

Calculation ends

ModifyH V by

E-Mmethod

Yes

No

Figure 6 The calculation step of the E-M method

119881119897 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119898= 0

119881119871 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119890= 0

(28)

E-M iteration method is to solve the nonlinear equations(27) and (28) to obtain the numerical solution For formula(27) Newton method is used to calculate the elevation of thetarget point then substitute the elevation error into (28) andmodify119867 and119881 finally substitute themodified119867 and119881 into(27) to do the new round of iterationThe steps are as follows

(i) Assume the values of 119867 and 119881 and substitute theminto the main cable shape equation and calculatethe shape of each cable segment through Newtoniteration method

(ii) Check whether the elevation error of ending pointand middle point elevation meet the precisionrequirements (|119889119910

119890| lt 119862 and |119889119910

119898| lt 119863) if so the

calculation ends otherwise execute step (3)

(iii) Modify 119867 119881 through E-M method and then substi-tute themodified119867 and119881 into step (1)The flow chartis shown in Figure 6

43 The Main Cable Shape-Finding in the Side Span For themain cable in the side span of SB the shape-finding can alsouse formulas (17) Unlike in the middle span 119867 is knownand equals the horizontal component of the main reactionin the middle span and in the iteration process 119867 remainsconstant and only modifies 119881 as shown in Figure 7 Manyscholars have done research on this method then there is nomore discussion here

V998400

H998400

H

V

T1

T2

T3

Anchor point

Main cable

Tower support

H998400 = H = horizontal force of themain cable in the middle span

Figure 7 The calculation scheme of the main cable shape-findingin the side span

5 Part III of Calculation ModelThe Deformation Compatibility Condition

In Part III SF and MCEF are substituted into the stiffeningequation and check whether the SF and MCEF satisfies theSSG if it does the calculation stops and the last calculationis the expected result otherwise modify the SF by matrix 119861

(Section 32) and repeat the iteration process

6 The Numerical Calculation Process ofthe Proposed Method

According to the theories introduced before an analyticalcalculation method considering the combined effect of themain cable-suspender-stiffening girder is programmedbyVBprograming language The main steps are as follows

(i) Under the condition of the known SSG (the shapeof the stiffening girder) calculate the suspender force1198790(119894) according to the known girder segments119873 seg-

ment length 119871(119894) elastic modulus 119864 cross-sectionalmoment of inertia 119868 uniform load 119902 design shapeequation of the stiffening girder 119910

0119894 and the error

precision requirement 119890(ii) With the suspender force119879

0(119894) solve out the horizon-

tal force1198670of the stiffening girder end

(iii) Substitute 1198670and 119879

0(119894) into the equation of the

stiffening girder then the influence matrix 119861 119889120575 and119889119879 can be obtained

(iv) Check whether |119889120575|meets the precision requirementsor not if |119889120575| lt 119890 the calculation ends otherwisemodify 119879

0(119894) (1198790(119894) = 119879

0(119894) + 119889119879) until |119889120575| meet the

precision requirements(v) Substitute the new 119879

0(119894) into step (ii) repeat steps (ii)

to (iv)

The flow chart is shown in Figure 8

7 Example Analysis

71 Example of the Proposed Method Figure 9 is the generallayout of a self-anchored suspension bridge with the span

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

In (13) 119879119894is unknown and differentiate 1199101015840

119894

1198891199101015840

119894+1(0) =

sin (119860119871119894+1

) minus 119860119871119894+1

1198602 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

119889119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

119889119879119895

1198891199101015840

119894(119871119894) = (

1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

119889119879119895

+1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

119889120575 = 1198891199101015840

119894+1(0) minus 119889119910

1015840

119894(119871119894)

=cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

119899minus1

sum

119895=1

120597119863119894+1

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

119899minus1

sum

119895=1

120597119863119894

120597119879119895

119889119879119895

+ (sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)lowast

119899minus1

sum

119895=1

120597119862119894+1

120597119879119895

minus(1

1198602minusctan (119860119871

119894)

119860119871119894)

119899minus1

sum

119895=1

120597119862119894

120597119879119895

)119889119879119895

(14)

Transform (14) into matrix form

119889120575 = 119870119889119879

119889119879 = 119870minus1119889120575

(15)

where

119861 = 119870minus1

119889119879 =

[[[[

[

1198891198791

119889119879119899minus1

]]]]

]

119870 =

[[[[

[

11987011

sdot sdot sdot 1198701119899minus1

d

119870119899minus11

sdot sdot sdot 119870119899minus1119899minus1

]]]]

]

119889120575 =

[[[[

[

1205751

120575119899minus1

]]]]

]

119870119894119895=sin (119860119871

119894+1) minus 119860119871

119894+1

1198602 sin (119860119871119894+1

)

120597119862119894+1

120597119879119895

+cos (119860119871

119894+1) minus 1

119860 sin (119860119871119894+1

)

120597119863119894+1

120597119879119895

minus (1

1198602minusctan (119860119871

119894)

119860119871119894)

120597119862119894

120597119879119895

minus1 minus cos (119860119871

119894)

119860 sin (119860119871119894)

120597119863119894

120597119879119895

(16)

H

VT

l

O X

h

Y

H998400

V998400T998400

Figure 4 The calculation scheme of the main cable segment

Matrix 119861 is the influential matrix of SSG to SF whichrepresents the relationship between 119889120575 (the first derivateincrement of the stiffening girder shape) and 119889119879 (the changeof suspender force) 119889119879 can be solved out through knowngirder shape and then modify 119879(119894) to adjust the suspenderforce

4 Part II of Calculation ModelThe Main Cable Analysis

Under the condition of the known SF calculated from PartI Part II is used to calculate MCEF which satisfies the shaperequirements of main cable by shape-finding

41 The Segmental Catenary Theory of Main Cable For theSSB the segmental catenary method assumes that the align-ment of the main cable is a catenary which is more approxi-mate with the actual situation In the main cable calculationthe segmental catenary method is adopted [12ndash19]

According to the differential equilibrium equations geo-metrical equations and the physical equations of cablesegment the relation between the shape and the internal forceof cable segment is obtainedThe calculation scheme is shownin Figure 4

119897 = minus1198671198780

119864119860minus

119867

119902ln (119881 + radic1198672 + 1198812)

minus ln(119881 minus 1199021198780+ radic1198672 + (119881 minus 119902119878

0)2

)

ℎ =1199021198780

2minus 2119881119878

0

2119864119860

minus1

119902[radic1198672 + 1198812 minusradic1198672 + (119881 minus 119902119878

0)2

]

(17)

In formulas (17) 119902 represents the self-weight of theunstressed main cable 119864 represents the elastic modulus119860 represents the cross-sectional area 119897 represents the spanlength of the cable segment ℎ represents the elevationdifference of two ends and 119878

0represents the unstressed

length of cable segment 119867 and 119881 represent the horizontaland vertical component of cable segment force in the left end

6 Mathematical Problems in Engineering

The calculation process of the main cable alignment is asfollows assume the value of119867 and119881 and calculate 119878

0 119897 and ℎ

of each cable segment according to the calculated suspenderforce and then check whether the result meets the alignmentrequirements or not If it does calculation stops otherwisemodify the value of 119867 and 119881 and repeat the above steps tillthe result meets the shape requirements

42 Improved E-M IterationMethod for theMain Cable Shape-Finding in the Middle Span Formulas (17) are nonlinearequations to solve the equations the value of119867 and119881 shouldbe assumed to get 119878

0 and then check whether ℎ of the target

point meets the requirements or not The solving process isan iterative process

The numerical iterative methods of the main cable in themiddle span of SBmay not converge in some casesThemainreason is that in the iteration process the elevation require-ments of end points and intermediate points are consideredindependently so that the mutual influences between themare ignored The proposed E-M iteration method consideredthe elevation requirements of both the ending points andintermediate points besides when modifying the horizontalforce 119867 and vertical force 119881 the influences of ending pointselevation error 119889119910

119890and intermediate points elevation error

119889119910119898are consideredAs shown in Figure 5 for any Point 119860 on the cable

segment according to the equilibrium equation of moment

119881119898 minus

119899

sum

119894=1

(119879119894119863119894+ 119878119894119908119862119894) minus 119867119910 = 0 (18)

where119867 and 119881 represent the horizontal and vertical compo-nents of the cable segment force in the left end 119879

119894represents

the suspender force 119878119894represents the unstressed length of

cable segment 119862119894represents the distance from the gravity

center of cable segment to Point119860119863119894represents the distance

from hanging point to Point 119860 119910 and 119898 represent thehorizontal and vertical distances from the left ending point ofthe main cable to Point 119860 respectively 119910

119890and 119910

119898represent

the elevation of the ending point and the middle pointrespectively and 119908 represents the self-weight of the maincable

Formula (18) can be transformed into

119910 =(119881119898 minus sum

119899

119894=1(119879119894119863119894+ 119878119894119908119862119894))

119867 (19)

where 119910 is a function of 119867 and 119881 and the differential of 119910 isgiven by

119889119910 =sum119899

119894=1(119879119894119863119894+ 119878119894119908119862119894) minus 119881119898

1198672119889119867 +

119898

119867119889119881 (20)

Substitute formula (19) into (20)

119889119910 = minus119910

119867119889119867 +

119898

119867119889119881 (21)

For the ending point 119910 = 119910119890119898 = 119871 then

119889119910119890= minus

119910119890

119867119889119867 +

119871

119867119889119881 (22)

where 119871 is the span length of the cable segment

S1S2

A

T1 T2T3

T4

T5H

VH998400

V998400

L

S3

m

C1

C2

C3

D2D1

Figure 5 The calculation scheme of the E-M method

For the middle point 119910119898

= 119891 then

119889119910119898

= minus119891

119867119889119867 +

119898

119867119889119881 (23)

where 119891 is the sag of the main cable at the middle pointBased on formulas (22)-(23) considering the elevation

error of the ending andmiddle points119867 and119881 are modified

[

119889119910119890

119889119910119898

] =

[[[

[

minus119910119890

119867

119871

119867

minus119891

119867

119898

119867

]]]

]

[

119889119867

119889119881] (24)

Then

[

119889119867

119889119881] =

1198672

119891119871 minus 119910119890119898

[[[

[

119898

119867minus119871

119867

119891

119867minus119910119890

119867

]]]

]

[

119889119910119890

119889119910119898

] (25)

For a three-span SB elevations of two pylons are equaland the middle point is at the mid-span of the intermediatespan where 119910

119890= 0 119910

119898= 119891119898 = 1198712 and then formula (25)

can be simplified as

[

119889119867

119889119881] =

[[[

[

119867

2119891minus119867

119891

119867

1198710

]]]

]

[

119889119910119890

119889119910119898

] (26)

where 119889119910119890= 0 minus 119910

119890 119889119910119898

= 119891 minus 119910119898

119891 (1198670 1198810 1198781) = 0

119891 (119867119894minus1

119881119894minus1

119878119894)

= minus119867119894minus1

119878119894

119864119860minus

119867119894minus1

119902

sdot ln(119881119894minus1

+ radic119867119894minus1

2+ 119881119894minus1

2)

minus ln(119881119894minus1

minus 119902119878119894+ radic119867

119894minus1

2+ (119881119894minus1

minus 119902119878119894)2

) minus 119897119894= 0

119891 (119867119899minus1

119881119899minus1

119878119899) = 0

(27)

Mathematical Problems in Engineering 7

Assuming the initial valuesof H and V

Substitute them into the maincable shape equation and

calculate the shape of each cablesegment

Check whether the elevation errorof ending point and middle point

elevation meets the precisionrequirements

Calculation ends

ModifyH V by

E-Mmethod

Yes

No

Figure 6 The calculation step of the E-M method

119881119897 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119898= 0

119881119871 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119890= 0

(28)

E-M iteration method is to solve the nonlinear equations(27) and (28) to obtain the numerical solution For formula(27) Newton method is used to calculate the elevation of thetarget point then substitute the elevation error into (28) andmodify119867 and119881 finally substitute themodified119867 and119881 into(27) to do the new round of iterationThe steps are as follows

(i) Assume the values of 119867 and 119881 and substitute theminto the main cable shape equation and calculatethe shape of each cable segment through Newtoniteration method

(ii) Check whether the elevation error of ending pointand middle point elevation meet the precisionrequirements (|119889119910

119890| lt 119862 and |119889119910

119898| lt 119863) if so the

calculation ends otherwise execute step (3)

(iii) Modify 119867 119881 through E-M method and then substi-tute themodified119867 and119881 into step (1)The flow chartis shown in Figure 6

43 The Main Cable Shape-Finding in the Side Span For themain cable in the side span of SB the shape-finding can alsouse formulas (17) Unlike in the middle span 119867 is knownand equals the horizontal component of the main reactionin the middle span and in the iteration process 119867 remainsconstant and only modifies 119881 as shown in Figure 7 Manyscholars have done research on this method then there is nomore discussion here

V998400

H998400

H

V

T1

T2

T3

Anchor point

Main cable

Tower support

H998400 = H = horizontal force of themain cable in the middle span

Figure 7 The calculation scheme of the main cable shape-findingin the side span

5 Part III of Calculation ModelThe Deformation Compatibility Condition

In Part III SF and MCEF are substituted into the stiffeningequation and check whether the SF and MCEF satisfies theSSG if it does the calculation stops and the last calculationis the expected result otherwise modify the SF by matrix 119861

(Section 32) and repeat the iteration process

6 The Numerical Calculation Process ofthe Proposed Method

According to the theories introduced before an analyticalcalculation method considering the combined effect of themain cable-suspender-stiffening girder is programmedbyVBprograming language The main steps are as follows

(i) Under the condition of the known SSG (the shapeof the stiffening girder) calculate the suspender force1198790(119894) according to the known girder segments119873 seg-

ment length 119871(119894) elastic modulus 119864 cross-sectionalmoment of inertia 119868 uniform load 119902 design shapeequation of the stiffening girder 119910

0119894 and the error

precision requirement 119890(ii) With the suspender force119879

0(119894) solve out the horizon-

tal force1198670of the stiffening girder end

(iii) Substitute 1198670and 119879

0(119894) into the equation of the

stiffening girder then the influence matrix 119861 119889120575 and119889119879 can be obtained

(iv) Check whether |119889120575|meets the precision requirementsor not if |119889120575| lt 119890 the calculation ends otherwisemodify 119879

0(119894) (1198790(119894) = 119879

0(119894) + 119889119879) until |119889120575| meet the

precision requirements(v) Substitute the new 119879

0(119894) into step (ii) repeat steps (ii)

to (iv)

The flow chart is shown in Figure 8

7 Example Analysis

71 Example of the Proposed Method Figure 9 is the generallayout of a self-anchored suspension bridge with the span

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The calculation process of the main cable alignment is asfollows assume the value of119867 and119881 and calculate 119878

0 119897 and ℎ

of each cable segment according to the calculated suspenderforce and then check whether the result meets the alignmentrequirements or not If it does calculation stops otherwisemodify the value of 119867 and 119881 and repeat the above steps tillthe result meets the shape requirements

42 Improved E-M IterationMethod for theMain Cable Shape-Finding in the Middle Span Formulas (17) are nonlinearequations to solve the equations the value of119867 and119881 shouldbe assumed to get 119878

0 and then check whether ℎ of the target

point meets the requirements or not The solving process isan iterative process

The numerical iterative methods of the main cable in themiddle span of SBmay not converge in some casesThemainreason is that in the iteration process the elevation require-ments of end points and intermediate points are consideredindependently so that the mutual influences between themare ignored The proposed E-M iteration method consideredthe elevation requirements of both the ending points andintermediate points besides when modifying the horizontalforce 119867 and vertical force 119881 the influences of ending pointselevation error 119889119910

119890and intermediate points elevation error

119889119910119898are consideredAs shown in Figure 5 for any Point 119860 on the cable

segment according to the equilibrium equation of moment

119881119898 minus

119899

sum

119894=1

(119879119894119863119894+ 119878119894119908119862119894) minus 119867119910 = 0 (18)

where119867 and 119881 represent the horizontal and vertical compo-nents of the cable segment force in the left end 119879

119894represents

the suspender force 119878119894represents the unstressed length of

cable segment 119862119894represents the distance from the gravity

center of cable segment to Point119860119863119894represents the distance

from hanging point to Point 119860 119910 and 119898 represent thehorizontal and vertical distances from the left ending point ofthe main cable to Point 119860 respectively 119910

119890and 119910

119898represent

the elevation of the ending point and the middle pointrespectively and 119908 represents the self-weight of the maincable

Formula (18) can be transformed into

119910 =(119881119898 minus sum

119899

119894=1(119879119894119863119894+ 119878119894119908119862119894))

119867 (19)

where 119910 is a function of 119867 and 119881 and the differential of 119910 isgiven by

119889119910 =sum119899

119894=1(119879119894119863119894+ 119878119894119908119862119894) minus 119881119898

1198672119889119867 +

119898

119867119889119881 (20)

Substitute formula (19) into (20)

119889119910 = minus119910

119867119889119867 +

119898

119867119889119881 (21)

For the ending point 119910 = 119910119890119898 = 119871 then

119889119910119890= minus

119910119890

119867119889119867 +

119871

119867119889119881 (22)

where 119871 is the span length of the cable segment

S1S2

A

T1 T2T3

T4

T5H

VH998400

V998400

L

S3

m

C1

C2

C3

D2D1

Figure 5 The calculation scheme of the E-M method

For the middle point 119910119898

= 119891 then

119889119910119898

= minus119891

119867119889119867 +

119898

119867119889119881 (23)

where 119891 is the sag of the main cable at the middle pointBased on formulas (22)-(23) considering the elevation

error of the ending andmiddle points119867 and119881 are modified

[

119889119910119890

119889119910119898

] =

[[[

[

minus119910119890

119867

119871

119867

minus119891

119867

119898

119867

]]]

]

[

119889119867

119889119881] (24)

Then

[

119889119867

119889119881] =

1198672

119891119871 minus 119910119890119898

[[[

[

119898

119867minus119871

119867

119891

119867minus119910119890

119867

]]]

]

[

119889119910119890

119889119910119898

] (25)

For a three-span SB elevations of two pylons are equaland the middle point is at the mid-span of the intermediatespan where 119910

119890= 0 119910

119898= 119891119898 = 1198712 and then formula (25)

can be simplified as

[

119889119867

119889119881] =

[[[

[

119867

2119891minus119867

119891

119867

1198710

]]]

]

[

119889119910119890

119889119910119898

] (26)

where 119889119910119890= 0 minus 119910

119890 119889119910119898

= 119891 minus 119910119898

119891 (1198670 1198810 1198781) = 0

119891 (119867119894minus1

119881119894minus1

119878119894)

= minus119867119894minus1

119878119894

119864119860minus

119867119894minus1

119902

sdot ln(119881119894minus1

+ radic119867119894minus1

2+ 119881119894minus1

2)

minus ln(119881119894minus1

minus 119902119878119894+ radic119867

119894minus1

2+ (119881119894minus1

minus 119902119878119894)2

) minus 119897119894= 0

119891 (119867119899minus1

119881119899minus1

119878119899) = 0

(27)

Mathematical Problems in Engineering 7

Assuming the initial valuesof H and V

Substitute them into the maincable shape equation and

calculate the shape of each cablesegment

Check whether the elevation errorof ending point and middle point

elevation meets the precisionrequirements

Calculation ends

ModifyH V by

E-Mmethod

Yes

No

Figure 6 The calculation step of the E-M method

119881119897 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119898= 0

119881119871 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119890= 0

(28)

E-M iteration method is to solve the nonlinear equations(27) and (28) to obtain the numerical solution For formula(27) Newton method is used to calculate the elevation of thetarget point then substitute the elevation error into (28) andmodify119867 and119881 finally substitute themodified119867 and119881 into(27) to do the new round of iterationThe steps are as follows

(i) Assume the values of 119867 and 119881 and substitute theminto the main cable shape equation and calculatethe shape of each cable segment through Newtoniteration method

(ii) Check whether the elevation error of ending pointand middle point elevation meet the precisionrequirements (|119889119910

119890| lt 119862 and |119889119910

119898| lt 119863) if so the

calculation ends otherwise execute step (3)

(iii) Modify 119867 119881 through E-M method and then substi-tute themodified119867 and119881 into step (1)The flow chartis shown in Figure 6

43 The Main Cable Shape-Finding in the Side Span For themain cable in the side span of SB the shape-finding can alsouse formulas (17) Unlike in the middle span 119867 is knownand equals the horizontal component of the main reactionin the middle span and in the iteration process 119867 remainsconstant and only modifies 119881 as shown in Figure 7 Manyscholars have done research on this method then there is nomore discussion here

V998400

H998400

H

V

T1

T2

T3

Anchor point

Main cable

Tower support

H998400 = H = horizontal force of themain cable in the middle span

Figure 7 The calculation scheme of the main cable shape-findingin the side span

5 Part III of Calculation ModelThe Deformation Compatibility Condition

In Part III SF and MCEF are substituted into the stiffeningequation and check whether the SF and MCEF satisfies theSSG if it does the calculation stops and the last calculationis the expected result otherwise modify the SF by matrix 119861

(Section 32) and repeat the iteration process

6 The Numerical Calculation Process ofthe Proposed Method

According to the theories introduced before an analyticalcalculation method considering the combined effect of themain cable-suspender-stiffening girder is programmedbyVBprograming language The main steps are as follows

(i) Under the condition of the known SSG (the shapeof the stiffening girder) calculate the suspender force1198790(119894) according to the known girder segments119873 seg-

ment length 119871(119894) elastic modulus 119864 cross-sectionalmoment of inertia 119868 uniform load 119902 design shapeequation of the stiffening girder 119910

0119894 and the error

precision requirement 119890(ii) With the suspender force119879

0(119894) solve out the horizon-

tal force1198670of the stiffening girder end

(iii) Substitute 1198670and 119879

0(119894) into the equation of the

stiffening girder then the influence matrix 119861 119889120575 and119889119879 can be obtained

(iv) Check whether |119889120575|meets the precision requirementsor not if |119889120575| lt 119890 the calculation ends otherwisemodify 119879

0(119894) (1198790(119894) = 119879

0(119894) + 119889119879) until |119889120575| meet the

precision requirements(v) Substitute the new 119879

0(119894) into step (ii) repeat steps (ii)

to (iv)

The flow chart is shown in Figure 8

7 Example Analysis

71 Example of the Proposed Method Figure 9 is the generallayout of a self-anchored suspension bridge with the span

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Assuming the initial valuesof H and V

Substitute them into the maincable shape equation and

calculate the shape of each cablesegment

Check whether the elevation errorof ending point and middle point

elevation meets the precisionrequirements

Calculation ends

ModifyH V by

E-Mmethod

Yes

No

Figure 6 The calculation step of the E-M method

119881119897 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119898= 0

119881119871 minus

119899

sum

119894=1

(119866119894119863119894+ 119878119894119908119862119894) minus 119867119910

119890= 0

(28)

E-M iteration method is to solve the nonlinear equations(27) and (28) to obtain the numerical solution For formula(27) Newton method is used to calculate the elevation of thetarget point then substitute the elevation error into (28) andmodify119867 and119881 finally substitute themodified119867 and119881 into(27) to do the new round of iterationThe steps are as follows

(i) Assume the values of 119867 and 119881 and substitute theminto the main cable shape equation and calculatethe shape of each cable segment through Newtoniteration method

(ii) Check whether the elevation error of ending pointand middle point elevation meet the precisionrequirements (|119889119910

119890| lt 119862 and |119889119910

119898| lt 119863) if so the

calculation ends otherwise execute step (3)

(iii) Modify 119867 119881 through E-M method and then substi-tute themodified119867 and119881 into step (1)The flow chartis shown in Figure 6

43 The Main Cable Shape-Finding in the Side Span For themain cable in the side span of SB the shape-finding can alsouse formulas (17) Unlike in the middle span 119867 is knownand equals the horizontal component of the main reactionin the middle span and in the iteration process 119867 remainsconstant and only modifies 119881 as shown in Figure 7 Manyscholars have done research on this method then there is nomore discussion here

V998400

H998400

H

V

T1

T2

T3

Anchor point

Main cable

Tower support

H998400 = H = horizontal force of themain cable in the middle span

Figure 7 The calculation scheme of the main cable shape-findingin the side span

5 Part III of Calculation ModelThe Deformation Compatibility Condition

In Part III SF and MCEF are substituted into the stiffeningequation and check whether the SF and MCEF satisfies theSSG if it does the calculation stops and the last calculationis the expected result otherwise modify the SF by matrix 119861

(Section 32) and repeat the iteration process

6 The Numerical Calculation Process ofthe Proposed Method

According to the theories introduced before an analyticalcalculation method considering the combined effect of themain cable-suspender-stiffening girder is programmedbyVBprograming language The main steps are as follows

(i) Under the condition of the known SSG (the shapeof the stiffening girder) calculate the suspender force1198790(119894) according to the known girder segments119873 seg-

ment length 119871(119894) elastic modulus 119864 cross-sectionalmoment of inertia 119868 uniform load 119902 design shapeequation of the stiffening girder 119910

0119894 and the error

precision requirement 119890(ii) With the suspender force119879

0(119894) solve out the horizon-

tal force1198670of the stiffening girder end

(iii) Substitute 1198670and 119879

0(119894) into the equation of the

stiffening girder then the influence matrix 119861 119889120575 and119889119879 can be obtained

(iv) Check whether |119889120575|meets the precision requirementsor not if |119889120575| lt 119890 the calculation ends otherwisemodify 119879

0(119894) (1198790(119894) = 119879

0(119894) + 119889119879) until |119889120575| meet the

precision requirements(v) Substitute the new 119879

0(119894) into step (ii) repeat steps (ii)

to (iv)

The flow chart is shown in Figure 8

7 Example Analysis

71 Example of the Proposed Method Figure 9 is the generallayout of a self-anchored suspension bridge with the span

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 1 The suspender force in each iteration step

Iteration times Suspender number Horizontal reaction of the main cable1 6 10 119879

112 17 22 26 119867 |Δ119867119867|

0 21196 17817 18132 18651 18132 17830 17368 17374 3000001 21196 17817 18132 18651 18132 17828 17215 17222 406663 262119864 minus 01

2 21196 17817 18132 18651 18132 17828 17218 17225 404287 588119864 minus 03

3 21196 17817 18132 18651 18132 17828 17218 17225 404340 131119864 minus 04

4 17900 17924 17843 17799 17816 17723 17640 17677 404338 495119864 minus 06

arrangement of 150m + 406m + 150m The suspenderspacing arrangement in the side span is 145 + 9 times 135 + 14

and is 14 + 28 times 135 + 14 in the middle span The theoreticalvertex of the main cable is 85m higher than the theoreticalanchorage point For the stiffening girder the moment ofinertia is 119868 = 4428675m4 the elastic modulus is 119864 =

2111986405MPa and the dead load in the first and second phaseis 119902 = 132 kNm for the main cable the cross-sectional areais 119860 = 01038m2 elastic modulus is 119864 = 19511986405MPa theself-weight is 119902 = 8145 kNm and the ratio of rise to spanis 119891119871 = 158 To simplify the calculation ignoring the self-weight of the clamp and suspender neglecting themain cableand saddle modification the calculation is just based on thetheoretical vertex of the main cable assuming that the leftanchorage of main cable is as the coordinate origin as shownin Figure 9

The stiffening girder linear is designed with a two-waylongitudinal slope of 1 and a round curve transition of 119877 =

9450m in the middle (Figure 10)As shown in Table 1 and Figure 11 the results of the

proposed method can meet the precision requirements (120575 =

Δ119867119867 lt 10minus5) with fast convergence rate (4 times of

iteration) And the final suspender force is more even withthe maximum of 21196 kN in 1 and 51 suspender

Figure 12 indicates that the calculatedmain cable shape inthis method is smooth without anymutation point and it alsosatisfies the requirement of rise to span ratio of the middlespan (119891119871 = 158)

Figure 13 shows that the bending moment of the stiff-ening girder is distributed uniformly with the maximumpositive and negative bending moment of 28236 KNsdotm and20573 kNsdotm respectively which occur near to the end of thestiffening girder

72 Example of the E-M Method in Part II Figure 14 is thegeneral layout of a main cable in the middle span The cross-sectional area is 119860 = 01038m2 elastic modulus is 119864 =

19511986405MPa the self-weight is 119902 = 8145 kNm and the ratioof rise to span is 119891119871 = 158 To simplify the calculationignoring the self-weight of the clamp and suspender thecalculation just assumes the left anchorage of the main cableas the coordinate origin (Figure 14)

Two examples are presented and compared to verify theconvergence of E-M method as follows

(i) In example 1 the suspender forces are distributedevenly and symmetrically with the value of mainly around1800 kN as shown in Table 2

Under the condition of theknown SSG the suspender

force T0(i)

Check whether H0 and T0(i)

meet the shape requirementsof the stiffening girder

Such as |d120575| lt e

Substitute H0 and T0(i)

into the equation of stiffening

Finish

Modify

dT

Under the condition of satisfyingthe rise to span ration fL

calculate the horizontalcomponent H0

No

Yes

Yes

NoCheck whether H0

meet the shape requirementsof the main cable

Check whether T0(i)meet the shape requirements

of the stiffening girder

Yes

No

T0(i) = T0(i) +

girder calculate B d120575 and dT

Figure 8 The flow chart

Table 3 shows that after 7 times of iteration the end pointelevation error |119889119910

119890| reaches level 10minus4 and the middle point

elevation error |119889119910119898| reaches level 10minus5 meaning that the E-M

method has fast convergence rate and high precisionIt can be seen from Table 4 that the unstressed length

and the elevation of point 119895 calculated by E-M method aresymmetric with the middle point It is mainly because thesuspender forces are symmetric with the middle point

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 10 9 8 7 6 5 4 3 2 1

Y

X00

850

150

850

00T1 T2

150 406 150

Figure 9 The general layout of a self-anchored suspension bridge

2585 189 2585

R = 9450

1 1

Figure 10 The design alignment of the stiffening girder (unit m)

0

500

1000

1500

2000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 3133353739414345474951

Susp

ende

r for

ce (k

N)

Suspender number

Figure 11 The distribution of the suspender force

0102030405060708090100

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750

Elev

atio

n (m

)

Longitudinal position (m)

Figure 12 The alignment of the main cable

minus2000

0

2000

4000

6000

8000

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 303234363840424446485052

Suspender number

Bend

ing

mom

ent o

fsti

ffeni

ng g

irder

(kNmiddotm

)

Figure 13 The bending moment diagram of the stiffening girder

Table 2 The suspender forces

Suspender number Suspender force (kN)1 179802 179863 179944 180025 180096 180167 180238 180319 1802310 1801611 1800912 1800213 1799414 1798615 17980

(ii) To verify the convergence of E-M method thesuspender forces in example 2 are distributed unevenly andasymmetrically as shown in Table 5

Table 6 indicates that although the suspender forces aredistributed unevenly and asymmetrically after 7 times ofiteration like example 1 the end point elevation error |119889119910

119890|

reaches level 10minus5 and the middle point elevation error |119889119910119898|

reaches level 10minus5 which further proves that E-Mmethod hasgood convergence and high reliability

Table 7 and Figure 15 indicate that the main cable shapein example 1 is smooth mainly because the suspender forcesare distributed evenly while the main cable shape in example2 is not smooth mainly because the suspender forces aredistributed unevenly In example 1 the lowest point is 119909 =

108m 119910 = minus36m while in example 2 the lowest point is119909 = 945m 119910 = minus3956m because the suspender force at 119909 =

945m is 1198797= 268023 kN However in either example 1 or 2

the shapes of themain cable allmeet the requirements of threefixed-points proving the high reliability of E-M method

8 Summaries and Conclusions

For SSB since themain cable is directly anchored on the stiff-ening girder it is significant to consider the combined effects

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 3 Reaction of the main cable and the elevation error of the fixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 4332581 1444882 158119864 minus 02 169119864 + 01

2 2298366 1444565 420119864 minus 01 148119864 minus 01

3 2294396 1440096 390119864 minus 01 140119864 minus 01

4 2297893 1444239 249119864 minus 03 124119864 minus 03

5 2297893 1444212 137119864 minus 03 767119864 minus 04

6 2297888 1444227 240119864 minus 05 100119864 minus 05

7 2297888 1444227 130119864 minus 05 600119864 minus 06

Table 4 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1590 minus845 9 1350 minus35452 1533 minus1577 10 1359 minus33763 1483 minus2196 11 1377 minus30954 1440 minus2702 12 1404 minus27025 1404 minus3095 13 1440 minus21966 1377 minus3376 14 1483 minus15777 1359 minus3545 15 1533 minus8458 1350 minus3600 16 1590 0

Table 5 The suspender forces

Suspender number Suspender force (kN)1 780002 235073 1004 950065 55636 12827 2680238 80319 11310 68211 335912 98213 521414 6389615 7880

of the main cable the suspender and the stiffening girderConcerning this issue we focus on the theoretical researchesandmethod improvementsThe following conclusions can bedrawn from this paper

(i) An analytical calculation method considering thecombined effect of the main cable-suspender-stiffen-ing girder which cannot be solved in the existingmethods is proposed in this paper The method helpsto understand the mechanical behavior of the wholestructure in the preliminary design by a simplermodel

Table 6 Reaction of the main cable and the elevation error of thefixed point in each iteration step

Iteration times 119867 1198811003816100381610038161003816119889119910119890

1003816100381610038161003816

10038161003816100381610038161198891199101198981003816100381610038161003816

1 8706386 2902817 136119864 + 01 216119864 + 01

2 5121170 3449246 185119864 minus 01 697119864 minus 02

3 5117904 3444853 212119864 minus 01 756119864 minus 02

4 5122226 3449876 374119864 minus 04 300119864 minus 04

5 5122242 3449867 186119864 minus 04 186119864 minus 04

6 5122229 3449872 300119864 minus 06 100119864 minus 06

7 5122229 3449872 200119864 minus 06 100119864 minus 06

(ii) Compared to the existing methods in the proposedmethod the main cable and the stiffening girderare all considered through the numerical analyticalmethod and the calculation procedure does notneed to be iterated manually which can simplifycalculation process and save calculation time

(iii) Using the proposed method the reasonable finishedbridge state satisfying the minimum strain energytheory of the stiffening girder can be obtained byensuring zero vertical displacement of the hangingpoint under the joint action of the suspender forcethe horizontal component of the main cable and thedead load

(iv) To verify the proposed method examples are intro-duced whose calculation results indicate that thismethod is reliable with fast convergence speed andhigh precision The results can meet the precisionrequirements (120575 = Δ119867119867 lt 10

minus5) with only 4 timesof iteration and the calculated main cable shape is

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Table 7 The unstressed length and coordinates of the main cable

Suspender number Unstressed length 119878 (m) Elevation of point 119895 Suspender number Unstressed length (m) Elevation of point 1198951 1622 minus907 9 1399 minus32182 1516 minus1606 10 1400 minus28343 1487 minus2239 11 1401 minus24454 1486 minus2869 12 1405 minus20445 1398 minus3245 13 1406 minus16376 1393 minus3604 14 1411 minus12147 1392 minus3956 15 1471 minus6198 1393 minus3600 16 1481 0

T2T3 T4 T5 T6 T7 T8 T9 T10 T11

T12T13

T14

T15

S2S3 S4 S5 S6 S7 S8 S9 S10 S11

S12S13

S14S15

S16

VL VR

HL HR

Y

XS1

216

36T1

Figure 14 The calculation schematic of the main cable of a suspension bridge

00 18 36 54 72 90 108 126 144 162 180 198 216

Example 2Example 1

minus42

minus36

minus30

minus24

minus18

minus12

minus6

Vert

ical

coor

dina

te (m

)

Longitudinal distance (m)

Figure 15 The calculation schematic of the main cable segment

smoothThe suspender force and the internal force ofthe stiffening girder are relatively uniform

(v) The E-M iteration method in Part II considers theelevation requirements of both the ending points andintermediate points and ensures the convergence dur-ing the iteration by improving the iteration equationof 119889119867 and 119889119881

Consequently we expect that the proposed analytical cal-culation method will provide a brand new way for designersin the SSB preliminary design

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research is supported by National Basic Research Pro-gram of China (2012CB723300) and Natural Science Foun-dation Project of CQ (CSTC2012jjB0118) The authors alsowould like to thank Ministry of Transport of China (MOT)for funding of the present research (2010-353-341-230)

References

[1] D B Steinman A Practical Treatise on Suspension BridgesWiley New York NY USA 1953

[2] A Jennings ldquoDeflection theory analysis of different cableprofiles for suspension bridgesrdquo Engineering Structures vol 9no 2 pp 84ndash94 1987

[3] G P Wollmann ldquoPreliminary analysis of suspension bridgesrdquoJournal of Bridge Engineering vol 6 no 4 pp 227ndash233 2001

[4] H Ohshima K Sato and N Watanabe ldquoStructural analysis ofsuspension bridgesrdquo Journal of Engineering Mechanics vol 110no 3 pp 392ndash404 1984

[5] M-R Jung D-J Min and M-Y Kim ldquoNonlinear analysismethods based on the unstrained element length for determin-ing initial shaping of suspension bridges under dead loadsrdquoComputers and Structures vol 128 pp 272ndash285 2013

[6] A Montoya R Betti G Deodatis and H Waisman ldquoA stocha-stic finite element approach to determine the safety of suspen-sion bridge cablesrdquo in Proceedings of the ASCE InternationalWorkshop on Computing in Civil Engineering (IWCCE rsquo13) pp1ndash8 June 2013

[7] H-K Kim M-J Lee and S-P Chang ldquoDetermination ofhanger installation procedure for a self-anchored suspensionbridgerdquo Engineering Structures vol 28 no 7 pp 959ndash976 2006

[8] H-K Kim M-J Lee and S-P Chang ldquoNon-linear shape-finding analysis of a self-anchored suspension bridgerdquoEngineer-ing Structures vol 24 no 12 pp 1547ndash1559 2002

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[9] D-L Tan ldquoDecision method on reasonable design state of self-anchored suspension bridgerdquo China Journal of Highway andTransport vol 18 no 2 pp 51ndash55 2005 (Chinese)

[10] YHan Z-Q Chen and S-D LuoCalculationMethod on ShapeFinding of Self-Anchored Suspension Bridge with Spatial CablesChangsha University of Science amp Technology ChangshaChina 2009

[11] C-X Li H-J Ke H-B Liu and G-Y Xia ldquoDetermination offinished bridge state of self-anchored suspension bridge withspatial cablesrdquo EngineeringMechanics vol 27 no 5 pp 137ndash1462010

[12] L Greco N Impollonia and M Cuomo ldquoA procedure forthe static analysis of cable structures following elastic catenarytheoryrdquo International Journal of Solids and Structures vol 51 no7-8 pp 1521ndash1533 2014

[13] A Andreu L Gil and P Roca ldquoA new deformable catenaryelement for the analysis of cable net structuresrdquo Computers andStructures vol 84 no 29-30 pp 1882ndash1890 2006

[14] W Paulsen and G Slayton ldquoEigenfrequency analysis of cablestructures with inclined cablesrdquo Applied Mathematics andMechanics vol 27 no 1 pp 37ndash49 2006

[15] J A Ochsendorf and D P Billington ldquoSelf-anchored suspen-sion bridgesrdquo Journal of Bridge Engineering vol 4 no 3 pp 151ndash156 1999

[16] M S A Abad A Shooshtari V Esmaeili and A N RiabildquoNonlinear analysis of cable structures under general loadingsrdquoFinite Elements in Analysis and Design vol 73 pp 11ndash19 2013

[17] H-T Thai and S-E Kim ldquoNonlinear static and dynamicanalysis of cable structuresrdquo Finite Elements in Analysis andDesign vol 47 no 3 pp 237ndash246 2011

[18] M-YKimD-YKimM-R Jung andMMAttard ldquoImprovedmethods for determining the 3 dimensional initial shapes ofcable-supported bridgesrdquo International Journal of Steel Struc-tures vol 14 no 1 pp 83ndash102 2014

[19] R Karoumi ldquoSome modeling aspects in the nonlinear finiteelement analysis of cable supported bridgesrdquo Computers andStructures vol 71 no 4 pp 397ndash412 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of