Simplified Analysis for Preliminary Design of Towers inSuspension Bridges

Dong-Ho Choi, Ph.D., M.ASCE1; Sun-Gil Gwon2; and Ho-Sung Na, Ph.D.3

Abstract: This paper proposes an equivalent suspension bridge model for the preliminary design of support towers of suspension bridges. Inthis simplifiedmodel, the bridge is composed of its support towers, with equivalent springs replacing suspension cables and girders. An explicitformula for the spring stiffness was derived for the horizontal stiffness at the end of an inclinedmain cable and comparedwith Ernst’s formula aswell as finite-element analysis results. For the analysis of the model, dead and live loads acting on the girders of each span were replaced withhorizontal and vertical forces calculated using the deflection theory and placed on the tops of the support towers. Equilibrium equations werederived from free-body diagrams for each tower and their solutions were obtained from boundary conditions at the ends of the towers. Thisresulted in the formulation of nonlinear load-displacement equations in matrix form for the simplified bridge model. An example four-spansuspension bridgewas analyzed for displacements at the tops of the towers andmoment reactions at the bottoms of the towers under six live-loadcases. A comparison of the model predictions with a FEM analysis showed excellent agreement. The results prove that the proposed analysismethod can be a useful alternative for the preliminary analysis and design of suspension bridge towers. DOI: 10.1061/(ASCE)BE.1943-5592.0000551. © 2013 American Society of Civil Engineers.

Author keywords: Suspension bridge; Tower; Preliminary analysis; Equivalent cable spring; Cable stiffness.

Introduction

The ability to analyze the behavior of suspension bridges has greatlyimproved with the rapid development of computational analysis forcable-supported structures. In particular, nonlinear FEM analysis isrequired for the design and analysis of modern suspension bridges(Malik 2006; Deng and He 2010). The preliminary design ofa suspension bridge requires numerous FEM analyses to determineoptimal design parameters. However, with increasing structural size,nonlinear analysis becomes more computationally intensive and thepreparation of input data for alternative designs can become quitetime consuming (Cheng and Xiao 2006). Therefore, a simplermethod for the analysis of suspension bridges is required for pre-liminary design.

Gimsing (1997) proposed various simplified design methods forstructural members of suspension bridges such as towers, cables, andstiffened girders under various load conditions. Gimsing (1997) alsoconducted various parametric studies and found useful results for theeffective design of suspension bridges. Grigorjeva et al. (2010)presented a simplified and adaptable procedure to provide design aidsfor suspension system evaluations and practical recommendations.Cobo del Arco and Aparicio (2001) presented dimensionless charts

from their analysis results using the deflection theory. These proposedcharts and explicit formulas for design are useful for understanding thestatic behavior of single-span suspension bridges under live loads. Inaddition, Decan (2007) discussed the stiffness characteristics ofa cable-supported bridge by using the concept of geometric config-uration stiffness.

Determining the stiffness of the main cable is important in theanalysis of suspension bridges (Gimsing 1997; Yoshida et al. 2004).For this reason, many researchers have studied cable stiffness. Ernst(1965) defined cable stiffness as the sum of the elastic and geometricstiffnesses. Irvine (1981) derived a closed-form expression for theflexibility matrix of an inclined cable to quantify the stiffness.Gimsing (1997) derived an exact solution for the tangent geometricstiffness of a horizontal cable and also derived approximateexpressions for the secant and tangent geometric stiffnesses of aninclined cable by using an approximation similar to Ernst (1965).While these approximations are sufficient for taut and nearly hor-izontal cables, errors may exist when the cable is slack or when thecable inclination is significant. Yoshida et al. (2004) defined thehorizontal stiffness of a cable as the ratio of the infinitesimal changein horizontal force to that in the length of the cable. However, elasticelongation of the cable was neglected in this derivation. DerKiureghian and Sackman (2005) derived an exact formula for thetangent geometric stiffness of an inclined cable under self-weight viathe catenary equation. They compared their formula to the Ernst(1965) approximate formula and showed how the accuracy ofErnst’s formula deteriorates with increasing slackness or cable in-clination. However, the previous studies all included a fundamentalproblem; i.e., the equivalent cable stiffness was calculated by as-suming a series connection of the elastic and geometric stiffnesses.This concept does not consider parameters resulting from thecoupling of the elastic and geometric stiffnesses.

The purpose of this paper is to propose a new, simplified methodfor the preliminary design of towers in suspension bridges under staticloads. In thismethod, the equivalent suspensionbridgemodel consistsof towers and equivalent springs that replace suspension cables and

1Professor, Dept. of Civil and Environmental Engineering, HanyangUniv., 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Korea. E-mail:samga@hanyang.ac.kr

2Ph.D. Candidate, Dept. of Civil and Environmental Engineering,Hanyang Univ., 17 Haengdang-dong, Seongdong-gu, Seoul 133-791,Korea. E-mail: verysoon@hanyang.ac.kr

3Research Professor, Dept. of Civil and Environmental Engineering,HanyangUniv., 17Haengdang-dong, Seongdong-gu, Seoul 133-791,Korea(corresponding author). E-mail: saintna@hanyang.ac.kr

Note. This manuscript was submitted on October 15, 2012; approved onSeptember 6, 2013; published online on September 9, 2013. Discussionperiod open until May 4, 2014; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Bridge Engineering,© ASCE, ISSN 1084-0702/04013007(12)/$25.00.

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girders. The stiffness of the equivalent spring is derived for a sus-pension cable and is compared with the Ernst (1965) formula, theresults of Der Kiureghian and Sackman (2005), and the FEMresults. Dead and live loads acting on the girders of each span arereplaced in the model with horizontal and vertical forces acting onthe tops of each tower, which are calculated using the deflectiontheory. Equilibrium equations, found by using free-body diagrams,are then derived for each tower. Finally, the equivalent suspensionbridge model is analyzed by solving a matrix-based equationdescribing the load-displacement relationships of the towers. Anexample four-span suspension bridge is analyzed to comparedisplacements at the tops of towers and moment reactions at thebottoms of the towers with the results of the proposed method andthe FEM analysis.

Equivalent Suspension Bridge Model

Fig. 1 shows a four-span suspension bridge and its equivalentanalysis model. In this model, the girders and main cables arereplaced with equivalent springs (kc), while the dead (d) and live (q)loads acting on each span are replaced with horizontal (H) andvertical (P) forces acting on the tops of the towers. In Fig. 1,EtIt is theflexural rigidity of the towers; EcAc is the axial rigidity of the maincable; EgIg is the flexural rigidity of the girder; L is the span length; his the height of towers; and f is the sag of the main cable.

For the model to work, the stiffness of the equivalent spring mustinclude the following cable characteristics: the horizontal force ofthe suspension cable, the length of the span, the sag, an axial stiffnessof the cable, and an inclined angle with its variation. This studypresents a new equation for the horizontal stiffness of the equivalentspring that considers these parameters. The horizontal and verticalforces acting on the tops of the towers are calculated to include theeffect of the dead and live loads acting on the girders, the flexuralrigidity of the girders, and the self-weight of the main cable andhangers. The details are described in the subsequent sections.

After determining the stiffness of the equivalent springs andsumming the vertical loads applied on the towers, force equilibrium

equations (from the free-body diagrams) for each tower are derivedby considering the flexural rigidity of the towers. These equationsare then solved by applying appropriate boundary conditions at eachtower. These derived equilibrium equations are written in matrixform; thus, they can be solved by assuming a nonlinear load-displacement relationship. The solutions are then found using themidpoint Runge-Kutta method.

Horizontal Stiffness of the Inclined Cable

Previous research for determining the cable stiffness by Ernst (1965)assumed a parabolic shape of the inclined cable and then evaluatedthe cable stiffness (kc) as a series connection of elastic and geometricstiffnesses as follows:

1kc

¼ 1ke

þ 1kg

¼ LEA

þ d2L3

12H3 (1)

where L 5 horizontal length of a cable; EA 5 axial rigidity; d5 horizontally distributed load; H 5 horizontal tension of a cable;and ke and kg 5 elastic and geometric stiffnesses, respectively.Eq. (1) can be rewritten as

kc ¼ 1ð1=keÞ þ

�1=kg

� ¼ 1ðL=EAÞ þ ðd2L3=12H3Þ (2)

Eq. (2) assumes that the direction of the force is the same as thedirection of the cable chord. However, the cable stiffness needs toincorporate an inclination angle and its variation.

Fig. 2 demonstrates the variation of an inclined cable shape undera uniform load (d) with a small increase in the horizontal force. Thedotted line shows the initial shape of the cable under a horizontalcable force H. The solid line shows the cable shape after the hor-izontal cable force is increased toH1 dH. In Fig. 2,a is the angle ofthe cable; f is the sag of the cable; h is the cable height; S is thecable length; H is the horizontal force; L is the horizontal length ofthe cable;DS is the increase in cable length; da is the variation of the

Fig. 1. Four-span suspension bridge and its equivalent analysis model

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cable angle; and dH and dL are the increments of the horizontal forceand length of the cable, respectively.

In this study, the horizontal cable stiffness, kc, is defined asfollows as the ratio of infinitesimal change in horizontal force to thatin the horizontal length of a cable:

kc ¼ dHdL

(3)

If the shape of the cable in Fig. 2 is assumed to be parabolic, theequation for the cable shape is expressed as (Irvine 1981)

y ¼ x tanaþ 4fL2

xðx2LÞ (4)

The length of the cable, S, can be calculated by

S ¼ðL

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

�dydx

�2rdx ≅ L

cosa

�1þ 8

3

�fL

�2cos4 a

�

¼ Lcosa

1þ d2L2

24H2 cos4 a

(5)

The elastic elongation of the cable, DS, can be calculated by

DS ¼ HEA

ðL

0

�1þ

�dydx

�2�dx ¼ HL

EA

�1þ 16

3

�fL

�2þ tan2 a

�

¼ HLEA

1þ d2L2

12H2 þ tan2 a

(6)

whereE5 elastic modulus of the cable; andA5 cross-sectional areaof the cable.

By total differentiation, Eq. (5) can be expressed as

dS ¼ dSdL

dLþ dSdH

dH þ dSda

da

¼

1cosa

þ d2L2

8H2 cos3 a

dL2

d2L3

12H3 cos3 a

dH

þ

sinacos2 a

L2 d2L3

8H2 cos2 a sina

da (7)

Eq. (7) can be rearranged as follows:

dH

dL¼ ð1=cosaÞ þ ½ðd2L2=8H2Þcos3 a�

ðdS=dHÞ2 ½ðsina=cos2 aÞL2 ðd2L=8H2Þcos2 a sina�ðda=dHÞ þ ðd2L3=12H3Þcos3 a (8)

The final length of the cable influenced by the horizontal force H is S0 1DS, and the unstrained length of the cable (S0) is constant. Dif-ferentiating Eq. (6) with respect to the horizontal force H results in

dSdH

¼ ∂DS∂H

dHdH

þ ∂DS∂L

dLdH

þ ∂DS∂a

dadH

¼ LEA

2 d2L3

12EAH2 þLEA

tan2 aþHEA

þ d2L2

4EAHþ H tan2 a

EA

dLdH

þ�2HLEA

sinacos3 a

�dadH

(9)

Based on the geometric relationship of the cable

tana ¼ hL

(10)

and the rate of change in the cable angle with increasing horizontal force H can be written as

dadH

¼ 2 sin2 aL tana

dLdH

(11)

Eqs. (9) and (11) can be substituted into Eq. (8) to determine dH=dL, which is the horizontal cable stiffness, as defined in Eq. (3)

kc ¼ cosaþ ½ðd2L2=8H2Þcos3 að1þ sin2 aÞ�2 ½ðH=EAÞð12 tan2 aÞ�2 ðd2L2=4EAHÞ½ðL=EAÞð1þ tan2 aÞ� þ ½ðd2L3=12H3Þcos3 a2 ðd2L3=12EAH2Þ� (12)

Fig. 2. Variation of an inclined cable with an increase dH in thehorizontal force

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If the angle of the cable, a, is zero for the horizontal cable, Eq. (12)reduces to

kc ¼1þ �

d2L2=8H2�½12 ð2H=EAÞ�2 ðH=EAÞ

ðL=EAÞ þ ðd2L3=12H3Þ½12 ðH=EAÞ� (13)

For taut and flat cables, d2L2=8H2 and H=EA can be ignored forspecific cables with small strains, such as steel cables (Irvine 1981).Hence, Eq. (13) attains the same form as Eq. (2).

To verify the proposed equation for horizontal cable stiffness,a FEM analysis is performed for a single cable under a uniformlydistributed load, as shown in Fig. 3. The displacement producedby the horizontal force H is calculated at the right end of the cable.Next, the horizontal cable stiffness is determined by the force-displacement relationship. In the verification models, the Young’smodulus of the cable is 2:03 108 kN=m2; the cross-sectional areais 0:1166m2; and the magnitude of the distributed load acting onthe cable is 88 kN=m, obtained from the self-weight of the cable,hangers, and girders. The cables are modeled by 31 and 11 trusselements for the center and side spans, respectively, with Ernst’sequivalent elasticmodulus using theMIDASCivil software (MIDASInformation Technology Company 2012). Finally, a geometricallynonlinear analysis considering large deflection is performed usingthe Newton-Raphson method with the convergence criteria of thedisplacement norm of 0.0001.

Fig. 4 shows plots of the horizontal cable stiffnesses for the centerand side spans calculated using the proposed equation, Ernst (1965)formula, Der Kiureghian and Sackman (2005) improved approxi-mate formula, and FEM analysis. From these results, it can beobserved that the cable stiffness of the proposed equation at thecenter span matches the solutions from the Ernst formula, DerKiureghian and Sackman formula, and FEM analysis, as shown inFig. 4(a). In the case of the cable at the side span, shown in Fig. 4(b),there is a difference between the results fromErnst’s formula and theother methods because the horizontal force is large. This error inErnst’s formula is present because it disregards the variation of theangle of the cable slope (a). In this study, Eqs. (12) and (13) are usedto determine the horizontal cable stiffness in the equivalent sus-pension bridge model.

Horizontal and Vertical Forces on the Tops of Towers

All of the loads of a suspension bridge, except the self-weight of thetower, are transferred to the tops of the towers through the mainsuspension cables. In the equivalent suspension bridge model, thehorizontal and vertical forces are calculated from the reaction forces

Fig. 3. Verification models for cable stiffness: (a) center span; (b) side span

Fig. 4. Comparison of horizontal cable stiffnesses: (a) center span;(b) side span

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at the supports by analyzing the suspension cable of each span usingthe deflection theory. Thus, the sum of the reaction forces from twoadjacent spans is the horizontal and vertical forces acting on a giventower. The following assumptions are made for the derivation of thedifferential equation describing the suspended stiffening girder: (1)the shape of the suspension cable is parabolic under dead load; (2)the dead load is uniform along the girders and carried solely by thesuspension cable; and (3) the hangers are continuously distributedalong the girders and are inextensible (Wollmann 2001).

From the assumption that the geometric shape of the suspensioncable under dead load is parabolic, as illustrated in Fig. 5, theequation of the geometric shape of the cable is derived as follows(Irvine 1981; Lewis 2003):

y ¼ x tanaþ 4fL2

xðL2 xÞ

y9 ¼ tanaþ 4fL2

ðL2 2xÞ (14b)

y99 ¼ 28fL2

(14c)

(14a)

where y 5 cable ordinate; y9 5 cable slope; and y99 5 cablecurvature.

Fig. 6 shows a suspended stiffening girder and the dead (d) andlive (q) loads acting on the suspension cable and the girder. As-suming that the dead load is uniform along the girder and onlycarried by the suspension cable, the basic relationship for the cableunder dead load is

Hdy99þ d ¼ 0 (15)

From equilibrium conditions on a differential cable element, theequation for the suspension cable under dead and live loads isexpressed as follows (Ulstrup 1993; Wollmann 2001):

�Hd þ Hq

��y99þ v99

�þ ðd þ sÞ ¼ 0 (16)

where d 5 uniform dead load; s 5 distributed force in the verticalhanger cables attributed to the live load per unit length acting onthe girder; Hd 5 horizontal cable force under dead load d; Hq

5 horizontal cable force under live load q; and v 5 vertical dis-placement of the main cable and the girder under the live load.

Furthermore, the differential equation of a beam under a distrib-uted load is derived from the load condition of the girder, as shown inFig. 6, and is given by

EgIgviv ¼ q2 s (17)

where EgIg 5 flexural stiffness of the girder; and q 5 live load.Eqs. (15) and (16) can be substituted into Eq. (17) to give

EgIgviv 2

�Hd þ Hq

�v99 ¼ qþ Hqy99 (18)

Eq. (18) has the same form as the governing equation of a simplebeam with axial tension (Hd 1Hq) under vertical load(q2Hq8f =L2), as shown in Fig. 7.

The solution of the vertical displacement of a simple beamwith axialtension N ð5Hd 1HqÞ under vertical load p ð5 q2Hq8f =L2Þ, asshown in Fig. 8, is expressed as (Rubin andVogel 1982; Peterson 1993;Wollmann 2001)

vðxÞ ¼�1e2

cosh½eð0:52 jÞ�

coshðe=2Þ 2 1

þ jj9

2

�pL2

N(19)

where e5LffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiN=EgIg

p; and j5 x=L and j95 12 x=L are the di-

mensionless distances from the left- and right-side supports for spanlength L, respectively.

To solve Eq. (19), horizontal cable force Hq must be known. Acondition to determine it is provided by the compatibility re-quirement that the horizontal projection of the change in the cable

Fig. 5. Geometric shape of a cable

Fig. 6. Suspended stiffening girder and loads acting on the cable and the girder

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length as a result of the live load equals the change in the horizontaldistance between the cable endpoints, as shown in Fig. 9. Thecompatibility equation is given by (Wollmann 2001)

DL ¼ Hq

EcAc

ðL

0

�1þ y92

�3=2dxþ y99

ðL

0

vðxÞdx ¼ dR 2 dL (20)

whereDL and EcAc 5 horizontal elongation and axial rigidity of thecable, respectively; and dR and dL 5 displacements at the right andleft supports of the cable, respectively.

A four-span suspension bridge is then modeled, as shown inFig. 1, where the stiffening girders are continuous at the towers andtheir rigidity is constant for each span. Dead and live loads areuniform over the full length of the span. Fig. 10 illustrates the loadingconditions and support reactions as a result of the dead and live loadsfor each span. Using the concept of the analogy between a suspendedgirder and a simple beam under tension, the horizontal forces, Hd andHq, as a result of the dead and live loads in each beam are calculatedby Eqs. (15) and (20) with Eq. (19), respectively (Choi et al. 2013),where dR and dL are set to zero because cables of suspended stiff-ening girders in each span are supported by immovable hinges.

The vertical force acting on the tops of the towers in the equiv-alent suspension bridge model is determined by multiplying thehorizontal force by the cable slope at the top. Tables 1–3 summarizethe calculation procedure for the vertical and horizontal forces in theequivalent suspension bridge model, which are determined by

analyzing a single-span suspended girder using the deflection the-ory. From this analysis, the horizontal and vertical reactions at thesupports of each span are obtained. The sum of these reaction forcesat two adjacent spans is used as the horizontal and vertical forcesacting on the tops of each tower in the equivalent suspension bridgemodel, as shown in Fig. 1.

Equilibrium Equations in the Equivalent SuspensionBridge Model

Fig. 11 shows the forces on the free-body diagrams for each tower ofthe equivalent suspension bridgemodel. The termsPn andHn are thevertical and horizontal components, respectively, of the cable forceacting on the top of the towers. The term kcn is the cable springconstant; hn is the height of each tower; dn is the horizontal dis-placement at the top of each tower; andMn is the moment acting ona specific point that is at a distance x from the bottom of each tower.Each tower can be analyzed as beam column members subjected toaxial and lateral loads.

From Fig. 11, the equilibrium equation of the left tower is

E1I1y99þ P1y ¼ P1d1 þ ðh12 xÞ½H1 þ kc2ðd22 d1Þ2 kc1d1�(21)

The general solution of Eq. (21) is expressed as

y ¼ A sin k1xþ B cos k1xþ d1

þ 1P1

ðh12 xÞ½H1 þ kc2ðd22 d1Þ2 kc1d1� (22)

where k1 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP1=E1I1

p. Constants A and B are solved using the

boundary conditions of yð0Þ5 0 and y9ð0Þ5 0. Hence, Eq. (22) canbe expressed as

y ¼�

1k1P1

½H1 þ kc2ðd2 2 d1Þ2 kc1d1��sin k1x

þ�2d12

h1P1

½H1 þ kc2ðd22 d1Þ2 kc1d1��cos k1xþ d1

þ 1P1

ðh12 xÞ½H1 þ kc2ðd22 d1Þ2 kc1d1�(23)

Eq. (23) can be expressed differently by using the boundary con-dition of yðh1Þ5 d1 as

�2kc2 2 kc1

k1P1

sin k1h1 þ

�21þ h1

P1ðkc2 þ kc1Þ

�cos k1h1

�d1

þ

kc2k1P1

sin k1h12h1kc2P1

cos k1h1

d2

¼2 1k1P1

sin k1h1 þ h1P1

cos k1h1

H1

(24)

The equilibrium equation of the center tower is

E2I2y99þ P2y

¼ P2d2 þ ðh22 xÞ½H2 þ kc3ðd3 2 d2Þ2 kc2ðd22 d1Þ� (25)

The general solution of Eq. (25) is expressed as

Fig. 7. Analogy to a simple beam for the suspended stiffening girder

Fig. 8. Simple beam with axial force under vertical load

Fig. 9. Compatibility conditions for a cable

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y ¼ A sin k2xþ B cos k2xþ d2

þ 1P2

ðh22 xÞ½H2 þ kc3ðd3 2 d2Þ2 kc2ðd22 d1Þ� (26)

where k2 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP2=E2I2

p. Constants A and B are determined by the

boundary conditions of yð0Þ5 0 and y9ð0Þ5 0. Hence, Eq. (26) canbe expressed as

y ¼ 1k2P2

½H22 kc2ðd22 d1Þ þ kc3ðd3 2 d2Þ�sin k2x

2

�d2 þ h2

P2½H22 kc2ðd22 d1Þ þ kc3ðd3 2 d2Þ�

�cos k2x

þ d2 þ 1P2

ðh22 xÞ½H2 þ kc3ðd32 d2Þ2 kc2ðd22 d1Þ� (27)

Fig. 10. Loading conditions and the support reactions for each span

Table 1. External Forces Acting on the Top of the Left Tower

Load Direction Equation

Dead load Horizontal direction Hd 5Hd2 2Hd1 5 dL22=ð8f2Þ2 dL21=ð8f1Þ5 0Vertical direction Pd1 5Pd11 1Pd12 5Hd1 3 y19ðL1Þ1Hd2 3 y29ð0Þ

Live load Horizontal direction Hq 5Hq2 2Hq1

Vertical direction Pq1 5Pq11 1Pq12 5Hq1 3 y19ðL1Þ1Hq2 3 y29ð0ÞSum Horizontal direction H1 5Hd 1Hq 5Hq2 2Hq1

Vertical direction P1 5Pd1 1Pq1

Table 2. External Forces Acting on the Top of the Center Tower

Load Direction Equation

Dead load Horizontal direction Hd 5Hd3 2Hd2 5 dL23=ð8f3Þ2 dL22=ð8f2Þ5 0Vertical direction Pd2 5Pd22 1Pd23 5Hd2 3 y29ðL2Þ1Hd3 3 y39ð0Þ

Live load Horizontal direction Hq 5Hq3 2Hq2

Vertical direction Pq2 5Pq22 1Pq23 5Hq2 3 y29ðL2Þ1Hq3 3 y39ð0ÞSum Horizontal direction H2 5Hd 1Hq 5Hq3 2Hq2

Vertical direction P2 5Pd2 1Pq2

Table 3. External Forces Acting on the Top of the Right Tower

Load Direction Equation

Dead load Horizontal direction Hd 5Hd4 2Hd3 5 dL24=ð8f4Þ2 dL23=ð8f3Þ5 0Vertical direction Pd3 5Pd33 1Pd34 5Hd3 3 y39ðL3Þ1Hd4 3 y49ð0Þ

Live load Horizontal direction Hq 5Hq4 2Hq3

Vertical direction Pq3 5Pq33 1Pq34 5Hq3 3 y39ðL3Þ1Hq4 3 y49ð0ÞSum Horizontal direction H3 5Hd 1Hq 5Hq4 2Hq3

Vertical direction P3 5Pd3 1Pq3

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Using the boundary condition of yðh2Þ5 d2, Eq. (27) is expressed as

kc2k2P2

sin k2h2 2h2kc2P2

cos k2h2

d1 þ

�2kc22 kc3

k2P2

sin k2h2

þ�21þ h2

P2ðkc2 þ kc3Þ

�cos k2h2

�d2

þ

kc3k2P2

sin k2h22h2kc3P2

cos k2h2

d3

¼2 1k2P2

sin k2h2 þ h2P2

cos k2h2

H2

(28)

The equilibrium equation of the right tower is

E3I3y99þ P3y ¼ P3d3 þ ðh32 xÞ½H3 þ kc3ðd32 d2Þ2 kc4d3�(29)

The general solution of Eq. (29) is expressed as

y ¼ A sin k3xþ B cos k3xþ d3

þ 1P3

ðh3 2 xÞ½H3 þ kc3ðd32 d2Þ2 kc4d3� (30)

where k3 5ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP3=E3I3

p. Constants A and B are calculated using the

boundary conditions of yð0Þ5 0 and y9ð0Þ5 0. Thus, Eq. (30) canbe expressed as

y ¼�

1k3P3

½H3 þ kc3ðd3 2 d2Þ2 kc4d3��sin k3x

þ�2d32

h3P3

½H3 þ kc3ðd3 2 d2Þ2 kc4d3��cos k3x

þ d3 þ 1P3

ðh32 xÞ½H3 þ kc3ðd3 2 d2Þ2 kc4d3� (31)

Applying the boundary condition of yðh3Þ5 d3, Eq. (31) isexpressed as

kc3k3P3

sin k3h3 2h3kc3P3

cos k3h3

d2 þ

�2kc42 kc3

k3P3

sin k3h3

þ�21þ h3

P3ðkc3 þ kc4Þ

�cos k3h3

�d3

þ

kc3k2P2

sin k2h22h2kc3P2

cos k2h2

d3

¼2 1k3P3

sin k3h3 þ h3P3

cos k3h3

H3

(32)

The matrix-form solution, consisting of horizontal forces and hori-zontal displacements at the tops of the towers, can be compiled fromEqs. (24), (28), and (32) as

Fig. 11. Forces on the free-body diagram for each tower of the equivalent suspension bridge model

Fig. 12. Geometric and material properties of the 2D suspension bridge model for the New Millennium Bridge

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Fig. 13. FEM model and the dimensions of the members (unit: m)

Table 4. Geometric and Material Properties in the FE Model

Member Area A ðm2ÞTorsional constant

J ðm4ÞMoment of inertia

Young’s modulusE (MPa)

Shear modulusG (MPa)

Unit weightw ðkN=m3ÞIy ðm4Þ Iz ðm4Þ

Suspension cable 0.05828 — — — 200,000 — 77.0Hanger 0.003097 — — — 140,000 — 82.5Girder 0.5465 1.1414 0.4879 9.0389 210,000 80,769 77.0Center tower (top) 12.6000 30.6914 26.0050 19.8695 28,571 13,235 24.5Center tower (bottom) 47.1232 690.2120 660.9390 368.7350 28,571 13,235 24.5Side tower (top) 12.6000 30.6914 26.0050 19.8695 28,571 13,235 24.5Side tower (bottom) 21.8000 160.8990 102.5510 120.5540 28,571 13,235 24.5Cross beam (top) 1.5000 52.2667 34.0625 41.2500 28,571 13,235 24.5Cross beam (bottom) 2.1960 104.3941 43.2400 184.7055 28,571 13,235 24.5

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8<:

H1

H2

H3

9=; ¼

24 K1 2kc2 02kc2 K2 2kc30 2kc3 K3

358<:

d1d2d3

9=; (33)

where

K1 ¼ ð2kc12 kc2Þsin k1h1 þ k1ð2P1 þ kc1h1 þ kc2h1Þcos k1h1sin k1h1 2 k1h1 cos k1h1

K2 ¼ ð2kc22 kc3Þsin k2h2 þ k2ð2P2 þ kc3h2 þ kc2h2Þcos k2h2sin k2h2 2 k2h2 cos k2h2

K3 ¼ ð2kc42 kc3Þsin k3h3 þ k3ð2P3 þ kc4h3 þ kc3h3Þcos k3h3sin k3h3 2 k3h3 cos k3h3

kcn ¼cosan þ

�d2nL

2n=8H

2n

�cos3 an

�1þ sin2 an

�2 ðHn=EAnÞ

�12 tan2 an

�2�d2nL

2n=4EAnHn

�ðLn=EAnÞð1þ tan2 anÞ þ

�d2nL

3n=12H

3n

�cos3 an2

�d2nL

3n=12EAnH2

n

� ðn ¼ 1, 2, 3, 4Þ

The midpoint Runge-Kutta method is used to obtain displace-ments from Eq. (33). Furthermore, the moment reaction Mrn at thebottom of each tower is obtained by the following equations:

Mr1 ¼ P1d1 þ H1h1 þ kc2ðd22 d1Þh1 2 kc1d1h1Mr2 ¼ P2d2 þ H2h2 þ kc3ðd32 d2Þh22 kc2ðd2 2 d1Þh2 (34b)

Mr3 ¼ P3d3 þ H3h32 kc3ðd3 2 d2Þh3 þ kc4d3h3 (34c)

(34a)

Numerical Model

The New Millennium Bridge (ENVICO Consultants Company2010), a four-span suspension bridge inKorea,was used to verify theproposed method. Fig. 12 shows the geometric and materialproperties of the two-dimensional (2D) suspension bridge model.The lengths of the center spans and the two side spans are 650 and225m, respectively. The type of anchorage for this bridge is an earth-anchored system, and the cable arrangement follows a 2D double-cable system. The heights of the side and center towers are 145.45and 160.09m, respectively, and the sags for the side and center spansare 9.9 and 81 m, respectively. The dead load per unit length, d, iscalculated by considering all types of dead loads, such as the self-weight of the girders, main cable, average length of the hangers, andadditional dead loads. TheYoung’smodulus,moment of inertia, andarea of the members are shown in Fig. 12. The moment of inertia foreach tower is assumed constant along its longitudinal direction,while the value in the finite-element (FE) model varies along theheight. The moment of inertia for the tower is determined from thehorizontal stiffness that gives the same deflection because of a unitload applied at the top of the tower in the FE model.

Fig. 13 shows the three-dimensional numerical model and thedimensions of the members in the FE model for MIDAS Civil(MIDAS 2012) used to verify the proposed method. Table 4 showsthe geometric andmaterial properties in the FEmodel. The stiffeninggirders in the numerical model were modeled with beam elementspassing through the centroids of the girder sections. The anchorpoints of the hangers at the stiffening girders were connected to thecentral beam by imaginary rigid beams. Because the cross-sectional

properties of the real towers vary with height, the model divided thetowers into several beam elements with cross-sectional averagesapplied to each element. In the actual bridge, the cross beams of thetowers support the stiffening girders by vertical elastic bearings; thisconnection is represented in the model by beam elements. Thesuspension cable and the hangers were modeled with truss elementsusing Ernst’s equivalent elastic modulus. All beam elements werebased on the Timoshenko beam theory, which considers shear de-formation effects. In the model, there were 788 nodes in total, 180truss elements for the suspension cables, 160 truss elements for thehangers, 136 beam elements for all towers, and 278 beam elements forthe girders. Finally, a geometrically nonlinear analysis consideringlarge deflection was performed using the Newton-Raphson methodwith the convergence criteria of a displacement norm of 0.0001.

Six live-load cases (LC1–LC6) were considered, as shown inFig. 14. The live-load value of 38.1 kN/m was determined based onthe design specifications for Korean highway bridges (KoreanMinistry of Land Transport and Maritime Affairs 2005). Table 5shows the horizontal forces of the cables for each span under thedead load and six live-load cases. The horizontal forces initiallywere73,300 kN under the dead load and increased up to the maximumvalue of 96,986 kN under several live-load cases. These resultssupport the use of the proposed formula as well as theDer Kiureghian and Sackman (2005) equation. However, Ernst’s

Fig. 14. Six live-load cases (38:1 kN=m)

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formula in this range of horizontal forces resulted in large errors inthe horizontal cable stiffness for the side span, as previously shownin Fig. 4(b). Table 6 lists the external forces acting on the tops oftowers in the equivalent suspension bridge model for the six live-load cases.

Figs. 15–17 show the displacements and moment reactions ineach tower for the six live-load cases. The results of the FEManalysis and the proposed method are in good agreement for all sixcases. For the left tower, the maximum displacement and momentreaction occurred under LC6. For the center tower, the maximumdisplacement and moment reaction occurred under LC3. For theright tower, the maximum displacement and moment reaction oc-curred under LC2.

Conclusions

This paper proposes a simple analysis model (equivalent suspensionbridge model) for the preliminary design of towers in suspensionbridges. In this model, a suspension bridge is composed of supporttowers and equivalent springs to replace the suspension cables andgirders. A new analytical equation for the horizontal cable stiffnessis derived, and then compared with the results from the FEM anal-yses and Ernst’s formula. The horizontal cable stiffness at the centerspan is in good agreement with the estimations via the Ernst (1965)formula, Der Kiureghian and Sackman (2005) equation, and FEManalysis. However, with the side span, as the horizontal forces in-crease, Ernst’s formula overestimates the horizontal cable stiffness

Table 5. Horizontal Forces of the Cables

Load case

Horizontal cable force (kN)

Left side span(Hd1 or Hd1 1Hq1)

Left main span(Hd2 or Hd2 1Hq2)

Right main span(Hd3 or Hd3 1Hq3)

Right side span(Hd4 or Hd4 1Hq4)

Dead 73,300 73,300 73,300 73,300LC1 89,615 96,986 96,986 89,615LC2 89,615 96,986 73,300 73,300LC3 89,615 96,986 73,300 73,300LC4 89,615 73,300 73,300 73,300LC5 73,300 96,986 96,986 73,300LC6 73,300 96,986 73,300 73,300

Table 6. External Forces Acting on the Top of the Towers

Load case Direction

Force on towers (kN)

Left Center Right

LC1 Horizontal 7,371 0 27,371Vertical 79,943 80,853 79,943

LC2 Horizontal 7,371 0 223,686Vertical 79,943 80,853 71,198

LC3 Horizontal 7,371 223,686 0Vertical 79,943 73,351 59,391

LC4 Horizontal 216,315 0 0Vertical 68,135 57,238 59,391

LC5 Horizontal 23,686 0 223,686Vertical 71,198 80,853 71,198

LC6 Horizontal 23,686 223,686 0Vertical 71,198 69,045 59,391

Fig. 15. Displacements and moment reactions in the left tower: (a)displacements; (b) moment reactions

Fig. 16. Displacements and moment reactions in the center tower: (a)displacements; (b) moment reactions

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because it neglects the variation of the slope angle of the main cable.Dead and live loads acting on the girders of each span are replacedwith horizontal and vertical forces acting on the tops of each tower.Equilibrium equations are derived from free-body diagrams of eachtower. Solutions are obtained by applying boundary conditions at theends of each tower, leading to nonlinear load-displacement equa-tions in matrix form for the simplified suspension bridge model. Anexample four-span suspension bridge is analyzed using the proposedequivalent suspension bridge model and with a FEM model fordisplacements at the tops of the support towers and moment reac-tions at the bottoms of the towers under six live-load cases to verifythe accuracy of the proposed model. It is then found that the analysisresults from the proposedmodel are in good agreement with those ofthe FEM analysis. The key conclusions of this study can be sum-marized as follows:1. This study derived a horizontal stiffness equation for an

inclined cable under horizontal forces.2. Design parameters for support towers in suspension bridges

can be easily obtained using the proposed method, whereasFEM analysis is more time intensive, especially in the pre-liminary design stage.

3. This study presents a simple nonlinear matrix to analyzesupport towers in a suspension bridge, with results varyingby less than 5% from a FEM analysis.

4. The proposed equivalent suspension bridge model providesa simple alternative to aid in the design of support towers inpractical applications for the design of suspension bridges. Theresults of this study can be extended to the design of founda-tions, splay saddles, and anchorages.

Acknowledgments

This work was a part of a research project supported by the KoreanMinistry of Land, Transport andMaritime Affairs (MLTM) throughCoreResearchProject 1of theSuperLongSpanBridgeR&DCenter.The authors express their gratitude for the financial support. Thisresearch was supported by the Basic Science Research Programthrough the National Research Foundation of Korea (NRF) fundedby the Ministry of Education, Science and Technology (NRF-2012R1A1A2007054).

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Fig. 17. Displacements and moment reactions in the right tower: (a)displacements; (b) moment reactions

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