New Procedure for Estimating Cable Force in Cable-stayed ...docs.trb.org/prp/13-2407.pdf · 1 New Procedure for Estimating Cable Force in Cable-stayed Bridge 2 3 4 Dong-Ho Choi, Ph.D.,

New Procedure for Estimating Cable Force in Cable-stayed Bridge 1

2

3

Dong-Ho Choi, Ph.D., Professor 4

Department of Civil and Environmental Engineering, Hanyang University 5

222 Wangsimni-ro, Seongdong-gu 6

Seoul 133-791, Korea 7

+82-2-2220-4155 (Phone) 8

+82-2-2220-4322 (Fax) 9

[email protected] 10

11

Wan-Soon Park *, Ph.D., Post-Doctoral Fellow 12

Department of Civil and Environmental Engineering, Hanyang University 13

222 Wangsimni-ro, Seongdong-gu 14

Seoul 133-791, Korea 15

+82-2-2220-4155 (Phone) 16

+82-2-2220-4322 (Fax) 17


19

Hani Nassif, Ph.D., P.E., Professor 20

Department of Civil and Environmental Engineering 21

Rutgers, The State University of New Jersey 22

623 Bowser Road, Piscataway, NJ 08854 23

(848)-445-4414 (Phone) 24

(732)-445-0577 (Fax) 25


27

28

* corresponding author 29

Submission date : July 30, 2012 30

Word count :7,142 words (4,892 words for text + 2,250 words for five figures and four tables) 31

32

TRB 2013 Annual Meeting Paper revised from original submittal.

Choi, Park, and Nassif 1

ABSTRACT 1

In structural health monitoring of the cable-stayed bridges, estimation procedure for the cable 2

tension influences the accuracy of monitoring system. Although some formulas are used in many 3

monitoring systems, the results from these formulas are not accurate because they neglect the 4

initial deflection, the natural frequency changing, and the initial curvature shortening of an 5

oscillating cable. This paper presents a new estimation procedure that considers these neglected 6

effects in the existing formulas and achieves the modified accuracy in estimating the tension 7

force in the cable. Considering the initial deflection of the cable at the static status derives a 8

clear explanation about the natural frequency changing for the change of the slenderness ratio of 9

the cable. And by considering the stretching force induced by the initial curvature shortening, the 10

additional forces acting on the supports of the cable conservatively are also investigated. In this 11

paper, the concept of total tension force is proposed to increase the accuracy in tension force 12

estimation for the cable. The re-estimation for the cable-stayed Alamillo Bridge, Spain shows 13

that the new procedure estimates tension force of the cables more accurately and that this can be 14

used as an alternative procedure in the structural health monitoring system. 15

16

Key Words: 17

Cable force 18

Cable-stayed bridges 19

Structural health monitoring 20

Tension force 21

22



INTRODUCTION 1

Over the last decade, the number of structural health monitoring (SHM) systems for cable-stayed 2

bridges has increased. These SHM systems have become more complicated (1, 2, 3) due to their 3

demand on understanding the behaviour of cable-stayed bridges. Until now, the exact estimation 4

of the cable tension force has been the main objective of these SHM systems. For this purpose, 5

some formulas were developed to incorporate the raw data collected from the SHM systems. But 6

the accuracy of the estimation procedure has remained as the unsolved problem. The simple 7

estimation formulas used in the vibration method for the bridge cable can be derived from the 8

transverse vibrations of a taut string with the assumption of no sag. However, bridge cables do 9

not behave as taut strings because of their flexural rigidities. The flexural rigidity is extremely 10

important in calculating the tension force of bridge cables just as it was investigated by many 11

Japanese researchers. 12

Shinke et al. (4) proposed their first estimation formula that can account for the flexural 13

rigidity effects in tension force calculations; this formula is very practical, but the flexural 14

rigidity of the deflected cable is needed in the calculation. For more practical use, Zui et al. (5) 15

proposed a modified formula in which the first, second, and high natural frequencies of deflected 16

cables can be used. Utsuno et al. (6) of Kobe Steel, Ltd. also proposed an estimation method that 17

determines the tension force and the flexural rigidity of deflected cables simultaneously. This 18

method is based on the same partial differential equations used by Zui et al. but it simplifies the 19

solution by assuming the boundary conditions. Since publication, this method has been widely 20

used owing to its convenience and practicality. Outside of Japan, Robert et al. (7) proposed a 21

simple estimation formula that considers the flexural rigidity in calculating cable tension, and 22

Mehrabi and Tabatabai (8) modified the aforementioned formula to consider both flexural 23

rigidity and sag effects in bridge cables. Ren et al. (9) proposed a practical formula for rapid 24

calculation considering flexural rigidity and the sag effect. In their formula, the estimation 25

procedure offers one of two cases: considering only the cable sag effect and considering only the 26

cable bending stiffness effect. To estimate the cable tension only using the first natural frequency, 27

Kim et al. (10) derived a new formula by using the variational method. 28

To correct the difference between the calculated results by the existing formulas and the 29



measured data from field tests, Ahn (11) proposed compensating formulas. To increase the 1

accuracy of the cable tension estimates, Park et al. (12) proposed the system identification theory 2

using the finite element method. For the thermal axial force or thermal bending moment applied 3

to the cable, Treyssède (13) proposed a new formula considering thermal stress effects. 4

Installation of a damper or tube at the support of a cable changes the characteristics of the 5

vibration. However, the existing formulas do not consider this effect. To solve this problem, Yan 6

et al. (14) proposed a simplified new formula that considers the support stiffness. This formula 7

has a similar form to Utsuno et al.'s formula. 8

The boundary condition of the cable is the factor that affects the natural frequency of the 9

cable. The existing formulas for estimating the tension force assume hinge support conditions. In 10

fact, the supports of a cable in a bridge are partially fixed. If the slenderness ratio of the cable 11

decreases, the support stiffness of the cable increases and the support conditions are closer to 12

that of the fixed support condition. However, if the slenderness ratio of the cable increases, the 13

support stiffness of the cable decreases and the support conditions are close to that of the hinge 14

support condition. Because the slenderness ratio of the cables in a bridge is greater than 200 in 15

most cases, the support stiffness of the cable is very small and the support condition of the cable 16

can be assumed to match the hinge support condition. 17

The free vibrations of axially loaded beams under gravity effects were first studied by 18

Shih et al. (15). They explained the vibration of axially loaded beam as the sum of static and 19

dynamic deflections. An approximate solution for the dynamic response was derived by 20

perturbation methods based on nonlinear partial differential equations, and the governing 21

equation for dynamic response in their study was very similar to the existing equation for cable 22

tension estimation. Hughes and Bert (16) found an error in the governing equation of Shih et al. 23

and proposed a new governing equation for the dynamic response, which has the same form as 24

the existing equation for cable tension estimation. Although Hughes and Bert proposed an exact 25

governing equation, they did not clearly explain the dynamic response in symmetric and anti-26

symmetric modes. 27

To consider the gravity effect, the nonlinear solution for the static deflection is needed to 28

transform the beam from the static status to the dynamic status. This problem is related to the 29



curvature shortening effect. If immovable ends are assumed at the supports, the curvature 1

shortening effect is restrained. Zaslavsky (17) investigated the curvature shortening behaviour 2

and the reaction at immovable ends for changes in the support position. 3

In this paper, a new practical procedure is developed to increase the accuracy in 4

estimation of the cable tension in the cable-stayed bridge. In the new, practical procedure, the 5

initial deflection of the cable by the gravity effect, which is not considered in the existing 6

formulas, is taken into account. By considering this effect, the dynamic characteristics of 7

nonlinear cable oscillations can be understood in depth. The problem of underestimating the 8

value of the cable tension when using the existing formulas results from ignoring a self-9

stretching force, which is induced from the boundary condition of the immovable ends is also 10

cleared. To verify the effectiveness of the new procedure, this paper included the re-estimation 11

of the tension forces for the cable-stayed Alamillo Bridge, Spain. For the re-estimation, the 12

research paper about the tension force estimation of the Alamillo Bridge's cable by Casas (18, 19) 13

was referenced. 14

15

CORRELATED NOLINEAR INFLUENCES 16

Initial Deflection 17

The differential equation for the static deflection of the bridge cable to account for the curvature 18

shortening effect can be described as follows (20): 19

� �24 2

4 20

d2

Ls s s

ad w dw d wEAEI N x q q gA

L dxdx dx�

� ��

(1)

where EI is the flexural rigidity of the undeflected cable, EA is the axial rigidity of the cable, L is 20

the undeflected length of the cable, Na is the pure applied axial load, q is the uniform weight per 21

unit length, � is the cable density, g is the gravity acceleration, and �s is the static deflection 22

when tension force is applied. The relation between the resultant axial load N and the pure 23

applied axial load Na is defined as follows: 24



2

0

d2

Ls

adwEAN N x

L dx� � � � ��

(2)

If Na becomes zero, then2

0

, d .2

Lo

s odwEAw w N x

L dx� � � �� and Equation (1) becomes 1

24 2

4 20

d2

Lo o od w dw d wEAEI x q

L dxdx dx

� ��

(3)

where �o is the static deflection when tension force is not applied, and we denote 2

2

0

d2

L

con rvo

sedwEAN x

L dx� � � ��

(4)

Equation (3) can be rewritten as follows: 3

4 2

4 2o o

conservd w d w

EI N qdx dx

�

(5)

If Na becomes very large, then 2

0

0, d 0.2

Ls

sdwEAw x

L dx� � ��

� � and Equation (1) becomes 4

4 2

4 2 ( )s sa a

d w d wEI N q N N

dx dx�

(6)

Under the immovable ends condition, the linear static deflection �s(�) becomes 5

� � � �2

sinh sinh2 2 2 2

2cosh

2

a a

sa a a

N NL x xEI EIq x L x EI qw x

N N NLEI

� � � ��

(7)

Changing of Nonlinear Natural Frequency 6

The total deflection of vibration �(�) consists of the initial static deflection �s(�) in Equation (7) 7

and the continuously changing dynamic deflection �d(�): 8



1 1

( ) ( ) ( ) sin( ) ( )sin( )s d i ni n

x xw x w x w x w i a t nL L� ��

� ��

(8)

where�i is the coefficient of the Fourier sine series for initial static deflection, and �n is the 1

coefficient of the Fourier series for dynamic deflections. To build the differential equation for the 2

vibration of the bridge cable, the inertia and dynamic stretching forces are added to Equation (6). 3

� � 24 2 2

4 2 20

,( , ) ( , ) ( , )d2

L

aw x tw x t EA w x t w x tEI N x A q

L xx x t�

� ��

(9)

For i=n, substituting Equation (8) in Equation (9) gives 4

2 31 2 3

1

''( ) ( ) ( ) ( ) 0n n n n n n nn

a t g a t g a t g a t�

� � � �

(10)

where 5

4 2 2

1 2 23 1 (for 1,3,5...)

4a i

nN L AwEI ng n

A L In EI�

� �

� �� (11)

4 2

2 2 1 (for 2,4,6...)aN LEI n n

A L n EI�

� �

� ��

4

23 (for 1,3,5...)

4i

nAwEI ng n

A L I�

��

� � � �� (12)

0 (for 2,4,6...)n

4

3 (for 1,2,3...)

4nEI n Ag n

A L I�

��

� � � �� (13)

By using the multiple scale method, the approximate solution of Equation (10) can be expressed 6

as follows (21): 7

� �2 2 2 2

32 2 00

1 1

cos(2 2 )cos( ) ( )

2 6n n n

n nn n

a g a g ta t a t O

g g� � � �

� � � ��

� � � �

(14)

where �n is the angular nonlinear natural frequency of the system and is calculated as follows: 8



� �2 2 23 1 2 3

1 21

9 101 ( )

24n n n

n nn

a g g gg O

g� �

��

(15)

where a is the parametric amplitude, � is the perturbation parameter, and O(�3) represents terms 1

equal to or higher than �3. By eliminating nonlinear terms from Equation (15), the angular linear 2

natural frequency �n0 is determined as follows: 3

0 1n ng� (16)

If EI is zero, Equation (15) results in the natural frequency of the taut string. For the nonlinear 4

natural frequency �n, the parameter a� is needed and dependent on the initial displacement and 5

velocity at the initial excitation. To determine a� , the initial displacement �n0 and initial velocity 6

�n0 of �n(t) are assumed as follows: 7

� � � �0 0 0 0 0 0 0cos , sinn n n n n n ns a v a� � � � (17)

where an0 and �n0 are the amplitude and the phase angle of the initial excitation, respectively. 8

Because of the disturbance due to nonlinear oscillation, the natural frequency of the system 9

changes from �n0 at the initial excitation to �n. Substituting the initial conditions of Equation (17) 10

in Equation (14) gives 11

� �2 21 0 1 0 0 2 0 2

0 1 0 0 20

2 20 0 2

3 8 43 3

(for 1,3,5...)(4 2 )

n n n n n n n nn n n n

n

n n n

g s g s a g s ga g s a

aa n

a s g�

� ��

� (18)

0na a� (for 2,4,6...)n

In the nonlinear vibration of the cable, the amplitude an0 of the initial excitation may be several 12

times the initial midpoint deflection parameter �i. In addition, the phase angle �n0 may be 13

considered as zero for convenient calculation. 14

15

Restraining of Initial Curvature Shortening 16

As the cable becomes heavier or longer, the stretching force due to initial curvature shortening 17

by self-weight increases. If the cables in cable-stayed bridges have small diameters and are light 18



(slenderness ratio < 200), this stretching force is much smaller than the applied axial load Na; 1

therefore, the tension force of the cable can be estimated with only a small error margin using 2

the existing formulas while neglecting this stretching force. If the slenderness ratio of the cable 3

is greater than 200, or if the weight of cable is further increased by grouting, this stretching force 4

should be considered in the tension force calculation. 5

If the support conditions of the cable is simply supported, this force moves the end side 6

of cable on roller support to the opposite side (large displacement occurs). In this case, the cable 7

cannot act as the structural member. To sustain the applied tension force and vibrate normally, 8

the cable should have the immovable-end conditions. And under this support conditions, the 9

stretching force due to initial curvature shortening acts as the conservative force restraining the 10

large displacement of the cable. This force can be calculated using a second order elastic 11

analysis or via the relation between the elongation of stretching force Nconserv and the static 12

deflection �o (�) by self-weight in Equation (4). However, to use Equation (4), an iteration 13

calculation is needed to find the stretching force Nconserv. This iterative procedure may be too 14

much trouble. The stretching force Nconserv can be also determined by the relation between the 15

nonlinear natural frequency and the applied axial load Na. As shown in Equation (18), there is 16

the border line to divide the imaginary natural frequency and the real natural frequency. If the 17

radicand in Equation (18) is positive, the nonlinear natural frequency �n of the cable becomes 18

the real number. This means that the cable will be vibrating under the given condition. However, 19

if it is negative, the nonlinear natural frequency �n becomes the imaginary number. In this case, 20

the cable would sway but the vibration of the cable would not occur. By equating the radicand in 21

Equation (18) with zero and solving for Na, the axial load to divide the imaginary natural 22

frequency region and the real natural frequency region can be decided. This axial load becomes 23

the stretching force Nconserv which can be described as follows: 24

1/2 2 2 2

2 4

32

conservEI EA L qN

L�

�

� ��

� �

(19)

25

26

27



NEW TENSION ESTIMATION PROCEDURE 1

Comparison with Existing Formulas 2

Figure 1 shows the difference between the linear natural frequency 10f and nonlinear natural 3

frequency 1f with the natural frequencies calculated by the existing formulas. In addition, the 4

sag-to-span ratio �, defined as �1/L, is plotted in Figure 1. For the nonlinear natural frequency 1f , 5

an amplitude n0a equal to the initial midpoint deflection parameter �1 is assigned (� = 1.0). The 6

frequency curves of the taut string and derived from the work of Zui et al. (5) and Mehrabi and 7

Tabatabai (8) are also plotted. 8

As the applied axial load Na increases, the nonlinear natural frequency 1f increases and 9

the sag-to-span ratio � decreases. If the applied axial load Na is slightly greater than the 10

stretching force Nconserv , the increasing rate of the nonlinear natural frequency 1f becomes large, 11

and the decreasing rate of the span ratio � becomes slow. If the applied axial load Na is much 12

greater than the stretching force Nconserv , the increasing rate of the nonlinear natural frequency 1f13

and decreasing rate of the span ratio � are constant. 14

As the applied axial load Na increases, all frequency curves approach the taut string’s 15

frequency curve. The curve obtained from the formula of Zui et al. (5) is slightly larger than the 16

taut string’s natural frequency curve. This is because in the formula of Zui et al. (5), some 17

constant frequency (related to the flexural rigidity of the cable) is reflected. However, this 18

formula does not catch the nonlinear phenomenon when the applied axial load Na is small. The 19

natural frequency curve from the formula of Mehrabi and Tabatabai (8) is similar to that of the 20

linear natural frequency 10f . Considering the fact that Mehrabi and Tabatabai (8) used parameter 21

�2 as proposed by Irvine (22) in their formula, this similarity may be expected. However, the 22

formula of Mehrabi and Tabatabai (8) gives incorrect results when small axial loads are applied 23

to the cable. 24



1

(a) Overall comparison (Na = 0 ~ 3000kN) 2

3

(b) High tension region comparison (Na=2500~3000 kN) 4

FIGURE 1 Comparison between new and existing formulas for tension force calculations (continued). 5



1

(c) Comparison for short span beam (L=10m) 2

FIGURE 1 Comparison between new and existing formulas for tension force calculations. 3

4

Total Tension Force and Estimation Procedure 5

The behaviour of an axial loaded cable with immovable ends is different from that of an axial 6

loaded and simply supported cable. If the applied axial load Na becomes zero, there is no tension 7

force in the cable, but the reaction forces at the immovable ends still exist. The value of the 8

reaction force is the same as that of the stretching force Nconserv. The stretching force Nconserv acts 9

as the force sustaining force on the cable, and induces the reaction forces at the immovable ends. 10

The stretching force Nconserv can be considered a conservative force for the change of the applied 11

axial load Na. If the applied axial load Na becomes greater than zero, the cable is tensioned and 12

the total reaction force becomes Na + Nconserv . If an axial loaded beam is placed on immovable 13

ends, the total tension force Ntotal of the sagged cable consists of an applied axial load Na and 14

stretching force Nconserv : 15

total a conservN N N � (20)



Table 1 shows the new estimation procedure for the cable tension using the first mode 1

nonlinear natural frequency. To estimate cable tension, only four constants ( �, EI, EA, L) are 2

needed. In calculating the nonlinear natural frequency of the cable, the assumption about the 3

initial condition of the excitation using Equation (17) is needed. Also, to calculate the stretching 4

force Nconserv , the iterative calculation is needed. If the iterative calculation is cumbersome, the 5

approximate method proposed by Zaslavsky (17) or in Equation (19) can also be used. 6

7

TABLE 1 Estimation procedure for the cable tension using the first mode nonlinear natural frequency 8

Step Procedure for each step Parameters to

calculate

1 Check the material characteristics of the cable , , ,EI EA L�

2

Calculate the initial midpoint deflection parameter

� �4

1 3 2 2

4

a

qLwL N EI� �

�

1w

3

Calculate the coefficients of nonlinear oscillations 4 2 2

1 2

4

2

4

3

31

4

34

4

a in

in

n

N L AwEIgA L IEI

AwEIgA L I

EI AgA L I

��

��

��

� ��

� � � ��

� ��

1 2 3, ,n n ng g g

4 Calculate the linear natural frequency

10 11g� 10�

5 Assign the initial conditions (recommend: 10 1 10, 0a w � ) 10 10,a �

6

Calculate the nonlinear natural frequency

� �2 2 213 11 12 3

1 11 211

9 101 ( )

24

a g g gg O

g� �

��

1�



7 Calculate the axial load from the measured first natural frequency aN

8

Calculate the stretching force due to self-weight 1/2 2 2 2

2 4

32

conservEI EA L qN

L�

�

� ��

� � conservN

9 Calculate the total tension force

total a conservN N N � totalN

1

2

VERIFICATION EXAMPLE 3

The Alamillo Bridge, Spain 4

5

FIGURE 2 Layout of cables from the inclined pylon to the deck (by Casas (19)). 6

7

To verify the new formula, the tension force of the Alamillo Bridge cables was re-estimated 8

using the reported data. As shown in Figure 2, the Alamillo Bridge is the cable-stayed bridge 9

with just one inclined pylon. The parallel cable system consists of 26 cables. Each parallel cable 10

line has two cables that connect on each side of the pedestrian passage. Figure 2 also shows the 11

layout of the 13 parallel cables from the inclined pylon to the deck. For the anchorage of the 12



cable, DYWIDAG DB-E61 is used. For the C1~C12 cables, the cable consists of 60 epoxy 1

coated strands. The C13 cable consists of 45 epoxy coated strands. The diameter of the epoxy 2

coated strands that satisfies ASTM A416 Grade 270 is 15.24 mm. 3

4

For the Alamillo Bridge, Casas (18) tested the natural frequencies of the cables and 5

reported the estimated cable tensions. The estimated results by Casas (18) show a clear 6

contradiction between the calculation results using the existing formulas and the direct 7

measurement by jack pressure and strain gauges. In this paper, for the length and the natural 8

frequency of the cables, the work of Casas (18) is referenced, and for the material properties of 9

the strands, the data from DYWIDAG is used. 10

11

Table 2, Table 3, and Table 4 show the measured first mode natural frequencies and the 12

estimated tension forces for 26 cables using the formulas of Zui et al. (5), Mehrabi and 13

Tabatabai (8), and the new proposed formula. The values used in tables are as follows: for 14

C1~C12 cables, density of 8,811 kg/m , axial rigidity �� of 1.64718�109 N, and flexural 15

rigidity � of 1.14326�106 N�m�, and for C13 cables, density of 8,853 kg/m, axial rigidity 16

�� of 1.23585�109 N, and flexural rigidity � of 646,744 N�m�. The letters L and R attached to 17

the cable numbers in Tables refer to the left and right cables, respectively, in the parallel cable 18

position. In the Alamillo Bridge, two parallel cables are installed on the same transverse line and 19

act as a one-cable element in the global structural system because they are very near to each 20

other. However, the measured frequencies for the two parallel cables were different. 21

22

The average tension forces Nz_avg estimated by Zui et al.’s formula (5) show the greatest 23

values. The average tension forces Nmt_avg estimated by Mehrabi and Tabatabai’s formula (8) 24

show the second greatest values. Finally, Na_avg estimated by the new formula shows the 25

minimum values. The estimated tension forces showed considerable differences compared with 26

the average design values Td.. As mentioned before, this difference can be complemented by 27

adding the stretching force Nconserv due to initial curvature shortening to the estimated tension 28

forces. The total tension forces Ntotal_avg which include the stretching force Nconserv estimated by 29

Equation (19), agree well with the average design values Td given in Table 4. �� are the 30



calculated values in the design analysis of the Alamillo Bridge. This values are checked by the 1

jack pressure. In design, the parallel cables are assumed to be equally tensioned (non-eccentric 2

tensioning) and have the same tension force in the analysis. 3

4

TABLE 2 Tension force estimates for the Alamillo Bridge (Zui et al. (5)) 5

Cable No.

Length [m]

Meas. first freq.

[Hz] (18)

Estimated tension by Zui et al.'s formula

[ kN ]

Stretching force [ kN ]

Total Tension force [ kN ]

Design value

[ kN ] (18) Nz Nz_avg Nconserv Nz_avg + Nconserv Td

8L 169.2 0.69 4133 4133 1036 5169 5121

8R 169.2 0.69 4133 9L 186.0 0.63 4169

4236 1104 5340 5013 9R 186.0 0.64 4303 10L 202.6 0.56 3911

3841 1169 5010 4640 10R 202.6 0.55 3772 11L 219.5 0.52 3962

3962 1233 5195 4562 11R 219.5 0.52 3962 12L 236.1 0.46 3588

3667 1295 4962 4199 12R 236.1 0.47 3746 13L 253.0 0.50 3700

3700 1024 4723 4365 13R 253.0 0.50 3700

6

TABLE 3 Tension force estimates for the Alamillo Bridge (Mehrabi and Tabatabai (8)) 7

Cable No.

Length [m]

Meas. first freq.

[Hz] (18)

Estimated tension by Mehrabi and Tabatabai's

formula [ kN ]



Design value

[ kN ] (18) Nmt Nmt_avg Nconserv Nmt_avg + Nconserv Td

8L 169.2 0.69 4009 4009 1036 5045 5121

8R 169.2 0.69 4009 9L 186.0 0.63 4018

4090 1104 5194 5013 9R 186.0 0.64 4163 10L 202.6 0.56 3695

3615 1169 4784 4640 10R 202.6 0.55 3535 11L 219.5 0.52 3709 3709 1233 4942 4562



11R 219.5 0.52 3709 12L 236.1 0.46 3182

3287 1295 4581 4199 12R 236.1 0.47 3391 13L 253.0 0.50 3541

3541 1024 4565 4365 13R 253.0 0.50 3541

1

TABLE 4 Tension force estimates for the Alamillo Bridge (New estimation procedure) 2

Cable No.

Length [m]

Meas. first freq.

[Hz] (18)

Esitmated Tension by New procedure

[ kN ]



Design value

[ kN ] (18)

Na Na_avg Nconserv Ntotal_avg

(= Na_avg+ Nconserv) Td

8L 169.2 0.69 3930 3930 1036 4967 5121

8R 169.2 0.69 3930 9L 186.0 0.63 3906

3985 1104 5089 5013 9R 186.0 0.64 4064 10L 202.6 0.56 3502

3408 1169 4577 4640 10R 202.6 0.55 3314 11L 219.5 0.52 3467

3467 1233 4700 4562 11R 219.5 0.52 3467 12L 236.1 0.46 2695

2846 1295 4140 4199 12R 236.1 0.47 2996 13L 253.0 0.50 3397

3397 1024 4421 4365 13R 253.0 0.50 3397

3

Figure 3 compares the total tension forces in the cables, in which the stretching forces 4

Nconserv due to initial curvature shortening given in Table 2 and Table 3 were added to the 5

estimated average axial loads Nz_avg , Nmt_avg.. The new procedure was more accurate, and the 6

stretching force Nconserv should be included for tension force estimation of the cable-stayed 7

bridge cables. Therefore, the application of other formulas to the cable-stayed bridges should be 8

considered carefully. 9

10



1

FIGURE 3 Estimated results for the total tension forces of the Alamillo Bridge cables. 2

3

CONCULUSION 4

In this study, a new estimation procedure for the total tension force of bridge cables is proposed. 5

The existing methods based on the taut string theory consider additionally the flexural rigidity of 6

the cables. However, they do not consider the initial deflection, the natural frequency changing, 7

and the initial curvature shortening of the cables. To incorporate these correlated nonlinear 8

influences in the tension force estimations, the new estimation procedure considers the total 9

tension force consisting of the applied axial load and the stretching force due to the initial 10

curvature shortening for the cable. A new approximation method for this stretching force was 11

also derived from the nonlinear natural frequency formula for the cable. Each step of the new 12

estimation procedure was clearly proposed in this paper. The reliability of the new method was 13

verified by re-estimating the tension forces of the Alamillo Bridge cables. 14



1

ACKNOWLEDGEMENTS 2

This work is a part of a research project supported by Korea Ministry of Land, Transportation 3

Maritime Affairs (MLTM) through Core Research Project 1 of Super Long Span Bridge R&D 4

Center. The authors wish to express their gratitude for the financial support. This research was 5

supported by Basic Science Research Program through the National Research Foundation of 6

Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A 7

2007054). 8

9

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New Procedure for Estimating Cable Force in Cable-stayed ...docs.trb.org/prp/13-2407.pdf · 1 New Procedure for Estimating Cable Force in Cable-stayed Bridge 2 3 4 Dong-Ho Choi, Ph.D.,