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Tension estimation of cables with

different boundary conditions bymeans of the added mass technique

A. Bellino, L. Garibaldi, A. Fasana, S. Marchesiello

Dipartimento di Meccanica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino

Abstract

When analyzing cable vibration, both the tension and the bending stiffness must be taken into

account; in fact, the cable can be considered as an intermediate case between the beam (no

tension) and the string (no bending stiffness). Moreover, one of the most important issues for

this approach is the right definition of the boundary conditions, since they can be very different

from one structure to another.

Taking into account the aforementioned issues, the paper presents an overview of all possible

scenarios, especially focusing on the relationships between the cable properties and its boundary

conditions, summarized by the introduction of the modal length concept.

Successively, a method for the estimation of the cable tension has been developed, just based ontwo simple measurements: one with the cable in its standard configuration and one with an

added mass on it.

This supplementary information allows to well estimate the modal lengths and consequently the

tension. The procedure is applied both to a numerical example and to an experimental test done

on a stay-cable bridge.

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Introduction

The monitoring of cables is becoming one of the most important issues in the field of structural

health monitoring, for example in stay-cable or suspended bridges as well as in tensile structures

and everywhere steel cables have a structural function.

From a theoretical point of view, the cable can be considered as an intermediate case between

the beam and the string, and therefore its equation of motion includes not only the bending

stiffness but also the tension. Moreover, the cable is affected by the sag-extensibility and other

secondary effects that can influence the modal parameters, especially when analyzing complex

structures. For example, Caetano and Cunha [1] studied the cables of the International

Guardiana Bridge and of the Braga stadium. They showed the complexity of the interaction

between the different subsystems (deck, towers, cables) and the presence of internal resonances.

Other aspects like veering, local modes and modal hybridization have been studied by Gattulli

and Lepidi [2]. Lardies and Ta [3], instead, proposed a time domain and a timefrequency

domain approaches for modal parameter identification of stay cables using output-only

measurements.

Over the last years, mainly two types of techniques have been developed for the estimation of

the cable tension: one based on direct measure of the tension, by adopting pre-installed sensors

in series with the cable, the other based on indirect methods, often adopting vibration techniques

or wave propagation along the cable. This second approach is the kernel of the proposed

technique.

One of the most peculiar aspects of the cables is that it is not possible to obtain an analytical

formula linking the natural frequencies, the tension and the bending stiffness, except for the

simply supported case. This limitation prevents us to directly estimate the tension because, in

real applications, the cable do not have simply supports at the end. For this reason, many works

tried to develop some methods to overcome this problem. One of the first important researches

in the field was conducted by Zui et al. [4]: they proposed some practical formulas for the

estimation of the tension from the identified natural frequencies, taking into account the sag-

extensibility. Ren et al. [5] presented a new version of the practical formulas, after having

explained the relative influence of the sag-extensibility and of the bending stiffness.

Ni et al.[6] proposed a method for the analysis of a suspended bridge and a stay-cable bridge,

while Kim and Park [7] developed a method for the simultaneous estimation of the tension,

flexural rigidity and axial rigidity of a cable system. Moreover, in [8], the same authors made an

overview of all the available methods, based on two real bridges.

Ceballos and Prato [9] introduced two rotational springs at the cable ends to represent all the

possible boundary conditions and the stiffness of the springs is extracted from the first mode

that cannot be estimated by the identification procedure. Successively, the axial force of the

cable can be calculated.

In order to take into account all the scenarios, in our previous work [10], the concept of

equivalent length was introduced. It is a length that corresponds to the extracted natural

frequencies and to the known properties of the cable (as the mass per unit of length), based on

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the formula for simply supported cables. The cable tension is successively estimated by

comparing the natural frequencies of the standard configuration with those of a new

configuration with a mass mounted on the cable.

Even if the concept of equivalent length is a good manner to simplify the relationships for the

cables, actually different theoretical lengths should be defined at each mode, because the length

should be seen as a parameter that quantifies the difference from the simply supported case.

They will be called modal lengths along the paper. For example, a high order mode is not

considerably affected by the boundary conditions and so the modal lengths will be close to the

real length; vice versa for the first modes the boundary conditions significantly influence the

cable dynamics and so the modal length will be quite different from the reference value.

The article presents a method based on two measurements: firstly the cable is monitored in its

standard configuration, by using one or two accelerometers to pick up its free vibration;

secondly, a mass is placed on it and the measure is repeated. The natural frequencies extracted

from the second configuration are compared with the former ones to obtain the modal lengths of

the cable and then both tension and bending stiffness are estimated.

Afterwards, a numerical example about a cable with non-conventional boundary conditions is

proposed to show the potentiality of the method. In particular, since the tension and the bending

stiffness are known, the theoretical modal lengths can be calculated. If compared with those

extracted from the method, it will be clear that they are very similar and the estimation of the

tension is very precise.

Conclusively, the cables of a stay-cable bridge near Aosta (Italy) have been analyzed to obtain

the tension of both the long and the medium cables.

Cable dynamics

The equation of motion of a cable includes not only the bending stiffness (as happens for a

beam) but also the tension and the sag extensibility:

0d

)(d)(

),(),(),(

2

2

2

2

2

2

4

4

=

+

x

xyth

t

txw

x

txwT

x

txwEI (1)

where EIis the bending stiffness, ),( txw represents the deflection in thez-direction, Tis the

static cable tension, is the mass per unit length of the cable, )(th is the dynamic tension and

)(xy is the geometric shape of the cable.

It has been verified that the effect due to the sag extensibility slightly affects only the first 3-4

in-plane modes and then it can be neglected when considering higher modes or the out-of-plane

frequencies [9].

When analyzing cables, two important aspects should be taken into account:

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the relative influence of the bending stiffness and the tension, expressed by the bendingstiffness parameter [6,7] or by the non-dimensional tension parameter [9], which

is the inverse of . L is the cable length.

EI

TL= (2)

the boundary conditions, because only for the simply supported case an analyticalexpression involving the natural frequencies, the tension and the bending stiffness is

known:

EIL

rTLr

fr 4

22

2

2

441

+=

(3)

Both the issues must be considered together in order to understand the cable dynamics.

The modal lengths

Consider a cable, of length L and mass per unit of length , with two rotational springs at the

ends, with stiffness equal to AK and BK respectively, as in Figure 1, in order to simulatedifferent boundary conditions. As done in [9], we consider the normalized stiffness, defined by

EILK

LKk

A

AA 4+

= EILK

LKk

B

BB 4+

=

When a stiffness k is equal to zero, then the end is a simple support, when 1=k it is a

clamping condition.

Figure 1. The cable with two rotational springs at the ends.

AK

BK

L

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By applying the mode superposition, it is possible to write

0)(

d

)(d

d

)(d 22

2

4

4

= x

x

xT

x

xEI

(4)

and the general mode shape can be written as

( ) ( ) ( ) ( )xDxDxDxDx coshsinhcossin)( 4321 +++= (5)

where the different parameters are

gga ++=

24

(6)

gga +=24 (7)

EIa

24 = (8)

EI

Tg

2= (9)

After some manipulations and approximations [9], the following relationships are obtained:

( ) +++= 222228 1 BABABA kkkkLkkA

( ) 224 2 ++ BABA kkkkL (10)

( )( ) 2248 22 +++= BABABA kkkkLkkB (11)

=

A

Bn

1tan (12)

and finally

22

12

+

==

nnnn

nT

L

nf (13)

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The value n is dependent on the frequency n , therefore an iterative process is necessary to

extract the frequencies, by starting from the value of 0=n , corresponding to the case of

simple supports.

Now, we can introduce the parameter n as

=

=

nL

n

L

n nnn 1 (14)

Since for a simple supported cable Lnn / = , then it is possible to imagine that the cable is

simply supported but with a length depending on the mode. This value is called modal length:

n

LL

nn

=

1

(15)

Indeed Eq. (13), after basic calculations, becomes of the same form of Eq. (3):

EIL

rT

Lr

f

nn

r

4

22

2

2

44

1

+=

(16)

Let's consider, for example, a cable with length 25=L m and diameter 05.0=d m . In

Figure 2, the modal lengths for different boundary conditions (it is assumed that kkk BA == )

are depicted for a tension of510=T Nand a bending stiffness of

4108 =EI2

mN

Figure 2. Modal lengths for the cable proposed, for different boundary conditions.

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If the cable is simply supported, then modal lengths are equal to the reference length L for all

the modes. In the other cases, the modal lengths are monotonically increasing and the

asymptotic value is given by the reference length. This situation is justified since higher modes

are clearly less affected by the boundary conditions.

Meantime, if the bending stiffness parameter is varying from 5 to , there is a significant

variation of the modal lengths, as it can be seen in Figure 3 by imposing 5.0=Ak and

7.0=Bk . If is very large, then the behaviour is similar to that of a string and then the effect

of the boundary conditions is almost null. On the contrary, if is small, then the boundary

conditions affect notably the cable dynamics and consequently the modal lengths start from a

value quite different from the reference length.

Figure 3. Modal lengths for the cable proposed, for different values of the bending stiffness

parameter.

With the aforementioned procedure, every configuration of boundary conditions can be treated

as the cable would hold simple supports with different modal lengths. This is a huge advantage

because it allows us to use the known relationships among tension, bending stiffness and natural

frequencies of the cable.

Since for a simply supported beam the modal lengths are equal to the real length for all the

modes, also the calculus of can be done very quickly, directly from the natural frequencies by

manipulating Eq. (16-17) in [10]:

2222

2424

qp

pq

fpfq

fqfp

= (17)

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where p and q are two different modes. For other boundary conditions, this formula can be

considered however a good approximation of the real value.

Moreover, if the modal lengths are known, then, by an iterative process involving Eq. (6-13), it

is possible to estimate the values of the rotational stiffness to characterize the real boundaryconditions of the cable.

Estimation of the cable tension

The key to well approximate the cable tension and its bending stiffness is to obtain a good

estimation of the modal lengths, in order to perform a classical regression on the identified

natural frequencies.

In this section, a four-steps algorithm is proposed to approximate the modal lengths and to

extract the cable characteristics. The basic idea, as in [10], is to compare the natural frequencies

of the cable in two different acquisitions:

1. the cable in its standard configuration2. the cable with a mass placed on it

This second test is very important because it allows having additional information on the

system.

Initial regressions

The first step consists of two regressions based on Eq. 3 to obtain a first attempt estimation of

the tension and the bending stiffness of the cable, by using the measured length of the cable.

In the first regression only the low order modes are considered, therefore the tension obtained

can be considered quite close to the real value (based on the of the cable). Vice versa, in the

second regression, only the high modes are taken into account and then the bending stiffness is

quite well estimated. In both the processes, the values inT and inEI overestimates the real

tension and bending stiffness because the length considered is L , and it is larger than the right

values, expressed by the modal lengths.

Even when is large, the estimation of the bending stiffness inEI with the aforementioned

procedure is usually quite precise because the tension and the boundary conditions do not affectsignificantly the higher modes.

Admissible modal lengths

Consider that the first modal length can vary in the interval [ ]LaL, , where a is a constant

parameter that can be chosen from 0 to 1. A good choice is 8.0=a , but if is very small

(look at Figure 3), then a should be decreased. Starting from different values of the first modal

length, it is possible to calculate, for each couple of modes q and p (from Eq. 16):

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+=

+=

EIL

pTpfL

EIL

qT

q

fL

p

pp

q

qq

222

2

2

2

2222

441

44

1

(18)

Since the term containing the cable tension is equal in both the equations, then

+

=

2

2

2

2222

22

4qp

qq

pp

L

q

L

pEI

q

fL

p

fL

(19)

In Eq. 18, the value inEI estimated in the previous step must be used. Since inEI

overestimates EI, also the modal lengths in the second member of the equations can be

substituted by a larger value, which is L in this case. With this approximation, there is a big

simplification of the equation with only a small error:

( )222

2

222

22

4qpEI

Lf

p

f

p

q

fLL in

pp

q

qp +

=

(20)

Summarizing, the iterative procedure allows to obtain a set of admissible modal lengths for the

case under study.

Calculus of the modal lengths

The third part permits to choose the most appropriate modal lengths, starting from the total set

obtained in the previous algorithm step. The decision is based on the configuration with the

added mass.

As explained in [10], the relationship between the natural frequencies of the case with and

without the mass can be expressed by the following formula:

),(,

mr

rmr

xmg

ff = (21)

2

sin2

1),(

+= mmr x

L

r

L

mxmg

(22)

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where ),( mr xmg is frequency variation due to the mass for the r-th mode, mrf , is the r-th

natural frequency for the configuration with the mass, m is the value of the mass and mx is the

mass position. These relationships hold if the added mass is quite small respect to the cable

mass, i.e. it is 12-15% of the cable mass at maximum. Indeed, for all the possible cases, the ratio

between the frequencies in both the configurations is given by

mr

rr

f

fc

,

= (23)

This value must be compared with the theoretical vale of ),( mr xmg . The modal lengths are

chosen from the total set as the values minimizing the difference between rc and rg ,

Nr ,...,2,1= .

Final regressions

The final part is similar to the first one, because two regressive processes must be applied to

obtain the tension (only low order modes considered) and the bending stiffness (only high order

modes).

The difference respect to the first part of the algorithm is that now the modal lengths are used in

the regressive procedure, according to Eq. 16, instead of the reference length.

Numerical examples

Let us consider a steel cable with characteristics listed in Table 1. The cable has different

conditions for the two ends, quite far both from the supported condition and the clamping

condition. The bending stiffness parameter is equal to 95.27= , therefore it is a case in which

the tension is not very high. The added mass is equal to 1/10 of the total mass of the cable.

The time histories have been created by using a Runge-Kutta method, according to [10], with

3% of Gaussian noise added to the accelerations. The sampling frequency is equal to 320 Hz.

In Figure 4, the power spectral density of the signal created (the measurement point is at 7/2L )

shows the first 20 natural frequencies. As it happens in real applications, not all the modes are

easy to identify.

In Figure 5, a comparison among the modal lengths obtained by means of the described method,

the theoretical modal lengths and the reference length is depicted. As previously said, the

theoretical modal lengths are monotonically increasing and they tend to the reference lengthL .

The modal lengths extracted are not very similar to the theoretical one for the first modes but

afterwards they become quite close. For this reason, in the third step of the algorithm, it is

advisable to neglect some modes and to start the comparison once modal lengths stabilize.

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Table 1. Cable characteristics.

Length 25=L m

Diameter 05.0=d m

Maas per unit of length 32.15= mkg/ Rotational stiffness, left 3.0=Ak

Rotational stiffness, right 5.0=Bk

Tension 510=T N

Bending stiffness 4108 =EI2

mN

Added mass 29.38=m kg

Mass position 94.9=mx m

Figure 4. Power spectral density of the signal created numerically.

Figure 5. Comparison between the modal lengths identified with the method, the theoretical

values and the reference length.

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Table 2. Results extracted by the method.

510149.1 =inT N90.14=

inTer

First regression410112.8 =

in

EI2

mN 88.18=inEIer

510015.1 =sT N

45.1=sT

er

Final regression4

10051.8 =sEI2

mN 64.01=Ter

Conclusively, in Table 2, the results of the method are presented. If only the first regression is

applied, then the results obtained overestimate the real value, as previously said. If all the

procedure is applied, then the results are very close to the real values.

Figure 6. The stay-cable bridge under study and a magnification on one of the long cables.

Figure 7. Different typologies of cable.

Long cable

Medium cableShort

cable

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Experimental example

The experimental case is a tensional study on the cables of a pedestrian stay-cable bridge in

Gressan, near Aosta, in the northern part of Italy. From Figure 6, it evident that the tests were

performed during the winter season, at a mean temperature of -3C.

As it can be seen in Figure 7, there are three different typologies of cables, but in this paper we

analyze only the long and the medium cables. In Figure 8, a map with the name and the position

of the cables is presented.

Figure 8. Map of the different cables. The red circles indicate the long cables, the blue circles

indicate the medium cables.

A single accelerometer has been placed on each cable. A block of steel (the mass is equal to

8.50 kg) has been placed on each cable to obtain the added mass configuration.

Long cables

The long cables have a nominal length of 02.42=longL m. In Figure 9, the power spectral

density of the signal produced by the accelerometer is visualized. Many modes are available, but

in our analysis only the first thirty modes are considered.

Figure 9. Power spectral density of the signal acquired on the first cable.

AOSTA

GRESSAN

C8

C1

C2

C3

C4

C5

C6

C7

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Table 3. Results about the long cables.

Cable Tension Bending stiffness Bending parameter

1C 51 10455.6 =T N4

1 10570.5 =EI2

mN 05.1431 =

2C 52 10432.6 =T N4

2 10541.5 =EI2mN 16.1432 =

7C 57 10408.6 =T N4

7 10265.5 =EI2

mN 60.1467 =

8C 58 10480.6 =T N4

8 10332.5 =EI2

mN 49.1468 =

In Table 3 the results obtained by the method are presented. Since the bending stiffness

parameter is quite large, the method cannot calculate modal lengths different from the

reference value, and then the results obtained are those extracted from the initial regression.Both the estimated tensions and the estimated bending stiffnesses are very similar for all the

cables. This fact underlines the good health of the bridge.

Medium cables

The medium cables have a nominal length of 41.23=mediumL m. In Figure 10, the estimation

of the modal lengths for the four cables is depicted, compared with the reference length. The

values increase until approximately the tenth mode, then there is a sort of stabilization and

successively a new increase towards the nominal length.

In Table 4, the tension and the bending stiffness of the cables are listed. The bending stiffness

parameter is around three time inferior respect to the long cables. The results about the first

cable are slightly different from the others and this is due probably to some problems in the

identification of the natural frequencies. Indeed, its modal lengths has a decrease in

correspondence of the ninth mode and therefore its trend deviates more from the results found in

Figure 2.

Figure 10. Modal lengths for the four medium cables.

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Table 4. Results about the medium cables.

Cable Tension Bending stiffness Bending parameter

3C 53 10934.2 =T N4

3 10560.5 =EI2

mN 35.533 =

4C 54 10661.2 =T N4

4 10004.6 =EI2

mN 28.494 =

5C 55 10309.2 =T N4

5 10201.6 =EI2

mN 17.455 =

6C 56 10532.2 =T N4

6 10194.6 =EI2

mN 34.476 =

Conclusions

The article deals with a new method for the estimation of the tension and bending stiffness of a

cable. The procedure is based on the manipulation of the extracted natural frequencies. A new

concept is introduced to understand the cable dynamics: the modal lengths. They represents the

theoretical cable lengths, for each mode, corresponding to the boundary conditions of the cable,

in order to use the relationships proper of the simply supported case.

The method proposed allows us to calculate the modal lengths starting from the natural

frequencies available, and finally the tension and the bending stiffness are obtained by means of

a classical regression procedure.

The results are very good both for the numerical example, where a cable with particular

boundary conditions is studied, and for an experimental test on a pedestrian stay-cable bridge.

In this case two different types of cables are analyzed. The long cables have a large bending

stiffness parameter and the modal lengths can be approximated with the nominal one; the other

cables have a more common value of the bending stiffness parameter and consequently the

modal lengths are significantly different from the nominal length, especially for the first modes.

Acknowledgment

The authors are grateful with the Mayor of Gressan town in Valle d'Aosta, Mr. M. Martinet, and

all the staff of Valle d'Aosta regional administration, with a special thanks to Drs. Clermont,Clarey and Piazzano, for their support along this project.

References

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th International Conference on Structural Dynamics

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[2] V. Gattulli, M. Lepidi, Localization and veering in the dynamics of cable-stayedbridges, Computer and structures, Vol. 87 (2007), pp. 1661-1678.

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[3] J. Lardies, M.-N. Ta, Modal parameter identification of stay-cables from output-only measurements, Mechanical System and Signal Processing, Vol. 25 (2011), pp.

133-150.

[4] H. Zui, T. Shinke, Y. Namita, Practical formulas for estimation of cable tension byvibration method, American Society of Civil Engineers Journal of Structural

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[6] Y.Q. Ni, J.M. Ko, G. Zheng, Dynamic analysis of large diameter sagged cablestaking into account flexural rigidity, Journal of Sound and Vibration, Vol. 257 (2010),

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[7] B.H. Kim, T. Park, Estimation of cable tension force using the frequency-basedsystem identification method, Journal of Sound and Vibration, Vol. 304 (2007), pp. 660-

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[8] B.H. Kim, T. Park, H. Shin, T.Y. Yoon, A comparative study of the tensionestimation methods for cable supported bridges, Steel structures, Vol. 7 (2007), pp. 77-

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[9] M. A. Ceballos, C.A. Prato, Determination of the axial force on stay cablesaccounting for their bending stiffness and rotational end restraints by free vibration

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[10] A. Bellino, S. Marchesiello, A. Fasana, L. Garibaldi, Cable tension estimation bymeans of vibration response and moving mass technique, Mcanique et Industries, Vol.

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