-~-

'';;

,

'

,

'

,

'

,

'

,

,-

,

:'

,!,

' ~

!'-r.

'C:~

d j

d '

PRACTICAL FORMULAS FOR ESTIMATION OF CABLE TENSION BYVmRATION METROD

By Hiroshi Zui; Tohru Shinke,zand YoshioNamita3'.

e 1

;1J ¡

,g i

ABSTRACT: The vibration metbod is usualIy used for field measurement of cable tension during tbe construc-tion of cable system' bridges such as an arch bridge stiffened witb inclined cables or a cable-stayed bridge.Practical formuIás for tbe vibration metbod are proposed herein taking theeffects of flexural rigidity and sagof a cable into account. The formulas are based on the approximate solutions of high accuracy for tbe equationof inclined cable witb flexural rigidity. Cable tensions are easily estimated by tbese formulas using measurednatural frequencies of l~-order modes. The practical formulas presented herein are applicable to various cables,regardless of lengtb ind tension as far as tbe vibration of first- or second-order mode is measurable. As to avery long cable tbat cannot be easily excited artificially, a formula is presented by using natural frequencies ofhigh-order modes obtained from statiónary microvibrations. The accuracy is confirmed through comparison oftbe values obtained by practical formulas witb measured values and calculated values by the finite elementmethod.

1 j

f :

t i

: INTRODUCTION

The cable forces must be adjusted during construction ofcable system bridges such as the Nielsen bridge (an archbridge stiffened with inclined cables) or a cable-stayed bridge

, so that cable forces and bridge geometry may be optimized.¡ Thereforc, accurate measurement of cable forces has practical! importanceand a simple, speeciy,and rcliablc metbod of mea-j surement is needed by the field engineers. The vibration1 method by which cable forces ~estimated from measured; natural frequencies is often used for tIlemeasurement of cable: forces due to its simplic:ity andspeediness. N81ural frequencies, of a cable are inftuenced not only by cable force but also by, flexural rigidity, sag-spari ratio, and inclination of tIle cable,

and tIlese effects shouldbe taken into account for tIle esti~',mation of cable forces. Practical formulas for this purposewere proposed by tIle writers' in Shin1ceet al. (1980). Cableforces can be easily estimated from tIlese formulas using mea- 'sured,natural frequencies of cables, and, tIlus, tIleljeformulasare widely used in Japan. These formulas, however, havc acertain limit of application and do not yield good results whentIlecable is not slender or not sufficientlytensioned.When weintroduce a dimensionless parameter ~=VTIEI'I, whereT,El, and 1are cable force, flexural rigidity of cable,and spanlengtll, respectively, tIle applicable range oí tIle formulas isspecified as' 3 s Earid 10 s ~for tIle first and second modesof vibration, respectively. Actually, Corexample, in case of theNielsen bridge, tIle formulas become inapplicable for certaincables. Therefore, more reliable formulas, which can be ap-plied to wide ranges of parameter ~, are needed from tIleprac-tical viewpoint. ,

In this paper, we pre:;ent new formulas exclusively of usefor small values of ~ and reformulated tIle practical formulasto cope with tIle aforementioned problems. The new formulaspresented herein are sufficientlyaccurate and applicableto anyregion of ~ so far as first- or second-order mode vibrationscan be detected or artificially caused. In tIle case of a very

I Prof.. Cepe. Civ. Engrg.; Setsunan Univ., Ikeda-nakamachi, Neya-gawa. Osaka, 572. Japan.

2Prof., Dcpt. Civ. Engrg.. Kobe Tech. Coll., Gakuen-higashimachi. Ni-shi-ku. Kobe. 651-21. Japan., .

2Prof.. Dept. Civ. Engrg., Setsunan Univ., Ikeda-nakamachi, Neya-gawa. Osalca, 572, Japan. ' ,

Note. Auoc:iate Editor: James M. Nau. Discussion open until Novem-ber 1, 1996. To exteDd the closing date one monm.a Wrltten request mustbe ftled witb tbe ASeE Manager of Joumals. The manuscript for thispaper was submitted for review and possible publicatioD on June S, 1995.This paper is part of the JDIU1ÚJlD/ Structurai E"peCI'ÚI', Vol. 122,No. 6. June. 1996. CASCE, ISSN 0733-944S1961OOO6-0651-06561$4.00+ $;50 per pap. Paper No. 10893.

slender cable such as tIlat used at tIle uppermost pan of acable-stayed bridge, it is not easy to excite tIlecable artificiallyin firsF or second-order mode vibration. In such á case, sta-tionarymici'ovibratióris are USed."in which high-frequencymodes are usually dominant. Shimada (l994)proposed an es-timating metllod using high natural frequeí1cymodes by tIleexact soluticin oí cable vibration considcring tbe effects offlexural rigidity, sag, and inclination of cable. However, in tIlismetllod, iti,s necessary to solve a nonlinear equation and useof a computer is unavoidable. In tIle present paper, a simplerformula is plcsented for tbe region ,ol.ZOO:s Ein high naturalfrequency uiodes,considering that ~becomes fairlylarge incase oí a slendercable. '

BA$IC GOVERNING EQUATIONS ANO SOWTIONS

Baslc G()vemlng Equatlons

Fig. 1 shows an inclined catile model and its coordinates.As shown in, tIle figure, tIle left support O is. the origin ofcoordinates and tIle direction OP is ta1C:enas tIle x-coordinate.The direction nomalto OP is tIle y-coordinate of which tIledescending direction is taken as positive. Inaddition, 9 and 1are tIle,angle 'Ofcable inclination and cable lengtll, respec-tively. '

The following assumptions are made in the analysis:

T

T

() -sIlo

.lo/2

,Y

FlG. 1. Inclln8d9.~I. 800 ha 'F..ur..

JOURNAL OF STRUCTUAAL ENGINEERING I JUNE 1996 /651, .>',J

I.The sag-to-span ratio 8 ==sIlo is sufficient1ysmall (8 =silo « 1).. .

2. The cable vib~atesonly within.the xy-plane and its mo-tion in the.x:-direction.isnegligibly small (v « y).

3. The geometric shape of the cable is e'tpressed by a sec-ond-order parabola.

The equation of motion inthe y-direction becomes (Shimada1994) .

El a.v(x, t).- T a2v(x, t) - h a2y + ~ a2v(x, t) = O (1)ax. ax2 (t) ax2 g af

where El =. flexural rigidity of cable; v(x, t) ==deftection inthe y-direction due to vibration; T =cable force in the directionOP as shown in Fig. 1; h(t) ==derivative cable force causedby vibration; w =weight of. cable per unit length; and g =gravitational acceleration. In (1), v« y is assumed frOIDas-sumption 2. The geometric shape of cable is expressed by aparabolic formula from assumption 3

4dY =f x(1 - x) (2)

where d =the cable sag defined in the x-y coordinates (Fig.1).

Substituting (2) into (1)

El a.v(x, t) -T a2v(x,t) + ~ a2v(x,t) =~ h(t) -(3)ax. ax1 g af. 12

When cable force T is small and sag becomes relatively Jarge,the effect of derivative cable force due to vibration h(t) cannotbe n~glected forthe first-order mode. However, the effect ofh(t) is negligibly small for second- or higher-order modes evenwhen cable force. T is smalL The effect of h(t) is, therefore,

. ignored for simplicity. Then, (3) becomes

El a.v(x, t) - T a2v(x, t) + ~ a2v(x, t) ==Oax. ax2 g at2

The accuracy and applicability of (4) is discussed later. Eq.(4) coincides with the equation of motion of a beam with axialtension T (Humar, 1990, pp. 653-656).

By means of variable separation of

v(x, t) = v*(x)q(t)

Eq. (4) can be transformed into the form

d.v*(x) d2v*(x) WEI--T--w2-=0cJx. . cJx2 g

d:~t) + (a)2q(t)V*(x)=O (7)

with introduction of parameter (a)2.Eq. (7) is an equation offree vibration of a single degree of freedom, and (a)indicatesthe circular frequency ofthe system.

Thc: general solution of (6) is

v*(x) =Al sinh(~) + A2 cosh(¡h) + A3 sin(ax) + A. cos(ax)(8)

where

a2 =(,' + -y.il2 - ,2; ~2= (t. + -y.)If1.+ t2 (9a,b)

with e = TI2El; and 'Y. == ww2IgE/.By assuming that the cable is clamped at both ends, the

following equation of free vibration is obtained:

.2(al)(~I)[1 - cos(al)cosh(~l)]

+ [(~1)2 - (ali]sin(al)sinh(~l) =O

where(10)

6521JOURNAL OF STRUCTURALENGINEERING/ JUNE 1996

"al = lv'v'C. + 'Y.- ,2

~l.. lv'V',. + 'Y.+ ,2

(11)

(12)

ApproxlmateFormulas

Since (10) is a tran~ndental equation, cable forces cannotbeobtained direct1yfrem measured frequencies. When the dy-namic characteristic of a cable is similar to that of a string,namely, the nondimensionalparameter E.. V'TIEI'1 is large,we introduce a nondimensional parameter TI"in the fórm

TI. ==flf~

f~ = !!.. /Ti21"V-;;

(13)

(14)

where f = <a>I2'11';and f~ =. theoretical values of the nth ordernatural frequency of a string (Humar, 1990, pp. 689-690). Bysubstituting these formulas into (11).and (12), al and ~l are trans.formed. into the forms

aJ=J. ~~l+ (~y-)

~z J. ~~1+ ('"'¡'"y + 1

(15)

(16)

. and by substituting these equations into (10), the followingnondimensional expression is obtained:

(4)

2n'll'TJ,,(1- cos al cosh ~l) + ~ sin al sinh ~l =O (17)

Though (17) is also a transcendental equation, the solution 11.for a givenvalueoí Ecan be obtained by an iterative method

. such as the Newton-RaphsonmethOd(Humar, 1990, pp. 342-. 344).The initialvaluesofiterative calculationsare givenby

(18)-(21). Within the limit of the fint- and second-ordermOde,approximate solutions of (17) can be expressed in theform

(5)~ .

111 =f=""2.2; (17:S~)

1J1=1.075VI + e~8r;(6:S ~ :S 17)(19)

(18)

(6)

~Tl2=~- 2.2; (6O:s~)

112=0.985 ~ -~3.1; (17:S ~:S 60) (21)

where (18) and (19) are the approximate formulas for the firsl-order mOdeand (20) and (21) are fer the second-order modc.These approximateformulas are obtained in the manner men-tioned in the following.

1. In case oflarge E,the solutionof (17), Tlnis nearly equalto land can be expressed as 1 +4. By substituting 1 -+:Ainto (17) and by using the relationsof 2n'l'l'TlnlE« 1 and sl~h~l COI:cosh ~l » 1, (18) and (20) are obtained. Eq. (21) ISformed after adjusting the coefficient so as to minimizc tbtdifference betwccn theoretical and approXimatesolutions.

2. In case of relatively small E, Tlnbecomes similar to tbtvalue of an axially tensioned beam. We get (19) by using tbtequivalentstaticbendingrigidityEl' = E/(1 + E2/4'11'2) for.th~axially tensioned beam, and also by adjusting the coeffiCICIIin the aforementioned way. Fig. 2 shows the comparisonbe-

(20)

~o!';'~'

~

1 '" 1.1

11/1 1.8

1.5

1A - ExactApprox.

1.3

1.2

1.1

~o' 10' 1rJ

,1;

FlG. 2. Exactand Approxlmate Solutlons of 11~versus EI: tween the exact and approximate solutions of (17). The ap-! proximate solutions coincide with the exact ones wi.thinthej errorof 0.4%. . .

I

When Ebecomes small, the solution of (17), 1'I~,increasesvery rapidly, and it becomesdifficult to obtain exact solutions.Thus, these equations ate not suitable for the region of small

1. values of E.When Eis Small,the characteristics of a cable are

l' similar to that of a beam and \Veintroduce another nondimen-

sional parameter:i! 'f~= tI/: (22)

wheret: =the theoretical value of the nth order natural fre-quency of a beam clamped at both ends and is given as follows . "

(Humar, 1990, pp. 672-674): .

t: = a~ fEii21flz -y-;- (23)

where

¡ al =4.730; az =7.853I When the cable forceapproaches zero (E =O), 'f~ becomes 1.

By substituting these equations into (11) and (12),al and~lare transformed into

Eal =V2

(25)

~1 + e;~'f~Y-l(24)

~1 + e;'~'f~Y + 1E

~l = V2

Substitution of these equations into (10) yields the followingnondimensional equation of free vibration:

2a~'f~(1 - cos alcosh ~l) + EZsin al sinh ~l =O (26)

The solution 'f~ for a given value of Ecan be obtained by aniteration such as the Newton-Raphson method.

Approximate solutions of (26) can be obtained as follows:

'ft = ~1. + 1~; 0:s E :S 8 (27)

'P2=~1 + :;; O:s E:S 18 (28)

where (27) and (28) are approximate solutions for the mst-.'

,..

10'

. . .

::: ~.'m:'1.41.2

1

......................

o 2 4 6 8 10 12 14 16;

18

FlG. 3. Exact and ApprQxlmate Solutlons of cp~versus E

order and the second-ordermode, reipectively. Eqs. (27) and(28) are obtained by using the equivalent static bending rigid-ity for the axially tensi.onedbeam and by adjusting the coef-ficient so as to maleeminimum the difference between theo-retical and aproxímate solutions. Fig. 3 shows the comparisonbetween the exact and approximate solutions of (26). The ap-

. proximate solutions' agree fairly well with the exact solutionswithin the error of 0.4% for the region O :S E:S 8 for the mst-order mode and O :S E:S 18 for the second-order mode.

When a cable is very long, it is not easy to. excite ii artifi-cially in the mst- or second-ordermode and stationary micro-vibrations are necessarily used. High-frequency. modes areusualIy dominant in the stationary microvibrations and ap-proximate solutions for high-ordet modes are practicallyneeded. In such a case, Etakes a large value and it is regardedas 200 :S E.ConsequentIy,the.same approximate solution as(20) can be used with sufficient accuracy, that is

~ = E _E2.2; (2oo:s E)(29)

Approxlmate Formulas for Cables wlth Large Sag

When a cable has a relatively large sag, it is necessary toexamine the accuracy and applicability of approximate solu-tions especia11yfor the first-order mode. Irvine and Caughey(1974) showed theoretical natural frequencies for an inclinedcable with relatively large sag considering derivative cableforces due to vibrations as follows (flexural rigidity of cableis not taken into account):

A ¡gt: =:; -va; (30)

where

{

n1f (n =1,2, ...) ~orunsymme.

tricmode

>..= A-tan>"solution of ,:i = fo for symmetric mode

(31)

f wlo =128E;A8J COS'!!

These equations show that extension. and contraction' of a cabledo not occur andthat orily cable sliape'changes, and the so-lutions coincide with those of a striog foe unsymmetric modes.On the other hand, the effect of extension and contraction of

(32)

JOURNAL OF STRUCTURAL ENGINEERING 1JUNE 1996/653

TABLE1. . Parametrlc Calculatlon8 of Natural Frequenclea

Note:E.. 20.0; r' .. 1.00;r .. U 75.

cp~

3.0

2.0

FEM

~o 5.0.. 10.0e 20.0t> 50.0. 100.0

{

FlRsT SYMO'TRIC 000'

. .. (~) -taD}~)=r2

(W~i)

1.0iO

F1RST UNSYMMETRIC MODE.1.0

.2.0

I I

3.0 4.0

{ J V1l 0.31~+ 0.5}

.

r = 128EA63 c:os6 B . 0.31~- 0.5

FIO. 4. Relatlonbetween((J:and r

a cable appears' significantly, and derivative cable forces dueto vibrations cannot be neglected for symmetric modes. Thenatural frequency oí symmetric mode is determined by non-dimensionalparameter ro,which isthe function of length, sag,weight, extensional rigidity, and inclination angle of cable.

To check these characteristics for a cable with flexural ri-gidity, parametric calculations were carried out by the finiteelement method taking into account the effect of flexural ri-gidity of cable, and results are shown in Table 1 (Shinke etal. 1980).

In Table 1, f' = -vr;; and ~ are constant (f' = 1.0; ~ =20.0), and length, sag-to-span ratio, and inclination of angleof cable are varied. In this table, as mentioned before, 11.isthe ratio of cable natural frequencies to those of string. Thenew parameter 19: expresses the ratio of cable natural fre-quencies to.those of beam with axial tension. Tab1e 1 showsthe following characteristics of the solution:

1. The values of 11.and 19: are kept almost constant, in-dependent of the values of length, sag-to-span ratio, andinclination angle of a cable so far as f' and ~ are con-stant.

2. For unsymmetric mode, all of 19: become 1, whichmeans the effects of sag-to-span ratio and inclination an-gle of cable are negligible for the unsymmetric mode.

The approximate solutions of the unsymmetric mode, 112and192,therefore, are applicable for cables with any sag and in-clination..

.."'" I 11"'1I1I'I d' n¡:: <::Tl'lllrTl 'I'Idl ¡::""(~I"'¡::CI:.I'''Ir:>, 11'Me:: .""'"

AIso, as te the solutions of symmetric mode, 111and 1910theregion of applicability can be decided by parameter f' an4 t.Let us, then, introduce a new parameter r including f' and t

f =f' 0.3l~ + 0.50.3l~ - 0.5

Transforming Irvine's equation by using (33)

(33)

(w;r)- bn (w;r)- r' - { }

'wl 0.31~+ 0.5

l28EA83 cos'e' 0.31E - 0.5

(34)

Fig. 4 shows the relation between 19r and r. A solid lineexpresses (34) and coincides well with calculated values forvarious values of~. This means that relationship betweencableforce and natural frequency can be well explained by (34) forvarious values of ~ and r.

However, (34) is nonlinear and, moreover, two values ofcable force give ttle same first-order (first symmetric) nanualfrequency [as shown later in Fig. 6 (f < 3)], and the value ofcable force is very sensitive 10the change of first-ordernanualfrequency in the region where the effects of cable sag andderivative cable force are large. This means that the slighlmeasurement error of natural frequency causes a large errorofGableforce. On the other hand, even in this region, the effeclSof cable sag and derivative cable force are negligibly smallfO;second-ordermode (first unsymmetric mode), and the valueo

"~.~.

..

I EA w El T(m) 8 (kN) (kNIm) (kN .mI) (kN) 8 "'11 .'It cp cp;(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)10.0 fY' 1.58 x 10' 0;0784 7.223 28.89 0.0034 1.252 1.164 1.113 1.00010.0 fY' 1.568 x 10'. 0.157 11.47 45.85 0.0043 1.252 1.164 1.113 1.00010.0 fY' 3.136 x 10' 0.157 9.096 36.39 0;0027 1.252 1.115 1.113 1.00010.0 3fY' 1.568 x 10' 0.157 6.557 26.25 0,0043 1.255 1.165 1.116 1.00010.0 45. 1.568 x 10' 0.157 5.734 22.93 0.0060 1.253 1.165 1.114 1.00010.0 6fY' 1.568 x 10' 0.157. 4.548 18.19 0.0107 1.248 1.164 1.110 1.00020.0 rf 1.568 x lOS 0.157 42.50 42.48 0.0043 1.252 1.164 1.113 1.00030.0 fY' 1.568 x 10' 0.157 125.2 60.09 0.0049 1.252 1.164 1.113 1.000

.~- cable force is not so sensitive to the change of second-order~ 1. natural frequency (r < 3 in Fig. 6). "lberefore, for the region

ii,,; j where the eff~~ of c.ablesag are large, namely, the values of"..,' 'r are small, lt lS desirable to use second-order mode for the...:: sneasurement of cable forees. Practically, it is necessary to set

J(- a limit of applicable regíon of the formula fOl the first-order/, 'snode. In this paper, the applicable regíon fOl the first-order

i snode is set as r ;;::3, because. in this region. the effects.oflo; ; cable sag and inclination are negligíbly small even for tint-

, ¡ order mode and it is easicr to excite a cable in tint-oIder mode.~ than in second-order mode.

PRACTICAL FORMULAS OF CABLE FORCE

First, parameters ~ and r are calculated &om cable prop-, erties and design conditions. When r is greater than 3. vibra-; tions of first-order mode are used based on the aforementioned: reasons. When r is smaller than 3, vibrations of second-order

1 mode 8!e used. 'H a cable is vcry long like,in a case of upperside cables of a long-span cable-stayed bridge, it becomes dif-ficult to excite it artificially in low-order modes. :aven if sucha cable can be excited in low-order modes, it talces much timetO measure natural frequencies due to their long periods ofvibration. Therefc;»re, it is desirable to use high-frequencymodes of stationary microvibratiODS. '

By considering theseconditions. the following practical for-mulas for the estimation of cable forces are derived from(18)-(21) and (27)-(29):

1. In the case of using the natural frequency of first-ordermode (cable with sufliciently small sag 3 :sT)

T =~(fal)2 [1 -2.20 f ..:. 0.550 (¡) 2]; (17:S E> (35)

T =~w(fa1)2[0.865 - 11.6(fYJ; (6 $ E$ 17) (36)

, T =4; (fal)2[0.828- 10.5(f) 2]; (0$ E $ 6) (37)

he : 2. In the case of using the natural frequency of seeooo-orderE. : mode (cable with relatively large sag r :S 3) ,

1E w [ ,C

(f'

)2

], i T =- (Jil)21_"":4.40,.- 1.10 - ; (60 $ E> (3_8)

3)' g .12 2, 2 '

, T =~ (IzI)2[1.03- 6.33X - 1.58~) ] j (17 $ E:S 60)

==

(39)

T =~ (Izli [0.882- 85.0~) 2]; (O$ ~$ 17) (40)

3. In the case of using the natural frequencies of high-order4) : modes (very long cable 2 :S n)

j

( )Re : 4w nC '

'or ¡ T = ;r (/.1)2 l' - 2.20T ; (200 $ E).Ie I g .Ot ¡ where f., Iz, and f,. are nieasured natural frequencies of first-

i order. second-order, ,and nth-order modes 'respectively; C =ol ¡ V(Elg)/(wJ4);r= V(wl)/(128EA83cosJ8)[(0.3IE+ 0.5)/'81 (0.3IE- 0.5)];EA=extensional rigidity;8 =sag-to-span ratioal defined in x-y coordinates (=sIlo);and 8 = inclination angle of

al cable (see Fif' ,1).~ (37) is obtained in the following way.Id From (27), ~ =42(cp - 1) and by substituting E2 =TIElht

al T =42~1 (192 - 1)u )

~ 'Substituting (22) and (23) ¡nto (42), we obtain (37). We getother equations inthe same manner.

r

(41)

(42)

"

1ñe results based on these practical formulas coincide withexact solutions oí the theoretical formula oí cable vibrationwith the accuracy of 0.4%. and cable forces can be calculateddirectly &om measured natural frequencies.

"lbough it is possible, in principie, to derive practical for-mulas fOl high-order modes in the regíon of ~ ::Sí200, Erarelybecomes less than 200 when a cable is very long and hardlyexcited artificially in low-order modes. Also, it becomes Com-plicated to maleepractical formulas of high-order modes in theregíon of E::Sí200, because it is inevitable to subdivide theregionof applicationof formulas depending on the values oíEand n (number ofvibration modes). '

As far as 200 :S E,(41) can be applied for any values of ~and n.

V.riflcatlon of Practlcal Formulas

Let us veófy the accuracy of practical formulas by com-paring the values obtained by practical formulas with experi-mental values and calculated values by' the finite elementmethodshown in Shinke et al. (1980)and Shimada (1994).Table 2 shows cable properties used in.verification.

Experiments in Shinke et al. (1980) were carried out in thefollowing way. Cable s~imens were placed horizontally andtensioned by a hydraulic'jack, where cable tensions were mea-sured by a load cell. Specimens were excited artificially in thefirst or second mode of vibration, and natural frequenciesweremeasured,by accelerometers.

Fig. S sOOwsthe relation between cable force T and naturalfrequency f in the case of short sufficiently tensioned cable(small E). In these cases, characteristics of cables are similarto those of beams. A solid line shows the values of practicalformulas and the eireles show experimental values. "lbelengthof cables are 3.40 m, 7.15 m, and 9.95 m. "lbe values ofpractical formulasagree very well with experimental values.

Fig. 6 shows the relation between cable force T and naturalfrequencyf in the case of cables of medium length (mediumE).All the cables are 31.5'm long. Numerals shown insideFig.

TABLE 2-

I(m)(1)

7.15,9.953.40,31.5301.9

Cable Propertle. U88d In VerltlC8tJon

w El(kNlm) (kN.m2)

(2) (3)0.118 23.50.144 34.51.028 380.3

T700

]¡N600

500

800

400

300

200

1001- MeasuredI

o,O 20. 30 l. Hz10

FIG. 5. Relatlon betw8en Cable Force T and Nllturat Fr.quency (Small ti '

,lnllRNAI n¡: C:::TRllrTIIRAI r::"I~I",r::r::P''''~ I "I"'C 100<: I ¡;c:c:

.1t -3 l.Sm II .MeasuredI

2 3 4f. Hz

FIG. 6. Relatlon betwaen Cable Force T .nd Natural Fre-quency f(Medlum~)

5

12000T

kN10000

8000

It -301.9m I

I . FEMI

1.5 2 2.5

In Hz

FIG. 7. Relatlon between Long Incllned Cable Force T andNatural Frequency f (Large C)

6 are numbers of vibration order. In the region where r isgreater than 3, the values of practical formulas agree well withexperimental ones of first-order mode. In the region where ris less than 3, the effects of sag and derivative forees due tovibration become larger for first-order mode, and the differ-ences between the values of practical formulas and experi-mental values become significantly large. On the other hand,the values of practical formulas agree well with experimentalvalues of second-order mode in the whole region.

Finally, Fig. 7 shows the relation between cable force T andnatural frequency f in the case of long cables (large ~). Thecable is 301.9 m long and has an angle of inclination of 23°.In this case, natural ftequencies of high-order modes are usedon the condition of using stationary microvibrations. Numeralsshown inside Fig. 7 are numbers of vibration order. Circlesshow calculated values by the finite element method. In Fig.7, the verification of practical formulas is made only for cal-culated values, since measured values are not easily availablefor very long cables. The values of practical formulas are in

656/ JOURNAL OF STRUCTURAL ENGINEERING / JUNF. 1996

good agreement with calculatedvalues for any order modes.NatUral frequencies of low-:order mode are small and it takesmuch time 10 measure natural ftequencies of low-order mode.Also, slight errors of measurement affect the accuracy of cableforcessignificantly in low-order modes. Hence, it is practicallydesirable to use high-order modes for the purpose of decreas-ing. the measurement time and increasing the accuracy of es-timated cable forces.

CONCLUSIONS

Practical formulas 10 estimate cable forces by the vibrationmethod are proposed. Since these formulas are in algebraicform, cable forces are calculated directly ftom measured nat-ural ftequencies. The formulas are applicable for any cableindependent of the length and the intemal force of the cableas lar as the vibration of first- or second-order mode is mea-surable. As to a very long cable that cannot be easily excite<!artificially, a formula is presented by using natural ftequenciesof high-order modes obtained ftom stationary microvibrations.The accuracy of the formulas are confirmed by the comparlsonof the values of practical formulas with experimental valuesfor short- and middle-Iength cables and calculated values byfinite element method for very long cables. Formulas proposedherein can be usefully and conveniently applied to the adjust-ment work of cable forces during the erection of cable systembridges.

APPENDIX l. REFERENCES

3

. .

Humar, J. L. (1990). Dynamics 01 structures. Prentice-Hall, Inc., Engle-wood Cliffs, N.J.

Irvine, H. M., and Caughey, T. K. (1974). "The linear tbeory of freevibration of a suspended cable." Proc., Royal Soc., London, England,Series A, Vol. 341.

Shimada, T. (1994). "Estimating metbod of cable tension from naturalfrequency of high mode," Proc., JSCE, 50111-29, 163-171 (in Japa-nese).

Shinke, T., Hironaka, K., Zui, H., and Nishimura, H. (1980). "Practicalformulas for estimation of cable tension by vibration metbod." Proc.,JSCE, 294, 25-'-34 (in Japanese).

APPENDIX 11. NOTATION

The following symbols are used in this paper:

e = V(Elg)/(wl');EA = extensional rigidity of cable;El = flexuralrigidity of cable;f = theoretical natural frequency of cable;

J. = measured natural frequencies of cable in nth-ordermode; 1

f: = theoretical value of nth-order natural frequency of abeam clamped at both ends;

f: =theoretical values of nth-order natural frequency of astring;

g =gravitational acceleration;h(t) =derivativecable force causedby vibration;

1 = span length;T = cable force;

v(x, t) = deftection in y-di.rection due te vibration;w = weight of cable per unit length;8 = sag-to-span ratio;~ = dimensionless parameter =VT/El'l;

'T}.= nondimensional parametef =flf:;e = angle of cable inclination; and

<P. = nondimensional parameter =fiJ:.

/

350T

/eN 300

250

200

150

100

50

OO

6000

4000

2000

OO 0.5 1