Blucher Mechancal Engneerng Proceedngs Ma 014, vol. 1, num. 1 www.proceedngs.blucher.com.br/evento/10wccm A NEW FREQUENCY METHOD APPLIED TO CALCULATE CABLE FORCE FOR CABLES WITH THE INTERMEDIATE MULTI-SUPPORT Pngje L Ronghu Wang Nujng Ma School of Cvl Engneerng and Transportaton, South Chna Unverst of Technolog(pngjel@163.com) Abstract. Ths paper proposes a new frequenc method appled to calculate cable force for cables wth the ntermedate mult-support from measured natural frequences. The proposed method has establshed new relatonshp between cable frequenc and tenson force, takng cable force, the ntermedate mult-support, flexural rgdt, boundar condtons of two ends nto account. Followng the classcal gudelnes of cable dnamcs, and b makng some smplfcatons, frequenc equaton has been obtaned n non-dmensonal form. When natural frequences have measured, cable force can be calculated b solvng frequenc equaton. Comparson wth Matharmethod and the proposed method n the feld test, the dfference of cable force result between two methods s just onl 1%, so the proposed method has been made to valdate. Kewords:frequenc method, cable force, mult-support, cable 1. INTRODUCTION Frequenc method s the wa to calculate the cable force b measured natural frequences of cables. In recent ears, Frequenc method has receved ncreasng attenton because of ts smplct and speedness. In practce, frequenc method s wdel used for cables wth boundar condtons of constrants on both ends, and rarel used for cables wth boundar condtons of constrants on both ends and the ntermedate. As cables are appled more and more wdel, the boundar condton of cables s more and more complex, such as the damper at suspenders n the suspenson brdge and ntermedate support at the te bar n the arch brdge. So a new frequenc method s useful for applcaton to calculate the tenson force of cables wth the ntermedate mult-support. Nowadas, there are a lot of researches on the vbraton of cables. Irvne H M [1] has proposed an analtcal soluton for the frequenc equaton for sagged cable wthout bendng stffness. For such long cables, wth such large dameters, flexural stffness cannot be neglected n determnng dnamcal propertes of cables. So Zu H [],Geer Rand Km B H [3-4] has proposed methods whch allow us to obtan cable force n a smple manner, whch are based on the dentfcaton of modal propertes of cables. Flamand O and Matsumoto M [5-6] have observed the nfluence of ran wnd acton to cables. These studes are mostl for cables wth the boundar condton of constrants on both ends, and there are rare for cables wth the boundar condton of 1
constrants on both ends and the ntermedate. R H wang [7] has put forward to a fnte element soluton for tenson force of cables wth elastc mult-support, whch has been expermentall verfed. However, there s not an analtc method for tenson force of cables wth elastc mult-support. Ths paper proposes a new frequenc method for applcaton to calculate the tenson force of cables wth the ntermedate mult-support from measured natural frequences. The proposed method has establshed new relatonshp between cable frequenc and tenson force, takng cable force, the ntermedate mult-support, flexural rgdt, boundar condtons of two ends nto account. Followng the classcal gudelnes of cable dnamcs, and b makng some smplfcatons, frequenc equaton has been obtaned n non-dmensonal form. When natural frequences have measured, cable force can be calculated b solvng frequenc equaton..dynamic MODEL OF CABLES.1Vbraton model for cables The cable wth the boundar condton of restrants on both ends and the ntermedate s equvalent for the mult-span beam wth axal force, when flexural stffness of cables has not been neglected., the calculated parameters can be descrbed as show n fgure 1: L s the undeformed cable length; H s the horzontal component of ntal cable tenson; n s the number of constrants on the ntermedat of cables; K (=1,,...,n) s stffness of constrants; X (=1,,...,n) s the dstance between the adjacent constrants. The boundar condton of constrants on both ends s consder as clamped. Fgure 1. the cable wth the ntermedate mult-support In accordance wth [1], the lnearzed equatons of moton are: 4 v d v H h() t EI (,) 4 v f x t mv (1) x dx x v s the transverse vbraton dsplacement of cables changng over tme, ht () s the addtonal horzontal component of tenson changng over tme, ( xt, ) s the transverse coordnate of cables, x s the coordnate along the axs of beam, t s
vbraton tme, E s the Young modulus; I s the nerta moment; s vscous dampng coeffcents per untar cable length, v s frst tme dervatve of the vbraton dsplacement, v s second tme dervatve of the vbraton dsplacement, m s the cable mass per untar length, f ( xt, ) s mpressed force changng over tme. The vbraton model of the cable has consdered elastc support onl, but hasn t consdered the effect of dampng, so 0.accordng to Irvne H M [1], It can be neglgble to the addtonal horzontal component of tenson [1], so ht () =0. Assumng v v x, t x qt ( ), f ( xt, )= f ( xqt ) ( ),the partal dfferental equaton (1) s transformed nto two ordnar dfferental equatons, one governng the tme evoluton and one governng the spatal dstrbuton of the moton: qt () qt () 0 (a) IV H m 1 ( x) ( x) ( x) f ( x) (b) EI EI EI Expresson (b) s the fourth order nonhomogeneous lnear dfferental equaton, whch can be solved as: 1 ( x) A1cos( x) Asn( x) A3cosh( x) A4snh( x) f ( ) x (3) m In whch Aj ( j = 1,..., 4) are constants and the followng quanttes have been ntroduced: ( ) ( H ) m H EI EI EI (4a) ( ) ( H ) m H EI EI EI (4b).The cable frequenc equaton Boundar condtons for clamped restrant at cable anchorages are: v( L/, t) 0 (5a) vl ( /, t) 0 (5b) v ( L/, t) 0 x v ( L /, t ) 0 (5d) x The boundar condtons of cables wth clamped ends, gven b Eqs. (5a)- (5d), 3 (5c)
substtute nto Eq (3); And then, an algebrac set of four equatons, wth the unknown quanttes Bj ( j = 1,..., 4), can be solved; And Bj ( j = 1,..., 4) can be substtuted nto Eq. (3) and the fnal expresson of the modal shapes s obtaned followng: fˆ ( ) ˆcosh( ˆ ˆ)sn( ˆ / ) ˆcos( ˆ ˆ)snh( ˆ x x x / ) ˆ ( xˆ ) 1 ˆ ˆ cosh( ˆ / )sn( ˆ / ) ˆ cos( ˆ / )snh( ˆ / ) (6) The parameters of expresson (6) can be obtaned b Eqs. (7a)- (7f). ( x x ) s the Drac delta functon. x xˆ L (7a) L ˆ (7b) EI / m L H / EI (7c) ˆ ( ˆ) ( ) ˆ (7d) ˆ ( ˆ) ( ) ˆ (7e) ˆ KL f ( x) ( x ) ( xx ) ( 1,,..., n) (7f) 4 n ˆ ˆ ˆ ˆ ˆ EI 1 So frequenc equaton of cables can be gotten b Eq (6) and Eq (7f), the smplfed expressons s wrtten as: C I 0 (8) The parameters of expresson (8) can be gotten b Eqs. (9a)- (9b) EI ˆ ( 1,,..., n) (9a) 4 KL ˆ cosh( ˆ xˆ )sn( ˆ / ) ˆ cos( ˆ ˆ )snh( ˆ x / ) Cj 1 ( j 1,,..., n 1,,..., n) ˆ cosh( ˆ / )sn( ˆ / ) ˆ cos( ˆ / )snh( ˆ / ) (9b) 3 CABLE FORCE CALCULATION 3.1 Cable force equaton Cable frequenc equaton (8) contans two non-dmensonal parameters, one s 4
the non-dmensonal frequenc ˆ, and the other s the non-dmensonal cable force. So cable force equaton can be obtaned b eqs (8), (7d), (7e): f ( ˆ, ) 0 As cable frequenc s tested out, the non-dmensonal frequenc ˆ can be gotten. So expresson (10) s a sngle parameter equaton as show n expresson (11). f ( ) 0 (10) (11) 3. Iteratve algorthm for the cable force equaton As we known, the Eq(11) s a transcendental equaton. It can t be solve drectl. So t must adopt numercal methods, lke Nowton-Raphson, for the purpose of obtanng the roots of Eq. (11).It s convenent to manpulate ths expresson to elmnate some dfferences, whch, when the are made between close approxmate numbers, can lead to round off errors as show n fgure. f mleik,,,, xˆ, ˆ, ˆ( ˆ), ˆ ( ˆ), fˆ ( xˆ),, C j H 1 ˆn f n H n 1 H n H H n 1 H n H f f f,suppose error as = 10 n 4 fn f 0 f f 0 n H x Fgure. flowsheet for cable force calculaton 4 APPLICATION EXAMPLE 4.1 background 5
Sanshanx Brdge s concrete flled steel tubular ted arch brdge. Te bar, consstng of hundreds of steel strands, s located on the both end of brdge under the brdge deck wth the length of 60m,whch s used for resstng horzontal thrust of man arch. A bunch of te ncludes 10 to 1 steel strands, and these steel strands are contact each other. There are 50 bunches of te separated. Steel strands have been separated to three parts b ponts slk plate. The length of parts of steel strands s 49.7m, 160.6m, 49.7m respectvel. Ponts slk plate s weldng nsde steel box, whch s usng for protectng the te bar from corroson. Steel strand and ponts slk plate contact tghtl, so each bunch of te bar can be consdered to be anchored at the place of ponts slk plate. It can t be completed to determne the axal force of te bar b applng frequenc method drectl f we have not solved the followng two problems. One s how to test frequenc of te bar. Steel strands are contact each other, so we can appl some fllng blocks to separate the steel strand, what s mportant s determne the compresson stffness of fllng blocks. The other s how to calculate the axal force of te bar b measured frequenc. There s changed boundar condton of steel strand b ncreased fllng blocks. So t s essental to deduce a new equaton of cable force and frequenc for cables wth the ntermedate mult-support. 4. Test parameter Te bar and steel strand both can be consder to be the cable, but t just can be test one steel strand frequenc, so the cable s stand for steel strand n the followng. The cable parameter s descrbng at table 1, E s the modulus of elastct, I s the nerta moment, s the mass denst, A s area, d s dameter of secton, L s the length of cables. E(pa) I(m 4 ) Table 1. test parameter (kg/m 3 A(m ) d(mm) L(m) ) 1.95E+11 9.69E-10 7.85E+3 0.000137 15. 49.7 The fllng block has three laers. There s steel plate at the top and bottom laer wth the thckness of 5mm and block rubber at the mddle laer wth the thckness of 40mm. The sze of fllng blocks s 5cm 5cm.The compresson stffness of fllng blocks s determned b traxal compresson test. Two fllng blocks have been made 6 and the compresson stffness of them are respectl : K 1.046410 N/m 1 K 1.0104 10 6 N/m. 4.3 Cable frequenc and cable force Cable frequenc cannot be tested drectl because of the contact between cables, so fllng blocks are used for solatng cables as show n fgure 3. And cable frequenc 6
can be tested b dnamc testng nstrument when optmum vbraton sstem set at the mddle of two fllng blocks as show n fgure 4, whch are nstalled under the cable as show n fgure 5. Fgure 3. The model for testng cable frequenc Fgure 4. The nstallng place of optmum Fgure 5. The nstallng place of the fllng vbraton sstem block It s fnd that the 4th order frequenc of cables can be eas tested as show n table. Table. Tested cable frequenc number 1 3 4 5 The 4th order cable frequenc(hz) 16.109 17.041 16.3086 16.598 16.3574 B flowsheet for cable force calculaton (fg.), cable force has calculated as show n table 3. The average cable force s 97.4kN, whch can evaluate the whole level of te bar force. Usng non-destructve testng (NDT) methods of Matharmethod [8], te bar force can be tested to be 98.3kN. So the dfference of te bar force tested b two methods s small, just wth the error of 1%. Table 3. Calculated cable force number 1 3 4 5 average cable force(kn) 94.55 105.01 95.75 95.15 96.36 97.4 7
5 CONCLUSION The computatonal formula of frequenc method cannot calculate cable force for cables wth boundar condtons of constrants on the ntermedate. so a new computatonal formula of frequenc method s establshed for cables wth the ntermedate mult-support. Some conclusons are made as followng: 1)The proposal computatonal formula can be useful for applcaton to calculate the tenson force of cables wth the ntermedate mult-support, such as the damper at suspenders n the suspenson brdge and ntermedate support beam at the te bar n the arch brdge. )The proposal computatonal formula s a transcendental equaton, so numercal method should be adopted, but t s eas to make a programme, followng fg,to calculatng cable force. 3)The feld test shows that the dfference of cable force, testng b the proposal method and the matharmethod,s small, so the proposal method s relable. Acknowledgement Ths work s supported b Program of Natonal Natural Scence Foundaton of Chna (50978105) REFERENCE [1]Irvne H M. Cable Structures [M]. Cambrge:The MIT Press, 1981. [] Zu H, Shnke T, Namta Y. Practcal formulas for estmaton of cable tenson b vbraton method [J]. J Struct Eng 1996, 1(6):651 665. [3] Geer R, De Roeck G, Flesch R. Accurate cable force determnaton usng ambent vbraton measurements. Struct Infrastruct Eng 006;(1):43 5. [4] Km B H, Park T. Estmaton of cable tenson force usng the frequenc-based sstem dentfcaton method [J]. Journal of Sound and Vbraton. 007, 304(3-5): 660-676. [5] Flamand O. Ran-wnd nduced vbraton of cables. J Wnd Eng Ind Aerodn 1995;57:353 6. [6] Matsumoto M, Dato Y, Kanamura T, Shgemura Y, Sakuma S,Ishzak H. Wnd-nduced vbraton of cables of cable-staed brdges.j of Wnd Eng and Ind Aerodn 1998;74 76:1015 7. [7]Wang Rong-hu,Gan Quan,Gong Lng-lng,et al.a fnte element soluton for tenson of cable wth elastc mult-support[j]. Journal of Vbraton Engneerng, 010, 3(5):567-571. [8]BA50/93. Post-tensoned Concrete Brdges. Plannng, Organsaton and Methods for Carrng out Specal Inspectons. Department of Transport. Jul 1993. 8