Free torsional vibration analysis of H-section hangers attached with horizontal wind-resistant cables

Transcription

1 Free torsonal vbraton analyss of H-secton hangers attached wth horzontal wnd-resstant cables * Ka Xu ), Yang Zhao ), Zhhao Wang 3) ), ), 3) Department of Cvl Engneerng, North Chna Unversty of Water Resources and Electrc Power, Zhengzhou 4500, Chna ) @qq.com ) @qq.com 3) wangzhhao96@6.com ABSTRACT The stffness contrbuton of wnd-resstant cables to H-Secton hanger s then smplfed to be horzontal sprngs support attached to hanger, and a par of horzontal sprngs support can be equvalent to a torsonal sprng support. Based on the Vlasov theory about the restraned torson of opened cross secton thn-walled beams, the governng dfferental dynamc equaton for torsonal vbraton of H-Secton hanger s establshed. The theoretcal equaton of torsonal natural frequency for H-Secton hanger wth a torsonal sprng s deduced from compatblty and fxed-fxed boundary condtons. Takng the equvalent torsonal sprng stffness and the support poston as varables, the characterstcs of frst two order torsonal natural vbraton for H-Secton hanger are analyzed. The results show that, wnd-resstant cables can greatly promote torsonal natural frequences of H-Secton hanger, and each frequency order could be better when the wnd-resstant cables near ther own node of mode ampltude. The theoretcal frequences of the cable-hanger system are fnally verfed by fnte element method. The research results can be used as a reference for the optmal desgn of wnd-resstant cables. INTRODUCTION The rgd H-secton hangers whch have more excellent characterstcs ncludng convenence of manufacture, connecton, constructon and mantenance are appled wdely n steel arch brdges. However, the H-secton hangers of ths knd, open thnwalled bars, usng steel materals frequently, whch possess bluff body cross-sectons, bgger slenderness rato, low dampng and frequences, lead to a varety of wndnduced vbratons, whch may potentally nduce severe fatgue damage, especally n the connectons between hanger and brdge grder (Keller et al. 05, Chen et al. 0). ) Graduate Student ) Assocate Professor 3) Assocate Professor

2 In engneerng felds, some effectve dampng measures are taken to control wndnduced vbratons for H-secton hangers, such that attached tuned mass dampers (Prasad and O'nel 984), aerodynamc measures (Matsumoto et al 008, Matsumoto et al 99, Maher et al 980) and horzontal wnd-resstant cables (Chen et al. 0, Ulstrup 980), et al. Among these dampng measures, the horzontal wnd-resstant cables could connect each hanger and form the coupled system, whch constran the nteracton of each hanger s vbratons and mprove the rgdtes to ncrease the natural frequences and control vbratons. In August 006, typhoon attacked a steel arch brdge n chna, whch the multple H-secton hangers n the brdge emerged ntensvely torsonal vbratons and leaded to damage flange plates at the both ends of the hangers, after that takng wnd-resstant cables to damp (Chen et al. 0). Smlarly, wndresstant cables are taken early n some steel arch brdges n other countres (Ulstrup 980). Although the wnd-resstant cables have been appled n vbraton control for H- secton hangers, but the correspondng theores were less, and the more researches about wnd-resstant cables are concentratng on cross-tes of cable stay felds n cablestayed brdge (Huang and Ncholas 0, Zhou et al 05, Javad and Cheng S 03). Due to the complcacy of coupled system, the free vbraton characterstcs of coupled system wth a hanger and wnd-resstant cables are not well known, and the optmal desgn solutons of wnd-resstant cables parameters haven t establshed, whch restrct the applcaton and development to some extent. Accordngly, the coupled system wth rgd H-secton hanger n steel arch brdge and wnd-resstant cables and the theoretcally analytcal solutons for ts free torsonal vbraton characterstcs are establshed (coupled system s called for short farther below). Here s the approach: frstly, the rgd H-secton hangers are smplfed to the beams wth fxed-fxed boundares, and the torsonal rgdty contrbuton produced by wnd-resstant cables to H-secton hanger s smplfed to the equvalent torsonal sprng s nteracton. Secondly, based on Vlasov theory about the restraned torson of opened cross secton thn-walled beam to buld the dynamc dfferental equaton of torsonal vbraton, combnng the compatblty condtons and fxed-fxed boundary condtons, the theoretcal model and torsonal natural frequences solutons of coupled system wth a hanger and wnd-resstant cables are establshed. Fnally, by numercal results and parameters analyss, the effects to promote torsonal natural frequences for H-secton hanger and the theoretcal solutons accuracy and stablty are verfed.. THEORETICAL MODEL. Coupled system wth H-secton hanger and horzontal wndresstant cables H-secton hangers are connected wth brdge grder and man arch rb by hghstrength bolts, whch boundares, between fxed-fxed and smple, are close to fxedfxed (Ruscheweyh et al 996). For computng convenently, H-secton hangers are smplfed to fxed-fxed beams subjected axal force and the constrant force of attached horzontal wnd-resstant cables s smplfed horzontal elastc support. The coupled system wth a H-secton hanger and wnd-resstant cables s consdered as n Fg.. It s assumed that a par of wnd-resstant cables s descrbed by 4 horzontal sprngs and located at ponts X such that 0 X L and wth stffness k, k, k3, k4,

3 respectvely. The entre hanger s now dvded nto segments wth lengths L L respectvely whch s separated by the poston of 4 sprngs. The free vbraton ampltudes of angle due to torsonal, of each segment s denoted by ( xt, ) on the nterval X x X, where the sub-ndex n the parentheses represents the th segment and (Ln and Chang et al 005). The H-secton hanger parameters ncludng a, b, t and t represent the wdth of flange slabs, the sze both web s hgh and flange slab thckness, flange slab thckness and web thckness, respectvely. P L L L ( k, k) ( k3, k4) S a t t b t / t / x X k k3 k k4 P Fg. The coupled system wth H-secton hanger and horzontal wnd-resstant cables. The dynamc dfferental equatons of torsonal moton H-secton hangers are opened cross secton thn-walled bars, whch are appled force as Fg. on the base of Vlasov theory (Ambrosn et al 000, Aleksandar and Dragan 0)

4 T Ts T P T P f I TP TP x dx P T T x Fg. Torsonal force n the mcro-unt of a H-Secton hanger () T T T s ( xt, ) Ts GJ x 3 ( xt, ) T EC 3 x PIP ( x, t ) TP A x ( xt, ) fi IP t Where T, T s,t, T P and f I represent total torture, free torture, constrant torture, axal torture produced by axal force and torsonal nerta force of cross-secton respectvely [5] E, G s shear of the modulus of the materal, G whch E s.6 Young s modulus of the materal, J s torsonal constant of the cross-secton, 3 J bs (Ulstrup 978, Lee et al 04) whch b, s are the length and wdth of 3 E cross-secton lnes respectvely, E s corrected modulus, E and s 3 abt Posson rato, C s warpng constant, C ( ab, are descrbed n Fg.), P 4 s axal force and the tensle force s postve, I P s the polar moment nertal of crosssecton, A s the area of cross-secton, s the densty of the materal. Torsonal forces are analyzed n Fg., whch can wrte the equlbrum equaton of the mcro-unt (Ulstrup 978, Mehrdad 05) for the H-secton hanger:

5 T T T T T T T T T f x x x s P s P s P I () Substtuton of Eq. ()nto Eq. () lead to the torsonal equaton of moton for each H-secton hanger s segment: ( x, t) PI ( x, t) ( x, t) E C ( GJ ) I 0 4 P 4 P x A x t (3) Where EC and GJ are torsonal and warpng rgdtes, respectvely..3 Equvalent torsonal sprung rgdty of wnd-resstant cables Cross-secton forces, subjected 4 horzontal sprngs whch n cables poston, are descrbed n Fg.3. As n Fg.3, the 4 horzontal sprngs n the same poton of crosssecton can be equvalent wth one torsonal sprng. Usng S to denote the equvalent torsonal sprng rgdty whch smplfed by a par of wnd-resstant cables, can be denoted as torsonal angular dsplacements. And the force n cross-secton of H-secton hangers subjected by 4 sprngs can be expressed: F F k x 3 3 F F k x 4 (4) Where k s one sprng rgdty of the lateral wnd-resstant cable, whch EA k and E, A, l represent the Young s modulus, cross sectonal area and l length of wnd-resstant cables, x and x3 represent the horzontal dsplacements of H-secton hangers n the drecton of sprngs, moreover x x3 x and x can be expressed: b x sn (5)

6 b F a F k x t F4 k x F 4 x x S F t k x3 F3 k x3 F x 3 F 3 Fg.3 Cross-sectonal torsonal force of a H-Secton hanger The equlbrum equaton of equvalent torson can be deduced: S ( F F ) b ( F F ) b kx b (6) 3 4 Whch sn can be derved when s tny, the equvalent torsonal sprng rgdty of wnd-resstant cables s derved by smplfyng Eq. (6): A S E b l (7).4 Method to fnd egensolutons of coupled system Usng the separable solutons (Gokdag and Kopmaz 005, M.Tahmaseb et al 04), the torsonal dsplacements can be wrtten: ( x, t) ( x) e t (8) Where ( x) s th torsonal dsplacements mode functon of each H-secton hanger segment wth,, s crcular natural frequences of coupled system wth a H-secton hanger and wnd-resstant cables. Where: ( x) A sn( cl ) B cos( cl ) C snh( dl ) D cosh( dl ) (9)

7 4 4 4 g g 4 g g c ( a ) d ( a ) (0) 4 4 I P GJ PI P A () 4 g EC EC a A B C and D are constants assocated wth th segment (, ). The compatblty condtons, ncludng the torsonal dsplacement, the angle of torson rate, torson-bendng b-moment and the total torque of the cross-secton, across the equvalent torsonal sprng of wnd-resstant cables can be expressed ( X, t) ( X, t) () D U ( X, t) ( X, t) (3) D U E C ( X, t) E C ( X, t) (4) D U D PIP D U U PIP U EC ( X, t) ( GJ) ( X, t) S ( X, t) EC ( X, t) ( GJ ) ( X, t) (5) A A U D Where the symbols X and X denote the locatons mmedately above and below the poston X, ( x, t), ( x, t) and ( x, t) represent the, and 3 orders dervatve of ( xt, ). The compatblty condtons from Eq. (9) nto Eq. () to Eq. (5): ( L ) (0) ( L ) (0) ( L ) (0) PIP EC ( L ) ( GJ ) PI ( L ) S ( 0) EC (0) ( P GJ ) (0) A A (6) For the case of the H-secton hanger fxed at both ends, the correspondng boundary condtons can be expressed (0, t) 0 (0) 0 (7) (0, t) 0 (0) 0 (8) ( L, t) 0 ( L ) 0 (9) ( L, t) 0 ( L ) 0 (0) Begnnng wth those at the left fxed end, Eq. (9), (7) and (8), leads to

8 B D 0 D B c ca dc 0 C A d () Satsfacton of the boundary condtons of Eq. (9) at the rght fxed end, Eq. (9) and (0) requre A sn( cl ) B cos( cl ) C snh( dl ) D cosh( dl ) 0 () A c cos( cl ) B c sn( cl ) C d cosh( dl ) D d snh( dl ) 0 (3) Whch Eq. (9) nto Eq. (6) and Eq. () to (3) can be expressed n matrx form as Where R 0 (4) R R R R R (5) R c R d R sn( ) cos( ) snh( ) cosh( ) cl cl dl dl c cos( cl) c sn( cl) d cosh( dl) d snh( dl) sn( cl) cos( cl) snh( dl) cosh( dl) c cos( cl) c sn( cl) d cosh( dl) d snh( dl) (6) c sn( cl) c cos( cl) d snh( dl) d cosh( dl) ( c c)cos( cl) ( c c)sn( cl) ( d d)cosh( dl) ( d d)snh( dl) 0 0 c 0 d 0 R 0 c 0 d 3 3 ( c c) ( d d ) Whch E C L GJ PI P GJ ( L)

9 and s non-dmensonal equvalent torsonal sprng rgdty of wnd-resstant SL cables, whch, s the vector of undermned parameters and GJ T A B C D A B C D. Thus, the exstence of non-trval solutons requres det R 0 (7) Ths determnant provdes the sngle equaton for the soluton of the egenvalue,. The coeffcents of the egenfunctons, ( x), are obtaned by back substtuton j nto Eq. (6) to get and then Eq. (9) to get each segment of H-secton hanger.. EXAMPLE OF NUMERICAL CALCULATION RESULTS. Basc parameters Fg. 4 denotes the prototypcal cross-secton of H-secton hanger n certan steel arch brdge, and ts parameters as n Table. Wnd-resstant cables are used the s techncal specfcaton of f 860MPa,7 5.mm, whch the Young s modulus pk 3 E.95e5MPa, Posson's rato 0.3 and densty 8600kg/m Y Z Fg.4 Cross-sectonal dmenson of a H-Secton hanger(mm) Table Basc parameters of a H-Secton hanger Geometrcal parameters Values H-secton hanger parameters /a 0.5m H-secton hanger parameters /b.8m Length of prototype/ L 40.m Area of prototype/ A m Moment of nerta n strong axal/ Z m Moment of nerta n weak axal / Y I I m

10 I 3 4 Polar moment of nerta / P m St.Venant s torsonal constant/j m Warpng constant /C m Young s modulus/e.0 0 N/m Shear modulus/g N/m Densty/ kg/m Axal force/p 07KN. Fnte element model Thn shell element of the SHELL63 type s smulated to H-secton model, because of the mnmum values of cross-secton szes comparng to length of prototypes. For smulatng axal force, the above end s fxed all degrees of freedom and the below end free axal dsplacement but constran others degrees, whch can greatly mtate the effects of axal force rgdtes on torsonal natural frequences. The type of Lnk0 element can be smulated to wnd-resstant cables whch are fxed n the anchorage felds, and the ntal tenson of wnd-resstant cables s mtated by real constant of prmary stran. Consderng the connecton of the flange plates n H- secton hanger and wnd-resstant cables at the real brdge usng rope clp, the connecton can be used common nodes ncludng H-secton hanger and wnd-resstant cables, whch neglectng the frcton of both attached feld. Fg.5 Fnte element model of the coupled system wth a H-Secton hanger and wndresstant cables.3 The torsonal natural frequences of H-secton hanger The coupled system wth H-secton hanger and wnd-resstant cables s degraded nto ordnary H-secton hanger when the non-dmensonal equvalent torsonal sprng rgdty of wnd-resstant cables 0. Takng 0 nto Eq. (7), the determnant can be expressed

11 ) cd( cos cl cosh dl ) ( d c sn cl snh dl 0 (8) For verfyng the accuracy of Eq. (8) whch can be solved by usng the standard Newton-Raphson teratons wth the mathematcal software MATLAB, table shows the resultant comparsons wth fnte element method and Ref. [Ulstrup 978] solutons. The lowest two torsonal natural frequences of three solutons n table have lttle dfference. Table Comparsons of natural frequences of the anger(hz) Lowest two natural frequences Frst order Second order Theoretcal solutons n ths paper Formula solutons n Ref. [Ulstrup 978] Fnte element method computaton FREE TORSIONAL VIBRATION CHARACTERISTICS OF COUPLED SYSTEM 3. The effects of wnd-resstant cables locatons and rgdtes for torsonal natural frequences K 403 s defned as fundamental value whch s non-dmensonal rgdtes of equvalent torsonal sprng for wnd-resstant cables, assocatng wth length l 00m, area A 7.8mm (0.5 tmes of one steel strand above-mentoned). Fg. 6 shows the lowest two torsonal natural frequences of H-secton hanger obtaned by the method presented n ths research as the non-dmensonal stffness ncreased and the locatons of wnd-resstant cables( X / L) changed from zero to one, whch gvng the fnte element method(fem) computatons to further verfy theoretcal method n ths research. Table 3 shows the lowest two maxmum torsonal natural frequences of H- secton hanger n dfferent values and wnd-resstant cables locatons.

12 Frst order torsonal natural frenquences /Hz Second order torsonal natural frequences/hz Theory K FEM K Theory 5K FEM 5K Theory 0K FEM 0K Theory 50K FEM 50K X /L (a)frst order Theory K FEM K Theory 5K FEM 5K Theory 0K FEM 0K Theory 50K FEM 50K X /L (b)second order Fg.6 The lowest two torsonal natural frequences of the coupled system wth H- secton hanger and wnd-resstant cables n dfferent locatons versus Table 3 The maxmum torsonal natural frequences of the frst two orders for the hanger(hz) Frst order Second order K(.4e 3) K(7.05e 3) K(.4e 4) K(7.05e 4) Increased percentage( 50K ) 03.35% 79.66% As shown n Fg.6 and Table 3, some rules can be deduced: () The theoretcal solutons are hghly anastomosed wth fnte element method computatons of lowest two torsonal natural frequences, whch proves the accuracy and stablty of theoretcal solutons to solve the torsonal natural frequences of coupled system wth H-secton hanger and wnd-resstant cables. Due to consderng the 0.5m length of hgh strength bolts feld at the end of the hanger n fnte element model, resultng n the effectve length of the hanger decreased partly, the fnte element method computatons have a lttle bgger than theoretcal solutons. () The lowest two torsonal natural frequences of coupled system are ncreased gradually along wth the values of added. Observng the axal coordnates where each curve reach the extreme values, the hgher torsonal natural frequences are obtaned when the wnd-resstant cables are n the poton of the bggest node coordnate values of the mode shapes, but the torsonal natural frequences are not affected by the wnd-resstant cables when t s located n model node of H-secton

13 x/l H-secton hanger's axs x/l H-secton hanger's axs hanger (when x/ L =0.5, the second order torsonal frequences are common wth the values changed n Fg.6(b)). Hence, consderng the frst order torsonal vbraton control n real engneerng, the wnd-resstant cables should be nstalled n the mddle length of the hanger, and t should be set n the thrd or two thrds length of the hanger when the second order torsonal vbraton need to be restraned. (3) As s shown n table 3, the frst order torsonal natural frequences can be mproved to the hghest 3.03 tmes of the orgnal value whch don t have wnd-resstant cables attached n hanger, and the second torsonal natural frequences have the hghest.8 tmes of ts orgnal value, whch have hghly mprovements of lowest two torsonal frequences. 3. The effects of wnd-resstant cables locatons and rgdtes for torsonal mode shapes Due to the secton 3., the torsonal natural frequences are mproved to the bggest values when the wnd-resstant cables are nstalled n the postons of maxmum mode shapes node, and the nfluence rules of lowest two mode shapes wth ncreased are analyzed when the wnd-resstant cables are n the / and /3 lengths of hanger, whch are shown n Fg. 7 and Fg =0 =K =5K =0K =50K =500K =0 =K =5K =0K =50K =500K L L Torsonal mode shapes (x) Torsonal mode shapes (x) (a) / / X L (b) / / 3 X L Fg.7 Frst order torsonal mode shapes of the H-secton hanger

14 x/l H-secton hanger's axs x/l H-secton hanger's axs =0 =K =5K =0K =50K =500K =0 =0K =K =50K =5K =500K L 0.4 -L Torsonal mode shapes (x) Torsonal mode shapes (x) (a) X / L / (b) X / L / 3 Fg.8 Second order torsonal mode shapes of the H-secton hanger From Fg. 7 and Fg. 8: () When X / / L and 0,K, the maxmum mode functon values of frst order torsonal mode shapes s n the poston of 0.5L, and ts node coordnates of mode shapes are decreased to 0 gradually wth ncreased, whch have two ampltude postons of mode shapes ncludng x/ L 0.5 and x/ L 0.75 respectvely. When X / / 3 L, the node coordnates of ampltudes for frst order torsonal mode shapes have emerged negatve values wth ncreased, and the torsonal dsplacement turn to be 0 at the poston of wnd-resstant cables ( (/ 3) 0 ). The more bgger values, the more stronger constrant force of the wnd-resstant cables to H-secton hanger, and the poston can be fxed fnally. () When X / / L, the second order torsonal mode shapes have no changng wth ncreased, ths can concde wth the results whch the second torsonal natural frequences are nvarant when the wnd-resstant cables are n mdpont of the hanger obtaned n secton 3. and Fg. 7. When X / L / 3 and 50 K,500K, the second order torsonal mode curves have two ponts of ntersectons wth H-secton hanger s axs. 4. DAMPING SOLUTIONS OF THE FIRST ORDER TORSIONAL VIBRATION Due to Eq. (7), the, non-dmensonal equvalent sprng rgdty of wnd-resstant cables, s determned modulus, length and area of wnd-resstant cables, whch can easly change the length and area of wnd-resstant cables by steel strand parameters. In real brdge engneerng, frst order torsonal natural frequences can be mproved

15 Lowest two torsonal natural frequences/hz preferentally, and the best nstalled postons of wnd-resstant cables s n hangers md-pont owng to secton 3.. Changng the length and area of wnd-resstant cables to make ncrease, Fg. 9 shows the curves of lowest two torsonal natural frequences wth ncreased, whch also descrbe the fnte element method computatons to compare a Frst-order of Theory Frst-order of FEM Second-order of Theory Second-order of FEM / 0 3 Fg.9 Natural frequences of the coupled system wth a hanger and md-pont wndresstant cables versus As s shown n Fg. 9, () The curves of the frst order torsonal natural frequences are growng rapdly at the begnnng wth the hgh amplfcaton, but latter the tendency become slow and turn nto stable fnally wth ncreased, whch n ths tme the fundamental torsonal frequences reach the maxmum values. Because the wnd-resstant cables are nstalled n mddle hanger, the second order torsonal natural frequences are kept the same values and nvarant. () The frst torsonal natural frequences wll exceed the second after the pont of a whch s a pont ntersecton of two curves wth ncreased. And the frst order torsonal natural frequences keep ncreasng and the second order emerge frst after the pont of a, whch change the maden orders. In the above, for dampng the frst order torsonal natural frequences, the natural frequency value correspondng the pont of a s treated as the objectve torsonal frequency, whch the correspondng rgdty s treated as objectve stffness. Then backng the objectve rgdty nto Eq. (7), the requste length and area of wnd-resstant cables can be calculated.

16 5. CONCLUSIONS Based on the open thn-walled sectons theory of Vlasov and the compatblty condtons n the poston of wnd-resstant cables, the theoretcal model of coupled system wth a hanger under the fxed-fxed boundares and horzontal wnd-resstant cables s establshed. The effects and rules of lowest two torsonal natural frequences and ts mode shapes are analyzed wth the locatons and rgdtes of wnd-resstant cables changed. The followng conclusons can be drawn: () The establshed theoretcal approach to solve the torsonal natural frequences of coupled system wth a hanger and horzontal wnd-resstant cables s verfed the greatly accuracy. () For mprovng frst order torsonal natural frequences of H-secton hanger, the wnd-resstant cables should be nstalled n the mddle length of hanger, whch n ths poston the fundamental torsonal frequences ncrease rapdly at the begnnng, and the ncremental trend s slow gradually, whch the hanger s fxed n the poston of wnd-resstant cables and turned to be two ndependent segments completely wth the torsonal rgdty of wnd-resstant cables ncreased fnally. And the maxmum value of fundamental torsonal natural frequency for H-secton hanger n ths numercal results s 3.03 tmes of ts orgnal value. Due to the specal poston whch n md-pont of H- secton hanger of the wnd-resstant cables, the frst order torsonal frequences surpass the second order when the equvalent torsonal rgdtes of wnd-resstant cables are greater than a certan threshold, whch can nverse the wnd-resstant cables parameters accordng to objectve torsonal frequency whch s the ntersecton value of the frst and second order frequences curves. (3) Wth the nvarant wnd-resstant cables rgdty, the best second order torsonal frequences can be obtaned when the cables are nstalled n /3 or /3 length of hanger. And there are two ponts wth hanger s axs to the second order torsonal mode shapes wth the wnd-resstant cables rgdtes ncreased. The maxmum value of second torsonal natural frequency for H-secton hanger n ths numercal results s.80 tmes of ts orgnal value. REFERENCES Keller P, Hggns C, Lovejoy S.C. (05), Evaluaton of torsonal vbratons n steel truss brdge members. Journal of Brdge Engneerng-ASCE, Vol, 0(9): Chen Z.Q., Lu M.G., Hua X.G., et al. (0), Flutter, gallopng, and vortex-nduced vbratons of H-Secton hangers. Journal of Brdge Engneerng-ASCE, Vol 7(3): Prasad S., O'nel T. (984), Vlasov theory of electrostatc modes n a fnte length electron column. Physcs of Fluds, Vol, 7(): Matsumoto M., Shrato H., Mzuno K., et al. (008), Flutter characterstcs of h-shaped cylnders wth varous sde-ratos and comparsons wth characterstcs of rectangular cylnders. Journal of Wnd Engneerng and Industral Aerodynamcs, Vol, 96(6):

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