Optimization of absorbers in the highway bridges due to traffic flow

Transcription

1 25 Optimizatin f absrbers in the highway bridges due t traffic flw P. Sniady, R. Sieniawska, S. Zukwski Institute f Civil Engineering, Technical University fwrclaw Wybrzeze Wyspiailskieg 27, Wrclaw, Pland 1. INTRODUCTION The prblem f reducing the level f vibratins in varius cnstructins is cnsidered fr many years and numerus ways and means f preventing unacceptable vibratins are knwn [1]. One f this ways are varius vibratin absrbers. Their applicatin play a special rle because they can be used during the cnstructin design and als when unsatisfactry dynamic prperties appear in a cnstructin during peratin. Absrbers are applied in varius engineering structures such as shipbuildings, steel chimneys, TV twers, bridges [2, 3], etc. Vibratins f sme f this structures are excited by lad f randm nature. In this cases damped vibratin absrbers (DVA) are used t mitigate the randm excitatin [3] and their parametres are ptimized n the cnditin f the main mass's. The minimum variance f the displacement crrespnds t the maximum reliability f the system. The traffic flw is such type f lad which causes cmplicated stress and displacement states in bridges and als causes the material fatigue that ultimately can damage the structure. The applicatin f absrbers can greatly extend its lifetime. In the paper the prblem f ptimal chsing f parameters fr damping absrbers reducing the randm vibratin f highway bridges subjected t traffic flw is studied. It is assumed that the traffic is a cmpsite f different types f vehicles Each type f vehicle is mdelled by ne r mre cncentrated randm frces [4,5,6]. In this idealizatin the vehicles are regarded as being f randm weight. The interrarival times f the vehicles are regarded as randm variables. As the ptimizatin criterin f absrbers the minizatin f the first tw prbabilistic characteristics (cumulants) f bridge displacements which crrespnd t the maximum f the bridge's reliability is assumed. The cnnected cefficient f skewness and kurtsis f displacements are als calculated. T slve the ptimizatin prblem the simulatin apprach is used. 2. FORMULATION OF THE PROBLEM Let us cnsider a beam with an absrber fitted at pint x 0 under a stream f cncentrated randm frces mving in the same directin alng the beam with cnstant speed v. The frces, which mdel the mving vehicles in a traffic flw, arrive at the beam at randm times t k. This R. Rackwitz et al. (eds.), Reliability and Optimizatin f Structural Systems Springer Science+Business Media Drdrecht 1995

2 242 Part Tw Technical Cntributins arrival times cnstitute a Pissn stchastic prcess N(t) with parameter Â.. The absrber is mdelled by a linear scillatr. Figure 1. The mdel f a brigde with an absrber. The vibratins f the cntinuus-discrete system cnsisting f the beam and linear scillatr due t the stream ffrces are described by a set f cupled eq11:atins 1 Elww (x,t)+ c:w(x,t)+ 711W(x,t)+ Mij(t)(x- x 0 )= LA~c(x- v(t- t~c) N._4) lc=l (1) and the vibratins f absrber ( scillatr) are where EI dentes the flexural rigidity fthe beam, c is beam's dampimg cefficient, m is the mass per unit length, dentes the Dirac delta functin, M 0, c. k 0 are the mass, the damping cefficient and the spring's stiffuess fthe mdel f absrber, respectively, Ale dentes the randm weight fvehicles. It is assumed, that the randm variablesa1c are mutually independent and independent f the arrival times tk and that their prbabilistic mments E[ A;] are knwn. The vertical deflectin f the beam can be presented in the mda! frm as (2) "" w(x,t) = LYn(tJWn(x), n=l (3) where W" ( x) are the nrmal mdes and y n are the generalized crdinates.

3 Optimizatin f absrbers in the highway bridges 243 By such frmulatin ne becmes a set f rdinary differential equatins in the dmain f generalized crdinates "" "" ij(t) + 2a 0 ij(t) +m:q(t)- 2a 0 LYn(t)W"(x 0 )-m; LYn(t)W,.(X 0 ) =O (6) where n=l n=l (4) (5) The set f equatins (5) and (6) can be presented in the matrix frm as fllws: BZ+Ci.+Kz=F (7) where lql lql lql r l.. 1 YI 1 1 Y1 1 1 y1 1 1 fj(t) 1 z=l.y2l i. =1 Y2 l z=l Y2l, F=l f2(t) l, l;n ;J lj"rt)jj 1 : J l;j l : 1 r 1 l 1 1 Amj(x) 1 J B=l MW2(x 0 ) 1 l, 1 : lmw~(x 0 ) ~J 1 : 1 r 2a -2a 0~(X0 ) -2a 0 W2(x 0 ) -2a 0 Wn(X 0 ) l 1 2a 1 C=l O 2a l, l ~ 2a J

4 244 Part Tw Technical Cntributins lw2 -w;~(x 0 ) -w;w2(x 0 ) -w;w"rxa) l 1 K=l ~ 1 wf 1 w2 1. L w2 2 The slutins f the abve equatins are stchastic prcesses and can be calculated by similar apprach as presented in [4, 5]. The prbabilistic characteristics fthe bridge's respnse are n 1 J K~(x,t)=E{Ar j}jhr(x,t-;)d; (8) where the functin H ( x, t - T) is the dynamical influence functin which describes the respnse f the structure under simple lad impulse with quantity Ak = 1 which is mving alng the beam with cnstant speed v. This functin is calculated frm the equatin (7) in which the right side has the frm 1 l 1-2 ~(v(t- <)) F=l mi 2 w;(v(t-<))1. 1 myl 1 l Y2 ; ~ (v(t- T ))j mrn The impulse influence functin has tw different shapes: the first ne describes the structure vibratins when the frce is n its ( t - T < l 1 v) and the secnd ne describes the free vibratins (t-' > 11 v). Nw we frmulate the prblem f ptimal design f absrber. Find: K~(x,t)=E[Ar ]JJW(x,t-r)dT which satisfies fllwing cnditins M 0.01 ~ m; ~ 0.04, C 0 >0kN mls,

5 3. NUMERICAL RESULTS Optimizatin f absrbers in the highway bridges 245 The prblem f reducing randm vibratins in bridge under traffic flw by applying a damping absrber with ptimal parameters described abve will be illustrated by an example. The bridge is mdelled by a simply supprted beam with a damping absrber lcated in the midspan. It has been assumed that the parameters f the beam are as fllws: span 1 =30 m, mass per unit length m = 3500 kg/m, the first natural frequency m 1 = 4s- 1, the damping rati an =a = 0.04s- 1. The prbabilistic characteristics f the vehicle weight are as described in [6] i.e. E[ A}=31 kn, the variance cr~=320 kn*kn, the minimum weight is 8 kn and the maximum weight is 80 kn. The cnstant speed f the vehicles varies frm v = 1 Om 1 s t 3 5 m 1 s. The steady state respnse f the bridge is cnsidered. In rder t chse the ptimum absrber's parameters the expected value f the bridge's midspan deflectin, its standard deviatin, cefficient f skewness and kurtsis are calculated fr different absrber's parameters. The values f this variables as functins f the damping cefficient are shwn in Figures 2, 3, 4, 5, and as functins f spring's stiffuess in the Figures 6, 7, 8, 9, respectively. The calculatins have been made fr M 0 = 2 t = 2000 kg, v = 25 rn/s, Â = 1. In aii Figures the curve N 1 is the curve pltted with cntinuus line, the curve N 2 is pltted with dashed line, the curve N 3 is pltted with dtted line and the curve N 4 is pltted with dashed-dtted line. In the first fur Figures fur curves are presented. The curve N 1 has been btained fr the spring's mass k 0 =10 kn/m, the curve N 2 fr k 0 = 15 kn/m, the curve N 3 fr k 0 = 20 kn/m and the curve N 4 fr k 0 = 30 kn/m. :0 ~ \ Q) f--"11 ~ ,.\ «S \'\ ~ ~~'~,\~;,,. il. 2 a 2 2 ~-~'"'~:-... -, \;,..-, l---t-----1, ~r---;...,. ---t----1 t.za2~--1~~~--~~--; Q) ',-.;: } ~0. 2 a 1a L------'---=-=-='-=-'=-=--=::...1.., 2 t... ~.. ~=- c Figure f---~ ~--- ~~~f----~--~ O.Zf---+'--+-~' ~ Ol ~... aj } ~--~~~-~~--~ ~ O.lf-T~ ~_,.~.t. ;~~-~~ ~ lf,//,.,~. s ;,;';:::,._..?- 2 c Figure 4... ~ ~ ~~:~= ~~ -= -j... = 6 8.r,~. c Figure ~-T-~: _,.~ 3. 6f t----'-;,,--,,--l, ---t--~ ~3.5f---+;--+--~.-,,-., l a e3.4f---~j--+~3 ~... ~ ~~~ l.. = / ~~::... ::.. :::..-.~:.~:~.-~-. l./... l, ~ c Figure 5. 6 a

6 .f/ 246 Part Tw Technical Cntributins The curves presented in Figures 6, 7, 8, 9 have been calculated fr C 0 = 2 kn s/m (the curve N 1), C 0 = 3 kn s/m (the curve N 2), C 0 = 4 kn s/m (the curve N 3) and C 0 = 5 kn s/m (the curve N 4). :a ~ \ ~ ~~~~.-.~ \. -~. : ~ tll y~ ' ry: >0.002B24r-~~v -~.. ~. ~... ~,2 ~--;---, ----"'.'' '.('~ ~ ~ )( ~ "--..". -...;~-... ~... aj...ij :,., ::;;:..: ~ ~ '------'-----'------''----"-'=-.l [&l k Figure ~0.018 \.,0.016r-~, ~ --r > -~ ~0.014, ~ "0.012 ~ ~ "' -. ~ 4.- 1! :- -.::- ~~:~--:.; 3 rffi ::::: ~ Ul k Figure ~ ::,',\ g. 15 _ r - ~ \ aj 1/~ ~...'~ 1! ;:y ~ ( ,-d 0.05 ~~,<J.,..r,r ' k Figure 8. k Figure 9. In Figures 10,11,12,13 the influences f the change f absrber's mass n the bridge's prbabilistic characteristics are shwn. This calculatins have been made fr C 0 = 2 kns/m, k 0 = 25 kn/m (the curve N 1),.C 0 = 4 kns/m, k 0 = 25 kn/m (the curve N 2), C 0 = 4 kns/m, k 0 = 15 kn/m (the curve N 3), C 0 = 2 kns/m, k 0 = 15 kn/m (the curve N4). :a ' t~ ~-~~~~~-~ ~ \'" ' aj,,.,.:: r-.lc\'-'~""\.-l, ----;-----lf-----'1 lll \. '' :'- 3 : ~ " :::;: 4 aj !1. 1 )<..,., ' -._, -"-1--' ~~: ~~~: ~: R : >< [&l : ::~;:.=: lilo Figure 10. lilo Figure 11.

7 Optimizatin f absrbers in the highway bridges 247 ~- -..;. 0.16~ /~-~~--~1~~--~ lll O. HI----t-,/~' ~0.12~----+,-~~~.,..::..t----'==;..;:"-,_,--~.. ~ 5. 1~----Y_-_,.,..., =<"" :-~-f'--...::: ~.~~-.---.~~"~. 1' :-"..,""!t.-... ::::... :--:1 ~0.08~--,~~~-~--~----~~ ~. ~0.06~-~~-~---~----~ ~ 0.04./... ~----~--~~--~----~ lllo Figure lll / -..~ -~ T-W'---l----4 "..,------l E 3.3 ' ""'- 1-l / / <~ - <:::.: ~ '//...:::;;:_ ~:. _;.. - l. 3.15r.F~ ~ ~ lllo Figure 13. T evaluate the level f vibratin reductin in the bridge due t applying an absrber the expected value f midspan deflectin, its standard deviatin, cefficient f skewness and kurtsis calculated fr different vehicle speed are cmpared with the results btained fr the case withut absrber and presented in Figures 14, 15, 16, 17. The curve N 1 has been btained fr the bridge withut absrber, the curve N 2 fr the bridge with an absrber with parameters C 0 = 2 kns/m, k 0 = 25 kn/m, M 0 = 2 t, the curve N 3 fr the bridge with an absrber with parameters C 0 = 2 kns/m, k 0 = 25 kn/m, M 0 = 4 t, the curve N 4 fr the bridge with an absrber with parameters C 0 = 2 kns/m, k = 10 kn/m, M 0 = 2 t ,0.007 IIJ0.006 ~0.005 lll >.4 "tl 2!. 003 ~ "" X 1\ \ '"' 10 \. ' :..;."., 30 V Figure 14. -r ~==~=;~~~==~==~ -:::.14 V ~ ~ _, "'--.., A f / ' " ''.. 3 '. "O 0.01 j_l ' / ~ t// li -g.a u{ ~0.006t~~~-~~~;~==t====±====jt====j V Figure V Figure 16. V Figure 17.

8 248 Part Tw Technical Cntributins Fr this numerica! example fllwing cnclusins can be made: - fr a given absrber's mass the change f its damping cefficient r spring's stiffuess have a little influence n the expected value f the displacement and a significant influence n the standard deviatin, - if fr sme parameters the standard deviatin has the minimum value, the crrespnding cefficients f skewness and kurtsis have bth the maximum values, - if the change f the absrber's damping cefficient r spring's stiffuess causes increasing f the standard deviatin, the crrespnding cefficients f skewness and kurtsis decrease. Acknwledgement This research was partially supprted by the Scientific Research Cmmitte in Warsaw under grant number REFERENCES l. B. G. Krenev, L.M. Reznikv, Dynamic Vibratin Absrbers - Thery and Technical Applicatins, J.Wiley & Sns, Chichester, S. El-Bagalaty, M. Klasztmy, Eliminatin f Excessive Frced Vibratins f Structures Using Dynamic Absrbers, XXXVI Cnf. KILiW PAN & KN PZiTB, Krynica, 1990, v.l, A. K. Das, S.S. Dey, Effects f Tuned Mass Dampers n Randm Respnse f Bridges, Cmp. & Struct., 1992, Vi. 43, N. 4, pp C. C. Tung, Randm respnse f highway bridges t vehicle lads, Prceeding f the American Sciety f Civil Engineers, Jumal fthe Engineering Mechanics Divisin (1967) 93, R. Sieniawska, P. Sniady, First passage prblem f the beam under a randm stream f mving frces, Jumal fsund and Vibratin (1990) 136(2), R. Sieniawska, P. Sniady, Life expectancy fhighway bridges due t traffic lad, Jumal f Sund and Vibratin (1990) 140(1), P. N. Takaka, Reliability Analysis in Highway Bridges, PAN, KILiW, Jablnna 1982.