Hybrid probabilistic interval dynamic analysis of vehicle-bridge interaction system with uncertainties

Transcription

1 1 APCOM & SCM h Deember, 13, Singapore Hybrid probabilisi inerval dynami analysis of vehile-bridge ineraion sysem wih unerainies Nengguang iu 1, * Wei Gao 1, Chongmin Song 1 and Nong Zhang 1 Shool of Civil and Environmenal Engineering, The Universiy of New Souh Wales, Sydney, NSW 5, Ausralia Shool of Elerial, Mehanial and Meharoni Sysems, Fauly of Engineering and T, Universiy of Tehnology, Sydney, NSW 7, Ausralia *Corresponding auhor: w.gao@unsw.edu.au Absra Hybrid probabilisi inerval dynami analysis of vehile-bridge ineraion sysem wih a mixure of random and inerval properies is sudied in his paper. The vehile s parameers are onsidered as inerval variables and he bridge s parameers are reaed as random variables. By inroduing he random inerval momen mehod ino he dynami analysis of vehile-bride ineraion sysem, he expressions for he mean value and sandard deviaion of he random inerval bridge dynami response are developed. Examples are used o illusrae he effeiveness of he presened mehod. A hybrid simulaion mehod ombining dire simulaions for inerval variables and Mone-Carlo simulaions for random variables is implemened o validae he ompuaional resuls. Keywords: Vehile-bridge ineraion sysem, probabilisi inerval analysis, random inerval momen mehod, random inerval dynami response. nroduion The oupled vehile-bridge dynami sysem has araed onsiderable aenions over he pas wo deades (Yang and in, 5; Ju and in, 7; Zhang e al., 8). The values of sysem parameers are given preisely in mos of sudies. Aually, vehiles moving on a bridge have nondeerminisi haraerisis beause he sysem parameers are no onsan. Probabilisi mehods are preferred when informaion of unerain parameers in he form of preferene probabiliy funion is provided. And hese have been widely used o predi he response and in he implemenaion of sruural sysem reliabiliy evaluaion of unerainy (iu e al., 11). n probabilisi mehods, unerain parameers are modeled as random variables/fields and unerainies of loads are desribed by random proesses/variables. However, someimes i is hard o ge he enough probabilisi informaion for sruural parameers as heir values are affeed by a lo of non-deerminisi faors. Meanwhile, loads of many senarios an hardly be modeled as random variables due o large hanges in heir magniudes. The inerval mehods an be used when he probabiliy funion is unavailable bu he range of he unerain parameer is known. n he pas deade, signifian progress in analysis and opimal design of sruures wih bounded parameers has been ahieved (Qiu e al., 9; Jiang e al., 8; mpollonia and Musolino, 11). is desirable o model sruural parameers/loads as random variables if suffiien informaion an be obained o form he probabiliy densiy funions. Meanwhile,

2 some sruural parameers/loads migh be bes onsidered as inerval variables if he informaion/daa are no enough o model unerain sruural parameers and loadings as random variables, espeially in he early design sages. Consequenly, hybrid probabilisi inerval analysis and reliabiliy assessmen of sruures wih a mixure of random and inerval properies has been ondued (Gao, 1). The random inerval momen mehod has been developed by he auhors o deermine he mean value and sandard deviaion of random inerval responses of sruures under sai fores (Gao, 1). As aforemenioned, some parameers of vehile-bridge ineraion sysem ould be onsidered as random variables and some of hem migh be assumed as inerval variables. For example, he hange range of vehile's mass is large due o he differen loading ondiions; herefore, hese an be aken as inerval variables. n onras, he hange ranges of bridge's parameers are small beause of he sri manufauring sandards, whih an be onsidered as random variables. Therefore, a hybrid probabilisi inerval analysis model for vehile-bridge oupled sysems needs o be developed. andom inerval momen mehod e X () be he se of all real random variables on a probabiliy spae (, A, P), x is a random variable of (). denoes he se of all real numbers. x (or x ) and x are he mean (deerminisi) value and sandard deviaion of x, respeively. y [ y, y], y y y, y is an inerval variable of () whih denoes he se of all he losed real inervals. y and y are he lower and upper bounds of inerval variable y, respeively. nerval variable y an also be wrien as y y y ; y [ y, y] ; y y y ; y y y y y F (1) y where y, y, y and yf represen he midpoin value, maximum widh (inerval widh), unerain inerval and inerval hange raio of he inerval variable y. Wihou loss of generaliy, random inerval variable Z is he funion of muliple random and inerval variables, whih are respeively represened by random veor X ( x1, x,, xn ) and inerval veor Y ( y1, y,, ym). The deerminisi values of X and Y are X x, x,, x ) and,,, Y y y y ). ( 1 n ( 1 The Taylor series o he firs-order of he random inerval variable Z f ( ) abou ( ) is expressed as Z f ( X, Y ) f ( ) n i1 f xi ( xi x ) i m

3 m f f ( ) y 1 y n f i x i X, where is he remainder erm. m f xi y 1 Y 1 y ( xi x ) i () From his equaion, and he higher order erms is ignored, he expeaion and variane of random inerval variables Z f ( ) an be alulaed as (Gao e al., 1) m f E( Z ) f ( ) y Z { } (3) y Z E Z E( Z ) f x k m 1 1 n i 1 n k 1 f x k y f xi m f xi y X Y, 1 y Cov ( x i, x k ) y n m f f y Varxi i1 x i XY, 1 xi y XY, n n m m f f f f, y y Cov x x ik1 k1 x i, 1 xi y x XY, i XY, 1 xi y XY XY, (4) Vehile-bridge ineraion model i k n he vehile-bridge ineraion sysem, he bridge is modeled as a simply suppored beam (Yang, 5) and he vehile is represened by a half-ar model as shown in Figure 1. Here, m v, m1 and m denoe he sprung and unsprung masses respeively; he suspension sysem is represened by wo linear springs of siffness k s1, ks and wo linear dampers wih damping raesc s1, C s ; he ires are also modeled by wo linear springs of siffness k 1, k and wo linear dampers wih damping raes C 1, C ;, E, and are he mass per uni lengh, elasi modulus, momen of ineria and lengh of he beam respeively. n his sudy, parameers of he vehile m 1, m, and m, are onsidered as inerval v variables, meanwhile, bridge s parameers,, E and, are reaed as random variables. The equaion of moion governing he ransverse vibraion of he bridge under he moving vehile wih unerain parameers an be wrien as 3

4 Figure 1. Model of vehile-bridge ineraion sysem 4 W ( x, ) W ( x, ) W ( x, ) 4 1 C E x ( f ( x, ) f ( x, )) ( xv) (5) f ( x, ) ( m a 1 1 m ) g k ( x ( ) (, ) ) ( ( ) (, ) ) v 1 v W x C x 1 v W x xv xv f ( x, ) ( m a 1m ) gk ( x ( ) (, ) ) ( ( ) (, ) ) v v W x C x v W x xv xv (6) where C is he damping of he bridge, W ( x, ) is he random inerval verial displaemen of he bridge, x () v is he random inerval verial displaemen of he moving vehile, f 1 ( x, ) and f ( x, ) are he random inerval ona fores, ( x v) is he Dira dela funion evaluaed a he ona poin a posiion x v, and v is he speed of he moving vehile. Using he modal superposiion mehod, he soluion o Eq. (5) an be expressed as in erms of he mode shapes ( x). n his paper, he Wilson s damping hypohesis is adoped. As vehile mass is muh less han he bridge mass and he ires damping is quie small. Using he Duhamel inegral soluion, he displaemen response of he bridge an be alulaed by W x ( x, ) sin 1 ( m m m ) g 1 1 v b E E b ( ) b E v e sin 1 ( ) sin d (7) n his sudy, he onribuion of ires siffness o bridge verial displaemen response is omied due o he assumpion ha he bridge mass is muh greaer han ha of he vehile(yang, 5). Addiionally, bridge damping is reaed as deerminisi beause he exising researh ouomes show ha he mehanism of sruural damping is sill no lear enough. Furhermore, he lower and upper bounds of he mean value of he bridge's displaemen ( W ) are given by 4

5 x v W x, sin ( m m m ) g S sin S sin d 1 v 3 S E g x v sin Ssin S3 sin d m1m m S E v x v W x, sin ( m m m ) g S sin S sin d 1 v 3 S E g x v sin Ssin S3 sin d m1m m S E v The lower and upper bounds of he variane of he bridge's displaemens W ( x, ) are W x, W x, W x, W x, W ( x, ) m 1 m m v E E E m E m E m 1 v W x, W x, W x, W x, m 1 m m v m m m 1 v W x, W x, W x, W x, m 1 m m v m m m 1 v (1) W x, W x, W x, W x, W ( x, ) m 1 m m v E E E m E m E m 1 v W x, W x, W x, W x, m 1 m m v m m m 1 v W x, W x, W x, W x, m 1 m m v m m m 1 v Numerial Simulaions n his paper, he vehile-bridge ineraion model is demonsraed as he Figure 1. The bridge's parameers are onsidered as Gaussian random variables. The parameers of vehile are reaed as inerval variables. The nominal values (mean/midpoin values) of sysem parameers aken in he numerial simulaion are lised in Table 1. The uni of he bridge displaemen response is meer in his paper. (8) (9) (11) 5

6 n his sudy, he bridge damping raios b for all modes are aken as.5. For he sake of simpliiy, he oeffiien of variaion (COV) of, E and is adoped o represen he dispersal degree of random variables. Meanwhile, he inerval hange raio (C) of m 1, m, and m v is used o desribe he saer level of inerval variables. vehile speed, v 5 m/ s is aken ino aoun o invesigae he influene of vehile veloiy on he bridge response. Table 1. Daa of he vehile-bridge model Daa of he bridge (mean value) Daa of he vehile (midpoin) E= The mean value of he random inerval bridge displaemen response a is mid-span is given in Figures (a) (COV(, E, )=.5, C( m, m, m )=.) and(b) 1 v (COV(, E, )=.5, C( m, m, m )=.1), when differen ombinaions of 1 v unerain parameers are aken. Figure shows he mean bridge displaemen response when he randomness of all random parameers and all inerval parameers are onsidered. From Figure, i an be observed ha he inerval widh of bridge response inreases when he inerval hanges of inerval variables beome larger. n summary, he mean value of he random inerval bridge response is independen of he dispersal degrees of random sysem parameers as expeed. The inerval widh of he mean value of bridge response is direly proporional o he unerainies of inerval variables and vehile speed. -.1 Upper bound(mm) ower bound(mm) Upper bound(hsm) ower bound(hsm) -.1 Upper bound(mm) ower bound(mm) Upper bound(hsm) ower bound(hsm) Displaemen(m) Displaemen(m) Time(s) Time(s) (a) (b) Figure. Mean value of random inerval bridge displaemen response The sandard deviaion (SD) of he random inerval bridge displaemen response a is mid-span is shown in Figures 3(a) and (b). an also be observed ha he inerval 6

7 widh of he sandard deviaion of he random inerval bridge response is direly proporional o he unerainies of random and inerval variables from Figures 3. To validae he auray of he random inerval momen mehod (MM) presened in his paper, a hybrid simulaion mehod (HSM) is employed. This hybrid simulaion mehod (HSM) ombines dire simulaion for inerval variables and Mone-Carlo simulaions for random variables. To show he differenes beween he resuls generaed by he MM and HSM in deail, he relaive errors of mean value and sandard deviaion of bridge displaemen are lised in Tables and 3. Given he maximum relaive error is 1.1%, while he oeffiiens of variaion for all random parameers are.5 and he inerval hange raios of all inerval parameers are., he mean values alulaed by he wo mehods are very losed o eah oher. For he sandard deviaion, he maximum relaive error is 6.45%, whih an be aeped beause he hybrid simulaion imes used in his sudy are no enough o provide onvergen resuls. 1, simulaions used in he wo rounds of HSM anno yield onvergen and reliable resuls alhough he oal simulaions are 1 6. The auray of he resuls obained by he HSM an be improved if more simulaions are implemened. SD of Displaemen(m) ower bound(mm) Upper bound(mm) Upper bound(hsm) ower bound(hsm) SD of Displaemen(m) ower bound(mm) Upper bound(mm) Upper bound(hsm) ower bound(hsm) Time(s) Time(s) (a) (b) Figure 3. Sandard deviaion of random inerval bridge displaemen Generally, he auray of hese resuls is saisfaory in praie. The presened random inerval momen mehod has muh less ompuaional work han he simulaion mehod. should be noed ha he auray of he resuls of random inerval momen mehod an be furher improved if seond or higher order Taylor expansions are used. Table. Comparison of mean values Time (s) Upper bound ower bound MM HSM Error MM HSM Error % % % % % % % % % % % % % % 7

8 Table 3. Comparison of sandard deviaions Time Upper bound ower bound MM HSM Error MM HSM Error % % % % % % % % % % % % % % Conlusions n his paper, sohasi dynami response of vehile-bridge ineraion sysem wih unerainies is invesigaed by exending he random inerval momen mehod o he dynami oupling sysem. The unerainies of sysem are modeled as random and inerval variables. The expressions for alulaing he bounds of expeaion and variane of he random inerval bridge response are derived. Using hese formulaions, he upper and lower bounds of mean value and sandard deviaion of bridge response an be very easily obained. The resuls obained by he presened random inerval momen mehod are in very good agreemen wih hose deermined by Mone-Carlo simulaion mehod. The relaive errors of hese wo mehods are quie small when he hange ranges of sysem parameers are no large. Aknowledgemens The work repored in his paper was suppored by he Ausralian esearh Counil hrough Disovery Proes. eferenes Gao, W., Song, C., and Tin-oi, F. (1), Probabilisi inerval analysis for sruures wih unerainy. Sruural Safey, 3, pp mpollonia, N. and Musolino, G. (11), nerval analysis of sruures wih unerain-bu-bounded axial siffness. Compuer Mehods in Applied Mehanis and Engineering,, pp Jiang, C., Han, X. and iu, G. P. (8). Unerain opimizaion of omposie laminaed plaes using a nonlinear inerval number programming mehod.compuers & Sruures, 86, pp Jiang, C., Han, X., i, W. X., iu, J. and Zhang, Z. (1), A hybrid reliabiliy approah based on probabiliy and inerval for unerain sruures. Journal of Mehanial Design, 134, 311. Ju, S. H. and in, H. T. (7), A finie elemen model of vehile bridge ineraion onsidering braking and aeleraion. Journal of sound and vibraion, 33, pp iu, N., Gao, W., Song, C. M. and Zhang, N. (11), Probabilisi dynami analysis of vehile-bridge ineraion sysem wih unerain parameers. CMES-Compuer Modeling in Engineering & Sienes, 7(), pp Qiu, Z., Ma,. and Wang, X. (9), Non-probabilisi inerval analysis mehod for dynami response analysis of nonlinear sysems wih unerainy. Journal of Sound and Vibraion, 319, pp Yang, Y. B. and in, C. W. (5), Vehile bridge ineraion dynamis and poenial appliaions. Journal of sound and vibraion, 84, pp Zhang, N., Xia, H. and Guo, W. (8), Vehile bridge ineraion analysis under high-speed rains. Journal of Sound and Vibraion, 39, pp