Numerical Calculation of Dynamic Response for Multi-Span Non-Uniform Beam Subjected to Moving Mass with Friction

Transcription

1 Egieerig,,, doi:.46/eg..548 Pubished Oie May ( 67 Numerica Cacuatio of Dyamic Respose for Muti-Spa No-Uiform Beam Subjected to Moig Mass with Frictio Abstract Jupig Pu, Peg Liu Coege of Cii Egieerig & Architecture, Zhejiag Uiersity of echoogy, Hagzhou, Chia E-mai: Receied Jauary 9, ; reised February, ; accepted February, I order to simuate the coupig ibratio of a ehice or trai moes o a muti-spa cotiuous bridge with o-uiform cross sectios, a moig mass mode is used accordig to the Fiite Eemet Method, the effect of the iertia force, Coriois force ad cetrifuga force are cosidered by meas of the additie matrices. For a o-uiform rectaguar sectio beam with both iear ad paraboic ariabe heights i a pae, the stiffess ad mass matrices of the beam eemets are preseted. For a o-uiform box girder, Romberg umerica itegra scheme is adopted, each coefficiet of the stiffess matrix is obtaied by meas of a orma umerica computatio. By appyig these eemets to cacuate the o-uiform beam, the computatioa accuracy ad efficiecy are improed. he fiite eemet method program is worked out ad a etire dyamic respose process of the beam with o-uiform cross sectios subjected to a moig mass is simuated umericay, the resuts are compared to those preiousy pubished for some simpe exampes. For some compex muti-spa bridges subjected to some moig ehices with chageabe eocity ad frictio, the computatioa resuts, which ca be regarded as a referece for egieerig desig ad scietific research, are aso gie simutaeousy. Keywords: Dyamic Respose, Muti-Spa Beam, No-Uiform Sectio, Frictio, Brakig Force. Itroductio Cotiuous beams are geera staticay idetermiate structures, ad hae broad appicatios i cii egieerig, mechaism, aigatio egieerig ad so o. Muti-spa cotiuous bridges hae bee widey used i highway ad raiway, there is a great dea of merit for the structures, for exampe, their exterior is beautifu, the hoistic structures stabiity is we, the spacia spa is bigger ad o which ehices ca pacidy pass oer. It is of great importace to study the dyamic characteristic of the bridge uder moig mass for egieerig desig ad scietific research. May egieers ad scietists hae cotributed to the soutio of the probem with their ioatios, ad sti the subject draws cosiderabe attetio from researchers by ow. Fryba [,] had gie a exact soutio o dyamic resposes of the simpe supported beam ad cotiuous beam uder moig oad. Cai, Cheug ad Cha [] iestigated the dyamic resposes of the ifiite cotiuous beam subjected to a moig force by usig the mode superpositio method to get a exact soutio. Howeer, for a great umber of bridge structures i egieerig practice, it ca ot be simpy regarded as a perfect state, so their theoretic soutios are ot existet i geera. hus, amog the umerica methods the fiite eemet method becomes a idea approximate approach to soe this kid of probems [4-]. Wu [,4] performed the dyamic aaysis of a icied beam ad a fat pate due to moig oads, ad preseted the moig mass eemet by takig accout of the effect of iertia force, Coriois force ad cetrifuga force iduced by the moig mass. Form metioed iteratures, it is show that, for the muti-spa cotiuous o-uiform beam, oe used a moig oad mode to obtai the umerica Copyright SciRes.

2 68 J. P. PU E AL. resuts i geera, but did ot cosider the effect of iertia force, Coriois force ad cetrifuga force. Some umerica exampes deat with aboe effect but just aayzed the simpe uiform beam. his paper has bee performed some compex probems (icude muti-spa o-uiform beam with moig mass). For a o-uiform rectaguar sectio beam with both iear ad paraboic ariabe heights i a pae, the stiffess ad mass matrices of the beam eemets hae bee deducted. For a o-uiform box girder which ca be widey used i egieerig structures, sice the itegra formua of stiffess matrix is extremey compex, it is difficut to write dow the expressio of the stiffess coefficiets, so Romberg umerica itegra scheme is adopted. Each coefficiet of the stiffess matrix ca be obtaied by meas of a orma umerica computatio. For some compex muti-spa bridges subjected to some moig ehices with chageabe eocities ad frictios, umerica resuts of dyamic resposes are aso obtaied, which ca be regarded as a referece for egieerig desig ad scietific research.. Forced Vibratio Differetia Equatio of Euer-Beroui Beam Forced ibratio differetia equatio of Euer-Beroui beam i pae bedig with dampig takes the form y y EI( x) c I( x) x x s x t x y y m( x) c( x) f ( x, t t where, the mass per uit egth of the beam is m(x) = ρa, whie ρ is the materia desity ad A is the area of the cross sectio of the beam, c s is dampig coefficiet of strai rate, c(x) is dampig coefficiet of dispacemet rate. Whe a cocetrate excited force P( was exerted at x i =t of the beam, the force ca be expressed, by usig the Dirac Deta fuctio δ, as () f ( xt, ) Pt ( ) ( x () i where is the moig eocity. Whe a moig mass M with the costat eocity moed o the beam, the actio force o the beam ca be cosidered as a commo effect of the graity ad iertia force accordig to referece iterature [5]. f ( x, M[ g y ( t, ] ( x () hus the formua () wi be a set of systems of secod order differetia equatios with time-ariat coefficiet, it eeds to be soed by meas of umerica method. Lee s iestigatio idicated, i iterature [6,7], that whe the moig mass M with the costat eocity moed o the Euer-Beroui beam, icudig the graity ad iertia force iduced by moig mass M, oe sti eeds to cosider the effect of Coriois force ad cetrifuga force o the beam, that is f ( x, M [ g y ( t, y ( t, (4) y( t, ] ( x Cosiderig the cotact frictio betwee the whees ad the beam, a axia force X(ξ, shoud be exerted o the beam, thus the actio force o the beam ca be expressed as foows M[ g y (, y (, f ( x, ( ) x (5) y(, ] X (, Whe a moig ehice has a brake o the bridge at a momet t, if the cotact frictio is big eough, the ehice wi stop at momet t +, the axia force X(ξ, ca be expressed as X (, Ma t t ), (6) ( X, M ( g a ) ( t t t ), (7) ( X (, ( t t ). (8) where a is the iitia acceeratio of the ehice, μ ad g represet the frictio factor ad that of graity, respectiey. If the moig ehice does ot brake o the bridge at ay momet, the axia force X(ξ, wi be expressed as X (, Ma ( t ) (9) he positio ξ of the moig ehice at the bridge ca be expressed as foows (see Figure ) x t. a t t t ), () 5 ( x ( t t ).5a ( t t ) ( t t ) t () x / a ( t t ). () where is the iitia eocity, the positio, eocity ad acceeratio at momet t are, respectiey, x x t. a t, () 5 at a, (4) a g. (5). Discrete Mode of Vibratio Equatios uder Moig Mass Accordig to the fiite eemet method, the forced ibratio differetia equatio of Euer-Beroui beam i pae bedig with dampig ca be writte, i matrix form, as foows M y ( Cy ( Ky( F ( (6) where F ( P (, P (, X (, (7) a Copyright SciRes.

3 J. P. PU E AL. 69 P, a x t, a, a t t t / a Figure. he positio ξ of the moig ehice at the beam. he actio force of the ehice o the beam ca be expressed by the oda oad ector F(, which is composed of the graity P i (ξ, =M i g, ad the mass force P i a (ξ, (icude iertia force, Coriois force ad cetrifuga force) of the ehice, as we as the axia actio force X i (ξ,, whie i=, deotes the umber of moig mass M i. Due to the ocatio of each moig mass was cotiuousy trasformed with time, whie the moig mass passed aog each beam eemet, the actio of the graity, at curret positio i each time itera Δt, ca be distributed to the eemet odes to become the eemet odes forces by usig a iterpositio fuctio. For Euer-Beroui beam eemet, oe ca adopt cubic Hermite iterpositio fuctio of two-odes, N j (ξ), whie j =,6, to get graity oad P e (ξ,, which is expressed as e P, F F F F 4 F 5 F 6 (8) where ( j i F (, N ( ) P (, ( j,,6) (9) j he actio of the axia actio force X i (ξ,, at curret positio i each time itera Δt, ca aso be distributed to the eemet odes. Now just the axia actio was cosidered, the shear ad momet were igored. e 4 X (, X X () where j i X (, N j ( ) X (, ( j, 4) () he mass force P i a (ξ, ca be discretized accordig to the form of eemet dispacemet iterpositio fuctio y = ΣN j y j = Ny (j =,6). Cosequety, oe ca obtai the additie moig mass matrix m a (, moig dampig matrix c a ( ad moig stiffess matrix k a ( from Equatio () e INPa (, m a y ca y ka y () where I is the uit matrix, P e a (ξ, is the equiaet oda force ector iducig by the moig mass force, the shape fuctio ector is a diagoa matrix N=diag(N (ξ) N (ξ) N (ξ) N 4 (ξ) N 5 (ξ) N 6 (ξ)). A detaied deducig process ca be refereced i iterature []. Istitutig the additie moig mass matrix m a ( ad stiffess matrix k a ( ito the etire mass matrix M ad stiffess matrix K of the origia beam structure, respectiey, the ew etire matrix M ( ad K () t are formed ad are time-ariat. M ( M m ( () a K () t K k () t (4) he oera dampig matrix C of the beam is determied by usig the theory of Rayeigh dampig, addig the additie dampig matrix c a ( ito C, the ew etire dampig matrix C() t ca be gotte by formua (5) C ( ( M ( ( K ( c ( (5) where ( t ) ( ) /( ) (6) ( t ) ( ) /( ) (7) he coefficiets α( ad β( are aso time-ariat with chagig of the atura frequecies ω ad ω of the beam structure at each time steps. Fiay, accordig to Equatio (8) oe ca umericay compute a etire system of ibratio equatio with Newmark direct itegratio method or with Wiso-θ method. M ( y ( C ( y ( K ( y( F ( (8) where the oad ector F() t icudes the graity oad of the moig ehice P (ξ, ad the axia force X(ξ, as we as the adscititious oad P( iduced by other causatios. 4. Stiffess ad Mass Matrices of No-Uform Beam Eemets 4.. No-Uiform Rectaguar Cross Sectio For a o-uiform rectaguar sectio beams with both iear ad paraboic ariabe heights i a pae, the stiffess ad mass matrices are deducted, respectiey. So oe ca aayze the o-uiform beam accordig to a coeiet mode (see Figure ). Accordig to Figures (a), (b) ad (c), the cross sectio height, area ad the momet of iertia of the beam ca be gie by expressios (9)-(), respectiey. h h(x) h h x h h I x bh h / h(x) h h (a) (b) (c) a h a h(x) Figure. Variabe cross-sectioa beam eemets. (9) h Copyright SciRes.

4 7 h x h h I x b h h / h x h h I x b h h / k k5 k55 Eb(7h h h h h 7 h ) / k k5 Eb(5h h h h h h ) / k6 k56 Eb(h h h h h 5 h ) / k Eb(h 5h h hh h ) / 6 k6 Eb(4h h hhh 4 h)/6 k66 Eb(h h h 5h h h ) / 6 k k k k k k k k J. P. PU E AL. () () where h ad h are begiig ad ed height of the beam eemets with o-uiform cross sectios, respectiey. ξ=x/, whie is the egth of the beam eemet ad b is the width of the cross sectio with rectaguar form. he eemet stiffess ad mass matrices ca be obtaied, respectiey, from M e e K B EI xb d x () A( x) N Ndx () Ve N NdV where B= N / x, whie EI(x) ad A(x) are the fexura stiffess ad the cross sectio area of the beam, respectiey, which are chageabe with the height chagig of the cross sectio. Adoptig aboe-metioed o-uiform beam modes ad a itegra procedure is worked out, the coefficiets of the eemeta stiffess ad mass matrices, k ij ad m ij, ca be obtaied as foows 4... Stiffess Matrix Coefficiets with Liear Varabe Heights (Figure (a)) k k k Eb( h h )/ Mass Matrix Coefficiets with Liear Variabe Heights (Figure (a)) m b h h / m4 bh h / m (h h /5) b/ 7 m ( h /8 h /6) b m5 9( h h ) b/4 m6 ( h / 6 h / 7) b m ( h /68 h / 8) b m ( h / 7 h / 6) b 5 m6 ( h h ) b / 8 m44 bh h / m55 (h /5 h) b/7 m56 ( h /6 h /8) b m66 ( h / 8 h /68) b m m m m m m m m Stiffess Matrix Coefficiets with Paraboic Variabe Heights (Figure (b), h >h ) k k44 k4 Eb( h h )/ k k5 k55 Eb(7h 4h h 9hh 9 h ) / k k5 Eb(6h h h h h 8 h )/ k6 k56 Eb(h h h 8hh 64 h ) / k Eb(7h 46h h hh 4 h ) / 4 k6 Eb(5h 4h h 9hh h ) / 4 k66 Eb(7h h h 7hh 96 h ) / 4 k k k k k k k k Mass Matrix Coefficiets with Paraboic Varabe Heights (Figure (b), h >h ) m b9h 6 h /5 m4 bh 4 h / m (97h 744 h ) b/ 4545 m (59h 56 h ) b / 99 m5 (55h 8 h ) b/ 99 m6 (7 h / h / 4545) b m (7h 56 h ) b / 4545 m5 ( h / 6 h / 75) b m6 (9h 896 h ) b /88 m44 bh 6 h / m55 (79h 4858 h ) b/55 m56 (9 h / 9 h / 45) b m66 (h h ) b /55 m m m m m m m m Stiffess Matrix Coefficiets with Paraboic Variabe Heights (Figure (c), h <h ) k k k Eb( h h) / L k k k Eb(9h 9h h 4hh 7 h ) / k k Eb h h h hh h k k Eb h h h hh h k Eb h h h hh h (64 8 ) / 6 56 (8 6 ) / ( )/4 Copyright SciRes.

5 J. P. PU E AL. 7 k Eb(h 9h h 4h h 5 h ) / 4 6 k66 Eb(4h h h 46hh 7 h ) / 4 k k k k k k k k Mass Matrix Coefficiets with Paraboic Varabe Heights (Figure (c), h <h ) m b6 h h / m4 b4h h / m (4858h 79 h ) b/55 m (9 h / 45 9 h / ) b m5 (8h 55 h ) b/ 99 m6 (6 h / 75 h /) b m (h h ) b /55 m5 (98 h / h / 66) b m6 (896h 9 h ) b /88 m44 b6h 9 h /5 m55 (744h 97 h ) b/ 4545 m56 (56h 59 h ) b / 99 m66 (56h 7 h ) b / 4545 m m m m m m m m No-Uiform Box Girder Sectio he box sectio is show i Figure with a up bottom width of B ad a dow bottom width of D. he thickesses of both up ad dow bottom board are. he etra shied thickess is C, ad the sectio height of the box girder is h(x). he cetroid distace w from cetroida axis z to z axis of sef-defied is wx ( ) Ch ( x) ( D C) h( x).5( B D) Ch( x) ( B D 4 C) (4) So the area ad momet of iertia ca be deoted by a sectio height h(x) ad a cetroid distace w(x) as foows A( x) Ch( x) ( BD4 C) (5) I( x) Ch( x) 6 hx ( ) Ch( x) w( x) B B w( x) ( D C ) ( D C) h( x) w( x) (6) Hypothesis that usig a iear ad paraboic ariabe mode such as Figure, the beam height h(x) of box girder eemet ca aso be deoted by formua 9, ad. Sice w(x) is a composite fuctio of h(x), so the momet of iertia I(x) is aso a ery compex composite fuctio, moreoer hae some ratioa fractios i it. It is difficut to get a fixed form itegra resut by a maua cacuatio, so Romberg umerica itegra scheme is adopted i this paper. he umerica itegra precisio is cotroed to be -6. Each coefficiet of the stiffess matrix of o-uiform sectio box girder eemet ca be obtaied by meas of Formua (7). d N ( ) ( ) i x d N j x kij EI( x) dx dx (7) dx Whe deducig the mass matrix of o-uiform sectio box girder eemet, sice the area fuctio A(x) is reatiey simpe, so we adopt a maua cacuatio fashio ad the fixed form itegra resut is gotte. Obserig Formua (5), ad comparig the area formua of a rectaguar cross sectio, a superfuous item (B+D-4C) is foud. hese coefficiets are costats. Form the geera Expressios () of mass matrix, it is easiy kow that the mass matrix of o-uiform sectio box girder eemet is gotte, as og as substitutig the parameter b ito C ad addig a item (such as Formua (8)) i mass matrix of rectaguar cross sectio beam eemet. M ( BD4 C) N Ndx (8) e For referrig ad usig coeiety, based o before-metioed three o-uiform sectio modes, the deduced resuts for a eemets m ij of mass matrix of box girder eemet are eumerated as foows Mass Matrix Coefficiets of the Box Girder with Liear Variabe Heights (Figure (a)) 4 /6 4 /6 4 /6 /7 D C 4 /4 m C h h BD C m4 C h h BD C m44 C h h BD C m C h h m C 5h 7h BD4 C BD4 C /5 / m5 9 C h h BD4C m6 C 7h 6h B 4 /4 m C 5h h 4 BD4 C /4 m5 C 6h 7h BD4 C /4 m6 C h h BD C Copyright SciRes.

6 7 J. P. PU E AL. C B D h(x) w z z m6 C h h BD C m C h h BD C m5 C h h 5577B D 4C m6 C h 9h 87B D 4C m55 C 744h 97h 67B D 4C m56 C h h 479BD4C m66 C h h 49B D 4C m m m m m m m m / / / / / / / 4545 y Figure. Geometry size of cross sectio of the box girder. m55 C h h BD4 C / 5 m56 C 7h 5h B D 4 C / m66 C h 5h 4 BD4 C / 4 m m m m m m m m Mass Matrix Coefficiets of the Box Girder with Paraboic Variabe Heights (Figure (b), h >h ) / / / 55h h BD C m C h h B D C m C h h B D C 4 m C h h B D C 44 m C 97h 744h 67 B D 4 C / 4545 m C 59h 56h 479 B D 4 C / 99 m5 C / 99 m6 C 865h 7h 5577 B D 4 C /88 m C 7h 56h 49 B D 4 C / 4545 m C 545h 4h 5577 B D 4 C /88 5 m C h h B D C /88 m C 79h 4858h 5577 B D 4 C /55 m C 855h 864h 479 B D 4 C / 99 m66 C h h 4 B D 4 C /55 m m m m m m m m 4.. Mass Matrix Coefficiets of the Box Girder with Paraboic Variabe Heights (Figure (c), h <h ) m C 6h h 7B D 4C / m4 C4h h 5B D 4C / m44 C6h 9h 5B D 4C /5 m C4858h 79h 5577BD4 C/55 m C88h 85h 57BD4 C/ m5 C8h 55h 58BD4 C/ Mass Matrix Coefficiets of the Box Girder with Paraboic Variabe Heights (Fig. (c), h <h ) m C6h h 7 B D 4 C / m4 C4h h 5B D 4 C / m44 C6h 9h 5B D 4 C / 5 m C4858h 79h 5577BD4 C/ 55 m C88h 85h 57BD4 C/ m5 C8h 55h 58BD4 C/ 99 m6 C44h 55h 859BD4 C/ 66 m Ch h 4BD4 C/ 55 m5 C7h 865h 5577B D 4 C / 88 m6 C 896h 9h 87B D 4 C / 88 m55 C744h 97h 67B D 4 C / 4545 m56 C 56h 59h 479B D 4 C / 99 m66 C56h 7h 49B D 4 C / 4545 m m m m m m m m Numerica Exampes 5.. Vaidatio I order to demostrate the feasibiity of the preset stiffess ad mass eemet matrices, a simpe exampe is performed, that is, a o-uiform cross sectio beam with the egth = m, the width of the rectaguar sectio is b =. m, the sma ed ad the big ed height are h =. m ad h =. m, respectiey. Youg s moduus is E =. N/m, the cocetrate oad is F = kn, the shape of the beam is show i Figure 4. h F h Figure 4. Shape of the beam. h Copyright SciRes.

7 J. P. PU E AL. 7 abe. Cacuatig resuts for the tapered catieer beam. Eemet stye Usig eemet i Figure (a) Usig subsectio uiform eemet Eemet umbers Vertica Dispacemets/mm Error ratio /% First, takig haf egth of the beam ad computig a catieer beam with big ed is fixed ad sma ed is existed a cocetrate force F = kn, the umerica resuts is show i abe. From the datum i abe, it shows that usig the eemet i Figure (a) to compute the catieer beam, oe ca obtaied a aaytic soutio by takig the oera beam just as oe eemet, whereas usig the subsectio uiform eemet oe eeds to diide the oera beam ito 8- eemets to gai the approximate soutios. Secod, takig oera egth of the beam ad computig a simpe supported beam with a cocetrate force F = kn at the mid-poit of the beam, the computatioa resuts are show i Figures 5 ad 6. I aboe metioed Figures, it is show that the umerica resuts, by usig the subsectio uiform eemet, is cose graduay to those of by usig the paper preset eemet, eertheess, the coergece rate is decreased graduay with icrease of the subsectio umbers. Accordig to the preset eemet i Figures (b) ad (c) to compute the paraboic catieer beam ad simpe supported oe with o-uiform cross sectio, the resuts are show i Figures 7 ad 8, respectiey. From the computatioa resuts oe ca kow that the accuracy of by usig the paraboic eemets is ot as good as oe of by usig the tapered oes for the tapered beam, ad the resuts are approximatie. Neertheess, comparig with usig the subsectio uiform sectio eemet, i the coditio of esurig defiite computig precisio, the eeded eemet umbers for the paraboic beam is aso sma tha that of usig the subsectio uiform eemet. Catieer ed fexibiity /mm usig the eemet i Fig. Figure usig the subsectio uiform eemet Eemet umbers Figure 5. Reatioship betwee ed-poit dispacemets ad eemet umbers for the tapered catieer beam. Spa-midde fexibiity / mm usig eemet i Fig. Figure usig subsectio uiform eemet Eemet umbers Figure 6. Reatioship betwee mid-poit dispacemet ad eemet umbers for the tapered simpe supported beam. Catieer ed fexibiity / mm usig eemet i Fig. Figure usig subsectio uiform eemet Eemet umbers Figure 7. Reatioships betwee ed-poit dispacemets ad eemet umbers for the paraboic catieer beam. Spa-midde fexibiity / mm usig eemet i Fig. Figure usig subsectio uiform eemet Eemet umbers Figure 8. Reatioship betwee mid-poit dispacemets ad eemet umbers for the paraboic simpe supported beam. Copyright SciRes.

8 74 J. P. PU E AL. By appyig the preset eemet to aayze the ouiform rectaguar beams with both iear ad paraboic ariabe heights, the resuts are beig approached the accurate soutios much more. A o- uiform catieer beam eemet ad a o-uiform simpe supported beam are preseted to aidate the eemet s reiabiity, ad the cacuatig resuts shows that, if usig the subsectio uiform fiite eemets oe eeds to diide more eemets to coerge the umerica soutio to the curret exact soutio. herefore, by appyig the preset eemet to aaysis the o-uiform beam, the aaysis ca be simpified disticty ad the computatioa resuts wi approach the accurate soutios. 5.. A hree-spa Cotiuous Hauched Bridge uder a Moig Load I this exampe a three-spa o-uiform cotiuous bridge is performed, the height of the beam is chageabe i the pae (see Figure 9). A sige oad aue of P = kn, moig at a speed of = 7 m/s, is cosidered. ota egth of the bridge is L = 6 m, Youg s moduus is E = N/m ad the mass desity is ρ = 4 kg/m. he acceeratio of the graity is g = 9.8m/s i a exampes. he dampig coefficiet is ζ = ζ =.5 ad the associated atura frequecies are ω ad ω, which were obtaied i the mai dyamic program. First, a case of which the hauched bridge with dampig subjected to a moig oad, is cosidered. I this case the atura frequecies are ot time-ariat, because i this exampe the effects of iertia force, Coriois force ad cetrifuga force iduced by the moig mass hae bee igored, so the mass, stiffess ad dampig matrices of the etire ibratig system are ot time-ariat yet. he fiite eemet mode of the bridge is composed of 6 uiform beam eemets ad 4 tapered beam eemets. he umerica resuts for the defectios at each midspa positio are i exceet agreemet with those aaiabe oes from the referece iterature [6,8]. Next cosiderig the bridge with dampig subjected to a moig mass, sice the effect of the iertia force, Coriois force ad cetrifuga force iduced by the moig mass is existet, the oera matrices(icude mass, stiffess ad dampig) ad the associated atura frequecies are a time-ariat at each computig time step. he umerica resuts for the defectios at each mid-spa positio of the bridge are show i Figure. From Figure it is show that the differece of the dispacemets, for the case with moig oad ad with moig mass, is o eidet. Sice the defectios of the bridge are sma, the effects of iertia force, Coriois force ad cetrifuga force iduced by the moig mass are aso sma to the dyamica resposes of the bridge. Neertheess, the ast effect may be sigificat for other cases. 5.. A Simpe Supported Beam uder a Moig Mass with Uiform Variabe Speeds Cosiderig a simpe supported beam uder a moig mass with uiform ariabe speeds (see Figure ), the egth of the beam is L = m, Youg s moduus is E =.5 Pa, the mass desity of the beam is ρ = 675 kg/m ad the sectio area is A =.4 m, the momet of iertia is I =.8 m 4, the moig mass m = 6. kg. We diided the beam ito 5 beam eemets, ad take the time step as Δt =. s, the iitia eocity of the moig mass is = m/s, ad the acceeratio are a =, ±, ±6 ad ±9 m/s, respectiey. he dyamica dispacemet resuts of the mid-poit of the beam hae bee computed by adoptig the Newmark method. Uder the coditios of beig differet acceeratios (acceeratig ad deceeratig), the dyamica dispacemet Dispacemet / m ML(S-) ML(S-) ML(S-) x / L ML-moig oad; MM-moig mass; S-spa MM(S-) MM(S-) MM(S-) Figure. Defectios at each mid-poit positio for each spa uder a moig oad/mass. Spa- m P= kn 6 m 6 m A A m 6 m 6 m Spa- m. m.6 m.5 m.-.6 m y t m, a EI, ρa m x Figure 9. A three-spa cotiuous hauched bridge uder a moig oad. Figure. Simpe supported beam uder a moig mass with uiform ariabe speeds. Copyright SciRes.

9 J. P. PU E AL. 75 respose cotrast cures of the midpoit of the beam are gie i Figures ad, respectiey. From Figures ad it is show that, uder the coditios of beig same iitia eocity, the dyamica dispacemets of the midpoit of the beam iduced by moig mass with acceeratio is bigger tha that of with uiform eocity, ad the arger the acceeratio, the bigger the midpoit dispacemets of the beam. Whereas the dyamica dispacemets of the midpoit of the beam iduced by moig mass with deceeratio is smaer tha that of with uiform eocity, ad the arger the acceeratio, the smaer the midpoit dispacemets of the beam. It is just opposite to the state with acceerated motio. It is because the moig mass was exerted by a frictio iduced betwee the cotact iterfaces. Whe the moig mass motio with acceeratio, it is gie a frictio, which is i ie with the moemet directio, by the beam. At the same time, the beam is subjected to a reactio force imposed by the moig mass. his force ca be regarded as a axia pressure actig o the beam. his pressure wi geerate a additioa bedig momet i the beam, so that the defectio of the beam wi be icreased. O the cotrary, whe the Dispacemet / m a= m a= m a=6 m a=9 m. 4 / s Figure. Mid-poit dispacemet of the beam with effect of acceeratio. moig mass deceeratio moemet, the beam wi be subjected to a axia tesio, ad this tesio wi geerate a additioa bedig momet withi the beam to reduce the defectio A hree-spa Cotiuous Box Girder Bridge with No-Uiform Sectio uder a Moig Vehice with Frictio he purpose of this exampe is to compute a three-spa cotiuous box girder bridge with o-uiform sectios uder a moig ehice with frictio, the bridge is composed of 7 box girder segmets, the height of the boxsectio is chageabe with both iear ad paraboic soffit shape i the pae (see Figure 4). he tota egth of the bridge is L = m, the mass desity is ρ = 4 kg/m ad Youg s moduus is E = N/m. he acceeratio of the graity is g = 9.8m/s, the momet of iertia I(x) is chageabe with the positio x. here is a ehice at eft ed of the bridge, the weight of the ehice is P =.6 4 N (whie the frot-whee ad rear whee are P = 8. N ad P =.6 N, respectiey), the space betwee the whees is. m, at the time of t = the frot-whee is just i the eft ed of bridge, the ehice traes at a speed of =5 m/s. Whe the ehice moes forward meters from the eft ed of the bridge at a uiform eocity, the it is broke, so the eocity wi be sowed dow or stopped by cotact frictio. I order to impose the brakig force, a ramp fuctio is assumed (see Figure 5). his is based o the test resuts o highway ehices coduced by the rasport ad Road Research Laboratory [8]. he brakig force icreases ieary to a maximum F b max = εp ad the stays costat uti the ehice either comes to a stop or crosses the bridge spa ad is writte as P P Dispacemet / m a= m a=- m a=-6 m a=-9 m m m m m m m m m. m. m m-.8 m. m. 4 / s Figure. Mid-poit dispacemet of the beam with effect of deceeratio. m Figure 4. A three-spa cotiuous box girder bridge with ariabe sectios.. m Copyright SciRes.

10 76 J. P. PU E AL. F F ( t t ), t t bmax ( )( ) b t t Fbmax, t t (9) where ε is the impact coefficiet, P is the ehice static weight, t is the momet of begi brakig ad is the momet of the brakig force icreases to the maximum aue of F bmax i time itera t t t +. For this exampe, 6 iear ad paraboic beam eemets are diided i oera bridge, the time step is Δt =. s. We take the frictio coefficiets as μ =.,.,. ad.4, respectiey. akig the impact coefficiet ε =., the umerica resuts for the midpoit defectios at mid-spa positio of the bridge are show i Figure 6. Figure 6 shows that the dyamica dispacemets were affected by frictio together with the impact coefficiet. he arger the frictio coefficiet, the arger the maximum dyamica dispacemets. he arger the frictio, the shorter the time of brakig ehice. I the coditio of beig same frictio force. he arger the impact coefficiet, the bigger the dyamica dispacemets of midpoit at the mid-spa of bridge. It ca be obsered from Figure 7 i eidece. Impact coefficiets.4... ε t (-ε 4 )( t + -t ) t+ / s Figure 5. Reatioship of brakig time s. impact coefficiet. Dispacemet / m without frictio =. μ =. =. μ =. =. μ = =.4 μ = / s Figure 6. Defectios of mid-poit positio at mid spa (ε =.). t Dispacemet / m without frictio = ε = =. ε =. =. ε =. =. ε = / s Figure 7. Defectios of mid-poit positio at mid spa (μ =.). 5. Cocudig Remarks Dyamica soutios of the muti-spa o-uiform beam with a moig ehice ca ot be obtaied by the theoretica meas, the mass of the ehice ca ot just be regarded as a simpe moig oad, the effect of iertia force, Coriois force ad cetrifuga force is eeded to be cosidered i geera. he use of box beam bridge with o-uiform cross sectio is fairy commo i the egieerig. Numerica aaysis for dyamic respose of this kid of beam is beeficia to uderstad the dyamic characteristics of the bridge, to proide the scietific basis for the safe use of the bridge, ad proided with a certai practica sigificace ad appicatio aues. Usig obique-shaped ad Hparaboic-shaped beam eemet with o-uiform cross sectio Hca improe the accuracy ad efficiecy i soig. o defie the dyamic respose effect of the bridge caused by ehice moig with ariabe speed as we as the frictio ad brakig impact force o the bridge deck eeds to cosidered the reatioships of arious factors sytheticay, ad the to fid a desig scheme coser to egieerig practice combied with the reeat scietific experimet. 6. Refereces [] L. Fryba, Vibratio of Soids ad Structures uder Moig Loads, Noordhoff Iteratioa Pubishig, Groige, 97. [] L. Fryba, Dyamic Behaiour of Bridges due to High- Speed rais, I: R. Degado, R. Cacada ad A. Campos Eds., Workshop Bridges for High-Speed Raiways, Facuty of Egieerig, Uiersity of Porto, Porto, 4, pp [] C. W. Cai, Y. K. Cheug ad H. C. Cha, Dyamic Resposes of Ifiite Cotiuous Beams Subjected to a Moig Force-A Exact Method, Joura of Soud ad Vibratio, Vo., 988, pp [4] J. S. Wu ad C. W. Dai, Dyamic Resposes of Mu- Copyright SciRes.

11 J. P. PU E AL. 77 tispa Nouiform Beam due to Moig Loads, Joura of Structura Egieerig, Vo., 987, pp [5] H. P. Lee, Dyamic Resposes of a Beam with Itermediate Poit Costraits Subject to a Moig Load, Joura of Soud ad Vibratio, Vo. 7, 994, pp [6] G. Michatsos, D. Sophiaopouos ad A. N. Kouadis, he Effect of a Moig Mass ad other Parameters o the Dyamic Respose of a Simpy Supported Beam, Joura of Soud ad Vibratio, Vo. 9, 996, pp [7] K. Hechi, M. Fafard, G. Dhatt ad M. abot, Dyamic Behaior of Muti-Spa Beams uder Moig Loads, Joura of Soud ad Vibratio, Vo. 99, 997, pp. -5. [8] H. C. Kwo, M. C. Kim ad I. W. Lee, Vibratio Cotro of Bridges uder Moig Loads, Computers & Structures, Vo. 66, 998, pp [9] D. Y. Zheg, Y. K. Cheug, F.. K. Au ad Y. S. Cheg, Vibratio of Muti-Spa No-Uiform Beams uder Moig Loads by Usig Modified Beam Vibratio Fuctios, Joura of Soud ad Vibratio, Vo., 998, pp [] Y. A. Dugush, M. Eiseberger, Vibratios of No- Uiform Cotiuous Beams uder Moig Loads, Joura of Soud ad Vibratio, Vo. 54,, pp [] A. Yaari, M. Nouri, M. Mofid, Discrete Eemet Aaysis of Dyamic Respose of imosheko Beams uder Moig Mass, Adaces i Egieerig Software, Vo.,, pp [] A. E. Martiez-Castro, P. Museros ad A. Castio- Liares, Semi-Aaytic Soutio i the ime Domai for No-Uiform Muti-Spa Beroui-Euer Beams raersed by Moig Loads, Joura of Soud ad Vibratio, Vo. 94, 6, pp [] J. J. Wu, Dyamic Aaysis of a Icied Beam due to Moig Loads, Joura of Soud ad Vibratio, Vo. 88, 5, pp. 7-. [4] J. J. Wu, Vibratio Aayses of a Icied Fate Subjected to Moig Loads, Joura of Soud ad Vibratio, Vo. 99, 7, pp [5] E. Esmaizadeh ad M. Ghorashi, Vibratio Aaysis of Beams raersed by Uiform Partiay Distributed Moig Massed, Joura of Soud ad Vibratio, Vo. 84, 995, pp [6] U. Lee, Separatio betwee the Fexibe Structure ad the Moig Mass Sidig o it, Joura of Soud ad Vibratio, Vo. 9, 998, pp [7] U. Lee, C. H. Pak ad S. C. Hog, Dyamics of Pipig System with Itera Usteady Fow, Joura of Soud ad Vibratio, Vo. 8, 995, pp [8] S. S. Law ad X. Q. Zhu, Bridge Dyamic Resposes due to Road Surface Roughess ad Brakig of Vehice, Joura of Soud ad Vibratio, Vo. 8, 5, pp Copyright SciRes.