Keywords: Sectional forces, bending moment, shear force, finite element, Bernoulli-Euler beam, concentrated moving loads

Transcription

1 Shock and Vibration 15 (2008) IOS Prss Finit-mnt formua for cacuating th sctiona forcs of a Brnoui-Eur bam on continuousy viscoastic foundation subjctd to concntratd moving oads Ping Lou Schoo of Civi Enginring and Architctur, Raiway Campus, Cntra South Univrsity, 22 Shao-shan-nan Road, Changsha, Hunan , P.R. China Fax: ; E-mai: pingou@mai.csu.du.cn or pingoucsu@126.com Rcivd 9 Novmbr 2006 Rvisd 2 January 2007 Abstract. In th finit mnt mthod, thr ar shortcomings using th convntiona formua to cacuat th sctiona forcs, i.., th bnding momnt and th shar forc, at any cross-sction of Brnoui-Eur bam undr dynamic oads. This papr prsnts som nw finit-mnt formua ovrcoming th shortcomings of th convntiona ons to cacuat th sctiona forcs at any cross-sction of a Brnoui-Eur bam on continuousy viscoastic foundation subjctd to concntratd moving oads. Th proposd formua can asiy dgnrat into th formua for cacuating th sctiona forcs of a simpy supportd or a continuous Brnoui-Eur bam subjctd to concntratd moving oads, and into th formua for cacuating th sctiona forcs of a Brnoui-Eur bam on Winkr foundation undr static oads. Fiv numrica xamps incuding static and dynamic anayss ar chosn to iustrat th appication of th proposd formua. Numrica rsuts show: (1) compard with th convntiona formua, th proposd formua can improv th cacuation accuracy of th sctiona forcs of bam; (2) on shoud us th proposd formua, not th convntiona formua, to cacuat th sctiona forcs at any cross-sction in Brnoui-Eur bam. Kywords: Sctiona forcs, bnding momnt, shar forc, finit mnt, Brnoui-Eur bam, concntratd moving oads 1. Introduction Th dynamic anaysis of a bam on astic foundation or a simpy supportd bam subjctd to dynamic oads is widsprad in nginring. Hr ony som of th rvant itratur is mntiond. Many rsarchrs [5,8,9, 11,12,15,16,20,22] appid th anaytica mthods to invstigat th dynamic probm. Howvr, for most of th nginring probms, on must ry on numrica mthods sinc anaytica mthods ar usuay not avaiab. In th numrica mthods, th finit mnt mthod (FEM) is powrfu bcaus it aows soution to compx probms in nginring. Som rsarchrs [10,13,14,18,19,23 26] appid FEM to study th forgoing dynamic probm. It is w known that in th dsign of a bam on astic foundation or a simpy supportd bam subjctd to moving oads, th most important considrations ar th vrtica dfction and th bnding strss, and th attr is givn by using bnding momnt dividd by th sction moduus of th bam. Rfrncs [14,18,23,25,26] did not car for th bnding momnt of bam. Rfrncs [10,13,19,24] rportd th bnding momnt of bam, but thy did not giv th formua to cacuat th bnding momnt of bam. ISSN /08/$ IOS Prss and th authors. A rights rsrvd

2 148 P. Lou / Finit-mnt formua for cacuating th sctiona forcs In th dynamic anaysis using FEM, if th foowing formua (1) (4) ar usd to cacuat th sctiona forcs of th cross-sction at any point in a Brnoui-Eur bam mnt on continuousy viscoastic foundation subjctd to concntratd moving oads, thn, rrors may appar in th numrica rsuts bcaus of shortcomings in using th formua (1) (4) to cacuat th sctiona forcs. M(ξ,t) = EI 2 y(ξ,t) ξ 2 = EIN q for th right-hand sid of cross-sction at any point (1) Q(ξ,t) =EI 3 y(ξ,t) ξ 3 = EIN q for th right-hand sid of cross-sction at any point (2) M(ξ,t) =EI 2 y(ξ,t) ξ 2 = EIN q for th ft-hand sid of cross-sction at any point (3) Q(ξ,t) = EI 3 y(ξ,t) ξ 3 = EIN q for th ft-hand sid of cross-sction at any point (4) whr M(ξ,t) and Q(ξ,t) rspctivy dnot th bnding momnt and th shar forc of cross-sction at any point and tim t, and th positiv dirctions of M(ξ,t) and Q(ξ,t) for th right-hand sid of cross-sction and for th ft-hand sid of cross-sction at any point ar shown in Fig. 1(a) and (b), rspctivy; ξ dnots oca coordinat masurd from th ft nod of th bam mnt; EI dnots th constant bnding stiffnss of th bam; N dnots th shap matrix of th bam mnt; q dnots th noda dispacmnt vctor of th bam mnt; th suprscript prim dnots diffrntiation with rspct to ξ; and y(ξ) dnots th vrtica dfction at any point in th bam mnt. y(ξ) can b xprssd as y(ξ) =Nq (5) If th cubic Hrmitian poynomias [3] ar usd as th shap functions of a bam mnt, th shap matrix N of th bam mnt can b writtn as N = [ ] N 1 N 2 N 3 N 4 (6) with N 1 =1 3(ξ/) 2 +2(ξ/) 3 N 2 = ξ[1 2(ξ/)+(ξ/) 2 ] N 3 =3(ξ/) 2 2(ξ/) 3 N 4 = ξ[(ξ/) 2 (ξ/)] whr dnots th ngth of th bam mnt. In this papr, th formua (1) (4) ar rfrrd to as th convntiona ons for cacuating th sctiona forcs of a Brnoui-Eur bam undr dynamic oads. Th shortcomings using th formua (1) (4) to cacuat th sctiona forcs of a bam mnt on continuousy viscoastic foundation subjctd to concntratd moving oads ar as foows. First, thr ar shortcomings using th formua (1) (4) to cacuat th sctiona forcs at two nods of a bam mnt. Th formua (1) and (2) for cacuating th sctiona forcs at th ft nod (i.., ξ =0)and th formua (3) and (4) for cacuating th sctiona forcs at th right nod (i.., ξ = ) in a bam mnt ony considr th sctiona forcs inducd by th noda dispacmnts of th corrsponding bam mnt, bcaus th formua (1) and (2) with ξ =0and th formua (3) and (4) with ξ = can b xprssd as th sum of th products of th corrsponding mnt stiffnss matrix and th dispacmnts of th corrsponding bam mnt nods, i.., fd = k bq, in which, fd dnots th sctiona forcs of th bam mnt inducd by th noda dispacmnts of th bam mnt, and k b dnots th mnt stiffnss matrix of th bam mnt itsf. Th sctiona forcs of th cross-sction at th nods of mnt inducd by th noda accrations and th foundation spring and damping forcs acting on th corrsponding bam mnt ar not considrd. In addition, th fixd-nd sctiona forcs prsntd by rfrnc [17] at th two nds of a campd-campd (C-C) bam mnt inducd by moving oads acting on th bam mnt at tim t ar aso not considrd. Thn, thr xist shortcomings using th formua (3) and (4) to cacuat th sctiona forcs at any point A, rathr than at a nod, of a bam mnt. Th formua (3) and (4) for cacuating th sctiona forcs of th ft-hand sid of cross-sction at any point A, rathr than at a nod, of a Brnoui-Eur bam mnt and at tim t can b

3 P. Lou / Finit-mnt formua for cacuating th sctiona forcs 149 ft nod ξ ξ, t ξ, t ( ) ( ) right-hand sid of cross-sction at any point right nod r ξ r (a) Sctiona forcs diagram for th right-hand sid of cross-sction at any point ft nod ξ ( ξ, t ) ( ξ, t ) ft-hand sid of cross-sction at any point right nod r ξ r (b) Sctiona forcs diagram for th ft-hand sid of cross-sction at any point Fig. 1. Sctiona forcs diagram for th cross-sction at any point in a Brnoui-Eur bam mnt in th dynamic anaysis. writtn as, rspctivy, M(ξ,t) ξ=ξa = EI [ ] [ q 6 + ξ A EI ] q (7) Q(ξ,t) ξ=ξa = EI = M(ξ,t) ξ=0 + Q(ξ,t) ξ=0 ξ A [ ] q = Q(ξ,t) ξ=0 (8) Th formua (7) ony considrs th ffcts of th bnding momnt and th shar forc at th ft nod of th bam mnt, which ar inducd by th noda dispacmnts of th corrsponding mnt, on th bnding momnt of th ft-hand sid of cross-sction at th point A. In th formua (7), th ffcts of th fixd-nd bnding momnt and shar forc at th ft nd of th C-C bam mnt, inducd by moving oads acting on th bam mnt at tim t, on th bnding momnt of th ft-hand sid of cross-sction at th point A ar not considrd whi shoud b considrd. In addition, th ffcts on th bnding momnt of th ft-hand sid of cross-sction at th point A ar aso not considrd of moving oads acting on th bam mnt btwn th ft nod and th point A, of inrtia forc in th bam mnt btwn th ft nod and th point A, and of th foundation spring and damping forcs acting on th bam mnt btwn th ft nod and th point A. Simiary, th formua (8) ony considrs th ffct of th shar forc at th ft nod of th bam mnt, which is inducd by th noda dispacmnt of th corrsponding mnt, on th shar forc of th ft-hand sid of cross-sction at th point A. In th formua (8), th ffct of th fixd-nd shar forc at th ft nd of th C-C bam mnt, inducd by moving oads acting on th bam mnt at tim t, on th shar forc of th ft-hand sid of cross-sction at th point A is not considrd whi shoud b considrd. In addition, th ffcts on th shar forc of th ft-hand sid of cross-sction at th point A ar aso not considrd of moving oads acting on th bam mnt btwn th ft nod and th point A, of inrtia forc in th bam mnt btwn th ft nod and th point A, and of th foundation spring and damping forcs acting on th bam mnt btwn th ft nod and th point A. This purpos of this papr is to prsnt nw finit-mnt formua for cacuating th sctiona forcs at any cross-sction of a Brnoui-Eur bam on continuousy viscoastic foundation subjctd to concntratd moving oads. Th nw finit-mnt formua can ovrcom th forgoing shortcomings of th convntiona formua and can improv th accuracy of cacuating th sctiona forcs of bam. Evidnty, th proposd formua can asiy dgnrat into th formua for cacuating th sctiona forcs of a simpy supportd or a continuous Brnoui- Eur bam subjctd to concntratd moving oads, and into th formua for cacuating th sctiona forcs of a Brnoui-Eur bam on Winkr foundation undr static oads.

4 150 P. Lou / Finit-mnt formua for cacuating th sctiona forcs 2. Thory and formuation 2.1. Fundamnta assumptions Th foowing assumptions ar mad whn on stabishs th formua for cacuating th sctiona forcs of a bam on viscoastic foundation subjctd to concntratd moving oads. (1). Ony vrtica dynamic oads ar considrd. (2). Axia dformations and th damping of th bam ar ngctd. (3). Th bam is modd as a uniform Brnoui-Eur bam. (4). Th viscoastic foundation is modd as cosy spacd, indpndnt, inar springs and viscous damprs. (5). Th cubic Hrmitian poynomias [3] ar usd as th shap functions of a bam mnt Equation of motion for a bam on continuousy viscoastic foundation subjctd to concntratd moving oads Considr a uniform astic Brnoui-Eur bam with constant bnding stiffnss EI rsting on continuousy viscoastic foundation with stiffnss cofficint k w and damping cofficint c w subjctd to a numbr of concntratd moving oads, as shown in Fig. 2. Th bam is dividd into a numbr of finit mnts with qua ngth of. Th soid circs ( ) dnot th nods for th bam mnts. Sinc th axia dformations of th bam ar ngctd, pr nod of a bam mnt has two dgrs of frdom, i.., a vrtica transation and a rotation about an axis norma to th pan of th papr. According to th nrgy princip, th quation of motion for th bam on continuousy viscoastic foundation subjctd to a numbr of concntratd moving oads at tim t can b writtn as M q + C q + Kq = n N T i P i whr M, C, and K with n n ordr ar th ovra mass, damping and stiffnss matrics, rspctivy; q, q, and q with n 1 ordr ar th accration, vocity and dispacmnt vctors, rspctivy; th suprscript T dnots transpos; N T i is th transpos of th shap functions for th bam mnt which ar vauatd at th position of th i-th concntratd moving oad P i at tim t; P i is th magnitud of th i-th concntratd moving oad; n is th tota numbr of th dgrs of frdom of th bam; and n is th tota numbr of concntratd moving oads. In Eq. (9), th ovra mass matrix M can b obtaind by assmbing th mnt consistnt mass matrix m. Sinc th damping of bam itsf is ngctd, th ovra damping matrix C ony incuds th ffct of th viscous damping of foundation. C can b obtaind by assmbing th foundation mnt damping matrix c w inducd by viscous damping foundation supporting th bam mnt. Th ovra stiffnss matrix K can b obtaind by assmbing th mnt stiffnss matrix k b of th bam mnt itsf and th foundation mnt stiffnss matrix k w du to th astic foundation supporting th bam mnt. Th xprssions of mnt matrics m, c w, k b and k w ar istd in th Appndix. In addition, N i with 1 n ordr in Eq. (9) can b writtn as N i =[0 0 N 1 N 2 N 3 N 4 0 0] ξ=ξi (10) whr ξ i dnots th distanc btwn th acting point of th i-th concntratd moving oad P i and th ft nod of th bam mnt on which th oad P i is acting at tim t, as shown in Fig. 2. It shoud b notd that N i is a row matrix with zro ntris xcpt for thos trms corrsponding to two nods of th mnt on which th i th concntratd moving oad P i is acting. N i is tim dpndnt as th oad P i movs from on position to anothr within on mnt. As th oad P i movs to th nxt mnt, N i wi shift in position corrsponding to th dgrs of frdom of th mnt whr th oad P i is positiond. (9)

5 P. Lou / Finit-mnt formua for cacuating th sctiona forcs ξ w w Fig. 2. Mathmatica mod for a uniform astic Brnoui-Eur bam on continuousy viscoastic foundation subjctd to a numbr of concntratd moving oads. ft nod right nod ξ r r Fig. 3. Shar forcs Q and Q r, and bnding momnts M and M r at th cross-sction of th two nods of a typica bam mnt Formua for cacuating th sctiona forcs of a bam By introducing th boundary conditions of th bam, Eq. (9) can b sovd by th Wison-θ mthod or simiar mthods [1], to obtain th gnraizd dispacmnts, vocitis and accrations pr nod of th Brnoui-Eur bam at tim t. Thn, th sctiona forcs at any cross-sction of th bam at tim t can b obtaind by using th foowing procdur. First, t us considr how to cacuat th sctiona forcs at th cross-sction of th two nods of a bam mnt. Th sctiona forcs at th cross-sction of th two nods of a typica bam mnt at tim t, as shown in Fig. 3, can b xprssd as h f = m q + k b q + c w q + k w q + fi 0 (11) whr f is th sctiona forcs vctor at th two nods of a bam mnt, f =[Q M Q r Mr ] T, Q and M ar th shar forc and bnding momnt at th ft nod of th bam mnt, rspctivy; Q r and M r ar th shar forc and bnding momnt at th right nod of th bam mnt, rspctivy; th positiv dirctions of Q, M, Q r, and M r ar shown in Fig. 3; q, q, and q with 4 1 ordr ar th noda accration, vocity and dispacmnt vctors of th bam mnt, rspctivy, obtaind by soving Eq. (9); h is th tota numbr of concntratd moving oads acting on th bam mnt at tim t; and f i 0 is th fixd-nd forc vctor [17] at th two nds of th C-C bam mnt inducd by th i-th concntratd moving oad P i acting on th bam mnt at tim t. It shoud b notd that fi 0 and th quivant noda forc vctor inducd by th i-th concntratd moving oad P i acting on th C-C bam mnt at tim t ar qua in magnitud but opposit in dirction. Thus, th xprssion of fi 0 can b writtn as fi 0 = [N 1 N 2 N 3 N 4 ] T ξ=ξ i P i (12) f 0 i wi bcom zro vctor if th i-th concntratd moving oad P i acts outsid th bam mnt. Th first and scond trms on th right-hand sid in Eq. (11) dnot th vctors of th sctiona forcs at th two nods of th bam mnt inducd by, rspctivy, th noda accrations and th noda dispacmnts of th bam mnt. Th third and fourth trms on th right-hand sid dnot th vctors of th sctiona forcs at th two nods of th bam mnt inducd by, rspctivy, th continuousy damping forc and th spring forc undr th bam mnt. Th fifth trm on th right-hand sid dnots th sum of th fixd-nd forc vctors at th two nds of th C-C bam mnt inducd by h concntratd moving oads acting on th bam mnt at tim t. Thn, t us considr how to cacuat th sctiona forcs at a point, rathr than at a nod, within a bam mnt. Th sctiona forcs can b obtaind by using th quiibrium condition of forcs acting in th vrtica dirction and

6 152 P. Lou / Finit-mnt formua for cacuating th sctiona forcs ft nod 1 ξ (ξ) S ξ A (ξ) (ξ) D A ξ A A Fig. 4. Fr-body diagram of portion of a typica bam mnt on viscoastic foundation, in which f I (ξ) = mn q, f S (ξ) =k wnq, and f D (ξ) =c wn q. th quiibrium condition of bnding momnts. For xamp, it is assumd that point A ocats at a position in th bam btwn two adjacnt nods, as shown in Fig. 4, and thr ar h concntratd moving oads btwn th ft nod of th bam mnt and th point A at tim t. Th shar forc Q A and bnding momnt M A at th point A can b givn by th foowing xprssion ξ A h Q A = ( mn q + c w N q + k w Nq )dξ Q P i (13a) 0 M A = Q ξ A + h ξ A P i (ξ A ξ i ) M (ξ A ξ)( mn q + c w N q + k w Nq )dξ (14a) 0 whr N =[N 1 N 2 N 3 N 4 ], and ξ A dnots th distanc btwn th ft nod of th bam mnt and th point A. Th positiv dirctions of th shar forc Q A and bnding momnt M A at th point A ar shown in Figur 4. It shoud b notd that Q and M in th formua (13a) and (14a) hav bn obtaind by soving th formua (11). If m, c w, and k w ar constant, th formua (13a) and (14a) can b writtn as, rspctivy, Q A = m N ξ=ξa q + c w N ξ=ξa q + k w N ξ=ξa q Q M A = Q ξ A + h h P i P i (ξ A ξ i ) M m(ξ A N ξ=ξa Ñ ξ=ξa ) q c w (ξ A N ξ=ξa Ñ ξ=ξa ) q k w (ξ A N ξ=ξa Ñ ξ=ξa )q with N =[ N 1 N2 N3 N4 ] (13b) (14b) N 1 = ξ ξ ξ4 3 N2 = 1 2 ξ2 2 3 ξ ξ4 2 N 3 = ξ ξ4 3 and Ñ =[Ñ 1 Ñ 2 Ñ 3 Ñ 4 ] N4 = 1 4 ξ ξ3 Ñ 1 = 1 2 ξ2 3 4 ξ ξ5 3 Ñ 2 = 1 3 ξ3 1 2 ξ ξ5 2 Ñ 3 = 3 4 ξ ξ5 3 Ñ 4 = 1 5 ξ ξ4 It shoud b pointd out that th formua (11) ony cacuats th sctiona forcs at two nods of a bam mnt and th formuas (13) and (14) can cacuat th sctiona forcs at any point in a bam mnt xcpt two nods. By

7 P. Lou / Finit-mnt formua for cacuating th sctiona forcs 153 =98000N/m Fig. 5. A simpy supportd bam undr uniformy distributd static oad. using th formua (11), (13) and (14), on can obtain th sctiona forcs at any cross-sction of a Brnoui-Eur bam on continuousy viscoastic foundation subjctd to a numbr of concntratd moving oads at tim t. Furthrmor, for th static probm, on has q = q = 0, and th formua (11), (13) and (14) rduc to, rspctivy, f =(k b + k w )q + 0 h f 0 i ξ A h Q A = k w Nq dξ Q P i (16a) M A = Q ξ A + h ξ A P i (ξ A ξ i ) M (ξ A ξ)k w Nq dξ (17a) If k w is constant, th formua (16a) and (17a) can b writtn as, rspctivy, 0 (15) Q A = k w N ξ=ξa q Q h P i (16b) M A = Q ξ A + h P i (ξ A ξ i ) M k w (ξ A N ξ=ξa Ñ ξ=ξa )q In addition, if k w = 0 and c w = 0 ar usd in th formua (11), (13) and (14), thn th rvisd formua of (11), (13) and (14) can b usd to cacuat th sctiona forcs at any cross-sction of a simpy supportd or a continuous Brnoui-Eur bam subjctd to a numbr of concntratd moving oads. (17b) 3. Numrica xamps Fiv numrica xamps incuding static and dynamic anayss ar chosn to iustrat th appication of th proposd formua. In th finit mnt anaysis, th quation of motion for th foowing dynamic anayss wi b sovd by mans of th Wison-θ mthod with θ = Examp 1. A simpy supportd bam undr uniformy distributd static oad Figur 5 shows a simpy supportd bam undr uniformy distributd static oad p = N/m. Th paramtrs of th bam takn from rfrnc [4] ar: ngth L = 0.2 m and cross-sction ara = m 2 (0.02 m dp 0.01 m wid) and E = N/m 2. For th cas of a simpy supportd bam undr uniformy distributd static oad, th formua (11) for cacuating th sctiona forcs at th two nods of a bam mnt can b rvisd as f = k b q + f 0 (18)

8 154 P. Lou / Finit-mnt formua for cacuating th sctiona forcs Fig. 6. Bnding momnt diagrams in a simpy supportd bam undr uniformy distributd oad. whr f 0 dnots th fixd-nd forc vctor at th two nds of th C-C bam mnt inducd by uniformy distributd static oad, which can b writtn as f 0 = [pl/2 pl 2 /12 pl/2 pl 2 /12] T (19) Th formuas (13) and (14) for cacuating th sctiona forcs at a point A, rathr than at a nod, in a bam mnt can b rvisd as Q A = Q p ξ A (20) ξ A M A = Q ξ A + (ξ A ξ)pdξ M = Q ξ A p ξ2 A M (21) 0 Bnding momnt diagrams and shar forc diagrams of th simpy supportd bam hav bn pottd in Figs 6 and 7, rspctivy, incuding cosd form soutions and finit mnt soutions givn by th proposd formua with 1 mnt and by th convntiona formua [4] with 2 mnts of qua ngth. It can b sn from Figs 6 and 7 that th finit mnt soutions givn by th proposd formua with 1 mnt ar th sam as th cosd form soutions; howvr, th sam agrmnt cannot b found btwn th finit mnt soutions givn by th convntiona formua with 2 mnts and th cosd form soutions. Th rasons ar as foows. Th convntiona formua usd in rfrnc [4] is f = k b q (22) Compard with th proposd formua (18), th convntiona formua (22) ngcts f 0 to cacuat th sctiona forcs at two nods of a bam mnt. In addition, compard with th proposd formuas (20) and (21), th convntiona formua (22) dos not considr (i) th sctiona forcs at point A contributd by th fixd-nd forc at th ft nd of th C-C bam mnt inducd by th uniformy distributd oad acting on th bam mnt, and (ii) th sctiona forcs at point A inducd by th uniformy distributd oad acting on th bam mnt btwn th ft nod and th point A, whn th convntiona formua (22) is usd to cacuat th sctiona forcs at th point A in a bam mnt Examp 2. A bam with fr nds on Winkr foundation undr a concntratd static oad Considr a uniform Brnoui-Eur bam with fr nds rsting on Winkr foundation undr a concntratd static oad P = 10,000 N acting on th bam midpoint, as shown in Fig. 8. Th paramtrs in this study ar givn: E (bam astic moduus) = 9, N/m 2, I (momnt of inrtia of bam) = m 4, L (bam ngth) =

9 P. Lou / Finit-mnt formua for cacuating th sctiona forcs 155 Fig. 7. Shar forc diagrams in a simpy supportd bam undr uniformy distributd oad mm A Fig. 8. A uniform Brnoui-Eur bam with fr nd rsting on Winkr foundation undr a concntratd static oad m, and k w (stiffnss cofficint of Winkr foundation) = N/m 2. Ths paramtrs of this xamp ar takn from rfrnc [7]. Tab 1 rports th ratios of th numrica soutions to th xact soutions of th sctiona forcs at th point A, with th numrica soutions givn, rspctivy, by th proposd formuas (16b) and (17b), and by th convntiona formua (22) adoptd in rfrnc [7] using th cubic Hrmitian poynomias as th shap functions of a bam mnt. Th distanc btwn th point A and th bam midpoint is m, as shown in Fig. 8. Th xact soutions can b obtaind using th formua prsntd in rfrnc [21]. It shoud b notd that xact soutions ar not dpndnt on th numbr of mnts. It can b sn from Tab 1 that th bnding momnt and th shar forc givn by th proposd formuas (16b) and (17b) ar nary xact if 20 mnts ar usd for th bam ngth L. Th rror of th shar forc is 3 pr cnt and th rror of th bnding momnt is ss than 8 pr cnt if ony 10 mnts ar usd. Howvr, th sam agrmnt cannot b found for th rsuts givn by th convntiona formua (22) adoptd in rfrnc [7]. This is du to th shortcomings of th convntiona formua Examp 3. A simpy supportd bam subjctd to a concntratd moving oad Considr a uniform simpy supportd Brnoui-Eur bam with a span L of 20 m subjctd to a concntratd moving oad P = kn with constant spd v = 60 m/s from th ft nd to th right nd of th bam. Th bam paramtrs ar: E = N/m 2, I = 3.81 m 4, and m = 34,088 kg/m. In th finit mnt anaysis, th bam is modd as 2, 6 and 10 mnts with qua ngth, rspctivy, and tim intrva s is adoptd. Th bnding momnts at th midpoint cross-sction of bam givn by th rvisd proposd formua (11), i.., aftr dting th third and fourth trms of right hand sid in formua (11), with 2, 6 and 10 mnts hav bn pottd in Fig. 9, aong with th anaytica soution givn by th foowing Eq. (23) takn from Timoshnko t a. [22] with i = M(x, t) = EI 2 y(x, t) x 2 (23)

10 156 P. Lou / Finit-mnt formua for cacuating th sctiona forcs Tab 1 Ratios of th numrica soutions to th xact soutions of th sctiona forcs at th point A in Examp 2 with th numrica soutions givn, rspctivy, by th proposd formua and by th convntiona formua in rfrnc [7] Grad of msh (mnts of qua ngth) Bnding momnt Shar forc Prsnt Rf. [7] Prsnt Rf. [7] 10 mnts mnts mnts mnts mnts Fig. 9. Tim historis for bnding momnt at th midpoint cross-sction of bam with anaytica soution, and th soutions givn by th proposd formua with 2, 6 and 10 mnts. with y(x, t)= 2PL3 mπ 2 sin(iπx/l)sin(iπvt/l) i 2 (i 2 π 2 a 2 v 2 L 2 2PL4 v ) mπ 3 a sin(iπx/l)sin(i 2 π 2 at/l 2 ) i 3 (i 2 π 2 a 2 v 2 L 2 ) (24) a 2 = EI m From Fig. 9, good agrmnt has bn achivd btwn th prsnt soution with 2, 6 and 10 mnts, rspctivy, and th anaytica soution. Th bnding momnts at th midpoint cross-sction of bam givn by th rvisd proposd formua (11) and th convntiona formua (1) with 2, 6 and 10 mnts hav bn pottd in Figs 10 12, rspctivy. It can b sn that th diffrnc btwn th soution givn by th rvisd proposd formua (11) and that givn by th convntiona formua (1) incrass with th incras of ngth of mnt. Th rason why thr is th diffrnc btwn th soution givn by th rvisd proposd formua (11) and that givn by th convntiona formua (1) is that th fixd-nd bnding momnt at th ft nd of th (N/2+1)-th C-C bam mnt inducd by th oad acting on th (N/2+1)-th bam mnt is not considrd in th convntiona formua, in which N dnots th tota numbr of bam mnts Examp 4. A simpy supportd bam rsting on Winkr foundation subjctd to a stationary pusating concntratd oad Lt us considr a simpy supportd uniform Brnoui-Eur bam rsting on Winkr foundation subjctd to a stationary pusating concntratd oad of P (t) = P sin ω t acting at a distanc x 1 from th ft nd point of th

11 P. Lou / Finit-mnt formua for cacuating th sctiona forcs 157 Fig. 10. Tim historis for bnding momnt at th midpoint cross-sction of bam with th soutions givn by th proposd formua and by th convntiona formua with 2 mnts. Fig. 11. Tim historis for bnding momnt at th midpoint cross-sction of bam with th soutions givn by th proposd formua and by th convntiona formua with 6 mnts. bam. Th xprssions of th xact vrtica dispacmnt y(x, t) and bnding momnt M(x, t) of th bam takn from Timoshnko t a. [22] ar as foows y(x, t) = 2 PL 3 [ sin ω t sin(iπx/l)sin(iπx 1 /L) m π 4 a 2 (i 4 + µ 4 ) ω 2 ω ] sin ω i t L4 L 4 ω i (ωi 2 ω2 ) (25) with M(x, t) = EI 2 y(x, t) x 2 (26) ( π ) 2 ω i = a i4 + µ L 4 i =1, 2,... a 2 = EI m

12 158 P. Lou / Finit-mnt formua for cacuating th sctiona forcs Fig. 12. Tim historis for bnding momnt at th midpoint cross-sction of bam with th soutions givn by th proposd formua and by th convntiona formua with 10 mnts. Fig. 13. Tim historis for bnding momnt at th midpoint cross-sction of bam in Examp 4. µ 4 = ( ) 4 L k w π EI Th paramtrs in this study ar: E = N/m 2, I = m 4, m = 50 kg/m, L = 100 m, x 1 = 50.5 m, k w = N/m 2, P = 98,000 N, and ω =3 rad/s. Th tim historis for th bnding momnt at th midpoint cross-sction of bam ar shown in Fig. 13, whr th soid, th dottd and th dashd ins dnot th tim historis with th anaytica soution givn by Eq. (26) with i = , with th finit mnt soutions (with qua mnt ngth 1.0 m and tim intrva 0.01 s) givn by th rvisd proposd formua (11), i.., aftr dting th third trm of right hand sid in formua (11), and givn by th convntiona formua (1), rspctivy. It is vidnt that th tim historis with th finit mnt soution givn by th rvisd proposd formua (11) ar vry cos to thos with th anaytica soution givn by Eq. (26). Howvr, th diffrnc btwn th tim historis with th finit mnt soution givn by th convntiona formua (1) and thos with th anaytica soution givn by Eq. (26) is high. This is du to th shortcomings mntiond in Sction 1 of Introduction.

13 P. Lou / Finit-mnt formua for cacuating th sctiona forcs 159 Tab 2 Prsnt soutions (with 100 mnts of qua ngth) and anaytica soutions of maximum bnding momnts at th midpoint cross-sction of bam, and th ratios of th soution givn by th proposd formua to th corrsponding anaytica soution subjctd to a moving oad with various spds Spd (m/s) Prsnt soution (N-m) Anaytica soution (N-m) Ratio of prsnt soution to anaytica on Examp 5. A simpy supportd bam rsting on viscoastic foundation subjctd to a concntratd moving oad A simpy supportd uniform Brnoui-Eur bam rsting on continuousy viscoastic foundation subjctd to a concntratd moving oad P = 98,000 N with constant spd from th ft nd to th right nd of th bam is studid in this xamp. Th damping cofficint of foundation, c w, is dfind by a non-dimnsiona paramtr [6] (th ratio btwn actua damping and critica damping) givn by β = 2 m cw m/kw. β = 0.1 is adoptd in this xamp. Othr paramtrs for th bam and th foundation ar th sam as thos in Examp 4. Initiay, th bam is at rst, and th moving oad is at th ft nd of th bam. In th prsnt anaysis, 1000 qua tim stps for th cas of vry spd of th moving oad ar adoptd. Th maximum bnding momnts at th midpoint cross-sction of bam givn by th proposd formua (11) using 100 mnts with qua ngth undr th moving oad with various spds hav bn rportd in Tab 2, aong with th anaytica soutions givn by th foowing Eq. (27) takn from Fr ýba [9] M(α) = with α = v m/ei 2λ λ =(k w /4EI) 1/4 1 (1 α 2 ) 1/2 [ α2 β 2 /(1 α 2 ) 2 ] P Cas with α<1and β<< 1 (27) 4λ It is obsrvd from Tab 2 that th prsnt soution agrs w with th corrsponding anaytica on, and th diffrnc btwn th prsnt soution and th anaytica on is about 1 pr cnt. Th maximum bnding momnts at th midpoint cross-sction of bam givn by th proposd formua (11) with 40, 60, 80, 100, 120, 140, and 160 mnts and constant spd v = 50 m/s hav bn rportd in Tab 3, aong with thos givn by th convntiona formua (1) at th corrsponding tim whn th maximum bnding momnt is achivd by th proposd formua (11), i.., whn th moving oad is acting on th bam midpoint. In addition, th two rsuts hav bn pottd in Fig. 14. From Tab 3 and Fig. 14, on can obtain that th diffrnc btwn th soution givn from th proposd formua (11) and th anaytica soution givn by Eq. (27) is ow, for xamp, th diffrnc btwn th two rsuts is 4.8 pr cnt vn though 40 mnts (i.., mnt ngth of 2.5 m) hav bn usd for th bam; howvr, th diffrnc btwn th soution givn by th convntiona formua (1) and th anaytica soution givn by Eq. (27) is high, for xamp, th diffrncs btwn th two rsuts ar 68.1 and 9.8 pr cnt, rspctivy, whn 40 and 160 mnts hav bn usd for th bam. This is du to th shortcomings of th convntiona formua mntiond Sction 1 of Introduction. It shoud b pointd out that Eq. (27) is basd on th foowing assumption [9], i.., an infinit bam on a Winkr foundation is subjctd to a constant oad P moving from infinity to infinity at constant spd v. Th diffrntia quation of th vrtica vibration of th bam givn by rfrnc [9] is writtn as d 4 y(s) ds 4 with +4α 2 d2 y(s) ds 2 8αβ 1 dy(s) ds +4y(s) =8 δ(s) (28)

14 160 P. Lou / Finit-mnt formua for cacuating th sctiona forcs Tab 3 Maximum bnding momnts at th midpoint cross-sction of bam givn by th proposd formua and thos givn by th convntiona formua with various mnt numbrs subjctd to a moving oad with constant spd v = 50 m/s Grad of msh (mnts 40 mnts 60 mnts 80 mnts 100 mnts 120 mnts 140 mnts 160 mnts of qua ngth) Soution givn by th proposd formua (N-m) Soution givn by th convntiona formua (N-m) Anaytica soution is N-m. Fig. 14. Maximum bnding momnts at th midpoint cross-sction of bam givn by th proposd formua and thos givn by th convntiona formua with various mnt numbrs subjctd to a moving oad with constant spd v = 50 m/s. s = λ(x vt) β 1 = ω b m/kw δ(s)=δ(x)/λ whr ω b dnots circuar frquncy of damping of th bam, v is moving spd of th oad, and δ(x) is Dirac dta function. Th rasons why Eq. (27) can b usd as th anaytica soution of this xamp ar as foows. On th on hand, if β 1 in Eq. (28) is rpacd by β = 2 m cw m/kw, thn th modifid Eq. (28) is th diffrntia quation of th vrtica vibration of an infinit bam on a continuousy viscoastic foundation subjctd to a constant oad P moving from infinity to infinity at constant spd v. On th othr hand, th dispacmnts and sctiona forcs of th bam on astic foundation dcin to zro vry fast as th distanc from point oad incrass [21]. Thrfor, on may choos a bam with finit ngth to tak th pac of an infinit bam. 4. Concuding rmarks This papr has prsntd nw finit-mnt formua ovrcoming th shortcomings of th convntiona formua to cacuat th sctiona forcs at any cross-sction of a Brnoui-Eur bam on continuousy viscoastic foundation subjctd to concntratd moving oads. Th proposd formua can asiy dgnrat into th formua for cacuating th sctiona forcs of a simpy supportd or a continuous Brnoui-Eur bam subjctd to concntratd moving oads, and into th formua for cacuating th sctiona forcs of a Brnoui-Eur bam on Winkr foundation undr static oads. Fiv numrica xamps incuding static and dynamic anayss ar chosn to iustrat

15 P. Lou / Finit-mnt formua for cacuating th sctiona forcs 161 th appication of th proposd formua. Numrica rsuts show: (i) compard with th convntiona formua, th proposd formua can improv th cacuation accuracy of th sctiona forcs of bam; (ii) to cacuat th sctiona forcs at th cross-sction of th two nods of a bam mnt, on shoud us th proposd formua (11), not th convntiona formua (1), (2) or (3), (4); (iii) to cacuat th sctiona forcs at point within a bam mnt rathr than at a nod, on shoud us th proposd formua (13), (14), not th convntiona formua (1), (2) or (3), (4). Acknowdgmnt Th author woud ik to thank Prof. Dani J. Inman, Editor-in-Chif of Shock and Vibration, Dr. Sondipon Adhikari, Associat Editor of Shock and Vibration and two anonymous rviwrs for thir vauab and hpfu commnts. Appndix Th xprssion of th mnt consistnt mass matrix m with 4 4 ordr [2] usd in th Eq. (11) is m = m symm. 4 2 whr m is th mass pr unit ngth of bam. Th xprssion of th mnt stiffnss matrix k b with 4 4 ordr [2] usd in th Eq. (11) is 12/ 3 6/ 2 12/ 3 6/ 2 k b = EI 4/ 6/ 2 2/ 12/ 3 6/ 2 symm. 4/ (A1) (A2) Th xprssions of th foundation mnt stiffnss matrix k w and damping matrix c w usd in th Eq. (11) ar simiar in form to th mnt consistnt mass matrix m of Eq. (A-1). k w can b obtaind by simpy rpacing m by k w in th mnt consistnt mass matrix m. Simiary, c w can aso b obtaind by simpy rpacing m by c w in th mnt consistnt mass matrix m. Rfrncs [1] K.J. Bath and E.L. Wison, Numrica Mthods in Finit Emnt Anaysis, Prntic-Ha, Engwood Ciffs, Nw Jrsy, [2] R.W. Cough and J. Pnzin, Dynamics of Structurs, (2nd dition), McGraw-Hi, Nw York, [3] R.D. Cook, Concpts and Appications of Finit Emnt Anaysis, (2nd dition), Wiy, Nw York, [4] C.S. Dsai, Emntary Finit Emnt Mthod, Prntic-Ha, Inc., Engwood Ciffs, Nw Jrsy, 1979, [5] D.G. Duffy, Th rspons of an infinit rairoad track to a moving vibrating mass, Journa of Appid Mchanics, ASME 57(1) (1990), [6] C. Esvd, Modrn Raiway Track, (2nd dition), MRT-Productions, Zatbomm, ISBN , [7] Z. Fng and R.D. Cook, Bam mnt on two-paramtr astic foundations, Journa of Enginring Mchanics ASCE 109(6) (1983), [8] A.L. Fornc, Traving forc on a Timoshnko bam, Journa of Appid Mchanics ASME 32(2) (1965), [9] L. Frýba, Vibration of Soids and Structurs undr Moving Loads, (3rd dition), Acadmia, Pragu, [10] L. Frýba, S. Nakagiri and N. Yoshikawa, Stochastic finit mnts for a bam on a random foundation with uncrtain damping undr a moving forc, Journa of Sound and Vibration 163(1) (1993), [11] A. Garini, Vibrations of simp bam-ik modd bridg undr harmonic moving oads, Intrnationa Journa of Enginring Scinc 44(11 12) (2006), [12] J.T. Knny, Stady-stat vibrations of bam on astic foundation for moving oad, Journa of Appid Mchanics, ASME 21(4) (1954),

16 162 P. Lou / Finit-mnt formua for cacuating th sctiona forcs [13] T. Kocaturk and M. Simsk, Vibration of viscoastic bams subjctd to an ccntric comprssiv forc and a concntratd moving harmonic forc, Journa of Sound and Vibration 291(1 2) (2006), [14] P. Lou, G.L. Dai and Q.Y. Zng, Dynamic anaysis of a timoshnko bam subjctd to moving concntratd forcs using th finit mnt mthod, Shock and Vibration, in prss. [15] S.P. Pati, Natura frquncis of an infinit bam on a simp inrtia foundation mod, Intrnationa Journa of Soids and Structurs 23(12) (1987), [16] S.P. Pati, Rspons of infinit raiway track to vibrating mass, Journa of Enginring Mchanics, ASCE 114(4) (1988), [17] R.L. Sack, Structura Anaysis, McGraw-Hi Book Company, Nw York, [18] V.N. Shah, R.D. Cook and T.C. Huang, Loads moving on bam supportd by ayrd astic foundation, Journa of Mchanica Dsign, ASME 102(2) (1980), [19] D. Thambiratnam and Y. Zhug, Dynamic anaysis of bams on an astic foundation subjctd to moving oads, Journa of Sound and Vibration 198(2) (1996), [20] S. Timoshnko, Mthod of anaysis of statica and dynamica strsss in rais, Procdings of th Scond Intrnationa Congrss for Appid Mchanics, Zurich, Swiss, 1926, [21] S. Timoshnko, Strngth of Matrias Part II Advanc Thory and Probms, VanNostrand Rinhod Company Ltd, London, [22] S. Timoshnko, D.H. Young and W. Wavr, Jr., Vibration Probms in Enginring, (4th dition), Wiy, Nw York, [23] J.J. Wu, Dynamic anaysis of an incind bam du to moving oads, Journa of Sound and Vibration 288(1 2) (2005), [24] J.J. Wu, A.R. Whittakr and M.P. Cartm, Th us of finit mnt tchniqus for cacuating th dynamic rspons of structurs to moving oads, Computrs and Structurs 78(6) (2000), [25] J.S. Wu and P.Y. Shin, Dynamic rsponss of raiway and carriag undr th high-spd moving oads, Journa of Sound and Vibration 236(1) (2000), [26] J.S. Wu and P.Y. Shin, Th dynamic bhaviour of a finit raiway undr th high-spd mutip moving forcs by using finit mnt mthod, Communications in Numrica Mthods in Enginring 16(12) (2000),

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