A Simple Model for Axial Displacement in a Cylindrical Pipe With Internal Shock Loading

Transcription

1 A Simple Model for Axil Displcemen in Cylindricl Pipe Wih Inernl Shock Loding Nel P. Bier e-mil: Joseph E. Shepherd Grdue Aerospce Lborories, Cliforni Insiue of Technology, Psden, CA This pper describes simplified model for predicing he xil displcemen, sress, nd srin in pipes subjeced o inernl shock wves. This model involves he neglec of rdil nd rory ineri of he pipe, so is predicions represen he spilly verged or low-pss filered response of he ube. The simplified model is developed firs by pplicion of he physicl principles of conservion of mss nd momenum on ech side of he shock wve. This model is hen reproduced using he mhemicl heory of he Green s funcion, which llows oher lod nd boundry condiions o be more esily incorpored. Comprisons wih finie elemen simulions demonsre h he simple model dequely cpures he ube s xil moion, excep p ffiffiffiffiffiffiffiffi ner he criicl velociy corresponding o he br wve speed E=q. Ner his poin, he simplified model, despie being n unsedy model, predics ime-independen resonnce, while he finie elemen model predics resonnce h grows wih ime. [DOI: / ] Inroducion Exensive experimenl observions of he srucurl response of pipes nd ubes o inernl shock nd deonion wve loding hve been crried ou over he ls decde he Explosion Dynmics Lborory, s documened in Refs. [1] nd [2]. These observions consis primrily of poin mesuremens of srin, which hve been exremely useful for quniive chrcerizion of srucurl loding s well s poin vlidion of simulions. However, due o he lrge number of poenil vibrion modes nd he siffness of ypicl piping sysems, i is ofen difficul o infer he globl response, due o he confounding presence of high-frequency oscillions in srin signls. Finie-elemen simulions nd reduced order models re priculrly useful ools o use in conjuncion wih experimens o gin insigh ino lrgescle, lower frequency response modes. This noe presens n nlysis of he xil displcemen induced by shock loding nd proposes simplified bu quniively predicive model. Numericl Simulion Resuls One exmple for which we hve crried ou n nlysis is finie-elemen simulion (LS-Dyn) of shock wve wih pressure rise of 2.52 MP rveling 20 m/s in 2-m long secion of 2-in dimeer, schedule 40 sinless seel pipe (see Tble 1). This exmple is relevn o he evluion of he srucurl response of piping in he Hydrogen in Pipes nd Ancillry Vessels evluion crried ou for he wse remen pln he Hnford Sie [3,4]. Typicl resuls re shown in Figs. 1 nd 2 for ime of 225 ls, before ny wve reflecions occur from he RHS end of he pipe. The shock wve propges wih consn speed, sring from he Mnuscrip received April 22, 2013; finl mnuscrip received July 30, 2013; cceped mnuscrip posed Ocober 16, 2013; published online Ocober 16, Assoc. Edior: Weinong Chen. origin he LHS nd propging owrd he righ. The fluid dynmics ssocied wih he shock wve re no considered, s we re only ineresed in he srucurl response of he pipe o n idelized rveling lod. The boundry condiions re zero xil nd rdil displcemen ech end of he pipe. The pipe is modeled s liner elsic solid wih Young s modulus E ¼ P, densiy q ¼ 040 kg/m 3, nd Poisson s rio ¼ The inner rdius of he pipe is mm, nd he wll hickness is 3.91 mm. For he membrne model of moion discussed subsequenly, he effecive membrne rdius ¼ 2.2 mm nd he rio /h ¼ 7.2. The rdil displcemen w nd xil displcemen u propge s progressive wve sysems, s shown in Fig. 3. The leding edgepof ffiffiffiffiffiffiffiffi he wve is moving pproximely he br wve speed b ¼ E=q ¼ 4900 m/s, nd he wveform is n pproximely biliner displcemen field u(x, ) wih mximum mpliude u p moving he speed of he shock wve fron. The enire srucure is self-similr if he shock speed is consn nd pressure is uniform behind he shock. The mximum mpliude of he displcemen increses in mgniude pproximely linerly in ime u p ¼ _u p, where _u p < 0. Smll oscillions in xil displcemen re superposed on he min biliner profile. These re primrily consequence of he lrge oscillions in rdil displcemen shown in Fig. 1 nd he Poisson coupling beween rdil nd xil srin. Addiionl conribuions re due o he exciion of sher wves in he region jus hed of he shock fron nd propgion of xil disurbnces hed of he fron creed by he periodic moion behind he fron. The rdil displcemen w nd hoop srin e h ¼ w/ show very subsnil oscillions behind he shock fron nd much smller oscillions hed. The response behind he shock fron is periodic in spce nd lso in ime, wih n oscillion frequency of pproximely sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f h ¼ 1 E 2p qð1 2 Þ nd spil wvelengh of k h ¼ /f h. The oscillory moion is deermined by he ube ineri nd he symmeric rdil moion being opposed primrily by he hoop sress ssocied wih rdil displcemen, s discussed in Ref. [5]. Numericlly, his frequency is 29 khz nd he ssocied spil wvelengh behind he fron is 72 mm. The xil displcemen oscillions re much more ppren in xil srin (Fig. 2), due o he mplifying effecs of differeniion used o obin srin from displcemen, e x ¼ =@x. The evoluion of he xil displcemen wve wih ime cn be more clerly visulized if he profiles differen imes re overlid, shown in Fig. 4. This mehod of presenion shows h, for he cse of zero displcemen he pipe LHS, he displcemen field o he lef of he pressure wve fron is in sedy se. Tble 1 Dimensions, meril properies, nd lod condiions of schedule 40 sinless seel pipe Densiy q kg/m Young s modulus E GP 193 Poisson s rio 0.3 Br wve speed b m/s 4900 Shell dilionl wve speed d m/s 5190 Pressure wve speed m/s 20 Pressure lod P MP 2.52 Pipe inner rdius R i mm Pipe wll hickness R o mm 3.91 Men rdius mm 2.2 (1) Journl of Applied Mechnics Copyrigh VC 2014 by ASME MARCH 2014, Vol. 1 /

2 Fig. 1 Simulion resuls for rdil nd xil displcemen on he ouer surfce of he pipe 225 ls; he shock wve fron is loced 0.47 m nd he br wve fron is 1.1 m Membrne Model Moived by he simple form of he xil displcemen wves shown in Fig. 3, we seek model of he xil displcemen h focuses on xil wve moion nd simplifies or elimines he high-frequency oscillions h re prominen in he xil srin resuls. The firs sep is o pproxime he pipe s shell wihou ny sher deformion or rory ineri. This will elimine he high-frequency oscillions observed in he xil srin jus hed Fig. 3 Simulion resuls for rdil nd xil displcemen wve sysems shown incremens of 100 ls. For clriy, successive rces hve he zero vlues offse by n moun proporionl o he ime incremen. of he pressure fron in Fig. 2. The second sep is o consider he shell s membrne wihou ny bending siffness or rdil ineri o elimine rdil displcemen oscillions (i.e., he rdil moion is qusisedy in chrcer). This will elimine he hoop mode oscillions observed in he rdil displcemen bu will cree disconinuiies in rdil displcemen he shock fron s well s he shell end poins. Including bending siffness would smer hese disconinuiies over he chrcerisic bending lengh of he shell, so his needs o be kep in mind when inerpreing he model resuls. Fig. 2 Simulion resuls for hoop nd xil srins on he ouer surfce of he pipe 225 ls Fig. 4 Seleced xil displcemens from Fig. 3 shown wihou offseing he zero srin / Vol. 1, MARCH 2014 Trnscions of he ASME

3 In doping qusisedy membrne model of rdil moion, we re effecively ime-verging he rdil displcemen oscillions behind he fron, replcing hese by men response. This is resonble pproximion, s long s he ime scles of he rdil oscillion re much smller hn he xil wve propgion imes. In he exmple discussed bove, he period of rdil oscillion is 34 ls nd he ime for pressure wve o rvel he lengh of he pipe (2 m) is 95 ls. So pproximely 2 hoop oscillion cycles will occur in he ime i kes he pressure wve o rverse he lengh of he pipe. The xil moion of he ube is modeled using he following xil force blnce: 2 x where N x is he xil sress resuln, defined s he inegrl of xil sress hrough he wll hickness. Since we re neglecing he rdil dynmics nd xil bending of he pipe nd blncing he hoop sress gins he pressure lod, he Lme soluion [6] for hick-wlled pressure vessel cn be used o model he vriion of hoop nd rdil sresses hrough he wll hickness. This model is vlid only in regions of he pipe where he xil srin is uniform in boh he xil nd rdil direcions, bu due o our neglec of xil bending nd rory ineri, hese condiions re me everywhere excep ner he lod fron nd he boundries of he pipe. As resul, he use of he Lme soluion is consisen wih he ssumpions lredy mde in developing his simplified model. Noe h, for hin ubes, one cn lernively use he hin shell pproximions r r r h nd r h ¼ P=h, bu in he presen formulion, he hick-wll effecs cn be included wihou ddiionl difficuly. The Lme soluions for he hoop nd rdil sresses in long cylinder re he following: r h ðrþ ¼ PR2 i R 2 1 þ R2 o i R 2 o r 2 (3) r r ðrþ ¼ PR2 i R 2 1 R2 o i R 2 o r 2 (3b) where R i nd R o re he inner nd ouer rdii of he ube. From Hooke s lw, he xil sress is hen r x ¼ Ee x þ ðr r þ r h Þ ¼ Ee x þ 2PR2 (4) i R 2 o R2 i The xil sress resuln N x is found by inegring r x hrough he wll hickness. Since, in his cse, r x is uniform hrough he hickness, he resul is Upon insering Eq. (6) ino Eq. (2) nd replcing e x ¼ =@x, we obin p ffiffiffiffiffiffiffiffi where b is he br wve speed E=q. This resul is wve equion for he xil displcemen u, which is forced by he grdien of pressure. For hin ube, he correcion fcor C ends o 1.0 nd he quniy P= is he qusisic rdil displcemen of he ube from clssicl hin shell heory. Thus, he forcing erm cn lso be inerpreed s he grdien of he qusisic rdil displcemen. In he conex of rveling shock wves, he pressure disribuion is modeled s Pðx; Þ ¼P o Hð xþ () where H is he Heviside sep funcion nd is he speed of he shock wve. Under his lod, Eq. (7) becomes 2 2 ¼ C (7) P o dð xþ (9) where d(x) is he Dirc d funcion. In his cse, he source erm vnishes everywhere excep he shock fron x ¼ nd he xil wve moion of he ube is driven solely by he disconinuiy in pressure (which resuls in jump in hoop, xil, nd rdil sresses) he wvefron. Observions on he Numericl Simulion Resuls The numericl soluion exhibis liner growh in ime for boh he spil exen nd mpliude of he xil displcemen u. The poin of mximum displcemen mpliude u p moves wih he speed of he pressure wve nd s shown in Fig. 5 nd grows linerly wih ime wih speed of _u p ¼ 0:035m=s. The xil displcemen lso shows nerly liner vriion wih disnce (Fig. 4) in he segmens of he wve hed nd behind he pressure wve fron. We will herefore idelize he srin in hese segmens s ime-independen, wih disinc vlues hed of, e x,1, nd behind, e x,2, he wve fron. From he coninuiy of he xil displcemen he pressure wve fron, we cn inegre he srin hrough he wve o rele he wo srin mgniudes o he wve speeds nd obin he following relionship: e x;1 ¼ b e x;2 (10) In erms of _u p, he idelized srins cn be compued o be e x;1 ¼ _u p b (11) N x ¼ e x þ 2PhR2 i R 2 o R2 i (5) I my be noed h he wll hickness is h ¼ R o R i nd he men rdius is ¼ 0:5ðR o þ R i Þ. Using hese definiions, he xil sress resuln cn be expressed in he form N x ¼ e x þ P R2 i 2 (6) The quniy in prenheses is he correcion due o hick-wll effecs, which will herefer be denoed C. As he wll hickness is reduced, his fcor pproches 1.0 nd he expression h remins is he resul from clssicl hin shell heory. In his pper, he correcion erm mus be reined, since we re nlyzing ube wih inner rdius R i ¼ mm nd men rdius ¼ 2.2 mm, which gives correcion fcor of C ¼ Fig. 5 Exremum xil displcemen ( x 5 ) s funcion of ime showing he liner relionship Journl of Applied Mechnics MARCH 2014, Vol. 1 /

4 e x;2 ¼ _u p (12) Using Eq. (12), we cn wrie he displcemen field s uðx; Þ ¼ _u p x for 0 x (24) Soluion of he Model Equions Bsed on he resuls of he numericl simulions, we cn divide he soluion domin ino hree segmens: Region 0. b x L. Ahed of he br wve fron, here is no disurbnce nd he xil displcemen vnishes. Once he wve reches he RHS of he pipe, x ¼ L, refleced wve propging he lef is creed in order o sisfy he boundry condiion. This wve cn be redily compued using he mehod of chrcerisics [7] bu, in he ineres of breviy, will no be considered in his noe. Wih his limiion, he soluion is simply uðx; Þ ¼0 for b x L (13) Region 1. x b. Beween he pressure wve fron nd he br wve fron, he xil displcemen propges s righ fcing wve wih consn xil srin e x,1 nd sress resuln N x,1. The mgniude of he srin ws given bove by Eq. (11). Since he pressure is zero in his region, he xil sress resuln in Eq. (6) reduces o N x ¼ e x (14) which is consisen wih he se of unixil srin hed of he pressure wve fron. From he heory of chrcerisics for wve equions [7], he xil displcemen in his region hs he generl form u ¼ f ðx b Þ (15) From he ssumpion of consn xil srin in his region, we conclude h uðx; Þ ¼e x;1 ðx b Þ (16) which is consisen wih our previous observions. This implies ¼ e x;1 b ¼ e x;1 (1) Using Eq. (11), we cn wrie he displcemen field s uðx; Þ ¼ _u p ð b x bþ for x b (19) Region 2. 0 x. Beween he LHS (x ¼ 0) of he pipe nd he pressure wve fron x ¼, he displcemen is sedy (independen of ime) nd linerly proporionl o he disnce from he LHS of he ¼ 0 ¼ e x;2 (21) This cn be inegred o yield he complee soluion uðx; Þ ¼e x;2 x (22) The xil sress resuln in his region is N x;2 ¼ e x;2 þ PC (23) In order o complee he soluion, we need one more relionship o deermine he vlue of he srin in eiher region 1 or 2 or else he mgniude of _u p. An ddiionl relionship cn be obined by considering he xil force blnce in he shell on ech side of he pressure fron. In generl, he xil sress resulns on ech side will be differen in order o hve blnce of momenum he fron h correcly ccouns for he jump in meril velociy V ¼ =@ hrough he fron. The simples wy o consruc he blnce of momenum is o consider fixed conrol volume h encloses he shell. A ime, he lef boundry of he conrol volume is in region 2, jus behind he pressure fron, nd he righ boundry is well in fron of he br wve, compleely enclosing region 1. Recognizing h he velociy in region 2 is zero, he chnge in momenum in ime D will be he sum of he loss of momenum in he shell meril h is brough o res by he moion of he fron nd he gin of momenum due o he moion of he br wve. This hs o be blnced by he impulse pplied o he conrol volume by he sress resuln in region 2, since he sress resuln hed of region 1 vnishes. The resuling momenum blnce is N x;2 D ¼ qhð b ÞV 1 D (25) Subsiuing for he xil sress resuln nd he velociy, we hve e x;2 þ PC ¼ qhð b Þð e x;1 b Þ (26) Subsiuions of expressions for xil srins in Eqs. (11) nd (12) led o _u p ¼ b P b þ h E C (27) b P e x;2 ¼ b þ h E C (2) b P e x;1 ¼þ h E C (29) From Hooke s lw, he hoop srin is given by e h ¼ 1 ½ E r h ðr x þ r r ÞŠ (30) A he ouer surfce of he ube, r r is zero, nd in region 2, r h nd r x re evlued using Eqs. (3) nd (4). Using hese equions, he hoop srin he ouer surfce in region 2 is found o be e h;2 ¼ 1 2 P b þ C (31) In region 1, boh r r nd r h re zero, so he pplicion of Eq. (30) gives e h;1 ¼ 2 b P C (32) As expeced, e h,2 is inermedie beween he limiing cses of zero xil srin, e x,2 ¼ 0, nd zero xil sress, N x,2 ¼ 0, which give he limis P 1 2 C e h;2 P C (33) Subsiuing he prmeers lised in Tble 1, we obin he numericl vlues given in Tble 2. The corresponding LS-Dyn simulion resuls were nlyzed o exrc men vlues for / Vol. 1, MARCH 2014 Trnscions of he ASME

5 Tble 2 Comprison of model resuls wih he LS-Dyn soluion Propery LS-Dyn Model _u p (m/s) e x,1 (lsrin) e x,2 (lsrin) e h,1 (lsrin) e h,2 (lsrin) comprison o he model. Given he simplificions of he model, he resuls compre fvorbly wih he vlues compued by he rnsien finie-elemen soluion. Green s Funcion Soluion The exc soluion for he xil displcemens cn lso be found using he mhemicl echnique of he Green s funcion. To do so, Eq. (7) will be solved on semi-infinie inervl [0, 1] wih he boundry condiion u(0, ) ¼ 0 he origin. The Green s funcion gðx; x o ; ; o Þ is found by replcing he inhomogeneous erm in Eq. (7) by spil nd emporl Dirc d funcions, 2 2 ¼ dðx x oþdð o Þ (34) The Green s funcion sisfying his equion is given by [] gðx; x o ; ; o Þ¼ b 2 H ð bð o Þ jx x o jþ ( b jx x o j < b ð o Þ ¼ 2 0 jx x o j > b ð o Þ (35) where H is he Heviside funcion. Soluions o he inhomogeneous wve Eq. (7) re found by inegring he Green s funcion gins he source erm, ð ð uðx; Þ ¼ gðx; x o ; ; o ÞC Pðx o ; o Þdx o d o o 0 1 We re only ineresed in soluions on he inervl [0, 1], bu in order o evlue his inegrl, i is necessry o specify he pressure P(x o, o ) for x o < 0. To sisfy he boundry condiion u(0, ) ¼ 0, he pressure disribuion is mirrored symmericlly bou he origin, i.e., Pð x o ; o Þ¼Pðx o ; o Þ (37) Equion (39) cn be inerpreed s he inegrl of he pplied pressure P long he C þ nd C chrcerisics, where inegrion long hese phs is prmeerized by o. This siuion is shown schemiclly on n x digrm in Fig. 6. The righ hlf of he digrm depics he domin of ineres, while he lef hlf is he mirror imge used o sisfy he zero displcemen boundry condiion. There re wo wves in he region of ineres. The fser wve, rveling speed b, is he leding chrcerisic, nd in fron of his wve (region 0), no moion of he ube occurs. The slower wve, rveling speed, is he pplied pressure lod. Wihin he shded region on his plo, he pplied pressure is consn nd equl o P o, while ouside his region, he pressure is zero. In his digrm, we re seeking he soluion u pir of coordines (x, ). As represened in Eq. (39), he soluion his poin is found by inegring he pressure long he C þ nd C chrcerisics, shown s dshed lines in he figure. If he poin of ineres lies in region 2, s depiced by poin (x, ), hen boh he C þ nd C chrcerisics pss hrough he shded region, where he pressure is nonzero. On he oher hnd, for poin such s (x b, b ), lying in region 1, only he C þ chrcerisic psses hrough he shded region. As resul, he form of he soluion is differen in ech of hese regions. Region 2 To evlue Eq. (39) in region 2, we need only o inegre from 1 < o < long he C þ chrcerisic nd from 2 < o < long he C chrcerisic, where 1 nd 2 re he imes which he chrcerisics inersec he pressure wve fron. Thus, Eq. (39) becomes ð b ð P o d o P o d o (41) b ¼ C 2 P oð 1 2 Þ From he geomery of he problem, he imes 1 nd 2 re found o be 1 ¼ b x (42) þ b 2 ¼ x þ b þ b Thus, he displcemen in region 2 is given by (42b) Since he Green s funcion is only nonzero when jx x o j < b ð o Þ, he limis of he spil inegrl in Eq. (36) re replced by finie vlues nd he soluion cn be wrien s b 2 ð ð xþbð oþ 0 x bð Pðx o ; o Þdx o d o o which is redily inegred o obin b 2 ð 0 Pxþ ð b ð o Þ; o Þ (39) Px ð b ð o Þ; o Þd o The bove resul is vlid for n rbirry pressure disribuion. We now specilize o he cse of shock wve rveling velociy, for which he pressure is sep funcion of mpliude P o, Pðx o ; o Þ¼P o Hð o x o Þ (40) Fig. 6 Wve digrm for shock wve rveling long ube speed Journl of Applied Mechnics MARCH 2014, Vol. 1 /

6 uðx; Þ ¼ C P b o x ðx < Þ (43) þ b Region 1 The soluion in Region 1 is found using he sme pproch; however, only he C þ chrcerisic psses hrough he shded region in Fig. 6. As resul, Eq. (39) becomes b 2 P oð 4 3 Þ (44) Referring o Fig. 6, 3 is excly he sme s 1 bove nd 4 is given by 4 ¼ x b (45) b Hence, he displcemen in region 1 becomes P b o ðx b Þ ðx > Þ (46) A plo compring he xil displcemen prediced by his simple model wih finie elemen simulions (using condiions from Tble 1) is shown in Fig. 7. The simplified model cpures he xil moion remrkbly well, given he level of simplificion involved. Srins nd Sresses By differeniing he displcemens, he srins re found o be C P b o x < þ e x ¼ b (47) þc P b o x > These expressions re excly he sme s hose deermined in Eqs. (2) nd (29) using mss nd momenum blnces. Mking use of he definiion in Eq. (6), he xil sress resuln is found o be N x < x þ ¼ b (4) P o C b x > A similr nlysis cn be performed for he cse of pressure lod h is supersonic relive o he br wve velociy ( > b ), nd expressions for he xil displcemen u under hese condiions re provided in Tble 3. We hve lso reproduced hese resuls by king he semiinfinie Fourier sine rnsform of Eq. (7) nd solving he resuling emporl ordinry differenil equion. Upon invering he rnsform, he following form of soluion is obined: P 1 b o 2 2 ½ðjx þ b j jx b jþ b (49) b ðjx þ j jx jþš This resul is equivlen o Eqs. (43) nd (46) bu convenienly encpsules he displcemens for regions 0, 1, nd 2 s well s he subsonic ( < b ) nd supersonic ( > b ) cses in single expression. Zero-Sress Boundry Condiions In ddiion o he boundry condiion u(0, ) ¼ 0 used in he preceding secions, second prcicl boundry condiion is one of zero xil sress: N x (0, ) ¼ 0. This will occur if he pipe end is no fixed bu free o slide long he xil direcion. An exmple is when glnd sels re used o connec pipe secions. Referring o Ref. [6], his boundry condiion is equivlen o ¼ P x¼0 C (50) Since ny liner funcion of x nd sisfies he wve Eq. (7), iis possible o dd such funcion y(x, ) o he bove soluion u(x, ) in order o sisfy zero-sress boundry condiions rher hn zero-displcemen. The funcion h sisfies his propery is yðx; Þ ¼ C P b þ ðx b Þ (51) Afer dding his funcion o u(x, ) from Eqs. (43) nd (46), he resuling displcemen field for condiions of zero xil sress is he following: x b P o x < þ b uðx; Þ ¼ C 2 (52) ðx bþ x > The corresponding srin is e x ¼ C P o < 1 x < : 2 x > nd he xil sress resuln is N < 0 x < x ¼ 2 P o C : x > (53) (54) Once gin, similr expressions cn be obined for he cse of supersonic lod ( > b ), which re given in Tble 3. The xil displcemen profile for he sress-free boundry condiion is compred wih finie elemen resuls in Fig.. Once gin, excellen greemen beween he simple heory nd he finie elemen mehod (FEM) is observed. Fig. 7 Comprison of xil displcemens from finie elemen simulions nd he presen simple model. Boundry condiion is u(0, ) 5 0. Simulion condiions re hose summrized in Tble 1. Comprison Wih Finie Elemen Simulions Figure 9 compres he xil srin prediced by he simplified model wih he resuls of finie elemen simulions over wide rnge of lod speeds. The condiions oher hn lod speed re hose lised in Tble 1. As his figure shows, he simplified model / Vol. 1, MARCH 2014 Trnscions of he ASME

7 Tble 3 Lis of expressions for he xil displcemen for severl differen boundry condiions nd lod condiions. The displcemen is zero if x/ > mx(, b ). Lod cse Boundry condiion Displcemen b u(0) ¼ 0 bx P x < þ uðx; Þ ¼C b b ðx bþ x > b u(0) ¼ 0 bx P x < b þ b 2 2 ðx Þ x > b b b N x (0) ¼ 0 x þ b P x < þ b 2 ðx bþ x > b N x (0) ¼ 0 x þ 2 P x < b þ uðx; Þ ¼C b b ðx Þ x > b b cpures he spilly verged xil srins quie well. The srins in his plo re normlized by he men hoop srin e h ¼ P o C =, which is he sme for ll lod speeds. This scling demonsres he subsnil mplificion h occurs s he br velociy b is pproched. I is ineresing o noe h he simplified model, despie being n unsedy model, predics pek srins h do no vry wih ime. In conrs, he finie elemen model predics pek xil srins h grow wih ime re very close o 2/3 (which is in good greemen wih he heoreicl predicions of Ref. [9]). The simplified model does no cpure his emporl growh, since his effec involves he coupling beween rdil, rory, nd xil modes of moion, which re negleced in he simple heory. To enble more quniive comprison beween he predicions of he simplified model nd he finie elemen simulions, he simuled srin rces were verged on he inervls [0, ] nd [, b ], corresponding o he primry nd precursor wves. These verge srins re compred wih he predicions of he simple model in Fig. 10. In boh he precursor wve nd he primry wve, he xil srin predicions of he simplified model re in very good greemen wih he finie elemen simulions. Only ner he resonn poin / b ¼ 1 re significn differences observed. One reson for hese differences is he fc h he pek Fig. 9 Comprison of xil srin profiles for severl speeds of pressure lod. For he cse of / b 5 1.0, he model predics funcion of infinie heigh nd zero widh. The boundry condiion is N x (0, ) 5 0, nd srins re normlized by he sic hoop srin P o /. Fig. Comprison of xil displcemen from finie elemen simulions nd he presen simple model. Boundry condiion is N x (0, ) 5 0. Simulion condiions re summrized in Tble 1. Fig. 10 Averge xil srins in precursor nd primry wve regions. Symbols correspond o finie elemen simulions nd solid lines o he simple model. Boundry condiion is N x (0, ) 5 0. Journl of Applied Mechnics MARCH 2014, Vol. 1 /

8 srin prediced by he finie elemen model grows wih ime. The pek srin observed in he simulions is herefore limied by he finie lengh of he pipe, nd closer greemen wih he simplified model would probbly be obined if longer pipe were simuled. Conclusions A simplified model ws proposed for predicing he xil displcemen, sress, nd srin in pipes subjeced o inernl shock wves. This model ws derived using boh physicl rgumens from mss nd momenum conservion s well s direc soluion of he governing pril differenil equion of moion. Comprisons wih finie elemen simulions show h his model dequely cpures he low-frequency behvior over wide rnge of lod speeds, p wih speeds vrying from zero o greer hn he br speed b ¼ ffiffiffiffiffiffiffiffi E=q. When he lod speed pproches he br speed, he simple model predics ime-independen resonnce wih unbounded sresses nd srins. In conrs, he finie elemen model predics pek srins h grow wih 2/3. However, for lod speeds differing by more hn bou 10% from b, he predicions of he simple model re quie deque. Nomenclure ¼ men rdius of pipe 0:5ðR i þ R o Þ C ¼ correcion fcor ðr i =Þ 2 due o hick-wll effecs E ¼ elsic modulus of pipe g ¼ Green s funcion for he wve equion h ¼ pipe wll hickness N x ¼ xil sress resuln N h ¼ hoop sress resuln P ¼ pressure of shock wve R i ¼ inner rdius of pipe R o ¼ ouer rdius of pipe u ¼ xil displcemen of pipe u p ¼ mximum xil displcemen long pipe V ¼ xil velociy of pipe meril =@ w ¼ rdil displcemen of pipe e x ¼ xil membrne srin e h ¼ hoop srin H ¼ Heviside sep funcion ¼ Poisson s rio q ¼ pipe densiy ¼ speed of shock wve pffiffiffiffiffiffiffiffi b ¼ br wve speed E=q pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ dilionl shell wve speed E=qð1 2 Þ References [1] Belmn, W., Burscu, E., Shepherd, J., nd Zuhl, L., 1999, The Srucurl Response of Cylindricl Shells o Inernl Shock Loding, ASME J. Pressure Vessel Technol., 121(3), pp [2] Belmn, W., nd Shepherd, J., 2002, Liner Elsic Response of Tubes o Inernl Deonion Loding, J. Sound Vib., 252(4), pp [3] Shepherd, J. E., nd Akbr, R., 2009, Forces Due o Deonion Propgion in Bend, Grdue Aeronuicl Lborories Cliforni Insiue of Technology, Psden, CA, Technicl Repor No. FM [4] Shepherd, J. E., nd Akbr, R., 2009, Piping Sysem Response o Deonions. Resuls of ES1, TS1 nd SS1 Tesing, Grdue Aeronuicl Lborories Cliforni Insiue of Technology, Psden, CA, Technicl Repor No. FM [5] Krnesky, J., Dmzo, J. S., Chow-Yee, K., Rusinek, A., nd Shepherd, J. E., 2013, Plsic Deformion Due o Refleced Deonion, In. J. Solids Sruc., 50(1), pp [6] Timoshenko, S. P., 1934, Theory of Elsiciy, 1s ed., McGrw-Hill, New York. [7] Kolsky, H., 1963, Sress Wves in Solids, Dover, New York. [] Hbermn, R., 2004, Applied Pril Differenil Equions, 4h ed., Person Educion, Upper Sddle River, NJ. [9] Schiffner, K., nd Seele, C. R., 1971, Cylindricl Shell Wih n Axisymmeric Moving Lod, AIAA J., 9(1), pp / Vol. 1, MARCH 2014 Trnscions of he ASME