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A Similarity Law for Stressing Rapidly Heated Thin-Walled Cylinders①
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
661
AS imi l a r i t o rS t r e s s i ngRap i d l yLawf y He a t e dTh i n -Wa l l e dCy l i nd e r s ①❋
②
H.S.Ts i en② andC.M.Cheng③ ③
( Dan i e landFl o r e n c eGugge nhe imJe tPr opu l s i onCe n t e r,
Ca l ifo rn i aIn s t i t u t eof Te chno l ogy,Pa s ade na,Ca l if.)
Whenat h i nc l i nd r i c a ls he l lo fun i f o rmt h i ckne s si sv e r a i d l a t e dbyho th i r e s s u r eg a s y yr p yhe gh p
f l owi ngi n s i det hes he l l,t het emp e r a t u r eo fma t e r i a ld e c r e a s e ss t e e l r omah i emp e r a t u r ea tt he p yf ght
i n s i d es u r f a c et oamb i en tt emp e r a t u r e sa tt heou t s i d es u r f a c e.Young ’ smodu l u so fma t e r i a lt hu sva r i e s. Thepu r s eo ft hep r e s en tp a e ri st or e duc et hep r ob l em o fs t r e s sa na l s i so fs uchacy l i nd e rt oan po p y e i v a l en tp r ob l em i nc onven t i ona lc l i nd r i c a ls he l lwi t hou tt emp e r a t u r eg r a d i en ti nt he wa l l.The qu y e i v a l enc ec onc e ti sexp r e s s e da sas e r i e so fr e l a t i on sbe twe ent hequan t i t i e sf o rt heho tc l i nde rand qu p y t hequan t i t i e sf o rt hec o l dcy l i nd e r.The s er e l a t i on sg i v et hes imi l a r i t aw whe r e bys t r a i n sf o rt heho t yl c l i nd e rc a nbes imp l educ e df r om me a s u r eds t r a i n sont hec o l dc l i nd e randt hu sg r e a t l imp l i fyt he y yd y ys r ob l emo fexp e r imen t a ls t r e s sana l s i s. p y
Thec l i nde ro fas o l i dp r ope l l an tr o cke ti ss ub e c t edt ove r a i dhe a t i ngdu r i ngi t ss ho r t y j yr p hough t l l,a l lwa a c i r nd i l nc i h het st s o r c ona i t bu i r t s i ed r u t a r empe on.Thet i t a r fope ono i t a r du y , a r ox ima t e l hes amei neve r e c t i on i sno tl i ne a r.Th i sc ond i t i oni smo s ts eve r ea tt heend pp yt ys
o ft hec ombu s t i ono ft hep r ope l l an tg r a i n.F r omt hepo i n to fv i ewo fama t e r i a l seng i ne e r,t h i s , c a s ei sd i s t i ngu i s hedf r omo t he r sbyt het imer a t eo fhe a t i ng wh i chi ss ol a r ea st ono ta l l ow g
s uf f i c i en tt imef o ra r e c i a b l echang ei nt hes t r uc t u r eo ft hema t e r i a l.Thes t r eng t ho ft hewa l l pp ma t e r i a lunde rt h i sope r a t i ngc ond i t i oni squ i t ed i f f e r en tf r omt ha tunde rs l ow he a t i ng.Th i s []
f a c ti sc l e a r l onc l u s i ve l hownbyR.L.No l andi nar e c en tp a r1 .F r omt hepo i n to f yandc ys pe , v i ew o fas t r e s sana l s t t h er a t i o n a ld e s i n o fas o l i d r o e l l a n tr o c k e tc l i n d e ri st h u s y g p p y
c omp l i c a t edbyt heve r a r et he rma ls t r e s sandva r i a b l eYoung ’ smodu l u so ft h ema t e r i a la c r o s s yl g , t h ewa l la sar e s u l to ft h el a r et emp e r a t u r eg r a d i e n t.Fu r t h e rmo r ee xp e r ime n t a ls t r e s sd e t e rmi n a t i on g
und e ra c t u a lf i r i ngt e s t si sr a t h e rd i f f i c u l tdu et ot h es ho r tt e s tt imea v a i l a b l ea ndt h eh i emp e r a t u r e. ght
I ti sbe l i evedt ha tf o rt he s er e a s on st heon l a s e wh i ch ha sbe enana l edbyr e l i a b l e yc yz r a t i ona lme t hodi st hec a s eo fr o cke tc l i n d e ru n d e ru n i f o r mi n t e r n a l r e s s u r e .T h eb e n d i n y p g ❋R e c e i vedFeb r u a r y14,1952.
144 149,1952. o u r na lo h eAme r i c anRo c k e tSo c i e t l.22,pp. J ft y,vo Th i sp a e ri sb a s edon p a r to fat he s i ss ubmi t t edbyt hej un i o rau t ho rf o rp a r t i a lfu l f i l lmen to ft he ① p r equ i r emen t so fPh.D.i n Me ch an i c a lEng i ne e r i ng,Ca l i f o r n i aI n s t i t u t eo fTe chno l ogy. ② Ro be r tH.Godda r dPr o f e s s o ro fJ e tPr opu l s i on. ③
Gr adua t eAs s i s t an ti n Me chan i c a lEng i ne e r i ng.
662
COLLECTED WORKSOF HSUE-SHEN TS IEN
l eon r c.,a t ug s,e ngl i t s,moun e r u s o l oendenc sduet e s s e r t hes s,t e l z z edno t an oc sduet e s s e r t s y e s t ima t edbyve r ough me t hod s.Thepu r s eo ft h i sp a ri st oimp r ovet h i ss i t ua t i onby yr po pe s ugg e s t i ngana r o a ch wh i ch wi l lr educ et he g ene r a ls t r e s sp r ob l em o fho tc l i nde rt oa pp y r ob l em o fane i va l en tc o l dc l i nde r.Thee i va l en tp r ob l em c ant henbes o l vede i t he r p qu y qu ana l t i c a l l hec onven t i ona lme t hodo rd i r e c t l r imen t a ls t r e s sde t e rmi na t i on.I n y ybyt ybyexpe , e i t he rcho i c e t he p r ob l em i sbe l i evedt o be g r e a t l imp l i f i ed.Th i sl aw o fe i va l enc e ys qu be twe enho tc l i nde randc o l dc l i nde rc anbec a l l edt hes imi l a r i t aw. y y yl
S t r e s s e sandS t r a i n so faTh i n -Wa l l e dCy l i nd e r Thef a c tt ha tt het h i ckne s so ft hec l i nde ri ssma l li nc omp a r i s ont oi t sr a d i u sandl eng t h y a l l owsag r e a ts imp l i f i c a t i oni nt hes t r a i nana l s i s.Towi t,t hede f o rma t i ono feve r i n to f y ypo
t hecy l i nde rc anbede s c r i beds uf f i c i en t l c cu r a t e l hed i s l a c emen t so ft hepo i n t sona ya y byt p s i ng l es u r f a c ewi t h i nt hewa l lo ft hec l i nde r.Th i ss u r f a c ei st hes o c a l l edmed i ans u r f a c e.The y
s i t i ono ft hemed i ans u r f a c ei ss ode t e rmi nedt ha tabend i ngo ft hemed i ans u r f a c ewi l lno t po , i nduc ene tex t en s i ona lf o r c e si nt hep l aneo ft he med i ans u r f a c e a c r o s st het h i ckne s so ft he , wa l l.When Young ’ s modu l u si sac on s t an t,a si st hec a s ef o rac o l dc l i n d e r t h e m e d i a n y s u r f a c el i e smi dwa twe ent heou t e randt hei nne rbound a r u r f a c e so ft hec l i nde r.When ybe ys y , Young ’ smodu l u si sno tac on s t an tbu tde c r e a s e s wi t hi nc r e a s ei nt empe r a t u r e t he med i an
s u r f a c ei sd i s l a c edt owa r dt hec o l ds i de,a swi l lbes e enp r e s en t l p y. , , Le tx θ zbet hec oo r d i na t es s t em wi t ho r i i nont hec l i nd r i c a lmed i ans u r f a c es ucht ha t y g y
xpo i n t si nt hea x i a ld i r e c t i ono ft hec l i nde r, θi si nt hec i r cumf e r en t i a ld i r e c t i on,me a s u r edon y t hemed i ans u r f a c e,andzi sno rma lt ot he med i ans u r f a c e,po i n t i ngt owa r dt hea x i so ft he , , ( , ) cy l i nde r.Le tU V andW bed i s l a c emen t so fapo i n t x θ ont he med i ans u r f a c ei nt he p d i r e c t i on sx, θ, andz, r e s c t i ve l r et hu sfunc t i on so fxandθ ;bu tno to fz.Thent he pe y.Theya above -men t i onedfund amen t a ls imp l i f i c a t i ono ft h i ns he l l sc an bes t a t eda sf o l l ows:i ft he d i r e c ts t r a i n si nt hexandθd i r e c t i on sa r eex andeθ,andt hes he a rs t r a i nγxθ , t hen
ex =
췍2W 췍U -z 2 췍x 췍x
ïü ï ï 2 ï 1췍V W z æ췍V 췍 W ö + 2 ÷ eθ = ý - - 2ç R 췍θ R R è췍θ 췍θ ø ï ï 2 췍V 1췍U z æ 췍 W 췍V ö ï ÷ï γxθ = + + -2 ç 췍x R 췍θ R è췍x췍θ 췍x ø þ
( 1)
whe r eRi st her a d i u so ft hemed i anc l i nd r i c a ls u r f a c e,o rt he “ r a d i u so ft hec l i nde r.”Th i s y y r e s u l ti ss ome t ime sde s c r i beda st heKi r chho f fbend i nga s s ump t i on:P l aneno rma lt ot hemed i an
s u r f a c ebe f o r ebend i ngr ema i n ss oa f t e rbend i ng.
Thes i i f i c an ts t r e s s e si nat h i ns he l la r et hed i r e c ts t r e s s e sσx andσθi nxandθ,andt he gn
s he a rs t r e s sτxθ .Al lo t he rs t r e s s e sa r esma l li nc omp a r i s ont ot he s et h r e e.Nowl e tT bet he , t empe r a t u r eo fwa l la bovear e f e r enc et empe r a t u r es a her oomt empe r a t u r e.Ti sa s s umedt o yt
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
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r nde i l hec ft ngo i t a hehe st fxandθ.Thu to tno ez,bu t na i d r so s ckne i h ft ono i t l beon y yafunc i sa s s umedt obeun i f o rmove rt heen t i r es u r f a c eo ft hec l i nde r.Th i si sg ene r a l l r l o s e l y yve yc y , a r ox ima t edi nr e a l i t fαi st hec o e f f i c i en to ft he rma lexp an s i on t het he rma lexp an s i on pp y.I s t r a i ni st henαT .By Hooke ’ sl awoneha s
1 ü ( σx -ν σθ)+αT ï ï E ï ï 1 eθ = ( σθ -ν σx )+αT ý E ï ï ( ) 2 1+ν ïï τxθ γxθ = þ E
ex =
( 2)
whe r eE i s Young ’ s modu l u s,and v i s Po i s s on ’ sr a t i o.E i s,o fc ou r s e,afunc t i on o f t empe r a t u r eo rafunc t i ono fz.v,howeve r,wi l lbea s s umedt obec on s t an tf o rl a cko fde f i n i t e
i nf o rma t i on.Bys o l v i ngf o rt hes t r e s s e s,oneob t a i n sf r om Equa t i on ( 2),
E( z) ü {( ex +ν eθ)- ( 1+ν) αT ( z)} ï ï 1-ν2 ï ï E( z) ( ) ) ( ) ( { } σθ = e ν e 1+ ν α T z + ý θ x 2 1-ν ï ï E( z) ïï γxθ τxθ = þ 2( 1+ν) σx =
( 3)
Fo rt h i n -wa l l edc l i nde r s,t hee i l i b r i ume t i on sa r eexp r e s s edi nt e rmso f“ s e c t i ona l y qu qua ave r a s”o ft hes t r e s s e sg i veni nEqua t i on ( 3).Tha ti s,ones ak so ft heno rma lf o r c e sNxand ge pe , , , , Nθ t hes he a r i ngf o r c eNxθ t hebend i ng momen t sMx Mθ andt hetwi s t i ng momen tMxθ . Theya r er e l a t edt oσx ,σθ,andτxθ byt hef o l l owi nge t i on s qua
∫ ∫ ∫ σzd z,M =- τ zd =- σzd ∫ z,M =-∫ ∫ z Nx = σxd z,Nθ = σθd z,Nxθ = τxθd z
Mx
x
θ
θ
xθ
xθ
( 4) ( 5)
Thei n t e r a l si nt hea bovee t i on sa l lex t enda c r o s st het h i ckne s so ft hewa l l.Bys ub s t i t u t i ng g qua Equa t i on s( 1)and ( 3)i n t oEqua t i on s( 4)and ( 5),oneha sf o rexamp l e
Wö æ췍U ν 췍V + -ν ÷Nx = D0 ç Rø è췍x R 췍θ ν 췍V ν 췍2W ö æ췍2W ÷- NT + D1 ç 2 + 2 è췍x R 췍θ R2 췍θ2 ø and
Wö æ췍U ν 췍V + -ν ÷- Mx = D1 ç Rø è췍x R 췍θ ν 췍V ν 췍2W ö æ췍2W ÷- MT + D2 ç 2 + 2 è췍x R 췍θ R2 췍θ2 ø
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COLLECTED WORKSOF HSUE-SHEN TS IEN
e r whe
∫
1 ü E( z) d z ï ï 1-ν2 ý 1 ï ( ) E z zd zï D1 = þ 1-ν2
( 6)
1 E( z) z2d z 1-ν2
( 7)
D0 =
D2 = and
∫
∫
α E( z) T( z) d z 1-ν
( 8)
α E( z) T( z) zd z 1-ν
( 9)
NT = MT =
∫
∫
Thei n t e r a l sa a i nex t enda c r o s st het h i ckne s so ft he wa l l.I ti sev i den tf r om t hea bove g g exp r e s s i on sf o rNx and Mx t ha tc on s i de r a b l es imp l i f i c a t i onc anbea ch i evedbychoo s i ngt he med i ans u r f a c ei ns uchawa ha t yt
D1 =
∫
1 E( z) zd z =0 1-ν2
( 10)
Th i si sa c t ua l l hec ond i t i ont ol o c a t et hemed i ans u r f a c e.S i nc eYoung ’ smodu l u sE de c r e a s e s yt , ( ) wi t hi nc r e a s ei nt empe r a t u r ei ti ss e enf r omEqua t i on 10 t ha tt hemed i ans u r f a c ei sne a r e rt o
t hec o l dbound a r u r f a c et hant ot heho tbound a r u r f a c e.Fo rar o cke tc l i nde r,ho ti n s i de ys ys y bu tc o l dou t s i de,t he med i ans u r f a c ei sne a rt ot heou t s i des u r f a c e.Wi t ht h i scho i c eo ft he med i ans u r f a c e,t hef o r c e sandt hemomen t sa r er e l a t edt ot hed i s l a c emen t sbyt hef o l l owi ng p : s imp l e re u a t i o n s q
Wö æ췍U ν 췍V ü + -ν ÷- NT ï Nx = D0 ç ï Rø è췍x R 췍θ ï ï 췍U ö æ 1췍V W ÷- NT ý - +ν Nθ = D0 ç 췍x ø è R 췍θ R ï ï 1-ν æ췍V 1췍U ö ïï ÷ Nxθ = + D0 ç þ è췍x R 췍θ ø 2
( 11)
ν 췍2W ν 췍V ö æ췍2W ü ÷+ MT ï + Mx = D2 ç 2 + 2 è췍x ï R 췍θ2 R2 췍θ ø ï ï 1 췍V 췍2W ö æ 1 췍2W ÷+ MT ý ν + + Mθ = D2 ç 2 èR 췍θ2 R2 췍θ 췍x2 ø ï ï 2 1 췍V ö æ 1 췍W ïï ÷ + 2 Mxθ = ( 1-ν) D2 ç 2 þ èR 췍x췍θ R 췍x ø
( 12)
I ti swo r t h wh i l et opo i n tou ttwof a c t s:F i r s t l hecho i c eo fr e f e r enc et empe r a t u r ei s y,t , i t ea r b i t r a r i ngt her e f e r enc et empe r a t u r e wi l lchang et heva l ueo fNT Equa t i on qu y.Chang
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
665
( sNx ,Nθ e c r o lf rma heno vet a e ll l nwi i a r t ls rma heno nt ti tmen s d u nga i s e r r o tac 8).Bu j pond i venbyEqua t i on ( 11)unchang ed.The r e f o r et hephy s i c a lp r ob l emi squ i t ei nde to ft he g penden
cho i c eo fr e f e r enc et empe r a t u r e.MT i si nde to ft her e f e r enc et empe r a t u r e duet o penden Equa t i on ( 10).Se c ond l hep r e s en ts s t emo fe t i on sr educ e st ot ho s eo ft hec onven t i ona l y,t y qua
t h i ns he l lt he o r ft empe r a t u r eg r a d i en ti sa b s en t,o ri fYoung ’ smodu l u si si nde to ft he yi penden
t empe r a t u r eo f ma t e r i a l.I nt ha tc a s et he med i ans u r f a c ei s mi dwa twe ent hebound a r y be y , s u r f a c e s andD2i ss imp l heu s ua lf l exu r a lr i i d i t ft hes he l l. yt g yo
Nond imen s i ona lQuan t i t i e sandEqua t i on so fEqu i l i b r i um I ti su s e f u lt oi n t r odu c enond ime n s i on a lqu a n t i t i e sbyt a k i ngRa st h er e f e r e n c el e ng t h.Thu s
ξ = x/R u = U/R, v = V/R, w = W/R nξ = Nx/D0 , nθ = Nθ/D0 , nξθ = Nxθ/D0 , nT = NT/D0
( 13) ( 14)
mξ = MxR/D2 , mθ = MθR/D2 , mξθ = MxθR/D2 , MT = MTR/D2
( 16)
췍v 췍v 췍u 췍u ü +v -nT ï -vw -nT , nθ = -w +v ï 췍θ 췍θ 췍ξ 췍ξ ý uö ï 1-væ췍v 췍 ç ÷ nξθ = + ï þ 2 è췍ξ 췍θ ø
( 17)
췍2w 췍v 췍2w 췍2w 췍v 췍2w ü +v 2 +v +mT , mθ = 2 + +v 2 +mT ï 2 췍θ 췍θ ï 췍θ 췍ξ 췍θ 췍ξ ý ï æ췍2w 췍v ö ÷ mξθ = ( + 1-v)ç ï þ è췍ξ췍θ 췍ξ ø
( 18)
( 15)
and The r e f o r eEqua t i on s( 11)and ( 12)be c omenow
nξ =
and
mξ =
Th ee a t i on so fe i l i b r i umi nt e rmso ff o r c e sa nd mome n t sa r eh e r ee x a c t l h es amea st h e qu qu yt []
2 c onv e n t i on a lt h e o r eon l nnov a t i oni st o wr i t et h em i nnond ime n s i on a lf o rm.Fo rt h i s y .Th yi , , , u r o s e o n eh a s t od e f i n e t h en o n d i m e n s i o n a l u a n t i t i e s a n d o f t h ed i m e n s i o n a l s e c t i o n a l θ q q p p p q ξ , , : s h e a r i ngf o r c e sQx Qθa ndno rma lp r e s s u r el o a d i ngP a a i n s tzd i r e c t i ona sf o l l ows g
qξ = Qx/D0 qθ = Qθ/D0 p = PR/D0
( 19)
Thent hee i l i b r i ume t i on so ff o r c e sa r e qu qua
췍 nξ 췍 nξθ æ췍v 췍2w ö æ 췍2v 췍w ö 췍2w 췍2v ÷-nξθ ç ÷ =0 + n + -qξ 2 -qθ ç θ è췍ξ 췍ξ췍θ ø è췍ξ췍θ 췍ξ ø 췍ξ 췍θ 췍ξ 췍ξ2
ïü ï ï ï 췍v 췍2w ö æ 췍 nξθ 췍 nθ æ췍v 췍2w ö æ 췍2v 췍w ö 췍2v ÷-q ÷ =0ý ( + 2 ÷+nξ 2 +nξθ ç + 20) + -qξ ç θ ç1+ 췍θ 췍θ ø è췍ξ 췍ξ췍θ ø è췍ξ췍θ 췍ξ ø è 췍ξ 췍θ 췍ξ ï ï 췍v 췍2w ö æ 췍 æ췍v 췍2w ö 췍2w qξ 췍 qθ ï ÷+nξ ç1+ ÷ =p + nξθ ç + n + +2 + θ ï þ 췍θ 췍θ2 ø è췍ξ 췍ξ췍θ ø è 췍ξ 췍θ 췍ξ2
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COLLECTED WORKSOF HSUE-SHEN TS IEN
e r sa t fmomen umo i r b i l i hee rt o sf on i t Thee qu qua
췍mξθ 췍mθ æ 췍2v 췍w ö 췍2v ü ÷+β + +mξ 2 +mξθ ç qθ =0 ï è췍ξ췍θ 췍ξ ø ï 췍ξ 췍θ 췍ξ ý 췍mξ 췍mξθ æ 췍2v 췍w ö ï 췍2v ÷-β +mξθ 2 +mθ ç qξ = 0ï þ è췍ξ췍θ 췍ξ ø 췍θ 췍ξ 췍ξ
( 21)
2 β = R D0/D2
( 22)
whe r e
2 ,bt st hu saquan t i t ft heo r de ro f( R/ b) het h i ckne s so ft he wa l l.The r ea r ee l even yo βi i nd i v i dua le t i on si nt hes s t emo fEqua t i on s( 17),( 18),( 20),and ( 21).Wi t hs c i f i ed qua y pe , , , , , , , , , , , l o ad i ngp t he r ea r ea l s oe l evenunknown s u v w nξ nθ nξθ mξ mθ mξθ qξandqθ.The
s s t emo fe t i on si st hu sc omp l e t e. y qua
I n f i n i t eCy l i nd e rUnd e rUn i f o rmI n t e rna lPr e s s u r e Thes imp l e s ts c i a lc a s ei nt hep r e s en tg ene r a lp r ob l emi st hec a s eo fve r ongc l i nde r pe yl y unde run i f o rmi n t e r na lp r e s s u r e.I ft her o cke tc l i nde ri sl ongi nc omp a r i s onwi t hi t sd i ame t e r, y
t hea c t ua ls t r e s ss s t em du r i ngope r a t i oni sa r ox ima t edbyt h i si de a l i z eds imp l ec a s e.I nt h i s y pp r ob l emo fi nf i n i t e l ongun i f o rml o a dedc l i nde r,t hef o r c e snξ,nθ,andmomen t smξ,mθ p yl yl y
a r ec on s t an t si nde to fξandθ.Thes he a r i ngf o r c enξθandt hetwi s t i ngmomen tmξθvan i s h. penden
췍u sac on s t an t,s a vvan i s he s.wi sac on s t an t,andi sne a t i vei n ui sp r opo r t i ona lt oξo r i yk1 . g 췍ξ t hep r e s en tc oo r d i na t es s t em,s a t i on s( 17),( 18),and ( 20)g i ve y y-k2 .ThenEqua nξ0 =k1 +νk2 -nT ü ïï nθ0 =νk1 +k2 -nT ý ï þ mξ0 = mθ0 = mT
( 23)
whe r et hes upe r s c r i t0deno t et hequan t i t i e si nt hep r e s en ts imp l es t r e s ss s t em.Whent he p y , ( ) t empe r a t u r ed i s t r i bu t i onandt he ma t e r i a lp r ope r t i e sa r es e c i f i e d E u a t i o n 2 3 i v e st he p q g s t r a i n sk1andk2i nt e rmso ft hei n t e r na lp r e s s u r ep,andt hea x i a ll o a dnξ.I ft hea x i a ll o a di s r oduc edbyt hes amei n t e r na lp r e s s u r e,t heni tc ane a s i l hownt ha t p ybes
nξ0 = p0/2
( 24)
I ti so fi n t e r e s tt ono t et ha tt hebend i ng momen tmξandmθa r ee lt omT anda r ei nde t qua penden o ft hec ond i t i on so fl o a d i ng. Bys o l v i ngEqua t i on ( 23)f o rk1andk2,oneha s 1 1 ü 0 ( nT ï nξ0 -ν p )+ 1+ν ï 1-ν2 ý 1 1 ï 0 0 ( ) k2 = n ν n + T p ξ ï 1+ν þ 1-ν2
k1 =
( 25)
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
667
he st ve i 25)g on ( i t henEqua l,t a i r e t he ma ft no i a r t imums x he ma st oni i t i ond i s hede ft I gnc c r i t e r i ond i r e c t l r omt hep r e s s u r eandt empe r a t u r el o a d i ng. yf
L i n e a r i z e dTh e o r o rGen e r a lS e c onda r i ng yf yLoad Ass t a t ed i nt he p r ev i ou ss e c t i on,t he a c t ua ls t r e s ss s t em i n ar o cke t chambe ri s y
a r ox ima t e l ha to fi nf i n i t e l ongc l i nde runde run i f o rm i n t e r na lp r e s s u r e.Th i ss t r e s s pp yt yl y s s t emc anbec a l l edt hep r ima r t r e s s e s.Thedev i a t i on sf r omt hep r ima r t r e s ss s t em a r e y ys ys y r e s u l t so fbend i ng duet oc an t ed no z z l e,t oendenc l o s u r e s,t o moun t i ngl ug s,e t c.The s e add i t i ona ls t r e s s e so rt hes e c ond a r t r e s s e sa r e,howeve r,on l r a c t i on o ft he p r ima r ys y af y
s t r e s s e s.The r e f o r ei ti sj u s t i f i edt oc on s i de rt hes e c ondo r de rt e rmso fa dd i t i ona ls t r e s s e sand de f o rma t i on sa ssma l li nc omp a r i s ont ot hef i r s to r de rt e rms,andt hu sne l i i b l e . I no t he r gg wo r d s,
u =k1ξ +u ′,
v =v ′,
w =-k2 +w′ üï ï , nθ = p +n nξ = n +n ′ ′ nξθ = n ′ θ, θ ξ ξ ï , mθ = mT +m′ mξ = mT +m′ ′ ý θ , mξ θ =m θ ξ ξ ï ï ï 0 , , þ ′ ′ ′ = = + qξ =q q q p p p θ θ ξ 0 ξ
and
0
( 26)
whe r ek1 andk2 a r eg i venby Equa t i on ( 25).Thep r imedquan t i t i e sa r et hent hes e c ond a r y de f o rma t i on sandt hes e c ond a r t r e s s e s,t heya r ec on s i de r edt obesma l li nc omp a r i s ont ot he ys
r ima r f o rma t i on sands t r e s s e s.F r om Equa t i on s( 17)and ( 18),oneha st hef o l l owi ng p yde , r e l a t i on sbe twe ent hede f o rma t i on sandt hes t r e s s e s
u′ ö üï 췍v′ 췍v′ 1-νæ췍v′ 췍 췍u′ 췍u′ ,n ,n ç ÷ + ′ ′ -νw′ +ν -w′ + θ= θ =ν ξ 췍θ ø ï 췍ξ 췍θ 2 è 췍ξ 췍ξ 췍ξ ï and ï ï 췍2w′ 췍2w′ 췍v′ , ï m′ ν ν + + ξ= 췍θ ý 췍ξ2 췍θ2 ï 2 췍 w′ 췍2w′ 췍v′ ï , m′ + + θ =ν ï 췍θ 췍θ2 췍ξ2 ï æ췍2w′ 췍v′ ö ï ( ) ÷ + m′ θ = 1-ν ç ξ ï è췍ξ췍θ 췍ξ ø þ n ′ ξ=
( 27)
Bys ub s t i t u t i ng Equa t i on ( 26)i n t ot hee i l i b r i um e t i on s,Equa t i on s( 20),and qu qua d r opp i ngou tt hes e c ondo r de rt e rmso ft hep r imedquan t i t i e s,as s t emo fl i ne a r i z ede t i on s y qua
i sob t a i ned.Th i ss s t emc anbefu r t he rr educ edbys ub s t i t u t i ngt heq ′ ndq ′ b t a i nedf r omt he θo y ξa
l a s ttwoe t i on so fbend i ng momen te i l i b r i umi n t ot het h i r de t i on.Thef i na lr e s u l ti s qua qu qua : t hef o l l owi ngs s t e m o ft h r e ee u a t i o n s T h ef i r s ti st h ee u i l i b r i u m o ff o r c e si nt h ea x i a l y q q d i r e c t i on;t hes e c ondi st hee i l i b r i umo ff o r c e si nt hec i r cumf e r en t i a ld i r e c t i on;andt het h i r d qu : i st heequ i l i b r i umo ff o r c e si nt her a d i a ld i r e c t i on
668
COLLECTED WORKSOF HSUE-SHEN TS IEN
′ 1+ν췍2v ′ 췍w′ ′ 1-ν췍2u 췍2u =0 + + -ν 2 2 췍ξ 2 췍θ 2 췍ξ췍θ 췍ξ
ïü ï ï 췍2v ′ 1-ν췍2v ′ 1+ν췍2u ′ 췍w′ ï =0 + + ï 2 췍ξ2 2 췍ξ췍θ 췍ξ 췍θ2 ý ï and ï 2 ï æ 0 mT ö췍2w′ 췍v′ö æ 췍w′ 췍4w′ 췍2w′ 췍 w ′ 췍4w′ 0 ÷ =-β ÷ ï -w′+ +2 2 2 + 4 -βçν +β p p′+βçnξ 4 2 췍 θø è 췍 è θ 췍 췍 θ 췍 θ2 þ 췍 ξ β ø췍 ξ췍 ξ ξ
( 28)
The s ee t i on sha vebe ens imp l i f i edont heb a s i st ha tβi sal a r equan t i t ft heo r de ro ft he qua g yo s u a r eo f t h e r a d i u s t h i c k n e s s r a t i o . q I nt hel a t t e re t i on,p ′i st hes e c ond a r o a dimpo s edont hec l i nde rexp r e s s eda sa qua yl y , d i s t r i bu t edp r e s s u r eove rt hes u r f a c eo ft hec l i nde rd i r e c t edr a d i a l l twa r d.I ft hel o a di sa y you
c onc en t r a t edl o a d,t heni tha st obeexp andedi n t oap r oduc to fFou r i e rs e r i e sandFou r i e r []
i n t e r r a la sdonebyS.W.Yuan3 i nh i st r e a tmen to fc onc en t r a t edl o a donac o l dc l i nd r i c a l g y ( ) s he l l.Ot he rt e s o f l o a d s c a nb e s i m i l a r l e x a n d e d . T h e nE u a t i o n 2 8 i s a s s t e mo f t h r e e yp y p q y equa t i on sf o rt h r e eunknown su′,v′ ,andw′.Thef o r c e sand momen t sa r er e l a t edt ot he s e ( ) d i s l a c emen t sbyEqua t i on 27 . p Thep r ob l emo fg ene r a ls e c ond a r o a d i nga sexp r e s s edbyEqua t i on s( 27)and ( 28)i sve r yl y
s imi l a rt ot hep r ob l emo fg ene r a ll o a d i ngonac o l dc l i nd r i c a ls he l l,andc ant hu sbet r e a t edby y t heknown me t hod sdeve l opedf o rt h i sc onven t i ona lp r ob l em.I nf a c t,t heon l d i f f e r enc e y
be twe ent heho tc l i nde randt hec o l dc l i nde ri st hea a r anc eo ft het e rm mT i nEqua t i on y y ppe ( ) , : 28 .Howeve r event h i sd i f f e r enc ei st r i v i a l Ther e a s oni st heve r a r ema i t udeo fβa s yl g gn
s hownbyEqua t i on s( 6),( 7),and ( 22).I nf a c t, i f,a si sg ene r a l l hec a s e,NTi so ft hes ame yt
0 , t sa fmT/ oo i t a her tt ha howt ss on i t ede t i c bove hea hent t sNx udea t i fma ro de r o qua gn βandnξi mo s to ft heo r de ro fb/R,whe r ebi st het h i ckne s so ft hes he l l.S i nc et hes he l li sc on s i de r edt o bet h i n,o rb/R ≪ 1,t het e rmsi nvo l v i ng mT i n Equa t i on ( 28)c an bed r opped wi t hou t 0
imp a i r i ngt hea c cu r a c ft he p r e s en tt he o r h i si sdone,t heni nt hes s t em o f yo y. Whent y equa t i on sg i ven by Equa t i on s( 27)and ( 28),t hee f f e c t so ft he rma ls t r e s s e sand va r i a b l e Young ’ smodu l u sa r eno texp l i c i t.Asf a ra st henond imen s i ona le t i on sa r ec onc e r ned,t he qua , r o b l e m o fh o t c l i n d e r i s i d e n t i c a l t o t h e r o b l e mo f c o l dc l i n d e r a n d t h eb a s i c e u a t i o n s a r e p y p y q nowe s s en t i a l l hes amea st ha ta dop t edbyL.H.Donne l lf o rh i ss t udyo ft hes t a b i l i t ft h i n yt yo []
cy l i nd r i c a ls he l l s4 .Th i si st heb a s i so ft hes imi l a r i t awd i s cu s s edi nt hef o l l owi ngs e c t i on. yl
S imi l a r i t o rGen e r a lLoad i ng yLawf I ft hep r ob l emo fs e c ond a r o a d i ngi st obes o l vedana l t i c a l l her e s u l t so ft hep r ev i ou s yl y y,t
s e c t i ons how t ha ti tc an ber educ edt oane i va l en tp r ob l em o fc o l dc l i nde rands o l ved qu y , a c c o r d i ng l v e r amo r eu s e f u la l i c a t i ono ft h ee i v a l e n c ec on c e tl i e si nt hepo s s i b i l i t f y.Howe pp qu p yo expe r imen t a l l t e rmi n i ngt hes t r e s sandt hes t r a i nont hee i va l en tc o l dc l i nde randt hen yde qu y
u s i ngt hes imi l a r i t awt ode t e rmi net hes t r e s sandt hes t r a i ni nt heho tc l i nde r.The r ea r e yl y
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
669
d l o sonac t imen r a)Theexpe ch:( a o la r a t imen r expe emi ss i h ft so a e t dvan l n i ma pp g ytwoa cy l i nde rc anbedone mo r ee a s i l r ea c cu r a t e l hanpo s s i b l eonaho tc l i nde r. yand much mo yt y Thet e s tpe r i odc anbea sl onga sde s i r edandno tl imi t edt ot hes ho r tbu r n i ngt imeo ft he r o cke t.( b)Thes t r e s s e si nduc edby moun t i ngl ug s,e t c.,a r eve r d i f f i c u l t t oa r o x i m a t eb y pp y s imp l el o a ds s t emsamena b l et ot he o r e t i c a lc a l cu l a t i on s.Fo ri n s t anc e,t hel o a d sf r om a y
moun t i ngl uga r eno tr e a l l onc en t r a t edf o r c eandac onc en t r a t edmomen t.Tot aket hema sa yac c onc en t r a t edf o r c eandac onc en t r a t ed momen twou l dg r o s s l o v e r e s t i m a t et h ea c t u a ls t r e s s. y Suchd i f f i cu l t i e sd i s a a ri ft hel o a d i ngi sdoneexpe r imen t a l l ppe y. , Wi t hs uchexpe r imen t a ls t r e s sde t e rmi na t i oni nmi nd i twi l lbec onven i en tt oha vet heho t
cy l i nde randt hee i va l en tc o l dc l i nde ro fs ameg ene r a ls i z e s.Thu st her a d i u sR andt he qu y l eng t hL wi l lbet hes amef o rbo t hc l i nde r s.I no t he rt ha tt henond imen s i ona ld i f f e r en t i a l y , equa t i on s,Equa t i on ( 28),bet hes amef o rt heho tc l i n d e ra sf o rt h ec o l dc l i n d e r t h e y y r ame t e r si nt he s ed i f f e r en t i a le t i on smu s tbet hes ame.Tha ti s,i faquan t i t ft hec o l d pa qua yo cy l i nde ri sdeno t edbyab a rove rt hequan t i t y
췍0 D D0 1 = β= 2 D2 R D2 0 0 췍 췍0 Nx Nx P0 P 0 nξ0 = =췍 , p = =췍 D0 D0 D0 D0
( 29) ( 30)
췍o Thec ond i t i ono fEqua t i on ( 29)c anbes a t i s f i edby mak i ngt het h i ckne s sb ft hec o l dc l i nde r y sma l l e rt hant het h i ckne s sbo ft heho tc l i nde r.Th i si so fc ou r s et obeexpe c t ed,s i nc eYoung ’ s y “ ” modu l u so ft heho tma t e r i a li ssma l l e randhenc ema t e r i a li ss o f t e rt hant hec o l d ma t e r i a l. 췍i 췍0c Whent het h i ckne s sb sde t e rmi nedf r ombbyu s i ngEqua t i on ( 29),D anbec ompu t ed.Then 0 0 췍0andt 췍x Equa t i on ( 30)g i ve st hei n t e r na lp r e s s u r eP hea x i a ll o a dN f r omt hes c i f i edP0andNx pe
f o rt heho tc l i nde r.The s es t e st henf i xt heg e ome t r ft hec o l dc l i nde randt hep r ima r y p yo y y s s t e mo f l o a d s . y Fo rt hea dd i t i ona ls e c ond a r o a d s,t hef a c tt ha tEqua t i on s( 27)and ( 28)a r el i ne a r yl
equa t i on sc anbeu t i l i z edt oi n t r oduc eana ddedf r e edomi ns c i f i ngt hel o a d s.L i ne a rr e l a t i on s pe y a r eno ta l t e r edby mu l t i l i ngt heva r i a b l ebyac on s t an t.The r e f o r ef o ra dd i t i on a ll o a d sa nd p y a dd i t i on a ld i s l a c eme n t s,t h enond ime n s i on a lqu a n t i t i e sf o rt h ec o l dc l i nd e ra ndt h enond ime n s i on a l p y a n t i t i e sf o rt h eho tc l i nd e rn e e dno tb ei d e n t i c a l,bu td i f f e rbyaf a c t o rε.Thu s qu y
췍 췍 췍′)=ε( ( u ′, v ′, w u ′,v ′,w′)
and
췍 췍 췍 췍′ 췍′ 췍′ ( , , ,′ ,′ ,′ n ′ ′ n ′ m m n ′ n ′ m′ θ, θ; θ, θ )=ε( θ, θ; θ θ) ξn ξ ξ m ξ ξ n ξ ξ m ξ m
( 31)
p′ =ε p′
( 32)
Then -
Bu tt henond imen s i ona lp r e s s u r el o a d i ngpi sr e l a t edt ot hea c t ua lp r e s s u r el o a d i ngbyEqua t i on
670
COLLECTED WORKSOF HSUE-SHEN TS IEN
췍′f ( r a ond c e hes randt nde i l dc l o hec rt o ng sP i d a o el r u s s r e a r ond c e hes et r o f e r 19).The y y yp
r e s s u r el o a d i ngP′f o rt heho tc l i nde ra r er e l a t edt h r ough p y
췍 췍′ = çæD0εö÷P′ P èD0 ø
( 33)
췍0ε/D0 .S Ther a t i oo ft hep r e s s u r el o a di st henD i nc et her a d i u sR andt hel eng t hL o fbo t h cy l i nde r sa r et hes ame,o t he rt so fl o a d ss ucha sc onc en t r a t edf o r c e,o rmomen tf o rt hec o l d ype
cy l i nde randf o rt heho tc l i nde rmu s ta l s obe a rt hes amer a t i o.Ne ed l e s st os a hel o a d sf o r y y,t t hec o l dc l i nde rmu s tbea l i eda tc o r r e s i ngpo i n t sf o rl o a d si nt heho tc l i nde r. y pp pond y Thea dd i t i ona lf o r c e sN′ ′ t hea dd i t i ona ls he a rQ′ andt hea dd i t i ona lmomen t sN′ x ,Nx θ, x, x,
Mx′ tt heend so ft hec l i nde ra r ec on t r o l l edbyEqua t i on s( 15),( 16),and ( 19).I ti se a s i l θa y y s e ent ha tbe c au s eo fEqua t i on ( 29),t her a t i oo ft he s equan t i t i e sf o rt hec o l dc l i n d e ra n dt h e y 췍 ho tc l i nde ri sa a i nD0ε/D0 . y g
The r e f o r e,knowi ngt hel o a ds s t emont heho tc l i nde r,onec anf i ndt hec o r r e s i ng y y pond l o ads s t em f o rt hec o l dc l i nde r.Thef a c t o rεf o rt hes e c ond a r o a d sc an becho s ena t y y yl , c onven i enc eo ft heexpe r imen t e r.Fo ri n s t anc e ε mi tbes ocho s ena st o maket her a t i o gh 췍0ε/D0 equa D lt oun i t . T h e n t h e s e c o n d a r l o a d s s t e mf o r t h e c o l d c l i n d e r i s e x a c t l t h e s a m e y y y y y a st he ho tc l i nde r. When t he p r ope rl o a df o rt he c o l dc l i nde ri ss e l e c t ed and t he y y , c o r r e s i ngs t r a i nont hec o l dc l i nde rde t e rmi nedbys t r a i ng a e st hei nve r s ee i va l enc e pond y g qu
r ob l emi st hent of i ndt hes t r a i ni nt heho tc l i nde rf r omt het e s td a t aonc o l dc l i nde r. p y y Takef o ri n s t anc e,t hea x i a ls t r a i nex ( z).Fo rt hec o l dc l i nde r,a c c o r d i ngt oEqua t i on s y ( 1)and ( 25),
췍′ z 췍′ ö 췍췍2w æ췍u 1 췍x ( ÷ nT + ç e z)=k1 R 췍ξ2 ø è 췍ξ 1+ν
( 34)
whe r ez 췍i st heva l ueo fz me a s u r edf r om t he med i ans u r f a c eo ft hec o l dc l i nde r,mi dwa y y 췍 be twe ent hebound a r u r f a c e s.Nowl e tex bet hea ve r a eo ft heme a s u r eda x i a ls t r a i n sont he ys g
췍x bet ou t e rs u r f a c eandont hei nne rs u r f a c eo ft hec o l dc l i nde r,andl e tΔ e hed i f f e r enc eo ft he y
me a s u r eda x i a ls t r a i n sont heou t e rs u r f a c eandont hei nne rs u r f a c eo ft hec o l dc l i nde r.Then y ( ) f r om Equa t i on 34
췍 췍x =k1 - 1 nT +췍u′ =k1 - 1 nT +ε췍u′ e 췍ξ 1+ν 췍ξ 1+ν
( 35)
and
췍 2 췍 2췍 췍x = b 췍 w′ = bε췍 w′ Δ e R 췍ξ2 R 췍ξ2 Fo rt heho tc l i nde r,t hea x i a ls t r a i ni sg i venby y
췍u′ z 췍2w′ ex ( z)=k1 + 췍ξ R 췍ξ2
( 36)
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
671
췍2w′ 췍u′ and 2 f r om Equa t i on s( 35)and ( 36),oneha s Bye l imi na t i ng 췍ξ 췍ξ z 췍ö 1ö æ 1 æ췍 1 ex ÷ nT + çe ex ( z)= ç1- ÷k1 + x췍Δ ( b εø è ø ε εè 1+ν)
( 37)
Theva l ueo fzi sme a s u r edr a d i a l l nwa r df r omt hemed i ans u r f a c eo ft heho tc l i nde randt hu s yi y i sl a r e r i nm a n i t u d ef o r t h e i n s i d e s u r f a c e t h a nf o r t h eo u t s i d e s u r f a c e . g g S imi l a r l y, 1ö æ 1 1 æ췍 z 췍 ö eθ ÷ nT + çe eθ( z)= ç1- ÷k2 + θ췍Δ ( εø b è ø 1+ν) ε εè
( 38)
and
γxθ ( z)=
z 췍 ö 1 æ췍 çγxθ ÷ 췍 Δγxθ ø b εè
( 39)
췍θi whe r ee st hea ve r a eo ft heme a s u r edc i r cumf e r en t i a ls t r a i nont heou t e rs u r f a c eandt hei nne r g 췍 췍 췍 , ; s u r f a c eo ft hec o l dc l i n d e r Δ e i st h e d i f f e r e n c e o ft h e s es t r a i n s γ a n d Δ γ a r et h e θ xθ xθ y
c o r r e s i ngquan t i t i e sf o rt hes he a r i ngs t r a i n.I nEqua t i on s( 37)and ( 38), k1andk2,andnT pond ( ) ( ) a r et hep r ima r t r a i n sc ompu t edf r om Equa t i on s 25 and 8 .The r e f o r et he s ee t i on s ys qua a l l owt hec a l cu l a t i ono ft hes t r a i n si nt heho tc l i nde rf r omt e s tr e s u l t sf r omc o l dc l i nde r,and y y
t hu sc omp l e t et hede s i r eds imi l a r i t aw. yl , Fo ras t r e s sana l s tt henex ts t e spe r ha st hec a l cu l a t i ono ft hep r i nc i l es t r a i n sa te a ch y pi p p
va l ueo fzi nt hes he l l,andexami newhe t he rt hel a r e ro ft he s ep r i nc i a ls t r a i n sexc e ed st he g p
de s i imi to ft hema t e r i a la tt het empe r a t u r ep r eva i l i nga tt ha tpo i n t. gnl
Examp l eo fDimen s i on i ngt h eEqu i v a l en tCo l dCy l i nd e r Asan examp l eo ft he p r o c edu r e ou t l i nei nt he p r ev i ou ss e c t i on,t he d a t ag i ven by []
No l and1 wi l lbeu s edt of i ndt hee i va l en tc o l dc l i nde r.Thet empe r a t u r ed i s t r i bu t i oni nt he qu y wa l li st akenf r omF i 2o ft ha tp a randi sr e r oduc eda sF i 1he r e.Thema t e r i a li sa s s umed g. pe p g. t obe19 9 DL,andt heva r i a t i ono fYoung ’ smodu l u swi t ht empe r a t u r ei sp l o t t edi nF i 2, g. a a i nu s i ng No l and ’ sd a t a.F i r s t,t hepo s i t i ono ft hemed i ans u r f a c ewi l lbede t e rmi ned,u s i ng g Equa t i on ( 10).Th i si sf oundt obe0. 588bf r omt hei n s i des u r f a c e.Nex t,byt ak i ngt hec o l d cy l i nde rt obea t100℉ ,t her a t i oo ft het h i ckne s s e so ft heho tc l i n d e r a n d t h e e u i v a l e n t c o l d y q cy l i nde ri sc ompu t edbyu s i ngEqua t i on s( 6),( 7),and ( 29).I ti sf oundt ha t
췍 =0. b 936b
Thu st h ee i v a l e n tc o l dc l i nd e ro ft h es amema t e r i a li s93. 6p e rc e n ta st h i c ka st h eho tc l i nd e r. qu y y 췍0/D0 . Ther a t i oo ft hel o a d sont hec o l dc l i nde randt heho tc l i nde ri sc on t r o l l edbyD y y
Th i si sc ompu t eda s
췍0/D0 =1. D 29
COLLECTED WORKSOF HSUE-SHEN TS IEN
672
F i 1 Temp e r a t u r e g.
F i 2 Young ’ smodu l u sa s g.
d i s t r i bu t i oni nt hewa l l
afun c t i ono ft emp e r a t u r e
Byt ak i ngα =10-5 /℉ ,andb =0. 095i n.,t heva l ue so ft he rma ls t r e s s e sa r e
NT =22400l b/ i n M T =192l b.
nT =0. 958×10-2 mT =-0. 124
I ft her ad i u so ft hec l i nde ri s2. 25i n,andi ft hei n t e r na lp r e s s u r eP0i s1500p s i,t hen y 0 44×10-3 p =1. 0 I ft heax i a lt en s i onNx i sduet ot hes amei n t e r na lp r e s s u r e,t hen
nξ0 =
p 72×10-3 =0. 2 0
Now æ R ö2 β =12ç췍 ÷ èb ø The r e f o rt her a t i oo fmT/ s βandnξi 0
0 mT/( 0224 βnξ )=-0.
Th i si si nde edsma l l e rt hant het h i ckne s s r a d i u sr a t i ob/R = 0. 0422.Henc et hes u rmi s et ha t 0 / u s t i f i edbynume r i c a lc a l cu l a t i on. mT βi l i i b l ea a i n s tnξi snowj sne g g g
Junc t i onS t r e s sBe twe enCy l i nd e randHe ad Thep r ev i ou sf o rmu l a t i ono ft hes imi l a r i t awf o rs t r e s s e si nt heho tc l i nde randt hec o l d yl y cy l i nde ri sb a s edupont hea s s ump t i ont ha tt hes e c ond a r o a ds s t emi ss c i f i ed.Th i si sno t yl y pe t ruef o rt hej unc t i ons t r e s s e si nduc edbyf i t t i ng,s a s r i c a lhe a dt ot hec l i nde r. y,ahemi phe y
Suchs t r e s s e sa r ede t e rmi nedbyt hee l i t fde f o rma t i on so ft hehe a dandt hec l i nde ra tt he qua yo y unc t i on.I ft hes emi s r i c a ls he l lha st hes amet h i ckne s sa st hec l i nd r i c a ls he l l,t hent he j phe y t empe r a t u r ed i s t r i bu t i oni nt hes r i c a ls he l lwi l lbet hes amea si nt hec l i nd r i c a ls he l l.An phe y
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
673
췍, sb s ckne i h rmt o f i fun oo s l dea anbema imenc c e s ts dt l o r”c a l imi s he “ tt ha howst ss i l s ana pe y de t e rmi nedby Equa t i on ( 29).Howeve r,t hes imi l a r i t fbo t ht he p r ima r o a d i ng and yo yl : s e c ond a r o a d i ngnow r e i r e sana dd i t i ona lr e s t r i c t i on ε mu s tnow beun i t he s e yl qu y.Whent c ond i t i on sa r efu l f i l l ed,t hes imi l a r i t fj unc t i ons t r e s s e si nt heho tc l i nde randt hec o l d yo y cy l i nde rwi l lbea s s u r ed,andt her e l a t i on sp r ev i ou s l l opedf o rc ompu t i ngt hes t r e s s e si n ydeve
t heho tc l i nde rf r omt e s td a t aont hec o l dc l i nde rr ema i nva l i d. y y
Ri ngS t i f f en e rAr oundCy l i nd e r Tos t r eng t hent het h i nc l i nd r i c a ls he l la a i n s tc onc en t r a t edl o a d sf r omt hemoun t i ngl ug s, y g
ar i ngs t i f f ene ri so f t ena t t a chedt ot heou t s i des u r f a c e.I tr ema i n sc o l d.Young ’ smodu l u so f “ ” t her i ng ma t e r i a lt hu si st ha to fc o l dma t e r i a l.Tode t e rmi net hed imen s i on so fa s imi l a rr i ng
f o rt hec o l dc l i nde r,t hec ond i t i on st obes a t i s f i eda r et ho s eo fEqua t i on s( 31).I no t he r y , wo r d sf o rar a t i oεo fde f o rma t i ono ft her i ngont hec o l dc l i nde rt ot ha tont heho tc l i nde r, y y 췍0/D0 .Th t hef o r c er e i r edmu s tbe a rt her a t i oεD i sme an st ha tt her a t i oo ft hes t i f fne s so ft he qu
췍0/D0 .I r i ngf o rt hec o l dc l i nde rt ot ha tont heho tc l i nde ri sD ft her i ngi sr e c t angu l a ri n y y 췍 , / s e c t i on t hec o r r e c tr a t i oo ft hewi d t ho ft hetwor i ng sf o rc omp l e t es imi l a r i t st henD0 D0 . yi
Ther i ngont hec o l dc l i nde ri st hu swi de rt hanont heho tc l i nde r.Whent h i sc ond i t i onon y y t her i ngd imen s i oni ss a t i s f i ed,t hes imp l es imi l a r i t e l a t i on sf o rt hes t r e s s e si nt hes he l la r e yr a a i nc o r r e c t. g
Re f e r enc e s [1 ] No l andR L.S t r eng t h so fSeve r a lS t e e l sf o r Ro cke tCh ambe r sSub e c t edt o Hi t e so f He a t i ng. j gh Ra
ou r n a lo fAme r i c anRo cke tSo c i e t 6):154 162. J y,1951,21( [2 ] Timo s henkoS.The o r fP l a t e sandShe l l s.McGr aw-Hi l lBookComp any,New Yo r k,1940:389. yo [3 ] Yu anS W.TheCy l i nd r i c a lShe l l sSub e c t edt oCon c en t r a t edLo a d s.Qu a r t e r l fApp l i ed Ma t hema t i c s, j yo 1946, 4:13 26.
[4 ] Donne l lL H.S t a b i l i t fTh i n -Wa l l edTube sUnde rTo r s i on.NACA Te chn i c a lRepo r t473,1933. yo
661
AS imi l a r i t o rS t r e s s i ngRap i d l yLawf y He a t e dTh i n -Wa l l e dCy l i nd e r s ①❋
②
H.S.Ts i en② andC.M.Cheng③ ③
( Dan i e landFl o r e n c eGugge nhe imJe tPr opu l s i onCe n t e r,
Ca l ifo rn i aIn s t i t u t eof Te chno l ogy,Pa s ade na,Ca l if.)
Whenat h i nc l i nd r i c a ls he l lo fun i f o rmt h i ckne s si sv e r a i d l a t e dbyho th i r e s s u r eg a s y yr p yhe gh p
f l owi ngi n s i det hes he l l,t het emp e r a t u r eo fma t e r i a ld e c r e a s e ss t e e l r omah i emp e r a t u r ea tt he p yf ght
i n s i d es u r f a c et oamb i en tt emp e r a t u r e sa tt heou t s i d es u r f a c e.Young ’ smodu l u so fma t e r i a lt hu sva r i e s. Thepu r s eo ft hep r e s en tp a e ri st or e duc et hep r ob l em o fs t r e s sa na l s i so fs uchacy l i nd e rt oan po p y e i v a l en tp r ob l em i nc onven t i ona lc l i nd r i c a ls he l lwi t hou tt emp e r a t u r eg r a d i en ti nt he wa l l.The qu y e i v a l enc ec onc e ti sexp r e s s e da sas e r i e so fr e l a t i on sbe twe ent hequan t i t i e sf o rt heho tc l i nde rand qu p y t hequan t i t i e sf o rt hec o l dcy l i nd e r.The s er e l a t i on sg i v et hes imi l a r i t aw whe r e bys t r a i n sf o rt heho t yl c l i nd e rc a nbes imp l educ e df r om me a s u r eds t r a i n sont hec o l dc l i nd e randt hu sg r e a t l imp l i fyt he y yd y ys r ob l emo fexp e r imen t a ls t r e s sana l s i s. p y
Thec l i nde ro fas o l i dp r ope l l an tr o cke ti ss ub e c t edt ove r a i dhe a t i ngdu r i ngi t ss ho r t y j yr p hough t l l,a l lwa a c i r nd i l nc i h het st s o r c ona i t bu i r t s i ed r u t a r empe on.Thet i t a r fope ono i t a r du y , a r ox ima t e l hes amei neve r e c t i on i sno tl i ne a r.Th i sc ond i t i oni smo s ts eve r ea tt heend pp yt ys
o ft hec ombu s t i ono ft hep r ope l l an tg r a i n.F r omt hepo i n to fv i ewo fama t e r i a l seng i ne e r,t h i s , c a s ei sd i s t i ngu i s hedf r omo t he r sbyt het imer a t eo fhe a t i ng wh i chi ss ol a r ea st ono ta l l ow g
s uf f i c i en tt imef o ra r e c i a b l echang ei nt hes t r uc t u r eo ft hema t e r i a l.Thes t r eng t ho ft hewa l l pp ma t e r i a lunde rt h i sope r a t i ngc ond i t i oni squ i t ed i f f e r en tf r omt ha tunde rs l ow he a t i ng.Th i s []
f a c ti sc l e a r l onc l u s i ve l hownbyR.L.No l andi nar e c en tp a r1 .F r omt hepo i n to f yandc ys pe , v i ew o fas t r e s sana l s t t h er a t i o n a ld e s i n o fas o l i d r o e l l a n tr o c k e tc l i n d e ri st h u s y g p p y
c omp l i c a t edbyt heve r a r et he rma ls t r e s sandva r i a b l eYoung ’ smodu l u so ft h ema t e r i a la c r o s s yl g , t h ewa l la sar e s u l to ft h el a r et emp e r a t u r eg r a d i e n t.Fu r t h e rmo r ee xp e r ime n t a ls t r e s sd e t e rmi n a t i on g
und e ra c t u a lf i r i ngt e s t si sr a t h e rd i f f i c u l tdu et ot h es ho r tt e s tt imea v a i l a b l ea ndt h eh i emp e r a t u r e. ght
I ti sbe l i evedt ha tf o rt he s er e a s on st heon l a s e wh i ch ha sbe enana l edbyr e l i a b l e yc yz r a t i ona lme t hodi st hec a s eo fr o cke tc l i n d e ru n d e ru n i f o r mi n t e r n a l r e s s u r e .T h eb e n d i n y p g ❋R e c e i vedFeb r u a r y14,1952.
144 149,1952. o u r na lo h eAme r i c anRo c k e tSo c i e t l.22,pp. J ft y,vo Th i sp a e ri sb a s edon p a r to fat he s i ss ubmi t t edbyt hej un i o rau t ho rf o rp a r t i a lfu l f i l lmen to ft he ① p r equ i r emen t so fPh.D.i n Me ch an i c a lEng i ne e r i ng,Ca l i f o r n i aI n s t i t u t eo fTe chno l ogy. ② Ro be r tH.Godda r dPr o f e s s o ro fJ e tPr opu l s i on. ③
Gr adua t eAs s i s t an ti n Me chan i c a lEng i ne e r i ng.
662
COLLECTED WORKSOF HSUE-SHEN TS IEN
l eon r c.,a t ug s,e ngl i t s,moun e r u s o l oendenc sduet e s s e r t hes s,t e l z z edno t an oc sduet e s s e r t s y e s t ima t edbyve r ough me t hod s.Thepu r s eo ft h i sp a ri st oimp r ovet h i ss i t ua t i onby yr po pe s ugg e s t i ngana r o a ch wh i ch wi l lr educ et he g ene r a ls t r e s sp r ob l em o fho tc l i nde rt oa pp y r ob l em o fane i va l en tc o l dc l i nde r.Thee i va l en tp r ob l em c ant henbes o l vede i t he r p qu y qu ana l t i c a l l hec onven t i ona lme t hodo rd i r e c t l r imen t a ls t r e s sde t e rmi na t i on.I n y ybyt ybyexpe , e i t he rcho i c e t he p r ob l em i sbe l i evedt o be g r e a t l imp l i f i ed.Th i sl aw o fe i va l enc e ys qu be twe enho tc l i nde randc o l dc l i nde rc anbec a l l edt hes imi l a r i t aw. y y yl
S t r e s s e sandS t r a i n so faTh i n -Wa l l e dCy l i nd e r Thef a c tt ha tt het h i ckne s so ft hec l i nde ri ssma l li nc omp a r i s ont oi t sr a d i u sandl eng t h y a l l owsag r e a ts imp l i f i c a t i oni nt hes t r a i nana l s i s.Towi t,t hede f o rma t i ono feve r i n to f y ypo
t hecy l i nde rc anbede s c r i beds uf f i c i en t l c cu r a t e l hed i s l a c emen t so ft hepo i n t sona ya y byt p s i ng l es u r f a c ewi t h i nt hewa l lo ft hec l i nde r.Th i ss u r f a c ei st hes o c a l l edmed i ans u r f a c e.The y
s i t i ono ft hemed i ans u r f a c ei ss ode t e rmi nedt ha tabend i ngo ft hemed i ans u r f a c ewi l lno t po , i nduc ene tex t en s i ona lf o r c e si nt hep l aneo ft he med i ans u r f a c e a c r o s st het h i ckne s so ft he , wa l l.When Young ’ s modu l u si sac on s t an t,a si st hec a s ef o rac o l dc l i n d e r t h e m e d i a n y s u r f a c el i e smi dwa twe ent heou t e randt hei nne rbound a r u r f a c e so ft hec l i nde r.When ybe ys y , Young ’ smodu l u si sno tac on s t an tbu tde c r e a s e s wi t hi nc r e a s ei nt empe r a t u r e t he med i an
s u r f a c ei sd i s l a c edt owa r dt hec o l ds i de,a swi l lbes e enp r e s en t l p y. , , Le tx θ zbet hec oo r d i na t es s t em wi t ho r i i nont hec l i nd r i c a lmed i ans u r f a c es ucht ha t y g y
xpo i n t si nt hea x i a ld i r e c t i ono ft hec l i nde r, θi si nt hec i r cumf e r en t i a ld i r e c t i on,me a s u r edon y t hemed i ans u r f a c e,andzi sno rma lt ot he med i ans u r f a c e,po i n t i ngt owa r dt hea x i so ft he , , ( , ) cy l i nde r.Le tU V andW bed i s l a c emen t so fapo i n t x θ ont he med i ans u r f a c ei nt he p d i r e c t i on sx, θ, andz, r e s c t i ve l r et hu sfunc t i on so fxandθ ;bu tno to fz.Thent he pe y.Theya above -men t i onedfund amen t a ls imp l i f i c a t i ono ft h i ns he l l sc an bes t a t eda sf o l l ows:i ft he d i r e c ts t r a i n si nt hexandθd i r e c t i on sa r eex andeθ,andt hes he a rs t r a i nγxθ , t hen
ex =
췍2W 췍U -z 2 췍x 췍x
ïü ï ï 2 ï 1췍V W z æ췍V 췍 W ö + 2 ÷ eθ = ý - - 2ç R 췍θ R R è췍θ 췍θ ø ï ï 2 췍V 1췍U z æ 췍 W 췍V ö ï ÷ï γxθ = + + -2 ç 췍x R 췍θ R è췍x췍θ 췍x ø þ
( 1)
whe r eRi st her a d i u so ft hemed i anc l i nd r i c a ls u r f a c e,o rt he “ r a d i u so ft hec l i nde r.”Th i s y y r e s u l ti ss ome t ime sde s c r i beda st heKi r chho f fbend i nga s s ump t i on:P l aneno rma lt ot hemed i an
s u r f a c ebe f o r ebend i ngr ema i n ss oa f t e rbend i ng.
Thes i i f i c an ts t r e s s e si nat h i ns he l la r et hed i r e c ts t r e s s e sσx andσθi nxandθ,andt he gn
s he a rs t r e s sτxθ .Al lo t he rs t r e s s e sa r esma l li nc omp a r i s ont ot he s et h r e e.Nowl e tT bet he , t empe r a t u r eo fwa l la bovear e f e r enc et empe r a t u r es a her oomt empe r a t u r e.Ti sa s s umedt o yt
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
663
r nde i l hec ft ngo i t a hehe st fxandθ.Thu to tno ez,bu t na i d r so s ckne i h ft ono i t l beon y yafunc i sa s s umedt obeun i f o rmove rt heen t i r es u r f a c eo ft hec l i nde r.Th i si sg ene r a l l r l o s e l y yve yc y , a r ox ima t edi nr e a l i t fαi st hec o e f f i c i en to ft he rma lexp an s i on t het he rma lexp an s i on pp y.I s t r a i ni st henαT .By Hooke ’ sl awoneha s
1 ü ( σx -ν σθ)+αT ï ï E ï ï 1 eθ = ( σθ -ν σx )+αT ý E ï ï ( ) 2 1+ν ïï τxθ γxθ = þ E
ex =
( 2)
whe r eE i s Young ’ s modu l u s,and v i s Po i s s on ’ sr a t i o.E i s,o fc ou r s e,afunc t i on o f t empe r a t u r eo rafunc t i ono fz.v,howeve r,wi l lbea s s umedt obec on s t an tf o rl a cko fde f i n i t e
i nf o rma t i on.Bys o l v i ngf o rt hes t r e s s e s,oneob t a i n sf r om Equa t i on ( 2),
E( z) ü {( ex +ν eθ)- ( 1+ν) αT ( z)} ï ï 1-ν2 ï ï E( z) ( ) ) ( ) ( { } σθ = e ν e 1+ ν α T z + ý θ x 2 1-ν ï ï E( z) ïï γxθ τxθ = þ 2( 1+ν) σx =
( 3)
Fo rt h i n -wa l l edc l i nde r s,t hee i l i b r i ume t i on sa r eexp r e s s edi nt e rmso f“ s e c t i ona l y qu qua ave r a s”o ft hes t r e s s e sg i veni nEqua t i on ( 3).Tha ti s,ones ak so ft heno rma lf o r c e sNxand ge pe , , , , Nθ t hes he a r i ngf o r c eNxθ t hebend i ng momen t sMx Mθ andt hetwi s t i ng momen tMxθ . Theya r er e l a t edt oσx ,σθ,andτxθ byt hef o l l owi nge t i on s qua
∫ ∫ ∫ σzd z,M =- τ zd =- σzd ∫ z,M =-∫ ∫ z Nx = σxd z,Nθ = σθd z,Nxθ = τxθd z
Mx
x
θ
θ
xθ
xθ
( 4) ( 5)
Thei n t e r a l si nt hea bovee t i on sa l lex t enda c r o s st het h i ckne s so ft hewa l l.Bys ub s t i t u t i ng g qua Equa t i on s( 1)and ( 3)i n t oEqua t i on s( 4)and ( 5),oneha sf o rexamp l e
Wö æ췍U ν 췍V + -ν ÷Nx = D0 ç Rø è췍x R 췍θ ν 췍V ν 췍2W ö æ췍2W ÷- NT + D1 ç 2 + 2 è췍x R 췍θ R2 췍θ2 ø and
Wö æ췍U ν 췍V + -ν ÷- Mx = D1 ç Rø è췍x R 췍θ ν 췍V ν 췍2W ö æ췍2W ÷- MT + D2 ç 2 + 2 è췍x R 췍θ R2 췍θ2 ø
664
COLLECTED WORKSOF HSUE-SHEN TS IEN
e r whe
∫
1 ü E( z) d z ï ï 1-ν2 ý 1 ï ( ) E z zd zï D1 = þ 1-ν2
( 6)
1 E( z) z2d z 1-ν2
( 7)
D0 =
D2 = and
∫
∫
α E( z) T( z) d z 1-ν
( 8)
α E( z) T( z) zd z 1-ν
( 9)
NT = MT =
∫
∫
Thei n t e r a l sa a i nex t enda c r o s st het h i ckne s so ft he wa l l.I ti sev i den tf r om t hea bove g g exp r e s s i on sf o rNx and Mx t ha tc on s i de r a b l es imp l i f i c a t i onc anbea ch i evedbychoo s i ngt he med i ans u r f a c ei ns uchawa ha t yt
D1 =
∫
1 E( z) zd z =0 1-ν2
( 10)
Th i si sa c t ua l l hec ond i t i ont ol o c a t et hemed i ans u r f a c e.S i nc eYoung ’ smodu l u sE de c r e a s e s yt , ( ) wi t hi nc r e a s ei nt empe r a t u r ei ti ss e enf r omEqua t i on 10 t ha tt hemed i ans u r f a c ei sne a r e rt o
t hec o l dbound a r u r f a c et hant ot heho tbound a r u r f a c e.Fo rar o cke tc l i nde r,ho ti n s i de ys ys y bu tc o l dou t s i de,t he med i ans u r f a c ei sne a rt ot heou t s i des u r f a c e.Wi t ht h i scho i c eo ft he med i ans u r f a c e,t hef o r c e sandt hemomen t sa r er e l a t edt ot hed i s l a c emen t sbyt hef o l l owi ng p : s imp l e re u a t i o n s q
Wö æ췍U ν 췍V ü + -ν ÷- NT ï Nx = D0 ç ï Rø è췍x R 췍θ ï ï 췍U ö æ 1췍V W ÷- NT ý - +ν Nθ = D0 ç 췍x ø è R 췍θ R ï ï 1-ν æ췍V 1췍U ö ïï ÷ Nxθ = + D0 ç þ è췍x R 췍θ ø 2
( 11)
ν 췍2W ν 췍V ö æ췍2W ü ÷+ MT ï + Mx = D2 ç 2 + 2 è췍x ï R 췍θ2 R2 췍θ ø ï ï 1 췍V 췍2W ö æ 1 췍2W ÷+ MT ý ν + + Mθ = D2 ç 2 èR 췍θ2 R2 췍θ 췍x2 ø ï ï 2 1 췍V ö æ 1 췍W ïï ÷ + 2 Mxθ = ( 1-ν) D2 ç 2 þ èR 췍x췍θ R 췍x ø
( 12)
I ti swo r t h wh i l et opo i n tou ttwof a c t s:F i r s t l hecho i c eo fr e f e r enc et empe r a t u r ei s y,t , i t ea r b i t r a r i ngt her e f e r enc et empe r a t u r e wi l lchang et heva l ueo fNT Equa t i on qu y.Chang
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
665
( sNx ,Nθ e c r o lf rma heno vet a e ll l nwi i a r t ls rma heno nt ti tmen s d u nga i s e r r o tac 8).Bu j pond i venbyEqua t i on ( 11)unchang ed.The r e f o r et hephy s i c a lp r ob l emi squ i t ei nde to ft he g penden
cho i c eo fr e f e r enc et empe r a t u r e.MT i si nde to ft her e f e r enc et empe r a t u r e duet o penden Equa t i on ( 10).Se c ond l hep r e s en ts s t emo fe t i on sr educ e st ot ho s eo ft hec onven t i ona l y,t y qua
t h i ns he l lt he o r ft empe r a t u r eg r a d i en ti sa b s en t,o ri fYoung ’ smodu l u si si nde to ft he yi penden
t empe r a t u r eo f ma t e r i a l.I nt ha tc a s et he med i ans u r f a c ei s mi dwa twe ent hebound a r y be y , s u r f a c e s andD2i ss imp l heu s ua lf l exu r a lr i i d i t ft hes he l l. yt g yo
Nond imen s i ona lQuan t i t i e sandEqua t i on so fEqu i l i b r i um I ti su s e f u lt oi n t r odu c enond ime n s i on a lqu a n t i t i e sbyt a k i ngRa st h er e f e r e n c el e ng t h.Thu s
ξ = x/R u = U/R, v = V/R, w = W/R nξ = Nx/D0 , nθ = Nθ/D0 , nξθ = Nxθ/D0 , nT = NT/D0
( 13) ( 14)
mξ = MxR/D2 , mθ = MθR/D2 , mξθ = MxθR/D2 , MT = MTR/D2
( 16)
췍v 췍v 췍u 췍u ü +v -nT ï -vw -nT , nθ = -w +v ï 췍θ 췍θ 췍ξ 췍ξ ý uö ï 1-væ췍v 췍 ç ÷ nξθ = + ï þ 2 è췍ξ 췍θ ø
( 17)
췍2w 췍v 췍2w 췍2w 췍v 췍2w ü +v 2 +v +mT , mθ = 2 + +v 2 +mT ï 2 췍θ 췍θ ï 췍θ 췍ξ 췍θ 췍ξ ý ï æ췍2w 췍v ö ÷ mξθ = ( + 1-v)ç ï þ è췍ξ췍θ 췍ξ ø
( 18)
( 15)
and The r e f o r eEqua t i on s( 11)and ( 12)be c omenow
nξ =
and
mξ =
Th ee a t i on so fe i l i b r i umi nt e rmso ff o r c e sa nd mome n t sa r eh e r ee x a c t l h es amea st h e qu qu yt []
2 c onv e n t i on a lt h e o r eon l nnov a t i oni st o wr i t et h em i nnond ime n s i on a lf o rm.Fo rt h i s y .Th yi , , , u r o s e o n eh a s t od e f i n e t h en o n d i m e n s i o n a l u a n t i t i e s a n d o f t h ed i m e n s i o n a l s e c t i o n a l θ q q p p p q ξ , , : s h e a r i ngf o r c e sQx Qθa ndno rma lp r e s s u r el o a d i ngP a a i n s tzd i r e c t i ona sf o l l ows g
qξ = Qx/D0 qθ = Qθ/D0 p = PR/D0
( 19)
Thent hee i l i b r i ume t i on so ff o r c e sa r e qu qua
췍 nξ 췍 nξθ æ췍v 췍2w ö æ 췍2v 췍w ö 췍2w 췍2v ÷-nξθ ç ÷ =0 + n + -qξ 2 -qθ ç θ è췍ξ 췍ξ췍θ ø è췍ξ췍θ 췍ξ ø 췍ξ 췍θ 췍ξ 췍ξ2
ïü ï ï ï 췍v 췍2w ö æ 췍 nξθ 췍 nθ æ췍v 췍2w ö æ 췍2v 췍w ö 췍2v ÷-q ÷ =0ý ( + 2 ÷+nξ 2 +nξθ ç + 20) + -qξ ç θ ç1+ 췍θ 췍θ ø è췍ξ 췍ξ췍θ ø è췍ξ췍θ 췍ξ ø è 췍ξ 췍θ 췍ξ ï ï 췍v 췍2w ö æ 췍 æ췍v 췍2w ö 췍2w qξ 췍 qθ ï ÷+nξ ç1+ ÷ =p + nξθ ç + n + +2 + θ ï þ 췍θ 췍θ2 ø è췍ξ 췍ξ췍θ ø è 췍ξ 췍θ 췍ξ2
666
COLLECTED WORKSOF HSUE-SHEN TS IEN
e r sa t fmomen umo i r b i l i hee rt o sf on i t Thee qu qua
췍mξθ 췍mθ æ 췍2v 췍w ö 췍2v ü ÷+β + +mξ 2 +mξθ ç qθ =0 ï è췍ξ췍θ 췍ξ ø ï 췍ξ 췍θ 췍ξ ý 췍mξ 췍mξθ æ 췍2v 췍w ö ï 췍2v ÷-β +mξθ 2 +mθ ç qξ = 0ï þ è췍ξ췍θ 췍ξ ø 췍θ 췍ξ 췍ξ
( 21)
2 β = R D0/D2
( 22)
whe r e
2 ,bt st hu saquan t i t ft heo r de ro f( R/ b) het h i ckne s so ft he wa l l.The r ea r ee l even yo βi i nd i v i dua le t i on si nt hes s t emo fEqua t i on s( 17),( 18),( 20),and ( 21).Wi t hs c i f i ed qua y pe , , , , , , , , , , , l o ad i ngp t he r ea r ea l s oe l evenunknown s u v w nξ nθ nξθ mξ mθ mξθ qξandqθ.The
s s t emo fe t i on si st hu sc omp l e t e. y qua
I n f i n i t eCy l i nd e rUnd e rUn i f o rmI n t e rna lPr e s s u r e Thes imp l e s ts c i a lc a s ei nt hep r e s en tg ene r a lp r ob l emi st hec a s eo fve r ongc l i nde r pe yl y unde run i f o rmi n t e r na lp r e s s u r e.I ft her o cke tc l i nde ri sl ongi nc omp a r i s onwi t hi t sd i ame t e r, y
t hea c t ua ls t r e s ss s t em du r i ngope r a t i oni sa r ox ima t edbyt h i si de a l i z eds imp l ec a s e.I nt h i s y pp r ob l emo fi nf i n i t e l ongun i f o rml o a dedc l i nde r,t hef o r c e snξ,nθ,andmomen t smξ,mθ p yl yl y
a r ec on s t an t si nde to fξandθ.Thes he a r i ngf o r c enξθandt hetwi s t i ngmomen tmξθvan i s h. penden
췍u sac on s t an t,s a vvan i s he s.wi sac on s t an t,andi sne a t i vei n ui sp r opo r t i ona lt oξo r i yk1 . g 췍ξ t hep r e s en tc oo r d i na t es s t em,s a t i on s( 17),( 18),and ( 20)g i ve y y-k2 .ThenEqua nξ0 =k1 +νk2 -nT ü ïï nθ0 =νk1 +k2 -nT ý ï þ mξ0 = mθ0 = mT
( 23)
whe r et hes upe r s c r i t0deno t et hequan t i t i e si nt hep r e s en ts imp l es t r e s ss s t em.Whent he p y , ( ) t empe r a t u r ed i s t r i bu t i onandt he ma t e r i a lp r ope r t i e sa r es e c i f i e d E u a t i o n 2 3 i v e st he p q g s t r a i n sk1andk2i nt e rmso ft hei n t e r na lp r e s s u r ep,andt hea x i a ll o a dnξ.I ft hea x i a ll o a di s r oduc edbyt hes amei n t e r na lp r e s s u r e,t heni tc ane a s i l hownt ha t p ybes
nξ0 = p0/2
( 24)
I ti so fi n t e r e s tt ono t et ha tt hebend i ng momen tmξandmθa r ee lt omT anda r ei nde t qua penden o ft hec ond i t i on so fl o a d i ng. Bys o l v i ngEqua t i on ( 23)f o rk1andk2,oneha s 1 1 ü 0 ( nT ï nξ0 -ν p )+ 1+ν ï 1-ν2 ý 1 1 ï 0 0 ( ) k2 = n ν n + T p ξ ï 1+ν þ 1-ν2
k1 =
( 25)
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
667
he st ve i 25)g on ( i t henEqua l,t a i r e t he ma ft no i a r t imums x he ma st oni i t i ond i s hede ft I gnc c r i t e r i ond i r e c t l r omt hep r e s s u r eandt empe r a t u r el o a d i ng. yf
L i n e a r i z e dTh e o r o rGen e r a lS e c onda r i ng yf yLoad Ass t a t ed i nt he p r ev i ou ss e c t i on,t he a c t ua ls t r e s ss s t em i n ar o cke t chambe ri s y
a r ox ima t e l ha to fi nf i n i t e l ongc l i nde runde run i f o rm i n t e r na lp r e s s u r e.Th i ss t r e s s pp yt yl y s s t emc anbec a l l edt hep r ima r t r e s s e s.Thedev i a t i on sf r omt hep r ima r t r e s ss s t em a r e y ys ys y r e s u l t so fbend i ng duet oc an t ed no z z l e,t oendenc l o s u r e s,t o moun t i ngl ug s,e t c.The s e add i t i ona ls t r e s s e so rt hes e c ond a r t r e s s e sa r e,howeve r,on l r a c t i on o ft he p r ima r ys y af y
s t r e s s e s.The r e f o r ei ti sj u s t i f i edt oc on s i de rt hes e c ondo r de rt e rmso fa dd i t i ona ls t r e s s e sand de f o rma t i on sa ssma l li nc omp a r i s ont ot hef i r s to r de rt e rms,andt hu sne l i i b l e . I no t he r gg wo r d s,
u =k1ξ +u ′,
v =v ′,
w =-k2 +w′ üï ï , nθ = p +n nξ = n +n ′ ′ nξθ = n ′ θ, θ ξ ξ ï , mθ = mT +m′ mξ = mT +m′ ′ ý θ , mξ θ =m θ ξ ξ ï ï ï 0 , , þ ′ ′ ′ = = + qξ =q q q p p p θ θ ξ 0 ξ
and
0
( 26)
whe r ek1 andk2 a r eg i venby Equa t i on ( 25).Thep r imedquan t i t i e sa r et hent hes e c ond a r y de f o rma t i on sandt hes e c ond a r t r e s s e s,t heya r ec on s i de r edt obesma l li nc omp a r i s ont ot he ys
r ima r f o rma t i on sands t r e s s e s.F r om Equa t i on s( 17)and ( 18),oneha st hef o l l owi ng p yde , r e l a t i on sbe twe ent hede f o rma t i on sandt hes t r e s s e s
u′ ö üï 췍v′ 췍v′ 1-νæ췍v′ 췍 췍u′ 췍u′ ,n ,n ç ÷ + ′ ′ -νw′ +ν -w′ + θ= θ =ν ξ 췍θ ø ï 췍ξ 췍θ 2 è 췍ξ 췍ξ 췍ξ ï and ï ï 췍2w′ 췍2w′ 췍v′ , ï m′ ν ν + + ξ= 췍θ ý 췍ξ2 췍θ2 ï 2 췍 w′ 췍2w′ 췍v′ ï , m′ + + θ =ν ï 췍θ 췍θ2 췍ξ2 ï æ췍2w′ 췍v′ ö ï ( ) ÷ + m′ θ = 1-ν ç ξ ï è췍ξ췍θ 췍ξ ø þ n ′ ξ=
( 27)
Bys ub s t i t u t i ng Equa t i on ( 26)i n t ot hee i l i b r i um e t i on s,Equa t i on s( 20),and qu qua d r opp i ngou tt hes e c ondo r de rt e rmso ft hep r imedquan t i t i e s,as s t emo fl i ne a r i z ede t i on s y qua
i sob t a i ned.Th i ss s t emc anbefu r t he rr educ edbys ub s t i t u t i ngt heq ′ ndq ′ b t a i nedf r omt he θo y ξa
l a s ttwoe t i on so fbend i ng momen te i l i b r i umi n t ot het h i r de t i on.Thef i na lr e s u l ti s qua qu qua : t hef o l l owi ngs s t e m o ft h r e ee u a t i o n s T h ef i r s ti st h ee u i l i b r i u m o ff o r c e si nt h ea x i a l y q q d i r e c t i on;t hes e c ondi st hee i l i b r i umo ff o r c e si nt hec i r cumf e r en t i a ld i r e c t i on;andt het h i r d qu : i st heequ i l i b r i umo ff o r c e si nt her a d i a ld i r e c t i on
668
COLLECTED WORKSOF HSUE-SHEN TS IEN
′ 1+ν췍2v ′ 췍w′ ′ 1-ν췍2u 췍2u =0 + + -ν 2 2 췍ξ 2 췍θ 2 췍ξ췍θ 췍ξ
ïü ï ï 췍2v ′ 1-ν췍2v ′ 1+ν췍2u ′ 췍w′ ï =0 + + ï 2 췍ξ2 2 췍ξ췍θ 췍ξ 췍θ2 ý ï and ï 2 ï æ 0 mT ö췍2w′ 췍v′ö æ 췍w′ 췍4w′ 췍2w′ 췍 w ′ 췍4w′ 0 ÷ =-β ÷ ï -w′+ +2 2 2 + 4 -βçν +β p p′+βçnξ 4 2 췍 θø è 췍 è θ 췍 췍 θ 췍 θ2 þ 췍 ξ β ø췍 ξ췍 ξ ξ
( 28)
The s ee t i on sha vebe ens imp l i f i edont heb a s i st ha tβi sal a r equan t i t ft heo r de ro ft he qua g yo s u a r eo f t h e r a d i u s t h i c k n e s s r a t i o . q I nt hel a t t e re t i on,p ′i st hes e c ond a r o a dimpo s edont hec l i nde rexp r e s s eda sa qua yl y , d i s t r i bu t edp r e s s u r eove rt hes u r f a c eo ft hec l i nde rd i r e c t edr a d i a l l twa r d.I ft hel o a di sa y you
c onc en t r a t edl o a d,t heni tha st obeexp andedi n t oap r oduc to fFou r i e rs e r i e sandFou r i e r []
i n t e r r a la sdonebyS.W.Yuan3 i nh i st r e a tmen to fc onc en t r a t edl o a donac o l dc l i nd r i c a l g y ( ) s he l l.Ot he rt e s o f l o a d s c a nb e s i m i l a r l e x a n d e d . T h e nE u a t i o n 2 8 i s a s s t e mo f t h r e e yp y p q y equa t i on sf o rt h r e eunknown su′,v′ ,andw′.Thef o r c e sand momen t sa r er e l a t edt ot he s e ( ) d i s l a c emen t sbyEqua t i on 27 . p Thep r ob l emo fg ene r a ls e c ond a r o a d i nga sexp r e s s edbyEqua t i on s( 27)and ( 28)i sve r yl y
s imi l a rt ot hep r ob l emo fg ene r a ll o a d i ngonac o l dc l i nd r i c a ls he l l,andc ant hu sbet r e a t edby y t heknown me t hod sdeve l opedf o rt h i sc onven t i ona lp r ob l em.I nf a c t,t heon l d i f f e r enc e y
be twe ent heho tc l i nde randt hec o l dc l i nde ri st hea a r anc eo ft het e rm mT i nEqua t i on y y ppe ( ) , : 28 .Howeve r event h i sd i f f e r enc ei st r i v i a l Ther e a s oni st heve r a r ema i t udeo fβa s yl g gn
s hownbyEqua t i on s( 6),( 7),and ( 22).I nf a c t, i f,a si sg ene r a l l hec a s e,NTi so ft hes ame yt
0 , t sa fmT/ oo i t a her tt ha howt ss on i t ede t i c bove hea hent t sNx udea t i fma ro de r o qua gn βandnξi mo s to ft heo r de ro fb/R,whe r ebi st het h i ckne s so ft hes he l l.S i nc et hes he l li sc on s i de r edt o bet h i n,o rb/R ≪ 1,t het e rmsi nvo l v i ng mT i n Equa t i on ( 28)c an bed r opped wi t hou t 0
imp a i r i ngt hea c cu r a c ft he p r e s en tt he o r h i si sdone,t heni nt hes s t em o f yo y. Whent y equa t i on sg i ven by Equa t i on s( 27)and ( 28),t hee f f e c t so ft he rma ls t r e s s e sand va r i a b l e Young ’ smodu l u sa r eno texp l i c i t.Asf a ra st henond imen s i ona le t i on sa r ec onc e r ned,t he qua , r o b l e m o fh o t c l i n d e r i s i d e n t i c a l t o t h e r o b l e mo f c o l dc l i n d e r a n d t h eb a s i c e u a t i o n s a r e p y p y q nowe s s en t i a l l hes amea st ha ta dop t edbyL.H.Donne l lf o rh i ss t udyo ft hes t a b i l i t ft h i n yt yo []
cy l i nd r i c a ls he l l s4 .Th i si st heb a s i so ft hes imi l a r i t awd i s cu s s edi nt hef o l l owi ngs e c t i on. yl
S imi l a r i t o rGen e r a lLoad i ng yLawf I ft hep r ob l emo fs e c ond a r o a d i ngi st obes o l vedana l t i c a l l her e s u l t so ft hep r ev i ou s yl y y,t
s e c t i ons how t ha ti tc an ber educ edt oane i va l en tp r ob l em o fc o l dc l i nde rands o l ved qu y , a c c o r d i ng l v e r amo r eu s e f u la l i c a t i ono ft h ee i v a l e n c ec on c e tl i e si nt hepo s s i b i l i t f y.Howe pp qu p yo expe r imen t a l l t e rmi n i ngt hes t r e s sandt hes t r a i nont hee i va l en tc o l dc l i nde randt hen yde qu y
u s i ngt hes imi l a r i t awt ode t e rmi net hes t r e s sandt hes t r a i ni nt heho tc l i nde r.The r ea r e yl y
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
669
d l o sonac t imen r a)Theexpe ch:( a o la r a t imen r expe emi ss i h ft so a e t dvan l n i ma pp g ytwoa cy l i nde rc anbedone mo r ee a s i l r ea c cu r a t e l hanpo s s i b l eonaho tc l i nde r. yand much mo yt y Thet e s tpe r i odc anbea sl onga sde s i r edandno tl imi t edt ot hes ho r tbu r n i ngt imeo ft he r o cke t.( b)Thes t r e s s e si nduc edby moun t i ngl ug s,e t c.,a r eve r d i f f i c u l t t oa r o x i m a t eb y pp y s imp l el o a ds s t emsamena b l et ot he o r e t i c a lc a l cu l a t i on s.Fo ri n s t anc e,t hel o a d sf r om a y
moun t i ngl uga r eno tr e a l l onc en t r a t edf o r c eandac onc en t r a t edmomen t.Tot aket hema sa yac c onc en t r a t edf o r c eandac onc en t r a t ed momen twou l dg r o s s l o v e r e s t i m a t et h ea c t u a ls t r e s s. y Suchd i f f i cu l t i e sd i s a a ri ft hel o a d i ngi sdoneexpe r imen t a l l ppe y. , Wi t hs uchexpe r imen t a ls t r e s sde t e rmi na t i oni nmi nd i twi l lbec onven i en tt oha vet heho t
cy l i nde randt hee i va l en tc o l dc l i nde ro fs ameg ene r a ls i z e s.Thu st her a d i u sR andt he qu y l eng t hL wi l lbet hes amef o rbo t hc l i nde r s.I no t he rt ha tt henond imen s i ona ld i f f e r en t i a l y , equa t i on s,Equa t i on ( 28),bet hes amef o rt heho tc l i n d e ra sf o rt h ec o l dc l i n d e r t h e y y r ame t e r si nt he s ed i f f e r en t i a le t i on smu s tbet hes ame.Tha ti s,i faquan t i t ft hec o l d pa qua yo cy l i nde ri sdeno t edbyab a rove rt hequan t i t y
췍0 D D0 1 = β= 2 D2 R D2 0 0 췍 췍0 Nx Nx P0 P 0 nξ0 = =췍 , p = =췍 D0 D0 D0 D0
( 29) ( 30)
췍o Thec ond i t i ono fEqua t i on ( 29)c anbes a t i s f i edby mak i ngt het h i ckne s sb ft hec o l dc l i nde r y sma l l e rt hant het h i ckne s sbo ft heho tc l i nde r.Th i si so fc ou r s et obeexpe c t ed,s i nc eYoung ’ s y “ ” modu l u so ft heho tma t e r i a li ssma l l e randhenc ema t e r i a li ss o f t e rt hant hec o l d ma t e r i a l. 췍i 췍0c Whent het h i ckne s sb sde t e rmi nedf r ombbyu s i ngEqua t i on ( 29),D anbec ompu t ed.Then 0 0 췍0andt 췍x Equa t i on ( 30)g i ve st hei n t e r na lp r e s s u r eP hea x i a ll o a dN f r omt hes c i f i edP0andNx pe
f o rt heho tc l i nde r.The s es t e st henf i xt heg e ome t r ft hec o l dc l i nde randt hep r ima r y p yo y y s s t e mo f l o a d s . y Fo rt hea dd i t i ona ls e c ond a r o a d s,t hef a c tt ha tEqua t i on s( 27)and ( 28)a r el i ne a r yl
equa t i on sc anbeu t i l i z edt oi n t r oduc eana ddedf r e edomi ns c i f i ngt hel o a d s.L i ne a rr e l a t i on s pe y a r eno ta l t e r edby mu l t i l i ngt heva r i a b l ebyac on s t an t.The r e f o r ef o ra dd i t i on a ll o a d sa nd p y a dd i t i on a ld i s l a c eme n t s,t h enond ime n s i on a lqu a n t i t i e sf o rt h ec o l dc l i nd e ra ndt h enond ime n s i on a l p y a n t i t i e sf o rt h eho tc l i nd e rn e e dno tb ei d e n t i c a l,bu td i f f e rbyaf a c t o rε.Thu s qu y
췍 췍 췍′)=ε( ( u ′, v ′, w u ′,v ′,w′)
and
췍 췍 췍 췍′ 췍′ 췍′ ( , , ,′ ,′ ,′ n ′ ′ n ′ m m n ′ n ′ m′ θ, θ; θ, θ )=ε( θ, θ; θ θ) ξn ξ ξ m ξ ξ n ξ ξ m ξ m
( 31)
p′ =ε p′
( 32)
Then -
Bu tt henond imen s i ona lp r e s s u r el o a d i ngpi sr e l a t edt ot hea c t ua lp r e s s u r el o a d i ngbyEqua t i on
670
COLLECTED WORKSOF HSUE-SHEN TS IEN
췍′f ( r a ond c e hes randt nde i l dc l o hec rt o ng sP i d a o el r u s s r e a r ond c e hes et r o f e r 19).The y y yp
r e s s u r el o a d i ngP′f o rt heho tc l i nde ra r er e l a t edt h r ough p y
췍 췍′ = çæD0εö÷P′ P èD0 ø
( 33)
췍0ε/D0 .S Ther a t i oo ft hep r e s s u r el o a di st henD i nc et her a d i u sR andt hel eng t hL o fbo t h cy l i nde r sa r et hes ame,o t he rt so fl o a d ss ucha sc onc en t r a t edf o r c e,o rmomen tf o rt hec o l d ype
cy l i nde randf o rt heho tc l i nde rmu s ta l s obe a rt hes amer a t i o.Ne ed l e s st os a hel o a d sf o r y y,t t hec o l dc l i nde rmu s tbea l i eda tc o r r e s i ngpo i n t sf o rl o a d si nt heho tc l i nde r. y pp pond y Thea dd i t i ona lf o r c e sN′ ′ t hea dd i t i ona ls he a rQ′ andt hea dd i t i ona lmomen t sN′ x ,Nx θ, x, x,
Mx′ tt heend so ft hec l i nde ra r ec on t r o l l edbyEqua t i on s( 15),( 16),and ( 19).I ti se a s i l θa y y s e ent ha tbe c au s eo fEqua t i on ( 29),t her a t i oo ft he s equan t i t i e sf o rt hec o l dc l i n d e ra n dt h e y 췍 ho tc l i nde ri sa a i nD0ε/D0 . y g
The r e f o r e,knowi ngt hel o a ds s t emont heho tc l i nde r,onec anf i ndt hec o r r e s i ng y y pond l o ads s t em f o rt hec o l dc l i nde r.Thef a c t o rεf o rt hes e c ond a r o a d sc an becho s ena t y y yl , c onven i enc eo ft heexpe r imen t e r.Fo ri n s t anc e ε mi tbes ocho s ena st o maket her a t i o gh 췍0ε/D0 equa D lt oun i t . T h e n t h e s e c o n d a r l o a d s s t e mf o r t h e c o l d c l i n d e r i s e x a c t l t h e s a m e y y y y y a st he ho tc l i nde r. When t he p r ope rl o a df o rt he c o l dc l i nde ri ss e l e c t ed and t he y y , c o r r e s i ngs t r a i nont hec o l dc l i nde rde t e rmi nedbys t r a i ng a e st hei nve r s ee i va l enc e pond y g qu
r ob l emi st hent of i ndt hes t r a i ni nt heho tc l i nde rf r omt het e s td a t aonc o l dc l i nde r. p y y Takef o ri n s t anc e,t hea x i a ls t r a i nex ( z).Fo rt hec o l dc l i nde r,a c c o r d i ngt oEqua t i on s y ( 1)and ( 25),
췍′ z 췍′ ö 췍췍2w æ췍u 1 췍x ( ÷ nT + ç e z)=k1 R 췍ξ2 ø è 췍ξ 1+ν
( 34)
whe r ez 췍i st heva l ueo fz me a s u r edf r om t he med i ans u r f a c eo ft hec o l dc l i nde r,mi dwa y y 췍 be twe ent hebound a r u r f a c e s.Nowl e tex bet hea ve r a eo ft heme a s u r eda x i a ls t r a i n sont he ys g
췍x bet ou t e rs u r f a c eandont hei nne rs u r f a c eo ft hec o l dc l i nde r,andl e tΔ e hed i f f e r enc eo ft he y
me a s u r eda x i a ls t r a i n sont heou t e rs u r f a c eandont hei nne rs u r f a c eo ft hec o l dc l i nde r.Then y ( ) f r om Equa t i on 34
췍 췍x =k1 - 1 nT +췍u′ =k1 - 1 nT +ε췍u′ e 췍ξ 1+ν 췍ξ 1+ν
( 35)
and
췍 2 췍 2췍 췍x = b 췍 w′ = bε췍 w′ Δ e R 췍ξ2 R 췍ξ2 Fo rt heho tc l i nde r,t hea x i a ls t r a i ni sg i venby y
췍u′ z 췍2w′ ex ( z)=k1 + 췍ξ R 췍ξ2
( 36)
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
671
췍2w′ 췍u′ and 2 f r om Equa t i on s( 35)and ( 36),oneha s Bye l imi na t i ng 췍ξ 췍ξ z 췍ö 1ö æ 1 æ췍 1 ex ÷ nT + çe ex ( z)= ç1- ÷k1 + x췍Δ ( b εø è ø ε εè 1+ν)
( 37)
Theva l ueo fzi sme a s u r edr a d i a l l nwa r df r omt hemed i ans u r f a c eo ft heho tc l i nde randt hu s yi y i sl a r e r i nm a n i t u d ef o r t h e i n s i d e s u r f a c e t h a nf o r t h eo u t s i d e s u r f a c e . g g S imi l a r l y, 1ö æ 1 1 æ췍 z 췍 ö eθ ÷ nT + çe eθ( z)= ç1- ÷k2 + θ췍Δ ( εø b è ø 1+ν) ε εè
( 38)
and
γxθ ( z)=
z 췍 ö 1 æ췍 çγxθ ÷ 췍 Δγxθ ø b εè
( 39)
췍θi whe r ee st hea ve r a eo ft heme a s u r edc i r cumf e r en t i a ls t r a i nont heou t e rs u r f a c eandt hei nne r g 췍 췍 췍 , ; s u r f a c eo ft hec o l dc l i n d e r Δ e i st h e d i f f e r e n c e o ft h e s es t r a i n s γ a n d Δ γ a r et h e θ xθ xθ y
c o r r e s i ngquan t i t i e sf o rt hes he a r i ngs t r a i n.I nEqua t i on s( 37)and ( 38), k1andk2,andnT pond ( ) ( ) a r et hep r ima r t r a i n sc ompu t edf r om Equa t i on s 25 and 8 .The r e f o r et he s ee t i on s ys qua a l l owt hec a l cu l a t i ono ft hes t r a i n si nt heho tc l i nde rf r omt e s tr e s u l t sf r omc o l dc l i nde r,and y y
t hu sc omp l e t et hede s i r eds imi l a r i t aw. yl , Fo ras t r e s sana l s tt henex ts t e spe r ha st hec a l cu l a t i ono ft hep r i nc i l es t r a i n sa te a ch y pi p p
va l ueo fzi nt hes he l l,andexami newhe t he rt hel a r e ro ft he s ep r i nc i a ls t r a i n sexc e ed st he g p
de s i imi to ft hema t e r i a la tt het empe r a t u r ep r eva i l i nga tt ha tpo i n t. gnl
Examp l eo fDimen s i on i ngt h eEqu i v a l en tCo l dCy l i nd e r Asan examp l eo ft he p r o c edu r e ou t l i nei nt he p r ev i ou ss e c t i on,t he d a t ag i ven by []
No l and1 wi l lbeu s edt of i ndt hee i va l en tc o l dc l i nde r.Thet empe r a t u r ed i s t r i bu t i oni nt he qu y wa l li st akenf r omF i 2o ft ha tp a randi sr e r oduc eda sF i 1he r e.Thema t e r i a li sa s s umed g. pe p g. t obe19 9 DL,andt heva r i a t i ono fYoung ’ smodu l u swi t ht empe r a t u r ei sp l o t t edi nF i 2, g. a a i nu s i ng No l and ’ sd a t a.F i r s t,t hepo s i t i ono ft hemed i ans u r f a c ewi l lbede t e rmi ned,u s i ng g Equa t i on ( 10).Th i si sf oundt obe0. 588bf r omt hei n s i des u r f a c e.Nex t,byt ak i ngt hec o l d cy l i nde rt obea t100℉ ,t her a t i oo ft het h i ckne s s e so ft heho tc l i n d e r a n d t h e e u i v a l e n t c o l d y q cy l i nde ri sc ompu t edbyu s i ngEqua t i on s( 6),( 7),and ( 29).I ti sf oundt ha t
췍 =0. b 936b
Thu st h ee i v a l e n tc o l dc l i nd e ro ft h es amema t e r i a li s93. 6p e rc e n ta st h i c ka st h eho tc l i nd e r. qu y y 췍0/D0 . Ther a t i oo ft hel o a d sont hec o l dc l i nde randt heho tc l i nde ri sc on t r o l l edbyD y y
Th i si sc ompu t eda s
췍0/D0 =1. D 29
COLLECTED WORKSOF HSUE-SHEN TS IEN
672
F i 1 Temp e r a t u r e g.
F i 2 Young ’ smodu l u sa s g.
d i s t r i bu t i oni nt hewa l l
afun c t i ono ft emp e r a t u r e
Byt ak i ngα =10-5 /℉ ,andb =0. 095i n.,t heva l ue so ft he rma ls t r e s s e sa r e
NT =22400l b/ i n M T =192l b.
nT =0. 958×10-2 mT =-0. 124
I ft her ad i u so ft hec l i nde ri s2. 25i n,andi ft hei n t e r na lp r e s s u r eP0i s1500p s i,t hen y 0 44×10-3 p =1. 0 I ft heax i a lt en s i onNx i sduet ot hes amei n t e r na lp r e s s u r e,t hen
nξ0 =
p 72×10-3 =0. 2 0
Now æ R ö2 β =12ç췍 ÷ èb ø The r e f o rt her a t i oo fmT/ s βandnξi 0
0 mT/( 0224 βnξ )=-0.
Th i si si nde edsma l l e rt hant het h i ckne s s r a d i u sr a t i ob/R = 0. 0422.Henc et hes u rmi s et ha t 0 / u s t i f i edbynume r i c a lc a l cu l a t i on. mT βi l i i b l ea a i n s tnξi snowj sne g g g
Junc t i onS t r e s sBe twe enCy l i nd e randHe ad Thep r ev i ou sf o rmu l a t i ono ft hes imi l a r i t awf o rs t r e s s e si nt heho tc l i nde randt hec o l d yl y cy l i nde ri sb a s edupont hea s s ump t i ont ha tt hes e c ond a r o a ds s t emi ss c i f i ed.Th i si sno t yl y pe t ruef o rt hej unc t i ons t r e s s e si nduc edbyf i t t i ng,s a s r i c a lhe a dt ot hec l i nde r. y,ahemi phe y
Suchs t r e s s e sa r ede t e rmi nedbyt hee l i t fde f o rma t i on so ft hehe a dandt hec l i nde ra tt he qua yo y unc t i on.I ft hes emi s r i c a ls he l lha st hes amet h i ckne s sa st hec l i nd r i c a ls he l l,t hent he j phe y t empe r a t u r ed i s t r i bu t i oni nt hes r i c a ls he l lwi l lbet hes amea si nt hec l i nd r i c a ls he l l.An phe y
AS imi l a r i t o rS t r e s s i ngRa i d l a t edTh i n -Wa l l edCy l i nd e r s yLawf p y He
673
췍, sb s ckne i h rmt o f i fun oo s l dea anbema imenc c e s ts dt l o r”c a l imi s he “ tt ha howst ss i l s ana pe y de t e rmi nedby Equa t i on ( 29).Howeve r,t hes imi l a r i t fbo t ht he p r ima r o a d i ng and yo yl : s e c ond a r o a d i ngnow r e i r e sana dd i t i ona lr e s t r i c t i on ε mu s tnow beun i t he s e yl qu y.Whent c ond i t i on sa r efu l f i l l ed,t hes imi l a r i t fj unc t i ons t r e s s e si nt heho tc l i nde randt hec o l d yo y cy l i nde rwi l lbea s s u r ed,andt her e l a t i on sp r ev i ou s l l opedf o rc ompu t i ngt hes t r e s s e si n ydeve
t heho tc l i nde rf r omt e s td a t aont hec o l dc l i nde rr ema i nva l i d. y y
Ri ngS t i f f en e rAr oundCy l i nd e r Tos t r eng t hent het h i nc l i nd r i c a ls he l la a i n s tc onc en t r a t edl o a d sf r omt hemoun t i ngl ug s, y g
ar i ngs t i f f ene ri so f t ena t t a chedt ot heou t s i des u r f a c e.I tr ema i n sc o l d.Young ’ smodu l u so f “ ” t her i ng ma t e r i a lt hu si st ha to fc o l dma t e r i a l.Tode t e rmi net hed imen s i on so fa s imi l a rr i ng
f o rt hec o l dc l i nde r,t hec ond i t i on st obes a t i s f i eda r et ho s eo fEqua t i on s( 31).I no t he r y , wo r d sf o rar a t i oεo fde f o rma t i ono ft her i ngont hec o l dc l i nde rt ot ha tont heho tc l i nde r, y y 췍0/D0 .Th t hef o r c er e i r edmu s tbe a rt her a t i oεD i sme an st ha tt her a t i oo ft hes t i f fne s so ft he qu
췍0/D0 .I r i ngf o rt hec o l dc l i nde rt ot ha tont heho tc l i nde ri sD ft her i ngi sr e c t angu l a ri n y y 췍 , / s e c t i on t hec o r r e c tr a t i oo ft hewi d t ho ft hetwor i ng sf o rc omp l e t es imi l a r i t st henD0 D0 . yi
Ther i ngont hec o l dc l i nde ri st hu swi de rt hanont heho tc l i nde r.Whent h i sc ond i t i onon y y t her i ngd imen s i oni ss a t i s f i ed,t hes imp l es imi l a r i t e l a t i on sf o rt hes t r e s s e si nt hes he l la r e yr a a i nc o r r e c t. g
Re f e r enc e s [1 ] No l andR L.S t r eng t h so fSeve r a lS t e e l sf o r Ro cke tCh ambe r sSub e c t edt o Hi t e so f He a t i ng. j gh Ra
ou r n a lo fAme r i c anRo cke tSo c i e t 6):154 162. J y,1951,21( [2 ] Timo s henkoS.The o r fP l a t e sandShe l l s.McGr aw-Hi l lBookComp any,New Yo r k,1940:389. yo [3 ] Yu anS W.TheCy l i nd r i c a lShe l l sSub e c t edt oCon c en t r a t edLo a d s.Qu a r t e r l fApp l i ed Ma t hema t i c s, j yo 1946, 4:13 26.
[4 ] Donne l lL H.S t a b i l i t fTh i n -Wa l l edTube sUnde rTo r s i on.NACA Te chn i c a lRepo r t473,1933. yo