ML20151T580
| ML20151T580 | |
| Person / Time | |
|---|---|
| Site: | South Texas  |
| Issue date: | 05/31/1977 |
| From: | Larson L, Stokey W CARNEGIE-MELLON UNIV., PITTSBURGH, PA, WESTINGHOUSE ELECTRIC COMPANY, DIV OF CBS CORP. |
| To: | |
| Shared Package | |
| ML20151T466 | List: |
| References | |
| NUDOCS 8808160411 | |
| Download: ML20151T580 (6) | |
Text
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.hq Reprinted from Siny 19M Vol. 99, Jemtl d Preneure Vene=1 Technology v.M h
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P q The Inelastic Deformation of Pressurized
- u. u= Stain ess Steel Tubes Under Dynamic Bending
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- i. u.m. a s u o Y W* F' $10KE1 A '"' P'*'" **'*"'"*' '"* *"'** "*"' '"P0 Me k hador of inh!'MUJ prenwri:ed Ides subpeted to dsnamic beruling and torsion loads. The basic esperi.
op[;
i; assoenste vro ressor, ment eras a der >p test its which the cantileser tube was 4jnamically loaded by a toretus u oeoartment es ueesen,ut ges.r eenes, arm vCh a coventra!rd ireight at the end. The experiwental dats consist of strain
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P Peen"N,Tn'?a' tt!*** time records arul peak ami permannt deflectierts. .t nalytical predictions of the uom, asun respone nrwre made using a tumped parameter beam model of the structure. This
(%*g timple represen!ation of the prenurited tube is made passible by uw of an approzimate arsalytical modelfor the inelastie deformation of a pipe element subpeted to preneure,
&jl a &nding and torsions. The predicted responses are compared scith measured responses ff to sness the accuracy of the models.
a.
b p m M lntrodMCtl0n combiwi loading. It vu shown that the mWel gave twi pro-dictions of deformation in Type 104 stainless steel tubm sub.
'M.
q7 A problem at intere.t to the nuclest industry in the evaluat.em jcetel to combinalloa, ling, appliel statically. The ir.centive for g of the cim einences of certain pn tulatel scrident conditions i' desetoping the simphnel m..lel n u for application to dy namie p s; the re-pon e of prewirient pipe, to dynamic lovis which are landing of piping systems into the plutie strain range. Beewe eere enouah to cait e lure inel 6 tic deformation.. Secten !!! uf the nonhnearity of uch v. tem 4, the e<piation. of motion must
.N(p of the M.\lE Ilotter an.1 l'res ure \ e el Code lllt provide, rules le integratel in a stepwine'fuhion, con.idering all loads simul-for the es alnation of the re ult. of "} sultel Conditgons," m hich tann>ualv. Conceptumily, eith the u e of digital computers, such are estremel). lou.probabili y po tubted esent . l ur ewaple, t
step by step integration can be accomplished. Ilowner, there y in a pipior system an mee -ment of the consequences of pipe is a practical limitatico to the number of decreca of freedom
- Q whip following a rupture in the pipe rnight be re.piircl. thnit can ba handlelin a given problem. For a three dunensional This parer pre ent the tr,uli of espenments that we,e per- pipina system it is desirable, at leut m prelimincy analysis, to to
- 4 forme! to inve tigats the inela< tie t=hauor of pre un,ned tubes able w ded with an.<< leam elemet.ts e in the ela4 tic analv<is.
.y under dpamic lomlina Tubes mm.le nf Type 3:14.taintee steel In thi. paper, analytical prediction 4 bam! nn tha simphned were precurised to stre levels rmtmally penaittet in the da- ntodei are compared with the experimental results obesbel from 9 sign of piping for reactor systems, and subjected to drep te te dpie IWig W pre.uriset enntileveral tube <.
m =hich cauard adnificant plutie deformation under the dy.
- namically inducal bendirig and torsiond loads. The testa p%
3 yieldet strain verms time records and mesaurements of peak and Test Pfocedure
, ;, r permanent dettectiens Analytir;d prnlirtion< acre male mint The tubes were loaded by irop te* tine. u.ing the apparatus
.;, a niep by step numerical integration of the equati.m of raution ,hown in Fig.1. The tube being extel L supported as a canti.
fi.e a lumped parameter repre*entation of the tube. les er by the clamping fisture mountel on the ta<t carrige, which
%'J In an earlier paper [21, the author de enbrd an appnnimate ;, g,iided by too vertical ruls on 24-in. (fio Wm) centers 4 model for the clutie.pla tic analpis c.f a pipe element under The quie's releue mechaniam at the top of Fig 1 is supportal
.,4l 4 by a line which pum through a pulley, no that the drop height
, i can te readdy adjusted. The tube is pressurisel with oil sup-
%(' iNessher. to brutete d ca R renee= se ese et sacer, plict through the fleuble .rcuore line. The torque arm a h;*h is clamped to the tube, hu a weight on its end, and applias the Coate,heted by the Pressmee YeeWe sad Pipase Diession sad pressened inertial bending and tors onal lon.lt. The motian of the te*t ah on she Petroensen MeessaneeJ l'.asineenas sad Presease Vessels sa4 Pisses resterence, Mesmo Oty, Meuse, unreteentee le so. Isra, ed Tme AussK.m carri.1ge is arrested at the bottom uf its fall by trualllead plates,
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14acists ce M acunawa Emesmosas Mausenps vossoved at 48M F tiehe-which stop the carrige motion suddenly, tith Yefy httle r8" gearsers, Mar s4, Is76. Paese No. To PVP.st. bound. Theme plates are placal at the nadal points for free free q.
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%,Y t and a length of 2 in. (5.08 em). The matcial wu annealled, stau i(tusti after machining, to remove residual straves. While the heat
- Ue t s tie r s treatment caused some bowing of the specimen, the wall thick.
, $,C,'h',', ness remainel uniform, since no machining was done af ter the heat treatment. Tensile testie were cor.d.leted on specimens, which had been subjectal to the name heat treatment, to deter.
- mine the strens. strain behavior. A plus with a hole in it, through staw i,onous anw, which h pressurising oil was supplied, wu placal in the sup.
,,, ported end of the tube. A solid plug wu placalin the other end.
j/ These plugs were tack welded to the tube to seal the oil.
Analysis ee. : o r ase e..e..is . 4.t ne use of b pipe element denerbd in reference (2) permiu l the structural model to be reduced to a simple beam structure.
In the dynamle analysis, the lumped parameter model shown l dbration of b carriage in its fundamental mode, u determined schematically la Fig. 2 wu unal to repreent the preasurised by teet, tube and the torque arm. The direction of shock is normal to the < !
I'ont yield re.intance strain gagns were mounted with various plane of the model he preuurised tube, reproiented by Ikam orientations on the outer surface of ahe tubs near its supportal I, w as subdivided into a number of maeless rigid links connected h end. The gasce mere connected to ampliners, N outputs of by Rexible joints, =hich repre=ent the nexibility of length L, which were recorded on a Visicorder using four galvanometers '"*"l' *S *
- P"i" t i- Fpacing of the joints was varial with 4
hadng a frequency re<ponse of 0-10.M cps, and nne having a smaller lengths nsal near the support where the higheit deforma-
, 9t, og o 44)0 epe. taunt occurred he mw m, concentrated at the Jomt i represcota To mauuro the peak dynamle de8ectiorts, pieces of pluticine the maw of 6 Lube and Guid contained mithin it, of the length ;
i were plarai on the test carriage arid on a reference arm locatal 8Mendant to the j int. De torque arm is repreuntal by a
- under the torque arm. The initial gaps between the pluticine single Semible element of length Lt. since its behavior wu j and the bottom of the torque arm were such that the pluticine elastie during til testa.
eaa contacted and deformal daring a te=t. l'ermanent de6ee, The forces acting un element free lulies are shown in Fig. 3.
I tions wm found using appropriate measurements. f*iDE "luilibrium of the mwlem links to obca'n the shear forces in terms of moments and torqueg and subatituting thene into N t ' *** "d 1** ' ' * "" Poin t yidda, for i < r, r Specimens being h number of elementa in Ikam 1:
i De test specimens were machined frosa pieems of extruded esarnisms tube having an inside diarneter of 1.007 in. (2.56 ern). A i,= [ "'
+
+' '
'"' \ (1)
\ la L* la . ' 4' +8 / - p, test leech of 4 in. (10.14 cm) ce 10 la. (25.4 cm) =ne machined l to a wall thichoem of atM in. (0.127 son) og 6.100 is. (0.254 cm). ,
De supported end which Attaj late b elamping Esture had j am outaide diarneur of 1.278 la. (3.34 cus) and a length of 3 la, p, . ace.neration of 6 ma. relative to W founda-(7.42 ern), while h needing and had b anme outside diameter Lion
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p = foundation acceleestion For element i, onutting coediciente chich aro a:ro, eyntions -
For the lut mus on the tute:
v5 Mus T r
dK.=,3 Vn+Ba We+ 7,gh (Bo).C W 1(
- m. 4. Le) _ v5 3
=here m. equ ds mw of lut element of tube plus additional ma+9 dNd " g (Bu).d.II. + g f A .e + B ).d T (106) of clamping 6sturN. or the element reptmenting the torque arm:
Solvirig equation (162) for d.tl., subtituting into aquadon (106),
and using epadons (6), (7), and 0) yields:
7 '
p... - y. (3)
""I E' d y. . - d y, _ 2 (Bu).L. .
where m,.i equals the mus of torque arm plus mus of weight L..i r , (A u + #a).
added to the end. dT = (11)
- Bt Le Integration of the equations of motion su performed udng 3 j a +ug, E' + EG the foUoming finite diNerence equatiims fli to esprms the de. .g "[i A **
- U" -
deetion and its ttmo derivauona, =here a is the nth time step:
y.(f.) = y,(f.a) + i.(f..i 1A L + 0.7 .W.(f..i) (02) Thus, the change of torque ran be found in ternu of known D"" ** " "** "I
- y tt.) = y (f.a) + 0 *s .iliyll.a + p (f.)] (th) theu found from equation (lN).
Immeihately prior to the application of the dinamic load, the A time step .it = 0 000ml see eu unal. tube is n'reued only by internal prwure. The @namic die It is numal that on impact at the end of the drop the found3 turbance is introducal in the form of a rmtangular decelera.
tion modon is arrected in a time inservsl I, by a con tant ac.
tion put,e of very short duration apphal to the f#mdstion. Thus celeration >ielding:
the velvcity, the seederation, and the stress state for asch tube
- ." - V2g4/is i<4 (.*=2 ) element are known at the start of the transient. The values of theos variables, and of the di< placement, after each suceennive
- y. = 0 i > Is (56) time step are found by the following sequence of operations:
The displacement configuration at the end of a time step is g g gg gg 4g g, known and the change of curvature at a bint itan be determined ggg ,
from the central di#erence equation: for the system.
dy, . i 2 From thh, the changes of the curvature an i the ansk of
( L. 4 L. . ) 1
+ ) teht are calculatM for each tube element.
~ (dy.a3~ L.L. . . L. .i ( L,).
3 ne new state of stree for each tule element is de' ermined pince rotational inertia of ne tube is neglected, the torpe from the approximate tube defornation mel, new Internal forcen are calculatal and the force system acting on each lump l T h considered to te uniform along Beam 1.
mm W updated.
The change in the snak of twist at the end of the tube is th' 4 A new acceleration and s clocity is calculatalInt evh mm, num of the changre for the elements: and the procms is repeated from Step 1.
Since the pipe element analytical mod.l of reference (21 is dj, = (dK e),(Lf ). (7) formulatalin terms of the strms re%ltants and their amciatol ;
- s'eneralized strains, the phy4 cal strains at any point in the croo section must be obtainal by swooming a remnable strain div e hich may aho te upruaal u' tnbution. The eleue strain incremants re redly obtain,1 from the (twical unumptions. In this anafy45, the plwic.
ErdT 2
1
- 0) strain vector is wumal to vary linearly from the neutral asis d$. = (dy,a - dy, - ,
to the outer Ibrs. The ptA tic bending stram increment.s then ehere the third term repreesnts the dedeetion eaum! by the vary linearly across the tube and are tension on one 4 le of the neutral asis and compien4an on the other. The h+p and tar.
bending of the elutie torque arm, The coeficients for the equations for the deformation of a tuba sional strain increments atu vary linearly but hwe the wne Man in toth regions. Tran formation matricm relating physical element, u derival in referwice (2), are gisen in the Appendit.
strains to the generali =1 strains are pre *ntel in reference f41 The equations are of the form:
,)
dX = ( A n + Bn>lV + (Au + Bo)dT (S2) Results Table 1 summarisee the tent variabica for nis ta u performal.
b 2 F r the prwuri2=1 tuben, the initial hep stree wu apprusi.
4K,e = 4R'4 ( A. + B,ip.tf + 2rR84 ( A.e + B., T (96) mately W*e of the statie yiell strms of the material. Fig. 4 ,
shows npecimental and calculatal strains for Tn.t 1 in which l abere K = curs ature of tie tute bending and tarsion we applial dynamically to a preu trined j K.e = angle of leist per unit ler.gth tube. De bottom eurves show the teding strin obtainal from j '
longitudinal strain rneuurements an the top and tottom sur.
A ** " 'I"86* ** dei *nts f aeve at a location 0.75 in. (1.91 cm) from the clampal end.
8 = plutie coettrienta Deme strain magnitudm di#cred by Ims than 10 percent frorn the bending strain rnagnitud = mhoen. ne bending strain to.
M " *PPlied l*ndtog mornent sponse shows the double peaks that are characteristie of systems T = spphed torque in which two natural malas of vibration prolominate. This is M/NW Transactions of the ASMC
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Totes 1 feet pecemetere Tube Walt Drop Tag length thicknese hight Preneure t an A n e On.) Ga.) On.) (pe0 l 10 0.050 24 2500 Beeding and tornian (25.4 cm) (0.127 cm) (60.96 cm) (17,200 kPa) 2 10 0.100 48 5000 Bending and torsion (0.254 cm) (121.92 cm) (34,500 kPa) 3 10 0.100 48 0 Bending and torsion 4 4 0.0$0 49 2500 Bending and torsion (10.16 een) 8 4 0.004 48 0 Bending and torsion 4 10 0.100 48 6000 3endang eu - ust n 0 O rtst 0' 7 : 57 4 a te st a a test s e" " Citti 3 P itst 6 MtfW 4$*
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in. 0 04 Os it 16 enteiCit0 ETRatm. Ptt Ctat e se -
OtN0i40 StRales flg. l Campetteen of predleled eed meesured atralse D.rto 4 0 vi cml
_ s 3.," 'e se ,-
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strains while hoop strains on the top side were quite arnall Dis I
'" F is due to the diferent strema states in the two regions. he top I
eurves la Dg. 4 show the reponsee of a gage locatal on b bot.
,, i i i e r e tom side at a location 64 8 dg from the tute midplane and om se en o ne en ou * *e oriental at 45 deu to the tube uis. The toponne rencets the torsional lue 1 and little evidence of the highee frequency mode is Fig. 4 Predleted and miessered respeese fee teet no. I prement in eithee the es!culated or emperimental twponse eurym.
8ince no damping was included in b analynia, the agreement beteeve the calculatai and esperimental renprese curves is reunnable only for the Brit eyele. Al.w the correction factor typical of b tests in which both bending and torsion were ap- applial to the static material properties to account for high phed, la this type test, initial yieldirst is eau al primarily by strain rate efect vu bued on the initial strain rate only and no b mus on h end af h tube ehirh mouw with a higher fro- attempt wu mm.le to une an tratantaneous strain rate erect
{
quency than & maan on h end af h toique arm. his mais varial spatially throughout h structure. Therek.e. a com. 'j toversee direction tdore h mee on h end of b torque arm pari >n of past and permanent deMection provide b t==t etusing entne unionding fewed by a second peak. The middle bade for n==ning & accuracy of the models.
eursm show & boop strain repon.es at an angular location A comparw>ei of measural and profic.at peak and permanent 64.5 dg frora the tute midplane. Iluop strain repon e curs en strains is showo la Ms.1. Peak and permanent de8ection et the on the bottom side um typically very Mat although owne "noiw" end of h tuht and at the end of the tocque arm are compared b preent la b experimental curse shoe a. Ten ile hep attains in Fig. 6. Theae comparisons show that the behavior is being to the bottom side (ehich is initially lunded la compeension predicted resannably well, although bre la considerable scatter WtudinaDy) ver, of about & same magnitude as b bending la the data.
Joumal of Pressure Vessel Technology u Ay isn / M1
F tastno: o test e O t e si ., References l 7 nu a a tan s O 1tst 3 1 Bedee end Prwevee resul Cait ASME, Secen !!!,1971
> ttsi e edition, p.101 par, NB-3225.
ocrts e cet= s,wsoo mcicaras etaa etattctio" 2 1, arson, L o., stokey, w. F., ud transen, w, E., "An
- t. sctio svuoct =cotts etamantet otettctio" Approumste Model for an Elastie. Plastic Pipe Dement Under s saanto svueot.moicates outterio= at two Combined Loadin ." Jot'anat, or Pazucas Vcaan Tscwmot,.
octomousanw oot 224 T aans. ASS E, Vol. 97, Senes J No.1, Feb.1973, pp.
e w=eamato 3, meet inoicatts outection at tho or eatssveitto tvet 3 Biggs, J. M., late.h.dio i le Strudural Dy'us asics, McGraw.
}idi Book Corppany,1964.
4 Larson,1,. D., ainclutie Reeponse of Pressurised Tubes 3 L i i l'ader Dynarnie Bendar and Torsional leadi
- PhD thesa, Mechanical Engineering utment, Carneci fellon Uruvet.
sity, 1973; Uruversity Micrordm Order No. 73-22C2.
5 Baser, W. E., "Validity of Mathemaucal Edels of D .
a ~ '" namie Response of structurce to Traan>+nt leads," De 4 1 and ribraries Bi,#ain U, Put 7, Dec.1971, pp. l'e-23, N72 16643.
O 4
5 ee -
9 .-e-j APPENDlX 3 _.
As developel in rderence (2) b equations for b bending a
{ o h and torsional ddormation of an eternent of tube whoo b bend.
los moment and torque are vuied, h pressare and uial force y being constant, are:
e4 -
8 hi - (Au + Bn)dQi + (Au + Ba)dQ. (12)
, , , , he = (As + Be)dQi + (A. + Be)dQ. (13) e es e. es se se 2 Patoecito otettction 04 I where gi = - RK rio, e ceaepartsee et preenseed e.4 sieeee ed seneesees R
pe = g Kes OlSCuslI0n N "88""id"8NIU*
l'oua!!y h data from dynamic testa, espeda!!y thnee in K = tube curvature which plastie ddortnation occurs, show gruter scatter than those K,e = egle of twist per unit length obtainal frorn statie testa. Based on survey resulta, Baker (S]
reportel a minimum and muimum ocatter of 2 29.4 percent ,,, ,8 _
and i 67.3 pernent, respectively, for premanent deformation rE e of edl-controllai canulevee beam testa. The spread in the data reportal here is generally within 2 25 percent of the pratictal du = Aa = 0 value although nome pointa are outaide this band.
N correlation of analytical prmlictions and experimental A= = 2(13g + e) resulta shown here dong with the correladone for stade loading previously reportal in reference (2) demonstrate that the baaie C Q, i approsimation for a tube or pipe elaneet used to reduos a com. 8" " p ca.:g, ples thrne dimensioast problem to a aimple bearn problan give 5 rm.onable results. Thre are neveral potendal applications for C QiQ.
the approsimate modd. Since the inelmatie dynarnie response of 8"
- Id " r co p, '{i a preeaurised piping system is an entrernely comples problem, the degrw and type of approsimation usad wul ultimately de- 3" , @
pend on b particular probian and the desir=1 result. For eom. Q,'
pies systeins, approximate models such u diaeuenal here could .
te very undul in idenufying what crose secuon in h structure , 3 .g O'
udl undergo grues yiekling during a trarteient. 'It.eee locatione Q, .
+ i (Q, Q,): 4 Q,t foe fe couk! then be studied in more detail by re/Laed methods if re- ,,
qulted. * ~ O' ~ b g, . .E jf,l < E. l 4 2 Summary and Conclusions N' ~ ~ O
Yl ~ 3 4
% approsimate model for analynia of & laelutie ddarmados f = angle dcAning loeation rA neutral amis of a tute demeni under combin=1 loading has tven applied to the pratacnon of the dpamle inelude behavior of tuben. The O' = Jit/4Rsh, generdis=1 bendais strms predictions of peek straise and deAcetaons agreed remonably Q, . y,f A, gee,talised boop stress i crell with the meesured rwults far a numter of ddwent tube danmaione and loadmg comismatians, On b buis of thee. Qi = - p/2, generalised rad 2al stras reautw it is mactudal that the approsimate model is a usedal tool for h analyus of dysarnie kieding tevelyseg a,ctu6casi g, , 47.jg,g,( genershed hu m inelasue deSection.
Qi = N./2rRA, geaaralind ulal streme 262 / M AY 1s77 Transactions of the ASME
w O. - seeeralleed dective strees at & crome emesi e , g, *, g [ g ge,-i Na - hoop ettes resultaat dQ. Ci \ Ci /
F = coe f. + f. ein f, p = laternal pressure Sutatituting the empresione for et, t., Qi, and Q. into equatione y . applied bendmg =-t (12) and (13) and airnplifying yields:
T = apphed torque dT N. - axial forse dK = (Au + Bu) h dM + (Au + Bu)
p 4 jg, Qbe ettaES hafdeOIDs sufYe Which C&G d 3 dT
""(!.E / be ate.ed frem a simple i.neio dKdi.:- (Aa + 8a) is dV + f ta + 8-) 3,3,,
I Journal of Pressure Vessel Technology u Ay 1in / 263
-