STRUCTURAL BESIGÏÏ PHILOSOPHY ANS CRITEHIAPOH COECRETE BEACTQK VESSELS - U.S. PRACTICE

W. Rockenhauser

Prestreased concrete reactor pressure vessels technology isvery young and codified. Standards have not yet been established.However, efforts are being made in the U.S., as well as in Europe,to write uniform "tentative criteria*' or "recommended practices"which could eventually become tue codes for design ani coastructionof these vessels. Much of the material presented in the paper isdrawn from the content of tentative criteria and from the designstandards for the first U.S. PCPV, the Ft, St. Vrain reactor, Itis followed by a comparison of European and U.S. standards and abrief discussion of the relative merits of the various approaches,

A PCRV comprises five major constituents: the basic concretestructure, the post-tens!oning system, the nonprestressed reinforcement,the liner and the thermal control system (insulation and ooo]ingsystem). The paper concentrates on the first three components.

19

1. INTRODUCTION AND DEFINITIONS

Concrète E.eactoj; Vessels as defined here, and as distinguishedfrom the secondary containaientG art structures which contain areactor or reactor (primary) system directly without the use ofanother intervening pressure barrJer. As such they are continuallysubjected to the pressure of the primary system whenever the reactoris in operation.

These vessels are commonly known as "piestressed concretepressure vessels for rp-îctoiV or more briefiy, as prestressed con-crete reactor vesseJs (PCEV's); hereafter this abréviation will beused. A PCRV comprises five major constituents: (1) the basicconcrete structure; (2) the post-tonsioning system; (3) the non-prestressed'reinforcement ; (f\ ) the Ijner; (5) tne thermal controlsystem (insulation and cooling system). This paper will be confinedto the first three», components.

PCRV technology is very young; and codified, across-the-boardstandards have not. yst been established. However, efforts are under-way in the U. S. 5 as well as in Europe, to write uniform "tentativecriteria" or "recommended practices'* which could eventually becomethe codes for design and construction of these vessels. Much of thematerial presented here is drawn from the content of these tentativecriteria and from thr> design standards for the first U. S. PCRV, theFt. St. Vrain reactor. It is followed by a comparison of Europeanwith U. S. standards and a brief discussion of the relative meritsof the various approaches.

2. PCRV STRUCTURAI, RESPONSE

The response of a correctly designed PCRV to an internal pressureload is illustrated in Figur? 1. The figure shiows three distinctregimes of structural behavior, which has in fact been observed onactual vessels. (Depending on the point of ineasurensent, the distinc-tion between the second and the third regime is not always as clear asshovvi in the figure).

The regimes can be explained qualitatively as follows:Regime 1. Starting from a compressive strain (Imposed by the

tendon system), the vessel overall response, i.e., strain, is linearlyelastic slightly beyond the point identified as RP on the ordinate.In this regime, minor cracking of the concrete may occur as a resultof thermal loads or local discontinuity stresses, but basically thevessel behaves as a monolithic structure.

20

Pressure

ULTIMATE STRUCTURAL STRENGTH

DeformationFigure 1 Idealized General Load-deformation relationship of a PCRV

Regime 2. With further increase in pressure, the general tensilestress in the concrete reaches the point where major cracking commences.The concrete continues to crack throughout Regime. 2 with the structuralsteel still responding elastically. The overall response of the vesselis, therefore, still elastic (although no longer linearly so) and thevessel deformations are more or less reversible.

Regime 3. At even higher pressure, concrete will be extensivelycracked and the structural steel is loaded into its plastic range.Vessel deformation increases very rapidly wiTb increased pressures untilthe ultimate, strength of the structure is reached.

The foregoing simplified description assumes a short term responsein which time dependent deformations are not a factor. During the designlife of a vessel, considerable irreversible deformation (primarilyresulting from creep and shrinkage of the concrete) occurs. This resultsin a concomitant reduction in prestress which inust be factored into thedesign. Creep and shrinkage deformation interacts with the short termresponse of the vessel only to the extent that (because of the reductionin prestress) it changes the point at which the concrete enters thetensile regime under increasing load. Il: thus results in the somewhatlower end-of-life response curve also shown in Figure 1.

Thermal loads (thermal gradients through the. walls and heads) donot result in significant deformations in a structure as highly redundantas a PCRV. They have little direct effect on the response of a vesselto internal pressure although the presence of elastic thermal stressesmay result in a slight lowering of the pressure at which the vessel ceasesto be linearly elastic. Temperature does have an indirect effect, how-ever, in that it influences creep and shrinkage rates and thus increasesthe long term changes.

3. DESIGN PHILOSOPHY; DESIGN/ANALYSIS APPROACH

Design Requirements, Definitions of FajLlure

As in the case of other structures, a PCRV, to perform its functionadequately, must carry the imposed loads "safely", and its deformationsmust remain within allowable limits ur.der these Loads. In addition, itmust remain impervious to the. fluid it contains.

These requirements, imply three definite types of failure: (1) struc-ture failure; (2) excessive deformation ; (3) failure by leakage. Theintent of the design process, and the intent of the criteria on the basisof which one judges the design, is then to ensure that none of these typesof failures can occur during the service history of the structure.

22

Bas ic PCR VjDe s i grt Philo s o phy

To accomplish these aims, PCRV'ss (at .least to date) have beendesigned to be fully prestressed against the highest pressure which •the vessel is expected to encounter in seivice. The question ofwhich normal or abnormal pres&ure conditions need to be consideredin this context, is, of. course , related to design of the reactorsystem used inside the vessel, the pressure or absence of pressurerelief devices, the overall plant design, and other factors. Itcan, therefore, be discussed meaningfully only as part of the overallreactor system design philosophy.

In addition, the design usually intends to furnish some factorof over-strength beyond the highest anticipated load levels, regard-less of any credibility considerations. This is accomplished byproviding more structural steel than would be needed for service loadsalone.

Within the range of anticipated loads, vessel deformations areusually small enough as not to affect significantly the functioningof the PCRV. On the other hand, long-term viscoelastic response mustbe evaluated to ensure that creep and shrinkage deformation does notrender the vessel unusable. Aside from the associated reduction inprestress forces, the most critical factors in this respect tend to bethe position, alignment, and dimensional accuracy of penetrations usedfor refueling machinery or removable primary-system components.

Thermal loads are,, only partially balanced by prestressing (overand above the prestressing required for pressure) ; the remainder istaken into account fay appropriate use of bonded mild-steel reinforce-•rnent. The ratio of over-prestressing to reinforcing varies considerablybetween designs. However, the liner (and, therefore, the concreteimmediately adjacent) is kept in compression under all foreseeablenormal and abnormal operating conditions throughout the design life ofthe vessel, this compression, may actually become rather pronounced,particularly toward the end of the design life, and appropriate pre-cautions are therefore taken to avoid the possibility of liner buckling.Design Methods

In current U. S. practice, the design philosophy of Section 3 isimplemented by the simultaneous use of several design/analysis procedureswhich can be categorized under three main headings.

Elastic Ana^lysis^Cworkinjg _jsj gss_j esi gp }

The elastic response of the vessel to mechanical and thermal loadsis evaluated by appropriate methods and the stresses thus calculated arejudged against working stress criteria to ensure that the vessel responsewill be essentially elastic, under all credible combinations of operating»accident, and environmental loads.

23

Liaiit .Analysis

>,oad factors are applied to predicted JcaKimunt loads; thenallovible limit criteria are s.ttisfied for the«e factored luads,usinp apsropriat<? limit design tecnniques. This 1-, done to ensurethat rh& structural bo»avirr of th*> vessel beyond th« range ^f

3-3*ds is, in fact, predictable «m«.l to pr'o«?idv« suitable over-aargjns.

Creep Analysis (Viacoelaitlc Analygif )• <n»«nn»^i*i mn«ii*. *inm •• •* «HMI*»MI 44»*. . •.••••i * *> , »»i ••»<•. i^mnimi n •JiiK^i •««•

Upjvcr aid lower bound evaluations art» »ado o" th* p«?rnaneutdéfera» irions which ca« be ocpectea during tî»t> servi e îife of vh<«veasc. , on the basis of creep analyse? which arc usually carried outin conjtujction with toe elastic oiaiysis

4, LOABMC UON^niOHS AV) «via»ÎK Bfk'D,*'& CONCiBEAiffl iN TtT

Loads

1*4»^ loadings oa a if7-^' '•#& b<2 "> tutorize* in (-vvera1 vay&; for," *: (1) Construe» ton Loada vs. «.ijeratiag. 2 »«<»»; (?) IOQ<!S

by tue ayct'j'n >)"$ «»i>r^roame"tjî l.)acs; O) afcnatt'csl vs.loads. Xhe cate4or*zôtioQi$ obvio»^lv» o\*-t 1 Ap» but /o*

tHelrss K^C to persil th« applicatJon of <H f forent allowables»to tt'jW.irv construe t ii»n ^ado t >an would be- usr-d f'^r r-Kr^ice icitis,and & that fhc-xiaa) lodd^ ts&v bA tr«aff«' dirt.rentJ.y if» t'i«? analysisfrom •'ethanic^l loads. /">so, jf^ie 'o^s m.iy »<«ed 10 "hoaly . the iim^w *.«*!«, n s .

Cable T lis^s fhe ty*>e<s »i l"$i»s, wj'«icui tog rd lu cri «which ar«" co««i^f»r»«i in tit. vôwst î ^»i^^. &ia'-î »xit«iu'*o.v remarks,

lax to soiae of these loadinp ^i-'^i.t-i^ C>îlo\ '{

Dead '.oad

aft» je«t4i«a ly \j"'.ta, inJ aro <• «n idcr*,* in fh#but fcv comoari&ou with r î» . oressui-» !>.,«*<:! tho> urc notSignî.u.snt, Hovev«-£» ue^d 'o<»ds ' • c- -i iaport.nt c<«»ri»t exit ion fotthe c?~ti<*l!y coj'v'pted v«*»s2* .it v«r*ous .t Ot * »

corre»poto«linr*' evi>-j-*.if «^ «ire r^dc

Loadgj P^pe ^ a d & and

Live loads, Pipe Lruul* aad Ra^cttcnv .11-2 significant factors»in th»-1 detail design 8tf»?t t^.es t-lativ^4y local ef»fo t t » i^ea ro a«consi îcred. mhia involves» priaoi»r.»ly, thi- dcsigt. ar«} analysid of pene-trations and penetration

TABLE I

LOADING O

a) Desd Loads

Weight of th% PGRV i t s e l f : we'lgbl of pur t s of v^ r sc ' . ^. variouscwjstjruct-iori sta[/fr !3 3no equipment subs:, ructurc-c; r-.;,ap->r^ri iy (e.g..,

).:i£ ccn-';i act.ion) or perwsïi^ni ly i,cn.,nu.d o^ or v*ti.ia the PCRV,

Caused by nquipineni moun ted 011 wr w i th in r h f t >'CRV - refuel ing machinery,ro ra f i rg TMchii.-af}' , pn r, ••,:"- I v

11

Caused by T"<enBài K^pavçlon , Cold Spr rig, P^pc ?di lure&,

sbj',.; vf»i"i Lu/, îrcffi va:u<.i%tt (ic;. Oc.rt«ii-. seacror sysGPia'j) to**

aused by t J2iparatur^ f - i y d ' ^ n l throughout 1 1' • PP

î, ^.on

lo irl.3

* Tne terra Loaaxn.fr Co^dltioi' AJ. <3et i.o.fxd )»fcre i"«.J ."iis ^T1. «i^ f«t-i; vh;i-.hin strefia' ^jid 'or ^tr;jir»<-.; and, t o - r a f o r t j i* net i - > " L t o i <.o r.^chfi'.u <J

bee text

Pressure LoadsCavity pressure may vary all the way from vacuum, to values above

the peak working pressure unc'er certain postulated abnormal conditions.European practice has been to design for a pressure level higher thanthe anticipated p*»ak working pr^ss ire by come arbitrary percentage.U. Sv design practice uses a reference pressure "TU''1 as the significantload. This is defined as the. highest pressure which uust he consideredin the design o£ trie pTrtJcu'ar reactor plant tx be housed inside thevesse). Up to this prefi^urt» rbe averr^p stress in the concrete acrossany section of the vess«.i nust be kept comnresBive.£;j irj&nineota.l__Condit. ions

In the U. S. A. :>as beeow» established practice in nuclear powerplant design to consi-iei fw- severity levels t-iher. >~<s storing environ-mental disturbances* into <•? d.vsiga. M^t*1 noterbl} , ftr.s applies toseismic effects. A pla^L (<>fd c h ' . t e f o r f c , th»? PC'1..'.'), wl?J be evaluatedfor an "operations.!"'' eaxthouake %-hich r>e"omes i> r *- -"'f the loading condi-tions considered in the working «tress c'esJvn; and .=-, "destga"* earthquakewhich is factored ••'nto the Huit design* T'>e d«»Jg*i eai thq»jiake or safeshutdown (Ksfu/'bancG is neutrally tak>'-n î9 nuit .'pie o£ c be def t i fu dis-turbancft "7hcre the n>uJ t ip] ic°t ion f-u*tT d«pcndp '»u xhe spec i f ic plantsite.

Locio[__Corcb 5 nat ion a

lu both the worKing srrecs d«si.go sad tîu. Jiuût. de*ipv>, certaincombinatirms of the .loading conditjn:;s of Tabit i. rMur,t be rcnbidered.In the working slxvss dcoif^» this iicludes ' l.i < oribf nations wbich canreasonably occur darL.g tne iiCe of the plant. Tht coc.birac ions are.lasted iu Tf»bJ.f ÏÎ. S imi lar ly» for lirait dostg.j purposes, specificload factor équations. wb3"ch wi iL be discussed in Section 1} h#-ve beententatively agreed upon. These s re intandoJ fc predict, the elasticlimit of the structure.

5. CRITERIA FOR ELASTIC BJ sWN

,a general., the cri teria for th« "Working *!trcs8 Lv*-3ign" areformulated fonventlonally in terms of «illuw bi«i stressée, expressed as.€r?ctions of standard ïiaterial ptrcper t i f tp . ïh-».se aïe c/Jincter or rubestrength for concrete azvd y ie ld or tensile atr^vi^lh lor P teoJ . Inlarge neasute, w*»tl established al.3owav1e "aiues «rf» u<?eJ, primarilytfaoae of. ACT. 318-63. (Cor respond ing to B.S. CFM15 in Britain). Som«deviatlonrs Iron these acceptée* sfandards t.^.v.t ncj^ssary however; andnew crirorJa had to b^ formoïaced . t/ »sïî:.:ît spr?s-?:ueru of f.^nperpture,•shrinkage aad creep effects.

The Individual val^fcj? >»sed for aJlov.nble stres:. «Vf diuri^seâ iusome detail beto.w. Tb*> values given, represent, curie it U. S. (>ta.t 11'°.

stip-ilatel hy D. S, regulating

26

TABLE IT

Load Combinations

• Period of Design Life

I,.- ^Construction

~*\ ^ iPressure Pr as tress Dead & PipeLive

| 1 Loads"~ """ '^' "•- \- '- ' - - -

2. Son-pressurized Vessel£_ter Completion ofPresrressiRgOpération

•

Reactions'

1

Wind- r • "

Tempera- Creep Shrinkageture______ : . • _ .-L. ._ _ .

/ / i /'j-.- _. _._... ..... . !..__.........

VF Effective »

* $3, Nou-pressurlaed Vessel j

ertcr Initî&I Heating V?. Effective /i

ColdSpring |

GoldSpring

*

. i /</

'

/

JEEarth-quake./

/

i

^i ! i ! i

1 ! i'A. Fressur:J2ed Vessel aftatr;Initiai HiaLin- j ÏJ.5 SPai Effective / i /

[5. Pressurized ''Vesafcl i - - . 1 .

i if \ // !

i j• at Time of Start of NW? Effective / ' / !

Operation

6. rjessurized Vessel atEnd of. Design Life

7. Son-pressurized Vesselat £IK? of Design Life

RP

</ \ •/

.„!: _ .__!_. ..„. L_ ._ . . .JEffective i /

, / ;. ;J _ . J . _ ' .

Effective ^''

/'

/1 ., J... L..

'

/

'

j

/'

ConcreteGeneral Consjlderationsof ConcreteAllowable concrete stresses quoted in AC1 318-63 generally are

expressed as .fractions of the 28-dsy cylinder strength f'c. Thisis du.;e for several reasons. One 1s just, time-honored practice,Another is the realization that structures are often fully loadedonly when they are considerably older than one month; the increasein strength over the 28-day value riien provides ? margin coveringuncertainties in analysis, material properties, loading conditionsand 30 on.

In the case of PCRVs, the luain service load (internal pressure)usually is applied only after the vessel is at least a year old; evenpost-tensioning may not occur for many months after fL;//;'! completionof the vessel. The- loads are known accurately, Moreover,, methodso£ analysis are much more refined than those used in everyday concretetechnology. Therefore, stresses are caJcalated with high accuracy;and prestress losses, particularly those occurring while the concreteis young, are calculated with great care. Consequently, tiui use ofthe 28-day strength as n clesjgu basis is o£ questionably validity andunnecessarily conversative. This is recognized, and ttept a^es UP to90 days have been specified (Section 8). The most ration»! basis wouJrtseem to be to use the strength of concrete at the time of post-tonstoo-ing.

Typical (28 day) concrète strengths for vessels completed or underconstruction f a.l 1 into the range of 4500 psi to 6000 pai.Types of Allowab 1 e Streajs es Jîpecil'ied

The stress analysis methods employed in the deuign of PCRV's(fini te-eietuent or f'inite-di f tecence calculauions) ^ Lve detailedresults on magnitude and orInatation of stresses at msny locationsthro'!.:.;hout the structure. ThereEora „ prlncipai-sf-esK criteria can bespecified directly and. no shear stras- allowables need 1o i>e given.

For the first PCRV's connp.rvrtti.ve .lilowafoîe ftveas values werespecified for "general stresses" (soiuewhat comparable to membranestresses in sheila) i.e., «tresses affecting an. area or voluisd largeenough to have overall structural significance. Different higher valueswere permitted for more- "local stresses" vrhich could be coiisidereidanologous to secondary or discontinuity stresses in stop! pressure vtigyodesign. The obvious difficulty in thiu Ls Kow to define quantitativelythe distinction between "general." and "locfjl" for a -struct.-or*? as coif-i>.lexas a PCRV. The trend now seems to be tc get away from "^e^craL strt-ss"criteria and to have only one limit., higher than the customary 0.45 f'c.which must be satisfied everywhere except at poinLs which are obviousstress concentrations.

Multiaxidlity EffectsMost of the concrete in a PCRV is subjected Lo multiaxial Load-

ing. It is well known that concrete strength increases with increasingdegree of confinement, and an effort is made to take advantage of thischaracteristic,

In a sense, even the early approach?*» which permitted higherlocal stresses can be viewed in this J ight , A more- systematic attemptis beinç m?de with the Pjirt St. Vrain vessel, wherr the. principal allowable v..jtnp res give stress </a!ue is adjusted by a multiaxial strengthfactor "C" in a manner presented in some detail Jn Appendix 1.

For a aniaxial stress lie Id C-l; lor tru«, î.ydroslat Je compressionC - 2.71. It is worth acting that C is nut a f aï Dure criterion, butsimply a design tool intended i:c gi.v<3 rerdleî Ic or const.1 votive results.

_ _ _ _Val_ues

PrlMcipajl Allowable Coiftpressive SLjcef»*> 0,45 C t*c

This al lowable app^ips, to 'ill opcrstiti£ &tr£'»ses including thosevhiiJh occur when the PCRV 1" s ' jbjsotttJ to pr« i>t re j j& ^ri'l thermal gradientduring reactor shutdown p^i^ i l . <n.i:d 10 nuera^i i>i"e«-ur'j^ up to Refereacc-Pressui e during plc-rt o^tLut ion . T'-c Liu-it uf U.^5 C f IF net to btexceeded when r.he ten-'^^ris a? f s*.ress"d to O.ô of thei*. guoianteed ulti-mate ceïislle strength. The dl j--»v j b J o p c i n c t p s i l'umpressive stress isderived fron the valu? gLve^ in ACI-31B, P^ra^japh 7605 , which is vjiidfor prestressed concrete çlroc-t 'uret j where In? Ptt i ioJfi oi andïysis sreprescribed by the Code.. The fac.or (", l<-i iuclaJs'J lor the previouslys ta ted raascus and i& to be c>L';f mined p^r Ap j i rndLx A. Whore high localstresses aie Incliceted adiacer,' r^ s r ruc tu r r l discun*- Jnui LifcS , and thusc ïear ïy result ivosi stress c OPO' LU. rat ion e f f e c t s , the values prediccsdhy eldstii. ana lysis are c i a t - regarded sxncc «..giiituu ing c^pcrjence hasshown tuat such local coiidltioiïo oc< or in Ail «•"••nctures wifhouL causingdetrimental e f f e c t s .

';ncrefe has a i-ensilo otreiigLh cf 1/12 Lo 1/8 nf the 28 day com-press, on strength For a 6000 p&J "Likrete, tlje I ensile sî.rt?ngf}> shouldtherefore bo 500 to 750 psi. ïyplrailv, the andésite concrete iut theFort St. Vrain PCRV has exhibited a Lenstie «freugth of around 600 osi,i.c», about__i/10 of the conp)es&iv*« streiigchs. éor f', - fOOO pal» thevalue S'T' . givos an a] lovable tensiJ*.- stress uf 233

29

Reliance is «oL pioced on tot. tensile res.! SB tance of r.he concretewhen the computed tenslif stresses exceed 3/t'c, In such cases, it isassumed that concrete cracks will develop and, therefore, reinforcementis provided to carry the entire tensile load.

Bearing Stress Under Tendon Anchorake«" ——— • ——— • —— •.——--- — — —— -— — —— —- ——— — -- —— — ' —— - O.fj f .A,/ 0but not greater than f'

*

j 'ils allowable stress Is taken dirent.! y from Uie ACT -.3 IS, Paragraph2605., where A'b and A,., ;-jre as defined in ACT-318. Paragraph 2 GOO, forlow A'k/Atj ti.is formula results in conservative permissible values. Forlarge Afj /AL values the ion-uila roarers cxcossivtpe.rmip.'ïibio concretestresses. Therefore } the iv.axi.aum limit "f. ff'c is also specified whichagain is a very conservât i 12 VHÎUC. Guyon reports in hi H hook "î'estress eCoiicî'et_«. ., Page 155. that t-,:r an inleriia.i t'rictjon angj.t- ol" 3'5a , a formuladeveJoped by 1-1. Cd.quof g'j.yc-c an uJ.tj'aate bearing nre.'î.surt- pqudl to about11 times tlu. uniaxiai coh>pr'.iss5ve H trees' tlu The c.rit crjc--1'» of 0.6 f „i,. _____ . ' '-vA!, /A., i.s wide.lv -.îaetl «j»d t'Uert- .-ire no kno-^a vea^oivw fo.' doubting itsc: ons erv& ti sm •.

Stetij

In general t.ÎHi convanLiOi i -^T prov-îBÎoi îS of AOT~^li< c;jn reodily beadopted. The- applicable paragr^pus ace a? follows:

/vppj .icdbj.i! i'ara.

Tensile 8 tresses

This implies

1003

For deformed bars No. 1! and wni.ilJcT" witha yield strength of 60,000 prf.i or mAllowable Strc-ôs . , . „ . . . .For "M otl'er reiuforc,au7ents :Allow^.blt; Sv.rt-.ss . . . . . . . . . . .

_Compre_s3ive_ Sj.rest.eR

"aragrauh .1102 i.s ime-decl ro -..icjiccompression of steal iu bendju^; .Tuni-bers. Although no! Jirocfi> Dralogous,che paragraph way be used i:i-re =IK veil.Thus, where steei is Latendo.d to fnucf-jo'.!as corapres s i ve r t; i nf or c emen l, :Allowable coaprcsuj on . . . . . . . . . .

2«i ,000 psi

?u>000 ps.I

1102

24,000 psi

s

Bond Stresses 1301Tendon_StressesTendon Wires - Allowable Mean Strèsses^The provisions of ACI-31S, Paragraph 2606» are met:

Temporary jacking forceAllowable stresses ...... r ... 0.8 £'

Immediately after transferor anchoringAllowable stress . . . . . . . . . . . 0.7 fg

Effective Prèstress 0.6 f1gAllowable stress . . . . . tb« smaller of

0.6 fsyIn addition, the maximum temporary jacking force is iiraited to 1.be

yield stress of (f }, and the maximum stress after anchoring is limitedto 0.9 f _t> These two limitations are intended to covet prestressingsteels which are not manufactured to the acquirements of AS'CM A-421, ystwould be acceptable for use in a PCRV, (ASTM A-A21 specifies a («Lnrwirnyield strength of 0.8 f5 }STendon Anchors

Ko; general criteria are available. It is specified that the actualend anchor assemblies must withstand, without failure, at least 1,2 xGUTS under static as well as cyclic loads; this strength to be demonstratedby appropriate tests.

6. CRITERIA FOR THERMAL LOADING AND THELASÏIC EFFECTS

Thermal loads (temperature gradients), creep aixd shrinkage areclosely interrelated. All three may cause stresses; in addition, creepmay relieve stresses caused by the other effects. To be most meaningful,a design calculation should, therefore, consider these, factor together,as well as their interaction, which is difficult at besi:.

Furthermore, relatively little quantitative information exists aboutthese phenomena, particularly at temperatures above the 150° - 200° Frange.

In view of this, several techniques are generally used simultaneouslywhen considering these factors in the design of a PGRV: i. Thermal loadsthemselves are Lirai ted to values known to be safe; ?.. Upper-bound esti-mates are made of the permanent deformations to be expected as a resultof creap and shrinkage;, 3. Stresses resulting from temperature gradients,creep and shrinkage are assessed on an elastic, nodifiad elastic, orviscoelastic basis and the calculated values are limitée* to values con-sidered safe and reasonable.

31

Criteria for Thermal LoadsFour quantities are usually specified by designers. Numerical

values vary somewhat from case to case, but fall within the rangesgiven below. The four parameters are:

o Effective concrete temperatureat (or close to) concrete-linerinterface 100 - 170 F

o Local maximum concretetemperature 130 - 200 F

o Maximum temperature differencebetween inside and outside ofconcrete 40 - 70 F

o Maximum rate of temperature riseduring startup Order of 5-20 F/week

Deformation CrjiteriaKo generally valid criteria can be given. Under t;he worst possi-*

ble combination of tolerance stack-up, anticipated load history, andcreep rates, deformations at. ead-of-life :nust b<= srn.aU enough for thereactor system to remain functional. In audition, the loss of initialprestrees must be no greater than the. value factored into the design.

thermal and_ "r>'gp 3i.regs J ^ L°_PJLEarly designs evaluated thermal {stresses on a purely elastic basis

without considerixîg creep effects. It was found that:., for the designto remain within practical and economic limits, adjustments had to be.made. Some designers calculated thermal stresses using a fictitiouselastic modulus whose numerical value was established empirically.Others relaxed the allowable concrete tensile stress value for combinedmechanical and thermal loads.

In U. S. practice, a strong effort is made to calculate actualstresses in the- vessel throughout its life and. load history as realistic-ally as possible, using the best creep and shrinkage data available torthe specific concrete to be used. These stresses are then limited tothe allowable values previously established as a basis for cilsstic design..LIMIT DESIGN CRITERIA

In the design of early PGRV's, elastic design-analysis was supple-mented only by an attempt to establish a gross structural overload factorbased on what was taken to be the ultimate load carrying capability ofthe structure.

32

ID U. S. practice, this has been replaced by a dual approach,both parts o£ which have been called limit design techniques. Actually,the first is a true limit design: the second is a variant of the pre-viously mentioned ultimate load analysis carried out for a different,and more meaningful purpose.

In view of this., it might have been more accurate to talk in thispresentation about four methods used simultaneously in the design ofa PCRV: elastic, viscoelastic, lirait, and ultiinate-joad design-analysis.

Limit Design (Prediction of E3-astic Limit)

As stated previous3.y, load factors are applied to predicted maximumloads; and stipulated limit criteria are then satisfied for certain com-binations of these factored loads (analogous to the manner in whichallowable stresses are satisfied ir\ the elastic design for all reasonablecombinations of actual loads),

The intent is to set the limit conditions and the correspondingcriteria such that compliance with them assures predictable (responseof the 'vessel. Local large deformations are uu.i-ike.ly and prematurefailure in the liner, piping., or internal equipment due to excessivevessel deformations should not occur.

The specific limit condition equations now in use are:a. BL H- RP •••>- £' -T XL

b. DL + 1.5 EP -f 1.5 TL

where

DL = dead loadKP =• reference pressureE' = load resulting from "design"

earthquakeTL - thermal load

Elastic response under Limit Conditions a is intended to ensure asatisfactory seismic design of the PCi'V. The reason for selection oflimi't Condition b is that some reserve load capacity should exist beyondthe maximum anticipated loads.

The allowable criteria corresponding tc the above limit conditionsare as f o3lows :

ConcretePrincipal Corspreasi.y^Stress 0.6 C f c

33

The use of an increased allowable concrete compression under LimitConditions a and b is justified by the philosophy implicit in the adop-tion of the limit design concept where it is intended to ensure elasticresponse at Reference Pressure and design temperature by demonstratingthat elastic response is to be expected even under higher loading condi-tions. Thus, the allowable level of concrete stress can be the elasticlimit in compression for concrete. îfewnisn^ ' states that concrete maybe P ^ected to respond elastically in uni axial compression up to about50 to 60% of die unLaxiaJ ultimate strength: hence, the use of the upperlimit 0.6 f'c sterna justified. ACI-318-63 aJso makes provision for auincrease in allowable stress of 33-1/2^ for improbable lord combinations.If this increase is applied to the concrete allowable used in elasticdesign (0.45 C f ), the quoted Ugure of 0.6 C F' is obtained.Bearing Stress Under TCP don Anchors 0.7 f 7 A.' /A,————— ——— —————————— —————— t. b b

The allowable stress permitted for bearing stresses io Increasedby only 17% over the elastic design allowable. This is a very conserva-tive approach considering tht maximum fotresset> possible ft these rein-forced and confined zones as. pointed out in the discussion on the allow-able bearing stress under tendon anchorages fcr operating lo«<ds.

SteelLiner: PrincinaJ tensile stress 0.8 fsyReinforcing bars: Principal tensile stre&s 0.9 fsyPrès tress inp; wire: 0,95 fb sy

The selection of 0.9 i< _ is thought to provide an adequate marginof safety against inelastic action In the iinnr and tjonu^d reinfoi'cement.The action of the bonded reinforcement is well understood and is moreakin to that o£ flexural r&in Lorcement than cjtber categories. The capacityreduction factor selected of 0.9 coulronno ! n the 4C1--3J3 recommendationfor such s tee.!.

A reduction factor of 0.95 is considered justifiable icr prestressingwire because of the stringent qua! Lty control required a_n its manvtfacture,and the in situ load testing vhich occurs during pr*"3tre£sing.

Ultimate Load EvaJv^L-iC;?,As regards j n tun t , this may be.st be corap<jrccl w t f c h the ul t imate load

capacity equation given IP ACi~318-63, S- jcLion 1506. }Iov;evcr 5 only asingle, large load i a< - fo r oi 2.1 is applied Co the load RP, Trie justifi-cation for tne d l f f p r e ^ ' G iie^ in the f^ct char the dead weight of thePCRV results in negligible f-tress compared vita the ir.teinal pressure,thus allowing the effect of dead load to be neglected. The higher loadfactor applied to the Load RP actually results in .1 nore conservativedesign than ACJ-318 would stipulate.

The ultimate strength criteria of ACI-318 are directed primarilyto framed structures, where yield of reinforcement implies the formationof plastic hinges and collapse mechanisms. Yield in the wall reinforce-ment of the PCRV does not mean that a collapse mechanism has formed and,provided the deflection characteristics are investigated under increasingload, there is no reason to state that ultimate capacity has been reachedsince the tensile failure strength of reinforcement is much greater thanthe yield strength. Conventional analyses do not poc^pps this ability,making the attaining of yield stress a convenient point to halt an analysisalthough a reserve of strength exists beyond this point.

The allowable stresses for this conditions are:Concrete Principal Compressive Stress 0.85 fTendons 0.95 GUTSBonded reinforcement GUTS

except in heads whore steelstresses will be limited to 0.9 fsy

ID areas exhibiting primarily membrane action, the concrete willof course, be fully cracked. Where structural action is more complex,e.g.» in the heads, compressive stresses are limited to 0.85 f , avalue which is accepted bv A,CI-318-63 (Section 1900) as being applicableto members subjected to combined axial compression and bending. Whereit car» he clearly demons treated that a condition of triaxial compressivestress exists and the material is confined, an increase of this compressivestress allowable is justified similar to that provided under Limit Condi-tion 1, i.e. , 0.85 C f .c

b . Tendons and Bonded ReinforcementAs the ultimate limit condition is a definition of the minimum

cavity pressure required to fail the vessel, the use of minimum guaranteedultimate tensile strength seems warranted for bonded reinforcement. Àvalue of 0.95 GUTS is used for tendons. The use of this value is justi-fied by the results of European tests on full scale tendons reported inreference 27. Their authors report ultimate load tests on B.B.R.Vw 'unitscontaining 121 wires of 7 mm diameter. The results of tests on straighttendons indicated the development of 99.7% GUTS and the curved tendonsgave strengths of 96.5% GUTS.

The allowable stress of 0,9 f for bonded reinforcement in the heads,combined with the concrete allowab?! of 0.85 f, ensures that the thickplate nature of the structural response of the fèeads is not lost by theformation of large cracks. It also provides a margin of safety in thehead analysis which is necessary since the action of the heads is amenableto precise analysis than other portions of the structure.

35

8. COMPARISON OF VARIOUS DESIGN BASES

If one compares such design criteria information as has becomeavailable from different design organizations, considerable-variationwill be found. Design evaluation and progressive changes have takenplace within each organization as well.

Therefore, a single overall comparative evaluation cannot meaning-fully be made.

Table III is a compilation of parameters for which a comparison mayhave value. Brief explanatory notes follow:Pressures

In European practice,, the design pressure DP .(at which static balancebetween tendon force and internal pressure force occurs) is selected at10% above normal working pressure (NWP); and in the most recent Frenchvessels (Bugey and Fessenheim) as low as 6% above the NWP. This comparesto a corresponding value of 23% in the U. S. Moreover, in the U. S.practice this pressure (called Reference Pressure, RP), is higher thanthe highest pressure which can actually occur in the system. The Europeandesigns permit the maximum pressure to exceed the design pressure byanother 10%, giving a true maximum pressure of 1.21 x NWP and correspond-ing with it the possibility of net tension in the vessel walls, which maylead to partial cracking of the concrete. ,Load Factors

A basic characteristic of the European ultimate load evaluations, aswell as the limit conditions factored into the more recent French designs,is the a priori assumption of a certain mode of structural behavior orfailure respectively. This is exemplified by the analytical, approachesgiven in references 28 and 29. The analysis offers no method to ascertainwhether the postulated structural behavior would actually occur. Carryingout similar calculations for the Fort St. Vrain vessel would give ultimateload factors between 3.0 and 3.3.

In general, one can conclude that this type of. load 'factor evaluationis of limited use and can, in fact, be somewhat misleading since the resultdepends strongly on the assumptions and calculation techniques used.

The U. S. approach has been to base limit calculations as well asultimate load factor calculations on methods which mathematically pi-edict(rather then assume) the structural response of the vessel.-1 A correspond-ing numerical load factor of "« 2.5 x NWP has been selected.Concrete Temperatures

All temperature values shown in the table are sufficiently low asto eliminate the likelihood of significant material deterioration. U.K.practice parallels the British approach; both are somewhat more conserva-tive then the French designers*

36

TABLE III

\

P C . R . V . DESIGN COMPARISON- "

1 PQE55UQËNQBMAL WORklNGPGE55U2E (WWP)

maximum crediblysystem -pressures

, PCES5UÎSE (D R)BEFEGEMCSPJ2E.35UEE (CJP)INITIAL PEOOFdPIR)TE5T PCE5SUEE

2LLOAD FACTORUMiTCOMDîTICW la.tlAAJTCOMDtTSOK! îbUMSTCOWDITiOW L

FT 51 V2AINPSIG. W.W.R688

O345„_

845 1.23972 1.41

DL*IÎ3N\VP*E'+TLDL+L84H\YP*i5TL^(5EE TEXT) .PAGE vfl~25

3%KZNG°U*f!*"E 600OpREAMR£*at age ^days)

j EFFECTIVE4-.-COWCEETE TÊMR(F"}

5 STRESS „.„ .,.ceiTeaîA (PS.I.)

TtJMPOSAKY PCIWC1PAL"COWPBESSION

. .___ __ ._. _ PBWCIPALTEMSfOW

OP£J2AIING PS^tCIPALCGMPBES5IOWCQWTOEiSiON

(LOCAL)PeiMCIPAL

~ TEWSIOW

- - TCMDWWEABEAE1MG ;

SJEA8 AWCUOSS

6 LÏNF-P.. TUtC^WESS

130°

„_» ^-

) *^~""

u6C*^T:<feO.6fc AVERAGE

-V4-*

OLDBU2YP 5 Î G NWP350

1.2'1385 I.IO_ ^

WYLFAP51G384

423_ _ .

443 I.Z65 |437

NWP

1.21I.IO

_ _

1.165j

___ . . . ...,

1 i 55~T 3.5-0

4700 (25)

131'

EDF-4-&P S I G385

426

469

NWPOUNGÊWES5ÔP5IG455

)

I.IO |478{ —

'l.ZI ip^O

WWP

1.21I.IO_ ,

1.265

JDL+UONfWP+TLl{DL+t3ZWWPH5TL|

! !25 1.92 j 1000 2.75

4-700 (56).. _ .

95s 6'FKMUMK

2

pSifc s X40OJ

0.4'ifc5 2000

5600 {90}-

^ 167°._.

05^cs 29 8O

II9Ô 2.75,

470O (28)

13!"*-

497 |035C-, (9ÔO

jassfi» 4OOO jojfc 5 s&o

NO CC!T£2iA

i/z"

LOCAL-,1 — 5OO

500

5500NO CQITEGIA

V4*

!9S

0.53C*'C

AT LIN£~K_ iOQAT OUTSIDE- 50O

FC.EWCH FRESTEES&EdCOWCEETE CODE 1

MO CCiTEPiA | NO C!2ITCC?IA

s* Vz"

&UGEY - 1PSiG NWP637

G76 --I.06._

NONE 1 - -

i6SO 265

580 O (28)

ISO"

NO CQITEKSA

FE5SENMEIMPSIG NWP435

464 I.O6.

NONE

a.«ID6NWP+TLDW3ZN\VW.25TLiiOO Z53

5400 (Z8)

.e?fc s 36OOZÔ4

Q4fé s Zi6OQ8fé s 432O

170

NO CC.ITEE1A"

1* 1

= Factor which takes ïnto account the principal conpressavcstresses at the location under consideration

£' = Cylinder strength at (days) specif.ted

Al lowab le. Stress es

In ail instances, it is recognized thai the established allovjabj.esof conventional s truc tarai technology cannot be used Verbatim in ameaningful manner. Higher values are permitted in compression Forlocalized stresses, or where concrete is confined. British designers ,are permitted to exceed the compressive allowables of Code U.S. C? 115in locations where two or three dimensional c oppressive stresses ar? pre-sent, provided the excess is justified "expérimentai. ly or in the light ofexisting knowledge". Also, calculated tonsile stressas up to 500 pai arepermitted provided the net force across tJie ve.ase.1 wall section regainsccrapressive.

In general, allowable tensile stresser, in unreinf creed areas rangefrom 'x . /T7" to 7.3/ET"".

Allowable comptess^va bearing stresse,--, at anchor a vary between0.6 f and 0.93 Ir ' 3; in conventional European practice/., In somePC8VT3crelatively large high-strength concrete anchor bearing blocksare used. While these would seem to reduce significantly the bearingloads ou the PCRV, the blocks th ems elves require considerable develop-ment, and no standards are available. ACT.-3iti (which is «adhered to byU. S, practice) appeals to be the only code prescribing allowable stressethat: account for tendon spacing and hole size.

Two other factors, uot listed on Table III, are worth

Relatively similar criteria appear to have bt?en used, by all designers,A summary comparison is given in Tabla TV. An obvions difference of opinionexists on the question of tendon grouting. The tendons in the Frenchvessels have been grouted. In British and U. S» practice, corrosion isachieved by other means, and tendons are J.&ft unjrroute:' so as to make tendonreplacement possible should it ever be required.

The U. S. criteria imply a conservative approach as regarda the usaof reinforcing steel, In. Great Britain, fch»? C.E.G.B. specifications con-tain no sf-ipulations regarding i:he use of bonded reinforcement. Actualpractice has varied considerably between different designs. The Oldburyvessels have very little bonded reinforcf.meiit and a h.iy.h degree of pre-s tress; the Wylfa vessels, oa the or.her hi-ind, coo tain a considerable ajcountof reinforcement:. Other cylindrical designs include a moderate proportionof bonded steel. The early French vessels alao had little rein forcement. .Later French designs have involved munh higher amounts of bonded niild steel,mainly for purposes of crack control,

38

TABLE IV

Crinaria for Prestre^sing Systems

LIMIT OR ITEM U, S.Practice

Européen

Average axial tensile stress duringjackingAverage, axial tensile stress afteranchoring% Elongation at failure in totaltendon length

30' of straight tendon10" wire length

Corrosion protection

TendonsType

EfficiencyStraight

Uirveci

Tubes

Anchor hardwareAnchor ultimate,percents oftendon ultimate

0.8 £'

0,7 £'

3.54.0

oil or greasesystem

unbonded

100%

95%Solid

0.85

0.7 £'

1.5 tc 3.0

cernant-sandgrout,and greasesystems

unbonded andbonded

100%

Solid andflexible

39

ConclusionAs staged in the introduction, professional groups, both here

and abroad, are working toward the- formulation of such criteria.Drafts are being circulated in France, the U. K. and the U. S. withinnuclear and PCRV oriented groups. It is anticipated that in the nextfew years, we will ^ee the emergence of reasonably unified PCRVcriteria standards.

40

APPENDIX A

Many attempts have been made to derive a single failurecriterion for concrete subjected to the many kinds of combined stresses, sofar without success. The leading candidate theories, Mohr's theory and theoctaii^dral shear stress theory, have been shown to be inappropriate byBresler and Pister (Réf. 1 and 2) and Bellamy (Réf. 3), respectively. Inthe latter work Bellamy makes the point that criteria based on the principalstresses themselves, rather than derivatives of them, are likely to be themost successful. The Mohr theory can be the basis of a satisfactory criterion(Réf. 4) for triaxial compression when the two lesser principal stresses areequal, but fails if one of them approaches zero (Refs. 2 and 5)(or becomesnegative). The point has been made by several investigators that the failurecriterion should be representable by a surface of revolution, or the like,in principal stress space whose axis is inclined equally with respect tothe three principal stress axes. No simply expressed surface of revolutionappears to be capable of representing results of triaxial compression testsin which the least principal stress varies from high values right down tozero, apparently because of a change in mode of failure in the low minimumprincipal stress r.egion which may be due to the large disparity between thetensile and compressive strengths of concrete.

For purposes of PCRV design, this difficulty can be over-come by use of a criterion which makes no pretense of indicating actualfailure throughout the range of its usefulness, but which does representfailure conditions reasonably well in regions where unnecessary conservatismwould be uneconomic, and which is conservative, or even extremely conserva-tive, in regions where economics are not affected. This approach basicallyresults in close prediction of failure in the region where the least principalstress is small and increasing conservatism as the value of the leastprincipal stress is increased. This characteristic is appropriate too inorder to make allowance for the somewhat uncertain effects of pore waterpressure on failure» the effect being small at the lower levels of confiningpressure, but becoming larger at larger confining pressures, as shown byAkroyd (Ref. 6).

Close prediction of failure in the region where the leastprincipal stress is small, demands careful consideration of the work thathas been done on failure of concrete in biaxial compression. Hilsdorf(Réf. 7) has reviewed and discussed quite comprehensively the existingknowledge on this subject and contributed one or two experimental results.Vile (Réf. 8) has also published work on this subject. Although Hilsdorfclaims that lower values of failing stress for specimens under biaxial com-pression will be found when more representative means of loading specimensare employed, it does not seem likely that this will be found to be asubstantial reduction in view of the awareness of this problem and the careta&en in this respect by Vile and Wastlund (Ref 9)

41

The best available claua on failure under biaxial compression;by Vile (Re£. 8}, Wastlund (Réf. 9) and Weigler and Becker (Réf. 10), areplotted in FÏR. V.I,A.I as full-ling curves. The curve labeled "Vile" is theone drawn by him, though the scatter of his results would permit one to drawa curve almost equally well about halfway between his curve and the curve ofWeigler and Becker. The ooints representing equibiaxial compression on thecurves of Vile and of Weigier and Becker are quite well confirmed by Glomb(Réf. 11).

Chirm and Zimmerman (Réf. 12) have summarized prior inves-tigations of failure of concrete under triaxial compression and presentedtheir own work on this subject. Theirs appears to be the only investigationof failure of concrete for both the cases of axial pressure predominatingand confining pressure predominating. They also present data on the uniaxialcompressive strength of concrete remaining after it has been subjected to'high triaxial stresses. The outcome of all these investigations, includingChinn and Zimmerman's, is that for axial pressure predominating the maximumprincipal stress at failure is conservatively given by f£ plus four timesthe confining pressure. Ackroyd (Réf. 6) has found that this is only trueup to about 3 f£ for water saturated concrete « For the case of confiningpressure predominating Chinn and Zimmerman found that the -maximum principalstress at failure is approximately fco plus three times the axial pressurewhere f is the equibiaxial failing stress as determined in their tests.

The criterion chosen for use in PCRV analysis, valid forboth biaxial or triaxial compression is the paraboloidal expression(a,2 + c 2 + a 2) - (a o + o o + a a ) - .25 f (a + a + a ) - .75 f'2 - 0i 2 3 } 2 i 3 2 3 C I 2 3 ^where validity is confined to a < 3 f.* \ c

o. - Maximum principal stresso_ = Intermediate principal stresso «= Minimum pri^o.i-oal strp-?c'3Compression is negative, tension positive

For the case of biaxial compression, a£ <* 0, the abovecriterion yields (o 2 + o 2) - c^ o3 - .25 ££ <Oj + o3) = ,75 f,!2 which is _plotted as the interrupted curve in Fig. .A.I. It will be seen that thiscurve lies slightly on the conservative side of the envelope formed by thecurves of Wastlund and Weigler and Becker.

42

For triaxial compression with axial pressure predominating,the criterion is plotted in Fig. .A.2 where it is compared with c^ = f£ H- 4 03,i.e., the two lesser principal stresses are equal. For triaxial compressionwith confining pressure predominating, i.e., the two greater principal stressesare equal. The criterion is plotted in Fig. ,.A.3 where it is compared witho as i.i5 f + 3 o3 where 1.15 f'c is tl-e equibiaxial stress predicted by thecriterion.

The assumed failure criterion represents the biaxial com-pressive strength of concrete fairly well and represents the triaxialstrength conservatively. The degree of conservatism increases as the con-fining stress, and thus the assumed failing stress, increases. The completecriterion is shown graphically in Fig. ,A.4.

ACI-318 sets an allowable stress value (0.45 f^) forextreme fibers without consideration of multiaxial stresses or the higherstresses that occur at discontinuities since such stresses are not predictedby the analysis methods used. The finite element methods used for the PCRVdesign predict in some detail the multiaxial stresses that occur throughoutthe vessel. These predicted stresses are then compared with acceptable ACI-318allowable values which are modified to take into account the multiaxial state•of stress by the utilization of the factor "C". The factor "C" is determinedas follows:

The failure criterion equation is solved for o , the maximumprincipal stress at assumed failure.

1. If all three principal stresses (c , o , op are in compression:

calculate -jr anc* TTc ca

determine C « -rf from Fig. V.I.A.4c

[a | - 0.45 C f£

2. If a « tension, assume a « 00 ^

calculate -77 and -rr * 0c c

determine C - -rr f^om Fig. V.I.A.4c

|a,| £ 0.45 C f'

43

3. If o2 and 0- tension, o, •» a, =• 0C = 1

Jo ! 0.45 C f^

It is common design pr .ctice for ordinary biaxially stressedstructures to use the full allowable concrete compressive stress (0.45 f^)when the stress in a transverse direction is tension. Under such conditionsconcrete cracks will develop which will reduce the tensile stress fieldsin the concrete. The compressive stresses are carried by "compression colur.ns,"which cannot buckle because of the confinement provided by adjacent concrete.Lateral restrain may also be provided where bonded reinforcement is used.

In triaxially stressed structures where tension occurs, asimilar condition exists. Uniaxial tension with biaxial compression produces"compression planes" which are loaded biaxially in compression. Biaxialtension with uniaxial compression produces "compression columns" which sus-tain full uniaxial compressive loads providing confinement is provided.

It is recognized that this criterion depends on the resultsof test specimens being directly related to the structure where confinementis provided, and that biaxial determinations used as evidence in Fig. .A.Iare not further influenced by platen effects yet uncvaluated by the referencedresearchers. Although these assumptions are perhaps chailengeabie, they areused here in an effort to be as realistic as current failure technology permits.

44

ILr

WE1GLER & 3ECKÊR

0 0.2 0.4 0.6 0.8 1.0 1.2 1 . i* 1.6

f 'c

Fig, STRENGTHS OF CONCRETE STîRJKCTpP TO RTAXIAT.STRESS IN TERMS OF UNÏAXIAL COMPRESSIVE STRENGTH f '

,- - o

f 'c

Fig. .A .2—TK1AXIAL COMPRESSION WITH AXIAL PRESSURBPKWITNATTNG

46

r-1- O

3 -

2 -

0,2 0.6 0.8 1.0

ffc

Fig. .A.3—TRIAXIAL COMPRESSION WITH CONFINING PRESSUREPREDOMINATING

47

•Til- <"

0.2 -

0 0.2 0.4 0.6 0.8 1.0 1.2 l.t 1.6 1.8 2.0 2.2 2.fc 2.6 2,

"sifc

PRINCIPAL STRESS CURVES FOR DETERXIMMG FACTOR C

48

REFERENCES

1. Bresler, B, and K, S. Pister, Faj lut e of Plain, Concrete,, under Combined.St_resses_, ASCE Trans. 122, pp. 1049-1068, 1957»

2. Bresler, B. and K. S, Pister, Str: ft&th oj; Concret ejjnder ..Stresses^ AGI Journal 30 5 No. 3S p-t>, 321-345, 1958.

3. Bellamy, C. J., Str e ng t h o f Cone re te un de r Comb i ne d _S t r e sjs , AGI Journal58, No. 4, pp, 367-380, 1961.

4. Balmer, G, G. , Shearing . ..Strength pj L Conerevi:e .under _ljljgh Triaxlal Stres^s-. Compuca t ion pj Kohr s Erive. ope ..aj__a_ Ji' IYJl» ^'" ^* Dept. of Interior,Bureau of Reclamation, Laboratory Report Ko. SP-23, pp. 1-13, 1949.

5. McHenry, D. and J. Karnis g t rerig th^ of n Corije re t e^ under Comb Ine.d Te ns_i l_e.rar.d^ompr_esslve.iiSt.res:s. Prôc. AGI -54, ppf 829-839, 1958.

6. Akroyd, T. S, W» , Concre te jande.r Tr iaxial^ , S t :res _s^ Magazine of ConcreteResearch 13, No. 39, pp. 111-118, 1901.

7. Hilsdorf, H., De ternunat io^-oj t^&^J^y^al^§trejn.gtln_ p f Cone re te ,Deutscher Ausschuss fur Scahlbeton, Heft 173, Berlin, 1965.

8» VilGj G, W. D. , Strength _o_f... Concrete under^ Short- Term Static Biaxial^S.treAg.1 Paper P-2, International Conf. on the Structure of ConcreteImperial College, 1965.

9. Wastlund, G. , New Evidence Regarding the Basic Strength Properties ofConcrete,, Betong 1937, Keft 3, pp, 189-205»

10. Weigler, K. and Gf Becker, Exper .ing_nt .s or\ .the k_7racture and De j pjrmaticmBehavior of Concrete at Biaxial Loading» Deutscher Ausschuss furStahlbeton, Heft 157, Berlin, 1963.

11. G 1 crab, Die Ausnutzbarkelt Zwaiacb- ; tger^^Drtick^s^igkelt Des Bétons i,nFlachentragwerken, Congress F.I.P.S.L. Pa, 1, Berlin, 1958.

12. Chinn} J. and R. M. Ziranerrnan, B ehavi or _iof_ PI a .in ._CQRC.re it e^ynde r Var lous,.Hi.gjh_T.r_i.ax.lal....Cp_mpjr_aj_sl._qni o_andjjnj ConjlAt o ns Air Force WeaponsLaboratory, Tech. Report WL TR 64-163, August 1965.

13. University of Illinois, Bulletin Sérias No. 405, Vol. 50t No. 29, 1952.

49

14. Heft 5/54, Vèro'f fentlichungen des Deutshen Stahlbau Verbandee, 1954,V

15. Heft 138, Deutsher Ausschuss flir Stahibeton, Uber die Grundlagen desVerbundes ZIrisehin Stahl und Bebern, Dr,-Juj. Callus Rehm,

16. England, G. L. , Lpns.._TcrTO 'Ifoe.mal.j gejAeA-lX'Striic.t.uresit Conf. on Prestr'essed Concrete. Pressura Vessels , London,1967, Group p, papsr 34 ,

17. 0' Connor, H. £. and J, L. M. Morrison, The. Effect .of. Meaai.iiStre5S_onLjbguInternational Conferenceon Fatigua of Metals, AShfi, 1956,

18.' Design Data ~ Nelson Concrete Anchor Studs, Manual No. 21, Nelson StudWelding Division of Gregory Industries, Inc.j Lorain, Ohio,

19. Da vis, H» S.> fJ c.ts_p ._Htgh __Tretiip.e.jr.aturg_.j|xpp.&ure. .Qn.._pQ.ncy.&te, AmericanKuclear Society Meeting, Gatlinburg, Tennessee, June 1965.20. Cottrell, A. H., Theg.ry_ of .Jrj. tl Fraetj.ira _in^Stsel and Its, App.l icatlon

to^ Rad 1 a'c i on. %b r 1 t_t 1 ement . " onferenc'e. on Brittle Fracture, p. 1,Culcheth Laboratories, England, November 1, 1957.

21. Pellinij W. S. and ?» P, Puzak, Practical Consideyatigns^inLaboratoryVessels., U. S. Naval Research Report 6030, November 1963.

22 » The_ TStru.c t ura 1 Us e of^Frest r e s sed... Concr e fcef in Buj. Id injgs.» The Council forCodes of Practice^ British Standards Institution, CP 115» 1959.

23 « Sjate. _cf. .;grh_e ^ Nuclear.À Critical Review of the Literature, ORNL-TM.-312.

24. General Atomic Staff, P.re.s_tr.es_s_ed__C_o.ncrGeneral Atomic report GA-7097, October 25, 1966.

23» General Atomic Staff, | 'res très s&ô i _ Co rj te React.or Ves _s e 1, Mode 1 2 1General Atomic report GA~7150, November 4, 1966.

26, Newman, K. , The Structure .A?J.-.. n 1 ee.?M J'XCB I --vP--CP.P.nS. eJ .eJ»Proceedings of luternational Symposium oa the Theory of Arch Dams,Southampton, April 1964.

27 . Ros, M, R. and ?, E< Speck, L£r _Tend_on.g__for jr e s ur e JVe.s.sjg I s_in.SiSiÊâL- SSSïLâSâÊiSSS.» Conference -on Prestreseed Concrete PressureVessels, Hfjrch 1967 1 Group E^ Paper 25.

50

28. Harris, A. J. and J. D. Hay, Rupture Design . of. _ jth,e._0j4bury. VesselsConference on Prestressed Concrete Pressure Vessels, Institution ofCivil Engineers, London, March, 1967, Group F, Paper 29.

29. Finigan, A. , .Ult_ima te__ Analy s 1 sx._p £f .the,, Dufig,e fi g ^ Vags. ! . Conferenceon Prestressed Concrete Pressure Vessels» Institution of Civil Engineers,London, March, 1967, Group F, Paper 31.

30. Anthony, R. D. , Development: o Sj:_a.î:.iat.oir JRe._( u._iTreiine_n_tsj_for. Redactor Vessels,Conference on Prestressed Concrete Pressure Vessels, Institution of CivilEngineers, London, March, 1967, Group Bt Paper 9

51