Batant, Z.P. (1978). "Inelasticity and failure of concrete: A survey of recent progress,· Proc. of Seminar on "Analisi delle Strutture in Cemento Armato Mediante iI Metodo degli Elementi Finiti,· held to commemorate 50th anniversary of School IH£USflcrn AND fAIlURE Of COIClfTE: I SUHm or mE NT raoGRESS by Z. P. BAfAIIT POLlTECNICO DI MILANO CORSO DI PERFEZIONAMENTO PER LE COSTRUZIONI IN CEMENTO ARMATO • FRATELLI PESENTI· Cinquanrenario della iscicuzione 1927/28 - 1977/78 ANALISI DELLE STRUTTURE IN CEMENTO ARMATO MEDIANTE IL METODO DEGLI ELEMENTI FINITI seminario 20 - 23 giugno 1978 Prof. P BAZANT Norchwcscern Uni\'ersicy, Evanston, Illinois Prof. William C. SCHNOBRICH Universicy of Illinois, Urban;l.,llIinois Prof. Alex C. SCORDHIS Uni"ersicy of California, Berkeley. Cal iforni<l
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Batant, Z.P. (1978). "Inelasticity and failure of concrete: A survey of recent progress,· Proc. of Seminar on "Analisi delle Strutture in Cemento Armato Mediante iI Metodo degli Elementi Finiti,· held to commemorate 50th anniversary of School
IH£USflcrn AND fAIlURE Of COIClfTE: I SUHm or mE NT raoGRESS
ANALISI DELLE STRUTTURE IN CEMENTO ARMATO MEDIANTE IL METODO
DEGLI ELEMENTI FINITI
seminario 20 - 23 giugno 1978
Prof. Zden~k P BAZANT Norchwcscern Uni\'ersicy, Evanston, Illinois
Prof. William C. SCHNOBRICH Universicy of Illinois, Urban;l.,llIinois
Prof. Alex C. SCORDHIS Uni"ersicy of California, Berkeley. Cal iforni<l
Z. P. Baaant. "Inela.ticity and Failure of Concrete. A survey of Recent Progre ..... Reprinted from . Proceedina. of Seminar on "Analiai delle Strutture in cea:nto Armato Mediante il Metodo degl! Element! Finit! (ADelySis of Reinforced Concrete Structures by Finite Element Method). held in ca..emoration of 50th Anniver.ary of School of Reinforced C Politecnico di Mil DO Hil oncrete. pp. 5-59. a. an. Italy. JUne 1978.
ltelZa .econda ",.t<t dlZ Pllbb7'fZiD 19Z8 inizim>anD aZ PoZit." .. iCD di NiZa7ID le p"':- Z .. a,;""i d .. Z COl"eo di p"l'r .. aw-..to pel' Z. C J.t",_i""i ill C_to ~to. ""ZlIto " patl'OCinato dazZa !~Zia P"aftti """ 10 aaopo di """tl'ibkiNl a .fOlWrU'tl dllgZi "i"f/"pa6l'i ape"iaZiuati i .. tlCtt" Z. appliaaai""i dlZ ..-..to • • • p.<:iau. .. t" ... zz. CDatl'lCa~i i .. ,,_to _to-. Ri"o..,.. CD.l i .. qk./tto <DIftO aaaad-nco il tri_t"NU"iD di _ato iatitIC
ziOPW!. Za ""i doria i l'iPOl'tata ... Z ""Z_ fIV (1977) d"l1ti StlCdi • R.,.di"""ti del C01'IlO.
Pl'a Z. "",Zu i .. iaiat':". po .. ibili pel" """ degrra .. "tebNsiDM di """ data "oei iIopo .. ta1Ite .i I .... Zta QlCelZa pill CIO""';~t. alto a<:opo istitICaionaZ" dll eo ... o. Nlaz,:uaMc i .. lII:VIillN liNN. pttbbZicuz lOt SIlIrril'I<Zl"io ehl d./tel"iua 10 "/ttato d.ZZ 'Ql't,," ""lto """li.i d.n" stl'lCttlCl". in e ....... to <U'fll2to ",.dianu it ... todo degZi .t_ti .'i .. iti. TaZ ... toda. di.eNti .. _*, 10 atl"kttleJoa i" IDI inai_ di el-..ti, pe,..,tt. di adDtea..e pel" cia.CIDID di qkll8ti ta dAraeris"'''' a>laliticuz pill appJ'Ol'riata. C;W.to I pal'tit:Oz.u-..ta ... tant<t MU. atl'lCttIC ... i .. ..-to _to. pIlN:hi """ a.,.t. di u... ... """to. in 1DI' .... icuz analiai, aia di .. ~ti diu.,. •• qkQJl
to a tipoZogia st"'ttleJoaZ •• aia del e"""" .. _to oZtl'llJ!lOdQ COIIpZa .. o deZ cal ..... t"" .. o di ...... to. _tt<tria~to da lego. ccatitICtilJD """ Zi".a.... d".fo,.,.,.z~i di tipo viaClOao " r""lO'GaiDne.
IZ SIlIrrinal"io • • tato id.az-nta divieo i .. Ilk. pal'ti. 14 prillca ri-I!' .. ~" ! ·.'!'l'siai"",. d"n" l'i .... ,...,.. ... tati"" aZta .'o.-tasiDne dli Z.~ eoetit .. tivi da iIIt~ neZta anali.i. c....,o ill ..... Z. poa.ibilitd analit""'" dlt _todD i .. OfIIIlltto .op~ di f1PaII lI.M{!Q Z4 t1DN1.t: ... :uz -1.1 ooat.riale daZ "....to di vi.ta .fi.iC!O. 14 .....,..aa ripo,.ta _ d .... ri.i"". delta _todDlcgia di analiei • dille .... appliaaai""i ai print:ipali tipi di .t,...ttICN ddta irl(l.l1""ria <:ivit., ta qwat .... ct. in .vid ...... I. OOI'II!izioni "",: dI"" .oddi.!al'Il it ..ad.Z 10 cmIllitit:o ar!iMhi poaaa riprodwl"N ,:z """""PtaIIIeIoto NaZ. delZe 8tl'lCttICN. A tau propo.ito .... to ,;z delle ... di l'i"fll'Q&iaN i proo' ... ol"i za .... k P. Bazant. I>'iZli(llll St:lmobricIJ • At" C. Scal'ddi •• .. "" SOJlO staH alii pioni.ri in qkIl8to c....,o di rit:.roca. pe .. 1 ·",t ... iaPfO t:01f cui hanno at:t:llttato di til ...... m."sto s...nJIIJrio, • PIll' 10 fIIJJli.N CIOat ad.,....tIl atlo .pirito i .. .folWltlto ... con cwi Jranno p"'pal'ato iZ t.sto d.ll. Zo,"", les~i. RingraaiD aMlt. iZ oriD t:aZZabol'ato,... irl(l. twigi CedoZi ... pe" atIIl ... avoZto cod .rlicat:_te it Za1IOJ'O di aool'di_"to dn diTJel'si i .. t.,..,.,,".
A1I<>M i" ",,".ta occo ....... ri"""'l10 alta latriglia P ..... ti • • PIll' ... a aZta Soci.td Ita~ti di /lel'gCllllO, Z ... p,.. .. ""'" delta grotit .. di ... pe .. it ..... "Aatiawoo """ cwi Ita da • .,." .... oue""to iZ Co ... o di P."'ezio_"to. "h. Ita ,...0 po •• ibi Z. aneh. OIIe.ta iniziatiTJa.
MiZano. 20 giugnD 1918
n prof"MlI' ZUrtI1c P. BdaIIt ha ric_to it dipZ- di dottorato daltd ~ d6lla S4Ji_ C._Z- lid II1J • di_ doc:a..t. al Poliutmic:o di Proga ... 1 1117. Dope _P riaopcrto i~i di w.,.,aIlW7lto al C.E.B.'/.P. di Parigi, aZZ'/h..,;lJ8raitd di '/Orollto'. aZZ'lIIIivaNitd di Catifo .... ia a Bft,.~e roil, .!aX 1.S~ .fI.i i""fl'lOl praaecl lG 'ort'-,u,.. uIIiI18Pait¥. dava ... 1 II'1J di_ prof .. .., ... oPdinario. E' _1 ... " di AI'fI_ llatio .. at LaboNtorr. Gb1c Ridge .rational i4b0l'lZto", • SaI'fIetIt aIIIf LIDIdJ, E1tgp •• IltJ PNaUdMto td ccarrl.nou ABa.."., ... lla Pl'Op1"Utd d6i "",t.rUlli • fa part. d6Zla diN.iorN aditoriala d6i ,w",tdi ASCE-"" • C_t aIIIf COOlON" R._h. n prof'''or IltJaaott l'icauatta ta -'aglia RILDt 1161 1176, ....... oitato pal' "brillaltti .lIilkppi ",zta ~ dai ""'t'rUlH, ",Ita t.~oa d6i f.-i iii_i • • ",ltd Uoria datta .tabilitd. fortdarldtJ ~ a~"-taZ • • fo_t<zaitww taDric:a" • .. , 117' riclHtu daZtd ASCE H pNlll'io II. L. likbal' ~r td ric:aroa ",Ita illfl'fllWN oillil •• • aundo oitato ~r "ric:aroa "'fIH 'ff.tti d6llG lIi_ntd, i ... taaeioitd • eo>tt ..... to di wrriditd ... 1 Cldlc:a.tnaao. """"",rt..wllto IIorI UVaN • dipartt/6rlta daZ t.,., ... Zla atl'kttkN, Uoria d61td atabiUtd e d61td /Nttk:J'Q" • ... 1 1177 it premio '1.r. Lill par it .:."j"to _to pre""""1'8 .. 0. par ta -"" "Vi..,oaitd • ritiro ... i ,...ci di eo>tt • .n-to di Hattori _laGri". EgU ha appAa ric:atlkto td pN.tigiQaa berea di atktHo ~ .... par td ric:aroa ~I' l'/DIJIO ~ao Zi'l'-'II.
....no. E' co ..... t ... t. di "",:t • • O<riaed di propttaaiorN ~I' ! '_lui di ael'l<ttkN a".ci4ti ill _to _to. E' """"'ro d6ltd ~aait1lV ASCE"., ~I' i ... todi IIIIZt_tici • d6U. atJIIWIi .. iorli ASCE-S'fD ~I' la .tl'l<ttkN .".ci4li, lG propttuiolla d6Ua atl'l<ttlJie a ,...t:io di a..llto _to a t '_lui ~I' d_ti finiH d6U. atl'l<ttkN ill _to _to. ,*,lta part. d61td actillitd di rioaroa 1161 prof."ol' S4JltIIobri4h • • ta;;a d6dieata al-10 mlkppO di ... todi <l>l:2litiai ~I' 10 atkdio d6ltd ri.poata .taticG • di-wa d6U •• tl'kttkN. Si • OOCt<pato ill IIIodo partioolaN ddl'_li.i ~I' at_ei !, .. iei di atl'kttkN ~ a a Q.aio, i .. .:_to ~to, ~I' la qk4ti Ita .IIiZkppOto .. todi di oalc:oZ<! capaci di t_N _to di l.pi 00-
atitktilli IIorI li .. eari .• f •• lkrdliolla ora .... iual'aa~t. adottati. R • .: ... -t_ta .i • OOCt<pato d6lla Nnau..aa aU. aaiorli .i.acM ili parati di "",,_to • torri tH Nfrig'1'GIIioIIa •. ad iparboloida. 2'ra la "",h. C1p8N
di .:..i • • tato _t ... ta pel' td propttaaiorN IIG I'ioordato it "~i .... aIIIf lfatatori...," 1161 Gaorga JoIill,- Colt., •• • at 1I?? Ita ri_to it p"'",u, A. Vorl IlIII/Iboldt ~I' la S4Ji ......
Prof. Atn C. SooPd.li,
II prof .. ..". Ala C. 5;Jorclalia ~tt. it dip'- di IaoIIelor of ~ daZZ'lhtiueNitd di CaZiforrria ",t ISH' ad it dipZ_ di Muter of ~ dat Nu--.a.tta I .. aeitku of r.clwloloflll ... t ISH'. 'dto .u .. o _ ~ ad i",8f"tZN pN610 l'lhtiuerlitd di CGlifomia, daI18 _ c:a.n_u .fi_ prof ... ON orcli>tarLo. E' _z...t. di ftue .... ,'Iocie-td d .. prog.ttaaiolla ad ag ....... fIOlI8,.,..,ti.,. par t '_Hai di "I'kttkN a p.84io • ponti i .. ~to CZl'IIIlto. £' pN.w...t. d6ttd ~ •• io ... ASCE ~r l • • tl'kttkN ill ~to ~to • IIIIIIbro dell. ~a.iorIi: ASCr-ACI • lASS ~r i fII.'oi i .. "_lito _to, ASC£-S!'D ~r t 'analiai ddla at"'ttkN i .. _to _to _ iZ .. todD dagli .l_ti filliti, ACI ~r i ponti ill "...... to _to. t.a ric:a_ avotta dal prof ... or Sc:orclaU.a • • uta .... N rivolta allo .tkdio taoric:o • a~ri ..... tal. 11611 • • tl'kttkN iJt _to __ to, ad i .. particot<zre d6i fIk.d • d61la tralli da poilU a __ • rill daflU --a: '60 ha ric:oIIoaoil<to l. poa.ibiZitd d6t .. todo II6f1Zi .. *-.. ti, ill uMlOiaaiorN _ to mlkppO d6i ordcoZatori .lattrootiai, ~P 1'_lin d6Zla .tl'l<ttkN a gwoio ad • atato tN i prUrri ad appZic:aN i 1 .... -todo d.,U .:-ti filliH alto .tkdio d.U. UI'kttkN ill _to _to. Pra 1 • ..,Zt. O~N di c:ki • atato """",z...t. ~r td prog.ttaaiorN IIG ricordata ta copartkra a parabot0id6 iperboH.:o dellG St. 1trrrt '. Cathedral SaIl F-raNriaco. Nel lI76 Ita ric'''''to dall' ASCE it premio l*Ii .. aiff ~r td -.oria "AIlaU.i '"'" till'ON di gk'c:i di C~"""to _to di fo- qk41.ia-8i", e ",I 1978 • • tato eletto ... ftlbro d.ttd Ac:caderwia Naiorrat. di IIIfIegneria degli StaH lIIIiti.
Table of Co.ce.u
I.eroductio.
Pall I
II£USnCITf AD FlILII£ IF C.ICIfT[: A sOlm If I£C£IT rlDCIESS
bJ z. P. JAfAIIT
Prof ••• or of Civil l",i •• eri.1 Iortbwe.t.ra Vaiver.itJ
lv_to •• l11iaob 60201. USA
CIW'TIIl Al DlClDllllTALLt LDIIAI TIIIOaIlS FOR ftlAXlAL I!IIAYIdl
1h1a coadlUoa .... raot ••• that the work 40Ge 00 tha t ial d 'ft_ - er ut_ a c,cl.
of ."lylq aad r_iq 401j be _PUy.. (Th18 ia oo1y tM _004-
order part of the work. aiDc. tbe flr.t-ordar work ~ ° dc 1 ." lj lj' caac. • wlth
the werk ot applied l0.4s aceocdiq to the priDeipl. of virtual work.)
UDd.c e vld. rao,e of .1tuetioaa. the aoo-o.,atlyeae •• of jV2
1. equivalent
to a .. terial atabUlt)' c0041tioa. lID_ar. th18 1& aot tR. vbea hictlOD
ia lDYolv .... AI hal been a"- by .1. lCeadel (l3] &ad C. llaier (14]. Ia
fact. a .1apl • ...,1. of a .priq-loaded fr1c:t1oa block llhic:h 'dolat .. 10-
etlul1ty (Al.') yet 1& .tebl. iD the tllpuaov .en.e Caa he ltv... It be.
beaD • .urpdae when - ._pl ..... dl&Cover'" (lS] for aa oppoUt •• 1tua
tioo; aa.ely, a ca.a Vbea the .. tacial .. tl.fl.a Drucker'. po.tul.t. tAl.,) ),et is UDstable la Ltapuaov ....... (Thls .. y be ob •• rved la ... terial vIIlcb
.... DO elastlc cupon ••• 1.a •• 11 lnela.Uc elso for UDlo..sla,. e. la tcu.
of coacr.te.) COIIHqu.otl,. the COftoactlon of Drucker'. pollulete vith .tabillty i. aot UD.ablluou.. levecth.le.. viol.tiDO of II L_' • ruc_r • postulate ..... ld b. acla1tted oal, vIIan th.re 11 a cl.ar ph,. leal ru.oo for 40iDi eo.
auch •• the effact of iDtamal fricUoa. whlch 1& abMot 10 utala but !aportaGt for cOIIcr.t. aad .011 ••
AdopU ... Drucker'. postulate (Al.,). c .... riq lt to laaquaUty tAl. 7).
..... ootiDl thet both 1DaquallUaa .. at oold for uy do... lt toU ..... tbet
.xpr ... the .t •••• l""r ..... t •• tb. _ of th. al •• tic aad l_lutic .t •• ir.
1Ilc:r_u. vbU. ill the f.Kturl,.. •• el It Is tb. eluU" .nd 1aalastic:
.cr ••••• r.ther thell .tralll. vbic:b .r. _ •• It Is _ca ... r, to Urst ill-
wart the iIl"r ... 8t.l plutlc .tre •• -.tr.iIl r.l.tlO1l. b .... Oil I •• (At.12). After doina this. tbe pl •• tic .... fr.cturina .tr ••• iDe ..... t. My b •• uper-
iapoa". This procedure l •• d. to the foUowina ralaUollS (19).
The rule for the v.ri.Uon of eaij ia the ceDtra! ..... tion which is hard to
elacid .... the beals of .v.il.ble t •• t data. • ... rthale ••• OD. cyp. of U ... ·
aatic hardeninl prov •• to b. appropriate for ualoacIiog and cyclic loadinl.
OIl puraly theor.tical IrOUDd.. Thla will be d1acua .... latIN".
The .v.ilabl. t •• t data have b.... fitted al8D.t ..... 11' well b, an
SDdo~hrooic for.ulatioo and b, a pla.tic-fracturiog for.ul.tioa. What i.
cben the diff.renc. b.eween th •• e CWO for.ulatioo.? Sioe. the .. t .... tical
for.. ar .... It. diff.reDt. where 1. tha dlfferenc ... olf •• ted? The •• qu •• -
tion. can be ao_red b, po.tuiatiog th. cooc:.pt of "in.lastic .tiff ••••
locu." [IS'. Thi. locus i. defin.d •• th. locus of th. tip. of all .train
1acr_t,dCiJ
(vactora) which liva tile _ aal.Uud. of inalastic .traio
incr_nt • I dC"I. {The .... ltud. aa, be •• Uoed, a.l. a. It I. (. • )" • , • -1j-lj - l"'lth of vactor 'lJ ta the dz-clt..oslooal str.ta .pac •• ) Th. eli.ceoc.
taD be 4 •• caly 11 the dlrectioD blj of "actor d£ij ie .pproat.at.1y t.ovD. and tb. liaeariz.UOIl vUl .. acc.pt.ble caly for atr.10ina 4!recti .... that
are clo .. to 4£lj' It can be ebovo that tb. U .... riz.tioo of the eo4ocbroo
ie lor..lattoa .. , lIa achl..,ed II, replacina dC vitll the azpr ... 100 (15.221:
tiber.. -~ ....................... (Al.17) 1j mt .. b ..
Alternativel,. an .... ivaleat 11neariaaUOIl can II. canled out dtrectly b
the .tiffoea. coeffie1eot. of . Uolt. el_ts (lOJ. Th. 11oeariz.tiOll 1 .. 4.
to • DOr •• ff.cti". oUMr1tal et.p-lly-step .1,ot1tba for .tructural anal,.is. Vitb the sub.titution of Eq. (Al.17). the for-.l.tl00 bac_ 14entical
to • pl •• tie-frKtur1n& fOrallat101l. an4 lt 18 .1ao po •• ibl. to azpr ... the
Al.ll II
teosor of taoleotW _dull C1j... Ob.erve. bowYer. thar tht. ter. 1a
41ffereot for each .tralolnl 41rection b1j
• Thls ls rather uopleaNQt
vbao tbe 10"'101 petb exbiblts •• barp coroer ( •• 1. u .... lly tbe c ... for
_terial 1oatabl11ty) becau.e .... 1004 e.tt.at. of b1j
can be -.s. and .aD, po.sible 41rect1oos b1j .. at be trie4. 501vina tba si&.ovalu. probl. of
10atabl1lty for ODe cho .... dir.ctton bij 11v •• DO iDfo.,..tloo on tb •• itu.
tioo for otb.r po •• lbl. bij • a fact whlch .. at be k.pt 10 olnd 10 1ot.rpre
tina fl01t •• l--.ot claculation. for -t.riala whoa. 1Dala.t1c atUfo ••• locus 18 oot atral,bt •
Th. circular .hape of the 1oel •• Uc .Uffo ••• locus 10 ..... ochron1c
tbeory cao be DOtb1na nor. thao an approllt.aUon. 111 ,.oeral 00 .... t
expect the 4btanc .. to tb. 1oelaat1c stUfo ... locu. 1a tl!e bacitvu4 direc
tion and the uo,.oUal directloo to be 41ffereat fro. that io the • tra1 Iht
ahead dir.ctloo. ... eodochrODie theory vb1c:b allow ..... to control the ..
two properties -y be obtalned by r.pl.c1o& 4t vith Eq. CAl.lS) or with
vb.r. dt' • "" 4.1jd.1j .......... (AZ.lI)
r •• pectiv.l,. The •• two azpr ... 100 •• r ...... 1 •• of po.aible Ura .. 10-
duced anisotropy of teuor Plj .. dafloi ... ehe btriD.lc t1De. The corra.-
pondiol 10el.stic stlffo ••• locus is .0 ecc:entr1c clrcl. or an .111p •• (IS)
(FlI. Al.6). The available t.st det •• "-wer, ... to II. iaauffieleot for
1deotif,101 COD.t.Ot. PI .od Po 10 E.a. GAl.16) end (AZ.17).
The concept of lotrlodc -tiae baa racentl, b .... ,.Deralized 10 • 411-
ferent "00.10 b, Val.ola (21). eoo.1der1n, the uoiu1al ca ... b. propo."
that 4t - 14t - k d£.,1 wher. 4,., • da/E._ and a~ that for k • 1 en
auccescfully to pradict cb. r •• poase of b.... co .. isaie-C". c,clic loedinl
112]. ~ exaaple of the aareaunt vlth t.st data 1. liyea 10 Fil. Al.l.
rbe theor, allow" 8Od.lial r.aUstically tb. effect of .• cirr .. p. OD duc
Ulit,.
Al.2 38
The probl .. of teasile er.ckiDl i. of particul.r 1Aport.oce to COD
crete tbat is re1Dforcad by • steal bar aet Whicb at.bilize. tbe cr.cke.
ril. Al.l rita of P.rk, IaDt sad s..paon'. r.at Data [60] (1972) OIl C,clSc a..peas. of ..... by IIldcebroaic Theory IIlbaDc" by eraclL1Dl 1111 .. (after lalaat aad lbat [13]).
Thb ,robl_ bas 1a tbe ,.st ...... .tudied 18 t • .,.. of 1111it dedp (.·1·,
Jef. ll). Sa which tbe ... albl. fdcC10a OD the cracke bas b •• a.llect".
......... lored [l4J sllll. curiously. it a,pea .. " tbat a beavier ratber tbaa
1ilbter re1Dforc_t 1& r .... ir .. vhe fr1ctlOD OD tile craeke is ace_t"
for. Thu •• a oell.et of frletloo 1& DOt be ... Oft tbe dole of safety. Thi.
i. becau.e tb. t .... U. forc •• 1a yieldlol reloforc_t _.t not oaly bal
aace the app11ed load .... t ... t alao provide tb • ., ........ dve force aero ••
the crack tbet 1& aead" to davelop fricUoa. This force is alvay. ioduced
bJ dUatloa oa the cracke .. pOll .Uppi,.. of roulh, iaterlocked crack surface ••
(0)
Cb)
2.0
1.0
o Fli. Al.2
1.0 20
a = 30° m=0.5
~et ' • a .... nfarced Slab Vith Friction a th and Corre.ponelin, Lt.it De n. erack. (.) Friction Coefficientl k. (;!~c!~v:!~~:'i!b~ :o~ ~~rlou, eailtinl th~ory tapli •• k. -.) (After Bat ~~. TsubaU r 14)). ant .....
Al.4 40
DestlD dlalr_s for thl8 type of aptia ... liait dea1ID h.ve beeD developed
(34] (Fil. Al.8), .Dd tbe r •• ultinl .nvelap. of safe delilDa i. liven by
«n _nO) - a (n _DO»)( (D _onO) - a (D -1I0)J • (a (1- .)ain 211]2 (#.l.l) "" Iyy yy 1ss 2
in which a l • (1- sin a)/(I+.in a), 82· l/(i+ ain ,), 8· "'Ile of
) • a friction on the crack (probably 8 • 1.4 to 1.7 , Ds· N;tNI' n • N-'N1 , • • Y Y
•• N2/NI ; N,,' Ny - yield farcu iD the st.el rdnfarc_nt in arthalonal
directianl " and y; III' N2 • principal iDternal fare •• d .. e to applied laed,
II •• nlle of N1 with rupeet to .,,11 ". CeOlUtr1caUy, Iq. (Al.l) repre
.eats • hyp.rbol. in the (D", ny) d •• ian plan., aDd it il DOtevortby that
ita ..... ar. inclined wbereas for aero friction con.idered thus far itl
.... 1 are ortholonal.
Al.2 Str.in-Lacalization Instability 0 ... to Strain-SafteniDi
It 11 w.ll known that, except at very rapid dynaaic loadiq, a conti ....
u ... caD .xhibit no .train .ofteniq bee ..... it 18 unltable. Yet, Itr.iD
aafteninl c.n b. ob •• rved exper1aelltally even und.r Itatic lo.diq, aDd all
Hoi.ture 'er. •• bllit cI Char.cteristic of C y aD Sorption laotheraa C.tter (481). ODerete .t IItlh T"perature
..
A4.10 S4
CONCUJD llIC IIlltAIlKS Aa is clear frOll the preced1l1, .urv.y. a rea11.tic and &ecurate de.-
criptioD of the 1I1el •• tic properti •• of CODcret. nece •• it.ted theories that cut .cro •• "0, diacipliae.. Thu •• .o.t .i~1ficant future .dvance • .u.t be ezpeeted frOll an iDterdiacipltDary approacb.
no. i ..... U.bl. tbeor.tical ... tur. of v.riou. qu •• UOIl. often reapt. ODe to eRP,. 111 reaearch for the aake of tbeor,. To ait1&.te this wid.spread .. lais •• it i •• ther.for •• of pera.ount 1JIportance to keep io a1ad that .U tbeoretical developeellu .ust he checked .,.iD.t esper wental dat •• aDd not just a fev cOllveni_t1y .. leeted fllYClr.b1. d.t •• but .U re1ev.nt data av.il.b1e in the 1it.ratur.. Vitb the advent of cOllput.r opt1aiz.tion technique. for dat. fittiDl. this ta.k ha •• fortunat.1,. been ,reat1, reli.v.d of the t.diou.n ••• it ODC. 1I19Olved. no. pool of te.t data av.ilabl. in tb. 11teratur. is v •• t and .u.t b. tapped. even tbou'" .. ny tuta b ... not be.n .s w.ll doc_ted aDd •• clo •• l, coatrolled •• OD. ailbt have d .. ired.
ACIQIO\ILIDQGHtS Mo.t of tbe researcb covered iD thi. surv., has been carried out .t IIort ...... t.rn UniY.raity with th. support of U.S. latiooal Sci.nc. Foundation (uDd.r Cr.nts CIt-26030. DIG 75-14848. DIG 75-14848 A01 aDd DIG 77-0676~~. P.rt of tb. re ... rcb surv.,ed ... also supported b, U. S. Ioer., Ie.ear and Developll8Dt AdaiDietr.tlon under contr.ct with UniOll Carbid. Corp. throulh oak W,. Jatiooa1Lahor.tor,. aDd anoth.r pert of tbe vork wa. carri.d out ""U. tb. writ.r va. vi.itina aci.ntiat .t Swedi.b Ce .... t aDd Concr.t. Institute (CIl) .t lD,al In.Utut. of TechnololY. Sweden. no. writer i. thankful to Pe1ict.nico di Milano for provid1l11 the .t18Ulu. for thie vork •• nd particul.rly to Or. LuiSi Cedo1iD of Pol1tecnico
for .. n, h.1pful sulle.tiOll ••
M.ll
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M.13 S7
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V. C. SCIINOBRICH
PUT ][
Film (l(IIEIT 1I1L1SIS Of IEllrORCEI COIICRETE mUCTUI(S
by
A. C. SCORDELIS
Professor of Civil Enaineering University of Illinois Urbana. Illinois 61801
Professor of Civil Enaineering University of California Berkeley. California 94720
Teble of Contents
CHAPTER B1 FINITE ELL~~, MODELING OF REI~rORCED CONCRETE
STRllCTURES
by A.C.Scordelis
CHAPTER 112 THE SOLL,IO/I OF THE liO!'iLlliEAR EQUATIONS
by V.C.Schnobrich
CHAPTER Bl FINITE ELEMENT STUDY OF REINFORCED CONCRETE
IIU.'1S VITH DIACO)lAL TENSION CRACItS
by •• C.Scorde1is
CHAPTER B4 SLABS PP.aIlLOtS
by V.C.Schnobrich
CHAPTER C1 P .... 'iEL A!ID lIALL PROBLEMS
by W.C.Schnobrich
CHAPTER C2 ANALYSIS OF SHELLS
by V.C.Schnobrich
CHAPTER Cl NOHL lNEAR .... 'iAL YS IS OF REI!>'FORCED .... 'fD
PRESTRESSED CONCRnE IIU.'tS AND FRA.~S by A.C.Scordelis
ClIAPtER C4 .... 'IIALYSIS OF CURVED SEGl!Elo,ALLY ERECTED
PRESTRESSED CONCRETE BRIDGES
by A.C.Scorde1is
61
115
129
155
177
211
245
295
61
C I NQUANTE!'lAR I 0 ~El CORSO D I PERFEZ I ONA~IENTO PER lE
COSTRutlONI IN CEMENTO ARMATO "FRATElll PESENTI" (1928-1918)
ANAL lSI DEllE STRUTTURE IN CEME!'lTO ARIIATO
IlEDIA~TE IL METenO CEGll ElEME~TI F'~ITI
• SE'lIHAIlIO 20-23 GIUG)iO 19i8 ..
... rtedi ~O Giu9no IOrS:
9.00 -
9,30 -
11,30 -
14.30 -
16,30 -
9.30
11,;)()
13,00
10,00
18,00
Prolusion. del Cirettore del Corso (Pr-of. A. D.i Pol i)
I ncr ... nta I Pla.ticity (Prof. z..p. Bazant)
PI •• tic F'racturing .. teri.r Theory (Prof. Z.P. a.:ant)