smRESS OllA;IGES :u~ REibJFORO'la!J •I

/'

QONCIU!lf E M!.Ffl13Ell$ JJtfJl.I TO QitEE.P

Thesis i;rubm1tted tQ the Grad:uate E1a.oulty c>:t th$

Virginia. .Pal;rt~onnio ll'tstitute

in candidata;Y :to~ the degree of

ns'.f :ER 0 I! so :CENOlll:

in

July l :96:;

Bla¢lcsbttrg 1 Virginia

1. ti. • • ' • • • J. •. • • j_

~#.Iii! ~

:tI;t_. Gen..eral !1scua:-$1on ()f t.he .Ci•e¢1r Phe:nomen~it . ., -• 9

A, ilt!f-in:ltiC>• o;f Q-l7tElp_ Mt B~let l>t~ctt.11JeJi.o;n @t Qr'Ge.p 'lb.$~~1.~s .,. ,, ~ !!' • • 4f • • .- .. • • 9

Th~ la.tu::re of tre•p ,i •.•. ,, ii: • • , • .. • t l. · Ett~nii Qf lf$te~- 'ltll'l'i !l'd.:t:tf.h • * 11 . , $ 9 2 • lffeot •t ~4 <;)f r'ega1M~ .; • • .,.. • ., ll '-• ·:;~~== ·:i$::e ~u~j • '.~r=~ ·• • • • • ll lt... J!l:tfee~ tJi 1.:tzf) ot le.Jilt, ih .... ti: .. Ii .. "' aa S.; l(f&~·t of * · ihe%1. ~:tit 'Loe.d.ed ·I!· " - • • 13 6 •. Su.mmar1 ~t tb,e Orceep '.tb,$noaat.:na.. . •. • • -~ 14

Jef'lt:ct.J.tiJ.tt$ tn lt'lnforted filn~reiie: :Start$. • 16 i. · Jrt:feot rt>t O-inpx-t.sr.dv'1 ·R$l.nfo-:roeme'lll:t, · · ·

on !t;am. 1$tl~eit1oss • , "' -" • .,... • ... • US: 2. ·· Hetliot · Qf Pre:rlict.1.».g Lt.l'lng ':tme ~~-tal

ltt&flet·tio»;s . • t ,. ... " • •. ; .- • • • .. 11 :;. llQsti• a:tltti C'.t'$Ql? :te(:ove17., •. , .-.. • • • 17

ih_e. lat. u.··n_· .... $f _ ftri11ui:s fhaJllZl'tS Xll leinf!!>Tee4 Oon~:r-et~ Memb$:r& • ., .. ..~. . '.• .. •. ,.- . • •· • ., · 19 l. $hr:111Ut&ge E.tfe·fJtt on !St res$ Chang$$•.· , " 19 2. Cr1i:u~p !ftf~t OI1 StJt~i'l·fZ!: Oh .. pges ... · , .. ,. io 21 ;. f:he it4.lits'J1tib'"t1ta. of· leadl.n' Mom1Utt$ • !! 4., l'en&.t1e1tal . axut kt:rtim~n'tral llfte~t;$ 0:t

St:r-••• Qlaanges on th• st~otuire • • • ., 24 ' ' . . : . '

IV. InvefJtisat11\"1 c;f Stra&$ Ohange$ ill. ile1ntorled Oon~:re1HiJ Me1ll'b•;i:$ - • . • ''* ·• ~ .. • ., ·• . ~ • . , ., ,. • ,. 36

'Iii--~& - .. ~-a~.,.. . . ' . . . .

\l'i'!· •• f. ill ft.:t'G$.S a\ (JI) .... s~a.n,. !4'ndln' Kt1 _ \!'~• _ _ _ •... ll .. _ ~t·e·tf l1~d.e•~. ,. l~ -p1l'!ittnatiQD. ~4 t>tmtT.\S$·1ta •t tll'apb;fh1 " ~-~ :tr~• .Qt ~1fai~h.m f~~- . llt$i$1\ l?i&bl•-•' -~ 'Ii .;. · ·

F+ f:*!a~i:i:h~=f.ft~=~ ~-~~ ; b.. IQ\14.bl~ le.inf:~11~tl SI _ • ., -" • •

2~ Oa~'ti:\;.cnre1: l\.t#1'1·-I~;];~~~'"-· ~;s~ga ·. ~- WI .. --~- ,.

;$•. - th,,11111t. l•l:nf~:i"O&dl ltttl~-.~ · • ' ,. ·•·

3. ~;'1~,.~~rr~;;,4 .~~~-~-)\:.·_ _ #.,,_. lX~H!l. .IJidtdl »l•ll ~l~'·J ~$18-~ ,1 •

... -~ -.Ii,' f,' ...

VlP.t- · .A,o:gm.~wl1t!g•s:atH~~ * , -~ 1 tt -~· 11-

_'tt' __ ;x.-_. ·'"'--4_:""' __ -._ ... _-_··_ ~ _ V~D~ • •• ~ •• • • • •

1~ JQ :;a JO 'i. 31 ,. ':llit. ";1:/.;j;'I

'' ,,, '}6 ,, Yt

~' 56 ~6

. 51' :i-9 ~' ~lHl 6J ,, 11· 1~

,..,,, Sc;

\ ,•.

1:tme -~ - ....

'~

. "i\!]; . ~ ..;,)f

4

' 6

'!: I

,, 9

. 4 ..

·. ,,J?ia.t

:e; :.,. ,. ·:i· · 1$: ·

18' -:Ela~'t l:cl M4 lO~~GIP· ltl•:r•~ 1~ll• i. · • .If .,, • 'J' • . ' . ' \ . ~ ' . . ' ' ' . . ' .

. . · $b.:rinitqe · tn : a llfll'J;ol"e':tf,• •~meat• ~· :• ·Ii ·• ·~ !' ll ·. . .

;s:&$stt.a ant\ G~-,~!'. s·tt~•ita. ~lt-tJt1'b\tt.1~a l?l. ·1t '°~aO'tj't$° J11.tlil'llit : -~ -~ -~ · • ··· -~ . -~ • -. .,. ii< ~-

· · ·. _. ,. . ', . ,

; : -,~Jt$llt" 'ittlnf•~••t 'l•1ti•• ·.~ . • * • -~ '-~ •· . 'ii' . • - . 23

:.:.j.'::z:::::~:: ~ ~~~.· .. ~ ... !t · .. 11·-.gt-•.· .~ ... • ~- 11 :~ '·<!i " ;••'::~~:iii,. iii, •••• ""·· ••. :~ .. ,.,,, ; .....

io : . ,_.: ~::~i:.~!t~i::;:t•6!•!:'":~:: :~·r•:,.,~ »:a!~~JJ: .. :,1 .

11 o~~al!gtt ·1• .· lt:r$e.l•l.,4.p-·· hfit;iQt.t•il , ~ • • ., ·· .,a· · 12 13 14

l' li

17 l.fl li It 21

24

29 ~6

27.

26

JI

I!

u

~i

fl

If

f;I ~i fl

M ·ii ~t

II ti 'ff·

.~.

• • • ~ • • .. .. • ~

.. .. • • ·• •

. '"

:n1inne:roiu.~ i!lV!tlst1gt;;ti;Oli'S both an~.l;rtie~,ll;r a.,nd. expe:r:tmentatiy over a, long perlqd cit t dating :ft:i;,r b;~t.!!)k ..

.! :reeen:t · liiilt't1d:.V Qf Mom.iant ..

1n. Qontin:aous Stl'R(')1h1're& of ''l\ell'il,:f0·~~e.d Qf11>.:no:tEl.'i·;~n by Ko ... €fh1

Hsu {6) led 'this w:ri:te:r t•Q p~X'§!l'tl.e t.~1e to1)l.~ f'u~t.h.tl;!;t., l;.$11

nevel~p$d g~n,~ral1~e4: rn~xn~n/t~~u;r>va:tU;t~ .:r:ela:fi1Q~$ .;fQ;r s~v~ral

rei:nf~r¢l.e¢i oon~::eete ax:ult .$tttlii~d.'. tl1$ re(;\i;tl¥)t:tJiJ>ut1en ~r

bend:tn,g mt:>rnen,t$ in d~tail in .~ t1l!ted ... e.nd beam lQa!d.ed. u::rd.f'OlJ'.m-

ly and sctfbje~te:d te Cf:t"ee:P•

the redi$triln..tt.io:n. d:ld

a;nd mQ:l?$ g,1gn1t.1e~nt · m,exa~:iat

A saor~ d'ta,1le!fk :p~:r~$$tl .. e;f th' l:,l$.~~o ~ree11 eq~at'.ton$

$'t t'$$~1 eb,an:,,;e~ $'!,S

¢1l'~~P tt>a:t.t$'i.'.!tl atress enang;es· :whi,Qh ~~re tnd~p$nd:ent ot m~~~n't ~.bansr~.s. ~ri:he:r;i in ....

ve$tig~:t:ton l~d tCll t..he 1'$~l.i~a'tli~l':I, t .c;cul'~;

where .. the. bend.$.na; m.crnien.t ~ein~inE!dl ¢():ia,ataat, as in a loiaded

c.aJlti:tever lH~aro:.

en.itl.iD..$d till d.ev•l~p

a vat>ie'br

~t bea:r11s of differe:o:h ge0.metry ~ material. pt<~p~rt!~s al1ld oie;ep

c oe f'fie 1¢:nt s.,

l!aslo f.()rmu.la$ were d:.er1ved, :ra,:nges e.:f para+nJ~tl?rs. ~sta'b­

lish~:Hfi and data d~V$lQ;psll\t, uait\I 'tl'l~ li~M 16~0 ::01.!51te.l OiJ1ifJ1n:ter. '. ' ; ' ' ' ,_, ' ' ' ' ' '' .·. . . .. . '.·,·

f:b.e ds,tt;t was used to produce ~~VIJ~$l forl'lla ~r g:i:"€1.fihica.l c1Jtrves,

'?he use of the e;urves is illnstrf/i:ted t,n typieal dEisign p;roble:ms.

7

A-c .Area of C:H.):UQ:tete

As Are~ o.f tension !lllteel

A's Area of compression eteel 'b ¥lid.th of beam

o O:teep eoe!fieient

d l>ista.11oe f~dm extreine eomp.!'aS$1:ve fiber t0 ee.ntrQi4 o:f: t e.nsile re:1nfc;,roen1ent

d' D:t~rtanee flNnn ex1rreme eom.pra·S!s1ve .f'ibar to eentroid of eompres1t11,.e reln:fti>l"C:H$1vn.ent

E0 Strain o:f tne eone:rete Es Strain o.f the tension stee.l

E:* s Str~d.n (:.)f the eo!iipre$s1oa steel !le Medulus of elastieivy: o.t tlte oo:norete

E8 Modulus of' e1ast1e·lty of the steel

f St:resa

t 0 Stress ila the eonQrete fs Stress in. the ten$l¢1l ~rf.Htel

f's Stre$s in t.he t::H)mpress:lon $'te·el

I Mor!lent ef inertia

3 d I.nternal moment a::rm

k :Proportion c;f depth. from. the compreea1-0:ta ta¢e to the neutral ~'ti$.

kd ·J)ia:tanoe fre>m extre1~1e eomnreB'sive fiber to ne:tltral <"iZis of· the see'tiGn · - ··

L Length of .structural member

K Jen.ding mement ;n Ra,tto ·lltf Es t0 ie n.~,. leutra,l ~:l$ p P1opo:rti0n t•:ni:d.o.l.i .1:1;lt~.el ... As/bd

if' l 3ropo:rtie1:l. of eOI$::&>~e~.s1otn $i.le$l .... A~/'bcl

. l\~ Xn:te:rnal. e0mpresa.il:~:e. fl\);t(t;fa Qll th$ ~Ol'l<ltrf3te et t'.ll& b~a.•

"'····· ""'.$

In,te.m~l •ts;.1ils .f<>?e• the 'bes.m

Intern.al ~Omi;rtess1v>$ fGJ."{;\j(I! an tn• ~om.p:r~s$1()n r~.1ntoro.em~11t et th$ J:>$q,iJ.n

A. })efi1lition ot O:t"e~:p

tfnder $ueta:i.:ned

Brief l1'.$ett$Slon o.t Ot'eep 1'he0:tles ..

pla1n. eon~:rete a.n.d oon~r.~te menil!>~rs

the t:tme ... dependent po:ttl~.n af ·.une t·ot~l st~ain l"f!lSttlt1:ng f;rom

a. giv~n load. \fhe $tl:e.a.ai-e~wa.1a tU..'.lM1'$ ~.J;l :f:Lgu.re. 1. ( 1) d<!!""

p1cts tbe cl"ee:p 1?r!!.1pert1 :t:ti 001l¢ret$ .•

!hel"$ are several th~Q:t>1~$ e:~11la.in.in1 tlti.s ph~nom&l1C)'!i•

paste<, cltnrn.t.re of :.tnt<::rnl'it:L vt\iid.s ,. a:n.d/o:r ~rfl11ta111tte :fl<tAvt in

atggregatea f but 1i:t is bel1$Ved. thll:d~ th~ xrrajor pe)rt1on 1.s

caused b7/ see pa,~~ af. $;1.btlG)t'lH:ili w~ta.:r

formed by bYdratlon of t:n~ eeme!:l:tj!. (l<})

~rhe~ee ar~ maJ!lY faot.0:11a r'lat:lng t4' 1\he ~U;alit;r of eon ...

cr~te t.:n.at ~.fteet tl1e magl'l:ttti~e. a:1t1~ ~a:te tilt ci?eep. ot

t.he$e fa(JtQl'·~ aX'~ t.he wat~:t"-.~el!l~nt J:'.'$.tlc1,. tbs t;rp~ of aggre,..

gate.,. the. eon.ait:to:n,~. 0t durln~, the· t~mpe1Na.ture eha.l'l:ge$., si,ze

10

water~oement ratios oause increases 1n the size of the pores

in the paste structure. Thus water may more readily escape

the concrete mass, larger voids will·result in the concrete

and the water of adsorption may be expelled more readily from

the gel when the concrete is subjected to compressive stresses.

Inelastio Range

Elastic Range

Insumtaneous Loading Loading

-Inelastic Deformation

Oreep Deformation

. UNIT DEFORMATION (STRAIN)

Figure 1. Stress-Strain Curves

Since the water-cement. ratio is a measure of the strength of

the concrete, it can be stated that low strength concretes

experience a faster rate and greater magnitude of creep than

do high strength concretes. (10)

--

~. Eff~'¢t !1 .. !{ .. inc~ !!. ~\~~r~a!,i:(f:. Pa:via, Davis and .i~,milton. (ll :tan sev~rs;l 't$ats <:Jil ~fi/Jll,..

eret~ specimens t¢> s"',uliiy tllle etreat~ lf)f .d:tfte:relt'l:t kinds ~t

agg.rega:tee Gn c;l?~ep. !n tb.E;11" t~ets, f&;ll aggregates 11ere

uniformly gr~Htled tr'o1i1 fine t ~· ¢oar.S$ a.:nd 111~re hartohad in fi!,

sa:tu:ratea,. sur!aof.1-dliey' ~o:nO.itio:n; th.$11" ~l'Jsr.'.rtption taetQl'.'$

were generally lo~. !he;t<efoI"~, e:r:tf va:r:ia:tions ln e:reep wel"$ ' . . .

not due to the n1J:>iai:lurs····Qond:ttt6tt$ of th~ agsregr,rt~~. Id:e.tt; ...

1:n the S:P®C$imens w~re. not dtu~~ t~ s.eepia,se trmn the gel .•

lt.U..n¢.~ the eo:n$:rete spe~im~laB 1tt~:rie4 o.nl;r as to the ld.tid of

aggregate • .it m&;v be statti?d t)iat orysta.ll.:t:ne f'llt}w 1111. t;.he

~g;g:itegates ll)~u$ed. the d.iffere:n:b. vtr1,ltt~~s ot c:teep, It 1..$ poa-

$1.ble t~t \ta:t1.a:ti()1tts :in p~1"\iele shaPfa; .a'U.rfae:e texture,

and pore ~rtruet'lll"e {)f the $~$reg~ti:a$ m.a,;r ha~ve half! some it!-

fltHiln~e •. From the tes'·ti d.e.<te,, it wallii 1Je>n.0lttd$:~ tha.t t\1.e

the in.f'lue1t~~ o:f ditfet>a~t

be s.orrsidered l'l.$l:l§S.li@;i'ble 1:n

· a.fi'ef'.l'!t ing creep •.

3. Ef':f~et St.! ~Jle JJ:11m1c11~tz :f.!a:r\u. o.,i,;riry:s ~·

tha:t <lu~ed und~r <Jtey ~onai.tiott$. me,sts oond.uet~d, by Davi.iii

and ~roxe:ll (lO) on e1onc~$te e;yliat!l~:ts ¢if e~,ual size and. un ....

de:r e<raa.l $Uetain~4. leads, s:nowea, l1ha:t eyl:inie:r$. cured in

To pe:rQ~:tat r~latl.ve h.tini.141ty ~:ltpe~ien¢e<!l abeli1$ ·two a:ild ()ti$ ...

half 'times the ~reap defo.\t':rfl'a.tlt~n of tll,()~e ~ured 1n water. ·An. i:ne1rt~e,a~ in the numtd.tt1 of thi a.tn1J">spn1;.rte: reduQ<a$ th~

10$~ 'Of moisirtu:>e to thl.e $U.rround1t.Li atmesp~,er~; th:l$ slows ·down th$ f'l.et~'l" of mo:\.$.t'Ure ti!) the au:t.~r $Utcrtaee Qf th~ een•

th~ gtll. A f'aoter a.tfeoting ~,om,pre.e.$;i:ve e.reep i:n eoncret$

membe"ts ts tba;t a! deyintat $1'.:rink.ai~ ~t O?" near t.l~,e li'Jul'itaee

of the member'. \t'h1$ dr;ril\e; ~l1.t':1nltage results tn r~d1letion

of the Qross ... $e~t1('}tl:.al, ~:r~a r~ma1n1.ng :1.]l o.Qmpl"eata1.$;ia; and

therfitere ()au.me$ high~r $'e::reu:i·~e~, ( a:nd $.l.~,s.t i~ st rain). on the

oe11trfl\l eer~ rJf t'.h.$ tona:rt;rt•·• !'.b.~ nigl:tJ.:tl" t.b,~ .rela.t1V$ · hum1.a.1 ... ty durin!! eur:t:r1g,, the 1ts$ d~yin,g .$ht1n;kagl$i tl'1$re will tie.

<lOl'l$:\.de:rable ef:r~ct ();p, tbJf'k e:reep ef tbe co.ner~te. From tl;te

l'$S'Ults ot ~reep t.ests , ¢e1nduet$tl b;!f Da.111$, I>a:'ifie a.nd. l{s,ft'l.11ton

( 3) • tt was shawm t11a:t under t1qual val~•$! Gf ~uatai:iied

etre$ses :fo:r- a g1v~n perio'l Qf t:trtt$ 1• 't:he Qree:ti d.eto:rm.~tien

·1n a 6-.:l.noh d1.a,mErter ~tlinder; .. bl)th ¢fl.inde:rs were the sam•;

len~th. ~he fa¢t tl\ts;t under eq11al sust@-11.te<l atreases e;reie:p

1$ le!3s in la'.t€$~r mas~es of' o~ner~te .1$ pl"Ob-at>.lt clue ta tl\t:~

redtto$d $!ee.pag~. :tn l.a;tg$ eanQ:lf'ei;~. tt.t2$Se1!.l the pa:ths tra:ve:i ....

led. b;r tll.e e~:r>~lled water w-111 be lQng~r with a :t:>e!~ultin:~ in ...

~:eea$e :t.n the triotion~l ·resis.t~~.e t~ the flew of water f~~m. the tnte·rior ()f the· .:tlltl;ss. l,ii<i».r e1!ll"l.~~ of st:mtll!l;r shaped, apeet ....

mens in d:ry ~lr, the smalla.r s1~Hi!~"trtten ba.s :1nere .area. pe:t" uait

vPlum' tnan does tne lal'g~:r 'l\J;p.e~iRt~n1 ~nil is theref()re m()~fl!)

likely t11> 'be lilffeQtedr br" $U.l:'t~ca etrfill!• ocinus, th.ei etfe't :0f

size ~n ureep mR? 'be~Qttte mere pta:a0uced '1'1Then th~ aon~rete

$~Sees are plae:M~' i.:tl. ~ dl?f a,tnt~SphE!~~.

~. Et!~at . t~ .·~~. ~e;:a 1ir$t ~aai~f.· !!?he magtt1tua:e of ere$p 111 .. e. Qo:nor~te me.ttther ttepends to

a larg~ e'.l[t~:111t ol'J;_ the a~$ ot 'he me1n.ber '1h$n it was f1r$t

lQa\\led.. tt!he 1Qnge$" tne eQnorE!t.e 1$ allQW~d to ag~ befor~ it is 1oe,1ied, the l~s.s the a.mount ef to·te.1 al'eep., An explan.a ....

tio1l for thi.t!? is that the expulsi,Qm of mo:1$ttu."e !:rom: tbe fii;:el

'beaome$ rnor~ d:lffiQ~lt ~A:zl tl'.3:~ p~l'<l);tal.tt 1$ deer&~$ei tb11t!'ough

1\_ydrati\>n., !he rate t;if o•.~eep is g;l'EH~tt~st &t ·t'J:ie t1tlle of

load a.p;plle~tif>n. attd dieo.:i;~at!f~$ rapifl;r a~ the l1la€;n.i:tn1d.e ot the o:t'ee;p il'1<'l'.l1Eil~tses fer al:J.ou:t the tiJt>l'tltt ~1~ 111Q·ntb$·~/, Atte:r

the· fi:t"Sii s:l~ mQntJrl.s the· ~ate of ~reep will eoJ'i!,tin~; 1H:> d~.,..

ereaae and the m$J$ttitu.de. ~.:r e~ee:p w:tll ~:ont:ll11ue tQ 1n¢r~ase

sl1ghtl.1 tnro1~ghQut t:he life tl:l$ l0a,.d~·d st:t:uct11r.e. 11'1.gu,r&

2 1s £:; typioal oreep ... time ¢tttrvei (fJ)

14

Figure 3 shows the percentage of long time creep ver-

sus duration of load (7) for cured-in-air portland cement

concretes at whatever age first loaded. It can be seen

from the graph that 10.5 percent of the long time creep

took place within the first day the concrete was loaded. In

about four years 98 percent of the long time creep will have

occurred.

6. Summary .£! 1h!, Creep Phenomenon.

From the above information explaining the nature of

creep, it can be stated that creep is dependent to a large

-• ~ H •

0:: ~ 0.0002 • ~ H -~ H ~ 0.0001 E-i Cl)

P-i ~ 0:: 0

90 120

TIME (DAYS)

,Figure 2. Oreep-Time Ourve

. \

150 180

extent on the age of coµcrete when it is first loaded, the

water-cement ratio or, s.~rength of the concrete, and on the

15

environment in which curing takes place. To a lesser extent

creep is affected by the size of the mass and the kind of

aggregate used. It follows that the magnitude and rate of

creep is greater for small members of high water-cement

ratio cured in a dry atmosphere and loaded at an early age.

The rate is greatest immediately after loading, but decreases

100

80

60

.----- 10.5% in one day

l 2 3 4 DURATION OF LOAD (YEARS)

Figure 3. Percentage Creep - Duration of Load

quite fast. The magnitude continues to increase indefinite-

ly but after a time becomes essentially negligible.

16

.1.

. . !he c:reep · phe:neunf>.tion ha.tit a mittrk~4 efte.~t on de.fleotions

in aon¢rete stru.et.ures. In e, series Qf long-i;il'.ne deflection

test$ eonduoted bf Wa$ha and Fluek (12) 1 ~nldittonal defleo ...

tio:ns due to orfiep in re1nforcett eono;rete J)ea:ms aub;jeeted to

a sustain.ad load ~for two ~d ~:ae-half ;Y'$a.:r$ lte:re observed tQ \ .

be one to t'W'o t1mes l1\$ much as tbe o:r:igtnal efastlc detleo-I

·f.iions. ~he test$ sb:olin~d tha.t the inolusion ef arbitra:cy-

amounts of oompresa.iQn steel in simply supported reinfol'Qe4 eoneret~ beams \f~s ver"t effe<Ybive 1:ll!l. red\1oi.ng ti:,t~.l de.flee ...

tion fineluding ()reep) by :p:reventiag e:N:oe$aive er~ep def1>r1M;i .... . .

t ior1. In $evere.l m:e.m.bers of the same 1fidth and depth, the

1:nalus1on of oompress1on l'\lt.eel. equal 1u anteunt t.o the te:rud.on

steel raduoed thel a:verage totei,l d~Lfleetton b'i S.J;iproxima.tel7

one•half. ~he inolus10n o.'f :half as muoh. ~ompressien liJteel · . . '

as tension. eteel oa;t~J.H~.d. a. reductia,Jil l);f' tblJ a:vera.~e total de•

fleetion by apprQ:~ctmatelf o:ne•third.

Large oreep d.a:('lect:t0ns i.n beams are due tQ the e0ntinued

strs.in:l:ng of the conqir$'t:H~ on th'3 eorri.preea1o:n side." Wh$n eont-

preission steel .is used in, the member; the oompre:ss:l,en :Steel

tak~$ a large amount of the oompl''taii!!sive fQxoe. lth:tch rtutiulta

:tn. $i reduct ion 1n oonoret~ compr~ssive str(l}11Sses an.d an er..;. feet.ive dEu~rlfutse in creep deto:i'm~tion.

11

tlrta!t de:flactlon ~J.~:1 be ~alcn;i.1,a:ted by modifying the ,f0rmula~

for e1as.·tie de:rleetiotns.

Creep in eon~ret e !i:ta:r 'be tl~ttnad as the inere~iith~ in ;.rtra.1n du.e to a 1B1usta:lw~d stress. fhe modulu.£4 of ~1@c1s.tieity

ot a 1nata:cial 1s the slOJlf.i Gf the iStrEH:11; ... st1·ai1c1. 011rve; t}:i~re­

fore1. aJ~ creep ·"tatli!e:a pliti:~e the slope Lm<>dulus of el~sticit;r)

becomes ismall$!'" !be· small.er mo(tulus o:t &ltil.atioit5t w-111 re-

aul·t in a amalle;r ealeulate(l V®ilue of the mom.e:nt o:f. 1nert1a. of ·tbe asaued e1"af!kEJd ii:ran.s:torin~cl seetif)n.. With th:a new

values of mod1:i.lus o:t' ~1aS:'tloi:ty end mem~nt of ine:rtia used

in t;he formula tGl~ el.astte (l~.fl$C'tit:r;l'l; a re·ugh value for

long-•time total d.ef'laet1on mat .he o.ai;1o·u:ta:t~d.

Since cq:nerete .de'v-elops both elastic e,n.d ar.eep deto~ma­

tion i'lhile unde:r~ load* flt tli'i).l Sb.ow bath elastic and cre~p

:recovel'1 after remov~ c;:f tne load,. 'fha af.t'ect e>t du.r~t:ion

of ·tJ1e loa.d tr1U.$t be eonsidered :'Ir.ti diseuss1ng ereep reoov~r;r

just. as i:t was Qotu~iderei.4 in diiiHiussing ereep .deformati,:ln.,

:iJhe rate o;t; e17aep reoove~t is muoh slowe.r at la'te load•

18

No Specimens Under Load Constant Load

o.6

o.5

-H Cl) Creep P-i o.4 Deformation p:: ~ P-1 Cl) :z 0.3 Creep 0 H Recovery j H ::?.'! Elastic

0.2 Deformation "' Nonrecoverable :<::!

0 Deformation H

~ ~ 0.1 0 IX! ~ A

0 0 2 4 6 8 10 12

AGE (MONTHS)

Figure 4. Elastic and Oreep Deformation

l)·~ ! h~ w~(iat;i:'e Me;mb$:t$.

fh~r~ th~~e p~s .. itlal. phlt.l:tt(J)m.~ite. 1fh~1t• 1'($1

stre$f3 ehang~s tiloneret$ m~In~~-,s.

r$dl.st.:tibu.tiott ~£ 'b~nd1ng ~tom@:t11.t$

·().X'·· e:t~ep. ~he etre;~t.$

r~:t .e\\ ®.l"J,alyilce;l.ly;

~reru~tnret .1 ·t

o:::r~ 1$lp <:1an

ii$tl,(1:r4

l. · Sht'il!:~$,~·~ Ef'f'edt. an·'";;.;;;~~~ ·~ ...•... 11 . ..: .. a.~.'-.. ~.·.i.· .. fl.· .. •.· ~if<:'"#f .. ~ .. '.M1.,~-~··.,.,.,,

taJre$

\l1hU.t? I t '.r;H) .

per:1ad of xn.1:l~'l!i $Jrc1ve

it:he, ~£f'$.(;i);

<l o:nt :r li'i&at t ti;!! J~rhe~ ·t b.e

velll.e .f:.IJ:l:r' tb$ ttu~l~m~l e!:l~:f'ft,~ie•U.t

o. 000006 1n.tl$be~/1no:l,i;/d~~te~

\}~.~~U,S$ X!latt\:;t~l

.. · .. ,. ~¢~~pt~d

ex:1:11!1lill!llli:l.On ~£ ~¢1~.,<,tre'fse. l~

the s:tr.ain11; d,e'lrelo.ped in ·~ t'e~tr&tn,eQ. rs.inf(JJ?~ed O$ll:(Jr.ete.

element d:~:te to sh1·.:t.r1lt~~e.. (2) If .the e1~~ient 't1l'e:re ttnl'.'.'e~,n­

i'orced an~ unr£Hstr~ined 1 it woiilt'l tend to ~11or·t~::n. f:rom ~ec• ·

<J..,q. Thill w.111 re:sitlt .111 th(; $tee:t. btlitng ~t.r$sse<il in ~am, ...

:p ress1on: fror.1 it* to a' ~1hil¢ t.l'le eonerete $1\l"'.t'Outtdi:ag the .,::

steel i1rill ~tr:eastr#d in ten$it!n tr:o·m ;s• to o• 1.f it doee

not ¢.:raelt . ., ~he: eol.'ler~t~ neaJr the t0p of . t.he elemeut 1µ~y b~

st:t"it:'S!S·ed ~J:'.t eom:pre~s~;IJ.n,.

Il'.1.d:ueed 1$t~~~lsea 1.1i ttie

shri!!.kage lftay· b• <:ri~l~ulatett. t:r nxu.l'tiply1:ag th~ "U:n1t shritilt,...

age the pciint in q:uE.tati..~n bif tli\~ tnodulu.s O·f ele.$ti(;.tty et the r~s:p~e·ttve :material., 1lt1.mtat$· v~.luea <l)f $b1--b1ka€fier defQ:t-

mat:t~'.11 ot ord;in;.a,:r;r iu,i.~ea, rel.s.:t:i:wely th.in membe;x-~~ qf

21

for unit shrinkage:

s = 0.00125 (0.90 - h)

wherein S is unit shrinkage and h is the relative humidity

of the surrounding atmosphere. For a fully saturated curing

condition in fog or water, h will equal one and the concrete

element will experience expansion.

Length Before Shrinkage

I

I - 1 ---n.a.

I IA'

JZ=z::::z::::z::::z::::z::::z:==::z:;::::z:::::ir;"2~~

B : A

l.Z .I. (o"'·E',I Oontracted Length If

The.re Was No Steel

Figure 5. Shrinkage In A Concrete Element

2. Creep Effect .2!l Stress Ohanges.

Oreep strain occurs in a reinforced concrete member when

the member is subjected to an applied stress. This applied

stress may be due to the external loads on the member, to

$'tir$!1n 0'.¢¢1J.l'S in · $ ;re'.ttn.fQ~"'Q~f ¢:0~l:trete m.e~bet·· \\t w:tll C~'U:$f

oha;n;~u · tl! t~• ai>iJi$l $lid ~r~ll.~W$'te. li!J1iw&.1$~~s i\.n. that tneria,0~11.

blt · Q:t'eep in. the ~on~l"~t.~. Se~1t:loil iJ!V W'1ll E'IXJl}l.aS.n b.0~1 th .. t't!me a;tre"$!e:es ~~ b~ cta~~tllat~4 •

. ~tl'~iln d.i$trf'buttli>n devsl~p~~ §.~. · :re·~ntl!).:t1Q<it.d. CQ~l~.t:et~ el~,,..,

ltt~nt rJ'1ab3~~ted· to tin ~;p1;;it;l.¢)d. ~t.rt~·s1.,. . ' - I

Whee st~:ffne~s ot ~· .~~~ttt,o'l.tuwal nl:$m'bei• m;af b$. 4.Gf'l.n:e.d as

t".11~· m~111~nt ~\to~$si&:.J:7:.t.a .. :t"<41tatt.? :(1)11;f1 sinI:Pl1' sitp;p~~·c~i $!1:._

th?t'o~~ll itt. tlnit a'fl.glt~ wn~n 'uh~ i:)th.tr>l" &!lti 1\..$ t:1;g;;~4~ p.att1$:llt

i1:8:td $lmp11' SU;p;port~dl, S$1f'tr•$·$ 1!$.k~s · tfie totn.l; OC\~f

'¥'fb~J?' er 1$ a t:J;on.:a•,eunt deptnli~rJ.t tJi:)!'r>• the end ~.¢tn;41t lQtl.$ ' st tlb~ :m!Smbet'• Jiif!J l"!.·::r.•·11:~"~n

ct.~~ . ~t.:tfm~$s CJ,£ t ne. ~~.~:'tif!a:'s f;.~ a. l"E?l.:riif()to·(IJ.d ¢Grit;J~~ts ·-nrtbt\:rJ<t

me:m.i'Jier d~ex1~:~1s.~s. ~bi.~ it: ctU.~ to ti.h~

f~.¢t ·t.'.hat e,$ Q.~ee~ ei~ot\~$ · iin ·~.4$ eiiiln~:ttete, th$ c~nor.~t~ m~tlu­

f)f. eliist1G1t1 (Jil} ~11. lesaj a:nli 'thtore.f~:oe thl;\i iltll)nt~n·~ 0:t· t:li::d:>tta ·(I) ls 1e.~·fli•

23

Initial Strain (Elastic)~~~~ Increase in Strain (Creep)

Initial Strain (Elastic)

Increase in Strain (Elastic)

~~~n.a.1

n.a. 2

Neutral Axis Moves From Position (1) to Position (2)

Figure 6. Elastic and Creep Strain Distribution In A Concrete Element

In an 1.nde'te:r1n1nste :tt.E?Jl:nf~r~ei<i a:.Qno:tet$ stI"ttatu.re, th~

stiffn..e:tis is not/ n~rxnally unito~m throu.gh<Jut the struet.u.:ce

because or th~ variati~n 1n r.elnf~reeine.nt, "!Kl'he11 the stiff ....

ness ot the ert?•uatur& de$re~M!if.E)$ dt;:i;('J to cracep ~£ the eoiti:)r~t~.

·tn:e leas $'t~f'f se0:t:J.<1n.s ~f the E!'t;fu.otu.re ~xpe:rieIH~e a lar~$'l."

the struot11re. ~his non,.~uttifQrm .$\l.t:fn~ss will produe' a

redistribution <:1,t bentiin$ !nom.ents: i:n ~rh:teh -the stiffer $e~­

tions or the l.'!lt:t·~(ftu.;;r$ will ~;icpeti~n~e $.:l:l :tn~l'~\Hi$e in bend•

i:ng moment and the le$1ii! stiff seetiCJ11s •ill a~pe;r1e.ne·$ a

dee:J?ease tn tiend:tng :moraent • fb~ ~nan(i!je of b~ndlne; moment at

a.n, section wtll J;'eSult 1.n a eo:rre$ponding ~hi:l;ng;.e of stress at that seet1on .•

aihr1:nkag~ all.tion in a re1nfl)~C:ed. eo:nct~t$ $truetu.r~ will

quite often. ~e:u!llle t.:rian.slatio11$ and/OJ\' rcrtatlons itl t.h~ 1.nd:! ....

v1dua.1 menfbera <:>f the $t:tuet'l,l::re.. Whe$e 'roOV~l11exrta ma;r itt.du<:le

a.dd.itional · be:adillfiS efl'.~o·es and tlu~::r?eby .Pl'tHtlJ.tce ~- redi.st~tbu;..,

tion of bending moment throughout the ~);l:t3..re strl:ii.et.nre.. ftJr

eve:r;r change o:ft hendin~ m~m/$.nt "&hA~'1:·e will be a eo:tw-es1H>n,tt:t:ng

mora~:nts in a. reinf(lro~d oone:x•e·t© s'cruotu.re all ee:ttse ehangef3

of str$$S tJ!U:'ougb.t'.>U:ti the struotu;v(t). Shrinltag~ a.¢tion :tn <>on-

cr~te S'tructu.rea usually he.~ e. det:r1m~n:tal .ef:fe.ot be¢s.,'t;i.$e Qf

tlae la:rge 1Jenti1ng motaents that al.:'$ 0ft~1'l indu.oed bY it• Jibt..,

eesa1ve .st:t"atn!3 and stresses due to lPhl"inl~age .m.ay be largE:J

enough to ea.use e:r~ektlt whi~h 1f ()~nt1nu.~u$lw ~ltp~$ed to

weat1:ver1ng eQ11ld evet1:tua.llf Clause uetertol"'€1:t1on and bE1H;1H~m;$ a

serious proble~, I11 0ne :t'$speei shrinkage in oonerete :ts b$xaefieial; it it d.o~~ no't :t>X-~d.u,ee l!Jl:r~clting in the memo~r, it

•,

11till eause the eon<tret.e to g:t:ip the a't.eel t:t.glttly, thus de•

!l:reasin~ the pass1bil:!.t1 sf oar slippase. Gt;)eas1o:nJlllf ex ...

esssive at:rain~ a11d st:tes~es ~u~ to o.!\l'•·eP may 1ndtt!l~ large

t\ef'laet1o.n$ wl1ieh al.st> betlQnle a p:i:?Qbl•m.. l!L'wev~r, both

$hr1n1t~ge an.t\l creep str~l.:1:1.s m.ay b.e et'teetiv~l.1 de~reased bf

the e;duit:ten. et eompr~$l!tliV'e r$1nf:e:r~~riiel':lt.

Q:tee'P 1~ C)enm;cetc$ l.$ us.uall;r ad:vant ag~H'>US ba~~;use it

p:rod'.u¢es a redtltl1tr·to11'ti.ion ~f stres~es 1!:!,lo~g tllt!J! l.l()ng:i.tudinal

al:lts of the mamb~rfiJ th'.f'Qui'.beut th~ conurete striicture.. !f'he

stre$se$ :ta adjacent nI.e111be:rs~ .o~ !:n s..· different seotion of

the SH.true member n1a;r b~ :i..u~raase4 a'b one pctin:t ~nd deoreaseti.

at anoth~:r,, but ·the Gve:r-~11 e.fr~ot ua1ia111 results in ti

. benefic:l.lil.1 Sln0oth1ng ~ut ()f the $ttes~ p$.tter:n thrrlUt~ht1·ru.'t

the stru.c.rt11re. !thus, oi·eep has a. b~nerf:i¢1.s.l t:n:t'lue.;nee upon

eontinuoua eC?no:rete s·i:crv.~tureu<;. t~ th~ .$$m.tili wq tn.at the

dttottl1tt et strt1:etural ~teel fa(),.lita.1HH~ beneficial ~tress

~djustmeta.t 1n cont1ntl.GU$ steel l&ltrt1otttr$.s., ·

I'D' :: W' ii

,· .•

A. Aseull:pid.onEf Vs!!iHt~ in. llitriv;~tiontl ti fo in'le$t taat e the·.·· 1hatl~~·ui of· .$'t~$$'1ll*"'' oali~el b;r e:r$e<:P·

111; :.r,ein:force4 •on~re·t• ni•mbet'ltl', s~v~ral afU:Jum:pt1Jt1ll$ w:t:ll be ·

mad~ as tell<iws. t

1., · St!'~$1Pli: re.lltl1'.I; ·wltb1~ the 31.l.,~e't.'ble w:o~k1nt

:r~@·l1J t·pt1H1~01tle~ bt 1he :19$~ 1 •. e .:t ~ oo•• •, !. it$-m erO&J$~$et)'t.:l~11's ·•~• pl$ll• ;,nd" '.tema11.t, pl~•~

att~'Jr ben411;g i!Jtf t~• beam.

· J. <Joue·t'ete l)·elc>:w the llU~Ut.l!"al. u1s. on tlul 1H'!n$l~l>.. · s~d.~ ot the t1Gl1!:!b~tt 'laktts ·ne ten1:1le $t:ttee+

fhe f#Oacn~~t• is e.1su1it4 'to be era;ke4 •. 4.; · fb.e: ooap:;!'el!$1V$ ili;liti~t~a. tll1st:r·tbut1ea tJ)l the ~~0$$""'

.s~ctt'°• 1.s lbe,ar .• 5~ ihe· reiafozroing · $tet.l ~.t0"1$'ll0'1$ llO :f#,~t•p.,

6. o·rEl$p :1.$ e:&su.•et t~ t:ake pl.a.ct :trte o,t e.h:rintage,

7. . · Q'.i'te"p •tra1:i'l, .ts d.ll"4H~tl1' p.r9pl:-t1i1o:na.l t ~ tla~t :tc

atra:1n at.ii 'Witllb t•e $.\llow:~b'l• wot"klng rrang,t

b\)th. ~t'~ 'J"Optt"'t•afAf1 t• the ·O~llprefnstv•' t\lt:t!'tse

on t 11& <H1n;(J::reiH1t.

Aa pl!'$Vif>UJ!Jl7 8tate4 1• ~lU?JCtto• O.;;.); re1nf.q>:refili·d tlOllVi*:$1Ht

m.tmberl!I: t;ub jeot~d to ~ ·<ttll1$t•nt bEin41ng uu:>111$nt. e;;s;:perit)t'tflt: $

r·e41st:i;t11iut1$ll or stt-eti1$eS 4~' to ~refp; ~h.$'.t(:lfore b, 1?Jt$;n. ...

iq lir1th m appl"'ed !M>meut on the or<>ss ... eeotiflJA <:>f a beamt

,.qut1or.l~ o.f .n:te,.•fli. p.$:\f :ua11) aQ14~i.~.1:tt, 'tit~ o.Jt. t:t. Q:l~i)ae:t~ tJf th~ o;t9$$•$eCt.toa :iuttl · .p;r:')pt~''' t;Jft · tht :mat~:rt.aa.11 wtll· · b~ te·;w1,-eta .•

· 1et11n:~$ thft1 1:\1'$ii$/tti~11e,'tlt eq~~tteni$· $ff · fle~i vt.4t lt l.$·

ntc~:tiui~17 to tt11r~·ve th& r$¢U~tlo»' t~f 1l ltM p:ro;p~tti,lll tJt ·

d.~pta fr~• tht &>0$p.7ffJ;~$.lt• f~~~ t:o. 1ih-t 1>•~'.tfo~tal a~lt.J .. :to:t· 1\i.,, ..... ' . . . .. . . . . .. ,

11le re1affJr••4 -$11~ <lt'l4~l• rttitf.:ret4 -•alll tat¢~.;tttt: . !att ~e~a­$1delra,t.l$1;. tb:e O.l1'$ep t:n '11' o~••r•'•~

. $!.no·111 thfl. ~t)t~llt.s · t.;!i .. tla$ti•l~~ ·O'f -~o•t:~:·, . btu-~••a . \.. . ' . .

:alllallettt' $;$ Qr$~.i,, t~$' pl~et.,.·:·'th• v$1.a -Q:f' ~-)1$ .• ~Q!nt;r$;t; .• : nt~4l'• ' - ,

lus will be O'Q~•·ti.el"et t$- be !ElJ_~ &tin; e:re•P .. ; :ts ~ero, • =- l •d Eo .i.$ .·'bh$- ela$~lif .• 4ti1lt$;~: - .. l.Js .·-· ~~· .nft-1·• 1)$0$8$·,·.t.be m0i~l~~ ~~t~:>-~ - - ·-- - - - ~ - fa,-~ . ,, ~, . : _---- .. ·-.. --·. -·- : .: .. -.- -

:: ·~

. 'i

. ' ~· . ; ' . '.' : '

28

1. Tensile Reinforced Section.

-n.a. As = pbd

cnAs = cnpbd cnpbd

Figure 7. Tensile Reinforced Section

Taking the moment of ·areas about the neutral axis

and solving for k

bkd ~ = onpbd [d - kd]

k2 + (2 cnp)k - 2 onp = O

k = V2 cnp + (cnp)2 1 - cnp

29

2. Double Reinforced Section.

A~ = p 1bd

(on - l)A~ = (on - l)p 1bd

~lb1d'=i r :d 1

~~(on - l)p 1bd d ~~~-

L c::::=:::;::=::~ onpbd

Figure 8. Double Reinforced Section

Taking the moment of -,area.a about the neutral a:x:is and

solving for k

bkd(.!&1!) + (en - l)p 1bd(kd - d 1 ) - onpbd (d - kd) = O 2

k2 + 2(cnp + onp' - p')k - 2 (cnp + onp' ~ - p' ~) = 0 d d

k = 2(cnp + onp' ~ - p' f> + (onp + cnp' - p 1 )2.

- (cnp + cnp' - p 1 )

30

o. Derivation of Stress/Moment Equations.

1. Tensile Reinforced Section.

en

(a) (b)

Figure 9. Tensile Reinforced Section and Stress Diagram

a. Ooncrete Stress/Moment Equation.

M = P0 jd

M = t f 0 b kd Q 1 ~ 3) ~ ~ = 2 M bd2 k(l - k)

3

b. Steel Stress/Moment Equation.

M = PS jd

M = f 8pbd d(l - :!&) 3 where Pa = Asf s pbd f .. fs = 1 ·- sJ M bd2 - ~) p(l

From Section B-1

k = 1/2 onp + (cnp)2 1 - onp

31

2. Double Reinforced Section.

A~ "

~M r-d' ll k:d ~ t p~

t Pc ~I d / k (d -I / d(l - "3)1

~--~_____,+_ ~ t t • Ps

As ~

d

1 y s en

(a) ( b) (c)

Figure 10. Double Reinforced Section, Stress Diagram, and Strain Diagram.

a. Concrete Stress/Moment Equation.

Summing moments about As (Fig. 10 (b))

M = p~ ~ - d ~ • Pc ~<1 - ~u

M = f~ A~ (d - d') + i f 0 b kd2 (1 - ~) 3

From the strain diagram (Fig. 10 (c))

€~ kd - d' - = €c kd

so€' = (kd - d') f s kd c

then f ~ = (kd - d' >!£..... Es kd Eat

and f ~

''1'w.;tb...~ ..• ··.·,,··.""'.·. ........ • ·1"l··· f!·l>,:. '."".· ... · ·Y1Ltlww u ~ s:;:.- :~ ~

~··'· .. •··. '!.·.··, ' -·.· ·-".•.· •.. ' •. \.·,' n.11> t•' l d ·1·· ' ~ .i. :"1-,ol ...,:iit])·· \ ·· .. ·· . ..,.. ·~ .. · . . ·~"" a.

1 ·~ ~· ··-":""---'j.Gi=.=-. ---:--: .. -.. ~.::y.-,, __ __:=---rl-. ·~ .-•. --.---k-. -O-

J.'\> < .. ·· l ... -.. u..· ·.···.·.··.·.> exip' ( 1 ... ..»....· .·•.··. ··•·· .). + .~k ( l ..... ·.~. ) .. ·.·.· .·.'·' kd. ', ···, it'' ·"' .;

Al J:f.· t ... -.. · .. · ·· •. )!~ .. ·· .. ·.' .' E.·.· .. ·.··"''·. &.. •·.· c·.. . .. · .kd ' £! ·~:nii . .!'.·.··~ ·t,:· \ktf fl .,_. it ~ k.;t" ;-JTJ .~ . •. - ·• ... CE!i!r $ ... ·"' . ~. . l!lU

~.. l:~ns1le St.~el .~tr$$a/1'1~ment ~qu.~,t1~$tl"··

( F1g1,. lO fb) .)

M .~ :P~ [t ... d'] 4• [dfl ·~· ~)]

r:: f~ A~, ·Gt· • 4~ + i·t~ 'lr»· kd [ct(l ·~ ~o

Frem ·the ;itta,1t!. d.t.a.§f;:l{'am (Fti. lC (.~))

E:e E; Q

.. ,.,. .. · .... .:i .. _ . . ·#1.·t'* .•. :.·· c·.·_··.· . k .. , :m ... c: ~. .... , .... k . ··.)· cu.-~ ¥..,. 1"'- ·J:t''i~c'l,,s ;;;! "l .... ·i·

·.'.P · .• 1.. ·(· .. ·.ltd -" q f·.\.: . ..t. ~~~I $ d .... kl:

!$· <if~ ....... ·:~!} a .. k<l ( p ' ~ ... ,, f ' j ' k ' \l ;i; k1h .~l\·C""'~!·~"' - ~1 ~.~ ' l ... 'll.t ,;;,

Q'" .. · d~ .. .·· .·.... ;{IL ··.A· ..... ·· .. ·· ... ·.' .. "".·.-.·.~· •. ··~.) .•. ·.• ·.})' (:t ""' -.I.ii.····.·.·. ' ~.l("'llli:;i_ "(r-fi._, a -l' :.., 11! ' ' ' (

. ' ' ' ' ' . . .

l i:)lliq __ .- -- ... , . _ :lliillt",_ iiioitJ!l? .. : .. ir.:;i(;lit!O~: a.

.... .... -.·- ·, .: ' · . . . :_· ·· ..

; . ~: . -

; ;!j·llJt.'. - . .. .' .- .. ::il:.ill·

- : ,, · ... '. .-... !.' ... '.' ·.

··~ .· 211.li).H !It .. ~~. . :-._;

't.54 .-..,., ... Ii.:r•••·;t· ••• ""l-••« •:P•• •• , •. :fJf&'U .. 1:0 •. · '. : ; ,, ••. >.:' ,._: ·' ', :·:/. : ·:.::/,···· ··•·:•'., ·.;' .i' ' ' '· .... ·· .. ·· we:.l'e p~•a~.ae.«. •~ t~ IJJ1 lf.lf.i1 .ll1.tt .. -1. liJtP•t•ll'* .

1~: .•. ·.·· ... ·· 81 •··--j I ·:>~·> '. d I;.·· .. ·.' ••• .a ... JI I~· I• .. · .. · ·.··. •w..111-4' :-: ........ ·.···-111.1111::111 ... --1•·~··· ...... -At •

.• >.';·~.,~~~~~!!.~,,·~~~. J "••lblt ~~-·' "t nt•t.1:1.1. •1~•···· ·-- b'•~·°''*'• · l'''''t.-· .i~~e:• .$;fat••'•'*· i~· ·.!l:~•tt,·~·· :_:tt)u~'.in\lt~~· ,,, t:ke. P~~t•·I · , ·.

· .·· ·•~·-~··. '4~l~~1)1.,.,: .·.,. tt.• ·~••1: ·.,. _t·s.·, ta•t·''* •t:t . ,, .. . · .. - ,. . :• ;.:... .·, . .·... ·- , ..

tt);;l~f.t ' .··.'' ' ' ; ... · : < ·.· .. ·· ' . ·•.· ' ·. ' ' ' ' .· .. ,.~.···•·ff·~·~··.·. ' 1~·--··.,~·· ·"·'J···l~;··· .t.,. ·_.·_.·. ·,· '

... ;

., .. I; -~•·1. J,.,,>1~• :,~ •. :'~· ·· ··. . ....... ·.

-.·,··'

·: ·. ~. ' .. ' J. , .-

' ' . '' ... '- :"

.''

' ' ' ~-· 1.,, ,, ~~. ••• '* ... / :r, . ~,:· t,~ ·la••t. -~ 't•at.t~t ~•ttitt.~t·t~1ni1 ·-~~" .,., trill, -..• -·· ... ·· .. · ·

... , ,' ;.,,,, .t••··· ·~1· ,_~*':" ~., •. 8tlr' j.,tl ·~·>"~·~ :i):f 4•i-~h; ••. :·;··· .·.; ,' .·'

. ~-· . . . :

.::.:. . .···' .·

·- : ...

to,r a ra1~,1.g$ be;r~xi:d (lit .:: 9 to, 0 = 1) :t~ q,rt\~r 'tQ b$1rte:J? tValu ...

ate e:ffecrts of .e~i'leJWsrive dreep., lraltte!W of t/IlfJ. Wfi>re ealcu-

lat ii<l J?tr1.. all opinl:)in8t.1.o~~ (J)f 1fh~· a't?Ov~ pal:"a!tiet.er$. and ~re

tabulated in a s.epal'ai;;e sttppl,~ment 'GQ thi.s tl1e~;i,$~ (9)

:s. ~l:'aphs. ol 1'$'.t•oett;t Gh~tl~ tn $t :i'$.s$ at· Oltlnsta.:nt '.Ben~i~g Mt)me:erl,J vs" ltiiep ~oiaffieltD;'tl.~.

'!he .gpaphs· (s.~~•':'rt~!;1tt·~a ,l'.l t.h~out;h 2$) $h~w t~t f~t ~· · ·

beam $Ubjected tQ. e.onsta.~t tti~llient cr~f!il,JJ ~m:tt.~$S. a, :r~.ci:t'llQtto.:n;

in l)Olll.:IJ.r~ssive l)ai:;;1~~$"t$ $·t:res$~~. ·~.n 1,n.crease it~ ten$1le .

ste~l $treas$~;; s:nd ~\then ·pre$~tl.t, a:n .1r:1;t)r~ase :tn j~rapt~H11$:;'Lv~

stegl', $tre$$es~,. '1het pex·e~n.:ti,i:f;e $.~~s 1~ '.¢oli~r&t~ $tl'EHif$ :ls:

al11a:y$ a;bot:rt f{)Ul ... or t!l.ve, time$ !l:re~t~r, the.n t.he pe:rc~11.tag~

$J;n.d t:n-e peree?>.tag~ g~i~ l,~ ·c~m.:p±~$a1v.~ ateel" stress 1$ ai ...

t~ays a;bout thr$e to ;fl~t time~ ~:ltEi~:tet:r tl:1an the p~·l?e~n:ttag$

loss i:n·· the eon:eret~· str~ss .. ( r:Je.e ·11~6Urei~ l.4 tb2otigh 22.~.·)

14 throµgh 2~} mh:~w t,1erve·l'al. li;i:11ind11?ed 1~re<llf:nt ga1:11 ill ~!lal~u ...

latect .stresses i.n the eomp;c·eaall1)n steal ·a,.~· e:re$p take~ ;pl$QJh

!n1$ i:rie:rea:se. m6!¥: not be p~y$f«t~-~ly p~s$;ilble it' the· cGrm"'."

all of tt>..:.eir i't?·alll"*.e .. - - . -.~ ~~· · ... ·_. . --w·-,.·•·

!l;h~. co:mpr~.s~~t~:n. a·teel eiu·f~$ ~an be s:a:i,d ·to ha'V~ ·e.·

f!'treas out oft pol:rrt at the f.l:$ld l}t.111'-'1.i:• St:r$i:n parall~l~

.b.l~ ·"' .\,"" .fi·<>·"I ·~"'· . ·;,, u, ....... "" If "":""'V.,Q •

2. t.rse st f17al<,h~ :t,~.t ~~pi~n fxo.b~~!!;t•.

;i:J:;te. tact tllQ't these .graphs ~.r~ no:n .... dimenaiQrt.~l makes

them UilJe:t'ul for tl.:~a1gn }i'li:t':poaes. Fo7~ ~~.s/iup.ie.~ it al'i e".ngiue~;r

dasign.:a. @. beam ai!!d' 1<ro·."l~ li~1e. t.o· pre~1et th~ .~.ong 't::J.m.e ~trts~

change$, he ean ~a$1'.l;;r do ~o. 'bY t

l~ :©statilisti. a.vialtte, ~t

v:ailue far

ehang~s .. )

foll~1'11:ng met'h~d l

·tl•i,e '.CO<ner:1:Yt.e

· 2. FQ:ii': th~ Q.~s1c1,gtl. valu,e ~;f'. a., tdr.t,C'/:Qse ·uu~

J+. le:g;t, . .m~ve l'l('.l!!t"ita~nt.~ll1 $,lld !:'~$d: th.~ e•t.i:r~e ...

Spii)~~~ng th{ii! p~;rQ:e:n"b chat1:ge ln

i:rt r~ss· • Ml.l).t :i p.l;,-i}J,g t ht~ val us b:!f · t .h,~

ai1tastt e ~t rest!> V'~'l itt lif 111. ~i~t~ 1au1 a'.!'l~u:tat

Qf ~~~nse due t.o ~\r$$~,

38

---n~ .o'S - _..- --r ---- ------ \' ----- n-:::.0 --

----- t" ------- -------- ---- --- p:::.002 - - --, _,..,...- - --~ ------ ---_,,,.,.- .,..- ---

JO

.........:::....- --+- _,,.,.::,.....--- --~-~ o~~----+---~~-+-----~+-~~~-+---~---t-------~

E a ~

+-c \) ~-IO

~ (_)

b (/) (/) -20 Q) !.....

ch ..::: CJ) QJ-30 c: 0

0

-50

FIG. l I

2 3 4 s b 7 Creep Coefficient

N=0 _:e_' - 0 JO -

--- Concnit·e

______ Tension Steel

CHANGE IN STRESS - CREEP COEFFICIENT

39

ID

\()::::.09 - -1- -- --------- - --- --- -- ----;;::: '01

--- ---- I -- -- ---- ·-- --~ --- ---p:::.oo.2 ~--- --~ ----:::::-- -- --.z--- --

~ (!) L

U5 .£

-20

<JJ -30 Q) c 0 6 +-~-1.fO (.) .....

cf

-50

FIG. 12

2 3 'f s Creep Coefficient

7

N=ID +=o --- Concrete

----- Tension Steel

CHANGE IN STRESS - CREEP COEFFICIENT

40

o:::.OI ---l-- ---· -- ---.----:::: -- --;s---=-.~n- 0.., --

~ --===-- r:._ - • ----10 ..--::::~ · ----p:-.oo')..

~ _;;::;:;- - -~~---........: r-__

~---~..:--/::.;::;--0~

+--JO I::

~ r.o c (5 -io-.20

Ii) {f) QJ s...

+-U)

-~ -..30

<U ~ c CJ 0

-'10

2 3 Cree,p

1-f- s b Coefficient

N=IS L-Q p -

--- C oncre.te

- - - Tensi"ori Stee.!

p::; .03

7

FJG. 13 CHANGE IN STRESS - CREEP COEFFICIENT

500

+-c <J)

E 0

2: 'fOO

+-t::

E CJ> c 0

l_) 300

t <./') V) Q.) <-+- 2.00

(J)

c

-IDO

N=G 1- 2 p-.

---- Concrete - -- - Tension Steel

41

Compression Steel

FIG. 14 CHANGE IN STRESS - CREEP COFFFICI ENT

+-!.:::: t'.1300 +-(/) c a u

+-a ti) (J) Q)

~ (/)

200

.S JOO

Q} CJ) c: Cl

0

N=IO 1-2 p -.

Concrete

42

Tension Steel

Compression Steel

JO

p::c.OI_==~ --- - ---= --= ~.OoJ.. - --------==== - -- - - -----1 :;;; :;:::::=- t. p = . oa

FIG 15 CHANGE" JN STRESS - CREEP COEFFICIENT

lfO{)

t: q)

E a ~·.

+-c: 0 .,_ O')

~

300

b 200

ll) Cl) Cl) L i-

(/) 100

.s Q.) O> c 0

43

N=f5 .E- 2 p - •

- - - - Concrete

- --- Tension Steel

Compression Steel 0~ 0

/. . '€~

10 .te=_.001 __ -=-- -----~=---- t.p=.01

----:::::::::: -=:::: -;:::::..--- l!. =- • 0 .3

.....c 0 u ~~"'"' 2. 3 't r " -..;;::~;:::::: c c 'I

+-c Q) 0 L QJ

0: -100

FIG.16

-- ~ ::::. .::: ==:: ::::. ::::. _ _ reep oef f 1 c1ent --===~~==~7==~--~

p= .DI _/!J ---p = . 001..

p =- .0.3

CHANGE IN STRESS - CREEP COEFFICIENT

+-c Q)

E

soo

0 L:lf(JO +-c: ~ r.J) c: 0

U.300

t

44

N== 6 E- 4 p - •

- - - - Concrete

- --- Tension Steel

Compression Steel

----==-------=::::::::= - ==--- --::::- ::::=::= --

ID

p=.01 - ---t

-~p= .ooi:" p= .09

FIG. 17 - CHANGE IN STRESS - CRErP COEFFICIENT

-I-er ti) Ci) C))

f: V)

200

c -- 100

<U O" c: 0

-C u

45

N=IO .£:- 4 p-.

----Concrete - --- Tension Steel

--- Compression Steel

1,

(0

~ o ..... ------i----------~-------1~----~-------+------~ c -...;;:-<i:S.,,_ 2 3 't s 6 .:::::;. --

~ :::::: ~ ~-:::::-~ = = = _ Creep Coefficient & ---~~~===-'J_J==~====

-100

FIG. 18

p=.OO:J.. !]-----~ p =- .01

p-=- .03

CHANG[ IN STRESS - CREEP COEFFICIENT

+- "100 c Q)

£: 0

L: +- .:300 I:: E. < CJ.) c::. CJ

c.,_) +-200

C5

ti) tJ) Cl.)

~ Cl) 100

.f Q) CP c l:J ~ u +-.c <l) t.) ~ Ql a:_,lO

FIG. I q

N==IS

46

E.- 4 p - .

----Concrete - --- Tens ion Steel

--- Compression Steel

p== . 03 /0

e,::.002. -----------"' .. e._ =_. 0_1_

£"-:p == .03 --+(,---=~

CHANGE IN STRESS - CR[EP COfFFICIENT

SOD +-c Q)

E 0

2= +-t:: E ti) t: Q u

'100

..300 b

Q) O'> c: o

....s:: 100 u +-t:: Q) C..) L

47

N=t; · f=.h - - - - Concre-te.

---Tension Ste.eJ

--- Con1pression .SteeJ

10

P= DI . -- - ~~ .0=0=2.====

-:::=::::::-:::: =- - t:p;-: 03

~ o~;;::;:._ __ ~------~------1~----~-=-------f-:------t ~~ 2. 3 't s ' --.:: ~:;;: :::: = = _ Creep Coefficient ·.

--~~~~!Irr-= ~~~=-===-~ p= .002 P= .01 ·

p= .03 -JOO

FIG. 20 CHANGE IN .STRESS - CRErP COfFFI CJ ENT

48

+- "loo N = I 0 -f = . fo c ~ - - - - Concrete

~ ----Tension Steel +-3 o0 Compression Ste.e.I c E ~ c o·

(..)

-1--200-cr If) Cf) Q)

~ V) 100

c

+-c <I) l) L Q)

a:._100

FIG. 21

:.,03

8 p = , OQ;;l..._ ---i -- '°::. _. 0_1 ---j

__ FJ= _.0_3_

---.; """"" 2 3 Lf 5 b ::::::~-- c c .. - :::: =:: ~ ::: .:::: ::::: .:::: _ re e p oeff1c1ent

- - - ::111£'" ::::..::::: ::::: - - - - - -Tr~------------p = 002 _j / T - - - - - - ..=. _:= = ..:::

f~ .DI_} I p:. . 03-J

CHANGE IN STRESS - CREEP COEFFICIENT

+-c: ~ (Q s:: a u b

.300

200

+-c Q) (.) L. ill

0:-100

49

N=:JS f = 6 - - - - Concrete.

----Tension Steel

--- Compression Stee.I

p::. 03 7

FIG. 22 CHANGE JN STRESS - CRfEP CD£FFICIENT

-50

f\1::::15 ------ - N .. JO --- --- --- N"'~---- -- -- -- ---- -- -

10

--...-- --===--+- ....---- ---c /---==:::::::-cl) ~ ;;..--E o~~~~i--~~--1~~~--t~~~--t~~~-t~~~--1 a

2= +-c: a +:- -JO CJ) !:'.: a

(_)

U) -J..O <f) QJ

f:: U)

-~ q) -.30 CJ' c iJ

......r: u +-c_'fO Q) 0 I.....

~

-so

2 3 'f- s 7 Creep Coef f 1'cient

p' -p =.002 p-- 0 --- Concrete-

Tension Ste.el

FIG. 23 CHANGE IN STRESS - CRffP COEFFICIENT

+-~

~ U)

10

c -10 a u +-c (fl V) Q} -20 '"-0)

.£ Q) CT1 §-30

...c. u +-c

OI (.)

~-'fo 0:

-50

51

-· ----- ::::::::::------ ----2 3 'f b 7

Creep Coefficient

p =.OJ -f-== 0 ---- Concrete.

- -- - Tension Ste e.I

FIG. 2 4 CHANGE IN STRESS - CREEP COEFF/CI ENT

52

+- JO c Q)

E 0

/ 2=

---=~- === ----= ~ -- ---- :::::-- -- -----. ;;;;;;;: :::::=::: - · N = b

~ ---~ . N= 10 ~

~ N"'IS ~

:;::?

+- 0 s;::

E ("/) c

2 3 't 5 Creep Coefficient

7

0 u -IO +-

p=.03 ~'= 0 CT --- Concrete-Vj CJ) - - - Tens.ion Ste el Q)

t: (/) -20

c:: <J) ~ c l1 cS-30

+-~ Cl) 0 L Q)

0::. -LfO

FIG. 2 5 CHANGE IN STRESS - CREEP COEFFICIENT

J;,; Design ~!'oble~$. Senne exa:m,ple have 'been ~'fQ.:rke4 to i.llu~t.rat~

the ll.s~ the dev~l!i>pe<t ~utt.e$ iHi1 ah~lt th\$ magni.tut\e o.f

st::re$$ ah~-ilges th&it ~eb"~ li1 t\JOl\O:t~t.~ beams.~ tll.t

ill.iti.al el$.$'hi~ $lire~$~$ &,:ce •&l~ul$.t~d &:t!d thG st.r~ss~$ a:rt

then eal(}1.4lated. Virith the E):ffeitJ'b~ Q:t."fjep ineluded. :Fro blettt l (·a) i~ a ca;:o,t te:naion lllt~el.

QlUY • nob:;t~m l{b) tlt~ ~~I ant ~~~~·Pt fer th$ inolutal<>n et OGmpressiO:tl $'te~l. ~fJl) i.s b~lir1~1u~ed d$e:tgn

oant1lever bea~ i1i~h ttitt~1le $teE1l onl;'tjl Pr~blem 2fh) ;\.$

th~ Sa>i:llti) t);S l?t"Ghl$m 2(a,) &lt~'Pt tl:t$ itHllUff!i.on (!'Jf C(l)tnpll'tl$•

sion $'t;e~l. Pro·li!lem$ $ ~'l:d 4'· wtth dlf.fe:r$at· ~t&el. qu.~rttitlea

are f1~ed . ..-$'n,ded b•th t~e l'~dl$'t:t>ibttti.on G:f

l!ie.ndi111g nioa1et1ts a?lti th~ r-e4ialr~.but~o~. ot $tr~saes t'fill b~

ea.loula:t~d.

54

10 1 ~I w

(a)

As rf1 '\ 1 '" Ps = Asfs

\ \

n.a1 \ jd 20 11 J_ _J kd

~ Pc = ~f c b kd

LrJ L1o'J (b) (c)

Figure 26. Tensile Reinforced Cantilever Beam

55

10 1 ~1

(a)

As rfsl '\ ~'.Ps = Asfs

'\ '\

20" '\

·.,

~2" r P0 = tr0 b kd

kd ~ _L_ ... P' - A'f' lrc s - s s

A~ cn(kd - d' )fo

kd

(b) (c)

Figure 27. Double Reinforced Cantilever Beam

1, a~nt,PAl1T~~ ili\n .... 9,i~,rjt~~;t'.~;c~si l}"s:tga. a. Te:nia:ile lie.infot"Qe.tS, ~iaQtion.

i'iith '.f~f'~~li::1ntit~ to J'1gttr$ 2$•

3000 pst eant:l~et.e . ·~. n ;il! lQ, . .a.;.. ~· O

p

liil)L + ~f;&L ~ llf;60 lb/ft , r • : .' ~ ~~.:\:

A$ ~. 4.,0,. Sq!! j~~~

M.::· .;,,. lf.·.·1!.·,... .\lf60,H.1,q).!. '_, m~.•.•···.1".~~ 't"" ·.·"';t; !~~ ~ ~ -~- . •c c • • • "'""" Gt~.,.V~~ ~"'·.Ulfj .. ~,\! ..• ,

. llZ; .I

:Wia.$t!(J ea;11i1 r iif : lJ .: . : ,,

t t;:V~cnp + (tt~p).a\~.·· ()np·

)/: ,,,--y ~(l f(lO) {.02) + ~(10)( oO!l] ! '

,!;,

··~

...,. ~·.

.....

P ~· ~QU~l.~ .. ~ .. , ... .,!lt'l ... l'lil'."t"t1~.i11

in.t11 l\'ef~;re~~~ S.a~,1.oa".

~t~U,t'~ fJ/f.,

.lb ./:r::t ,,

.4'·. ·t.:9 ""lQ.,,1

$;~el!l •.. ltt'~$ft'!li ~~4';,~,

11$,~~

J.fi.644

11192

.1.~ i!ihl" l•···

. ·.· .. •. '. l . Q~'P ....

""'·· fe:t\'tp l);np '.' · .... :p• ) k •. ~~lli4

~lttti1!;,;trtl· 1a~~~nt.$ ~b*H~t ~~ t1) .. tind ~·«

~~~~~~~ .. ,.J~.P!!!,111 1'f,,Q4 151. :ta 1~

.A~··

)!$"Qp.~;.w g~~p~. . .~S$ • 11$ ''lil,:;ty~

1~ ~ .. ,;. .. 0!1 v~lta$$ fJ>.:£ pereint ·~~a:~i~~·!'I i~ st:t~$$· ~~~

i..:!llt er11~l 8:li,.t ii .•

. 2

·.J· ...

4

Q.(1.t;m:tf't,tf .Jn!$p·1 '·'

•!t:.9;

-34.J -41 .• 1

fl'·~"'"•""""'· eiunaa.~ In 1i;;1;; ···. ·. ·· -~"' ·- .. ··Q('}.Q~e$t,oa ~i~i:,,-l1tr£-~·tl . ' .~1ft,Si _.~1;il"l',Si

~bl·

·~··· ·""·'

fa .• 4 iaa.1 l:.$~ .•. !)

St:re.•i1~;1··1a ::11,am Aft•r O"tltp ···•() .. . -e .· . ·.. . . t·=~-'~!• . (I,~~::.•:~•• i·Ul .. :ia .,,-!II~) ,u.M·L .... 1

-~)T 1$.S!:J l$lG4 . 19·fl· 115ot$ loiJISO

' '

~'itil~'­~-~~Q-,ili

a. f$JUl1.1• letl).fO'tt-G·''; •••iit••· With. r.$f$!'-•cH> · •• Jlgu:~e !fi #· .

JOOO -\· ~o».e.rt"b.f

n # .to, .!! .. • er .p

p ~· <'>t<llif::w ,. # 0

. Wl)I + Wit):, ;: .l49J ll>./ftt~ As ~- -2 .• -.34 $:Q;_.1 ••. ; ,,(£. .. ;e t

As S.n. Jri$0l1f6:• "lhdt K =i 141,,67 lb~.-ft. k ~ o.'31,

· graph t~ us~!t i:Ul~a. th~ii.1'~ 1$ ¥1.Q tJu:ne to1! p ;®· o •. Ol.:1.1.

v·~l\lea or p~:re.e.:nt o:nai~g~ 4.·1\1. $t~e~~ will he :i:n:t.erpo ... la.i1ed.

~·

4

l"$~'.t(.l,:n.t ... ~ha.ll~t ]£~ ~~l:llGJ:$.~i .. ,itftSi,

~t~-~·

j.;t$al .,;$t,;te~1

4 ..• ) 1.~

~,.4

Att,1~ ~:r~~JJ

l0,5 £H:37

S#J~f. ... ~~e,a 2i~?4..

~ .. ~.t!'\·!!\li.··.O.· .. ·•.• &JJ7':;:1!'Jl·

b. Dc>ubl~

W:i,:tl'.t :V.i'tfe:r~~~~

'ooo ~~~e.;ir·~t~ 1

·la ;:;. io, • ·::: C>.~v. p.. · ... > ·'

.0046$

""''"'"'>' •ill fuq "./f't . ., ~·.ti•·'~.ttl•.•·.,. .A's .~· 0.94 •·q~. lrJ.111

61

As 1n:Proble:m 2(b)-M:::: 76, 667 lb.ft,

:t = o.~58 to s l19l ps;t

:r; =: 8584 pst ta ~ 21666 :psi .

For n = 10 $nd ri • /p = o .• 4,, ,1gur$ 18 1.$ th~

proper graph te use. $1uee thel"e is 11.0 QU:t"Ve fol'

p = 0.0111 .• iralu~s of p~aro~.nt {)ht:;tf:l.!$e ln st1·ea$ will

h.e 1nteJ::p<llate4.

0 -3 4

0 -3 4

tlone:r:·et~ Stress

•t'flfr(<rW/l'j )'!'@IP,¥.

-29.a ... 3a.9

Pe:ccent Ohattge l.n fension OomvrtHss:ton

fl.lte:el S'bress Steel ~"tresa 1"(_11' Ttf ."lf\l«V , ¥ .. ." :uo!M41

2i113

3).4·

. ; .• 3

5S.$ 103.0· 1$3.S

St:re'Sses 1n :Seam Ute:r Qre$p rl$.nitllltlh . eo:mpre~s1ttm

!Q_;ucrat! ... !i~~f!.'.l.. ,._ Slftel ...

22,93 22496

1;628 1?424 I

20473

rlO" As

20 11

~2" A' s

(b) Seo. AA

Figure

62

B 20 1 ~~~~~~~~--

(a)

rlo" I I

20 1•

As

(o) Seo. BB

28 •. Fixed-Ended Beam

:3. ~:tx~d~~'tta~I .~ea1m • ~ca~iA;CQrcil Jesl1.tu~

tU.th. retei"ei1x:$ :Ftg"~~¥ ~$:~

1iOOO p$l, <ilen.o;tei,j.$

n ::i: io., .$.17;ot:lQt>t ·.AA

As ~ a.oo 1~ .• · :i.lif.,. ·A.•$ .;ii;:.· 1 •

. fleflt lGl'l: · J.a~

• . i'it. '"l"ltt . , __ ._.,_)·_. ;t '".!'!.·.·.·,.· .• _· ........ ·.·.···_-•. _ ... _ .. · _rJ_· __ · · ___ • -.~.·_.· .. _· .. ·.. .J!'!'t ___ •.·. z:- v~.\#t>'.u • v -- v ~ - ..,, p

"Ii_-_&· _:.1J>_·,_._ ·: ·."fZj.· •• •• ·_. -~-• ··. 1.· __ .• ?_•_,,._ .. _-.·~-~ ._? __ i __ Z \'}.··•· ... _-.~-~-"-·· <_.·.·_._ 1_··•.-•.-~_·•.-·•-.·.'.··.•_·· .c_·.-.· •. ~.-~ ..•.• l_•-~-·-·_o_ .. ·• :i_·_. _· •. _ •• ~ .... "'. -... _· •••. i·..1:».u .~ :a. if,\;~,, t. .. -'-~ - ,.fJ6T J.l;). !m,,. ~4

As in l?r:c;hl.em, i ( b) * · te ~ f$ :;t

f •s = G3Q,'f ·· p~! ·

•£,.·.!··· '· -~~

~~~~1~~~t~s t'it& ll¢:s--~~t••:•~~~-." &t t•tri1• ~'. . . . . . . •, ..

'•eerttf5rtf 'AA alii Ill, 'I • :· . '

. ' .·, I•

tu.~ ~t;t~ • *14$ (4~)- ~ llA~11{!1iil • 4•1t

IM '* . .· r,,:i + llOH2>Uo..t1.int

.: .i, t-t~~.(·l···• .· o·~'(t!~:~~ ~· ~·,.t~ .• 4~~"'i'. •,,....,l """ \•\ii'I . .... f(1/;/ \1\11 .. ,.,~v ~. .: .0""1.:l ~~"""'.

t~. :tt~ll+ ·~ t<t--k4~a· . ·. . . :e;. J· .. .. . . . . I Mlll ""'< "' ·. · · ~ (l()f(l,~0 (t~ .,,. s.I))' . . ~· ·' l(·lf .

. = ,oJ~. b~.i .· Jr t:-• ·•~~~-~t.1 ,,_m ·~ltbn1~•t ed ·(iQta:$ilfiiu?11.1. t:Mt ·-~.:n~utd .. t-t!Jl .•Q·ms~l ·t:i· t•1t:nta., t·h• :tt$.\llt:i1t1·.'beiu1~ ·

~il:l'$$$•1 '···· ~~·-1·1:¢>~ Ii . . J\S: in -•'blt• 1 fb)

te • · 11'4 p~d. · f·$ ~ a?4TO~:. ;s$ t;~·~ &~.!I p$l

,.

· A• 1a :fr~:ble:m 1 .(•)

t4 1#, t;1 pt.i '£ . . '\ 8'lliH'~r.it .. 1 . Jt ~ . 11ii. • ;;,7."3' JS '

. . . . . . ' •' : . · · .. 3 ........ , .... '. :· :8.: .D .,." .'.~ Jj . i". ' . .', ·x:~ .. .r··_'·~ -- :· , f . ." J.:· .

* f(l)::r ~· mPrG· 4etail1~Ht tlf.pl..Gatloa ·•t a $l1il.~at11 t$e~lttl\tie. ;••• ·Mt:t~e•e• 'fi~.

.... aa.o -4l.J ... 49.5

... ~2.:; -;~.{}

le:t!ti!tt!n u. :!?ere$~'b. Oha:ti1e

!'ens1on iSt.e e:4J. J~lt· :t'e rss

'r!P'l¥ . .-"· .- ... L-... -~~.11'-~

la 0Gm:preei~io:n

!1'.ieff.',J .~t.:test1t s~.a

94,S 1~5.6

lettt11f.1n EB l'er~ent.eh~ge Xn

fe.rl.s1 on. · ste$l Stres$

........ • "'!If. _i},. : .. 4·11 _I, __ '@!'._II*

6;.0

111<.i

i.a~b.. Q~,S$ fl£ t;t.r~~.p ()a:1J;ta.$S a eh&:nie .tn the :mom.ent$ et i~•rti~. of the two ~ect1o:ruif and t:he~e·

by ~®'U$~$ a r~distrlbu.tiol.'l l'Jif 'bent\1~ :momen:ta. ~l1tll eh~ingGis ln .stre$S due to the :r~di$t:?':tbut1Gl\ o:t 'be!ff1dtn1: mo1nerit$ a.re ill;PlUd~d :tn the $t:re:ss

q1tt.:iouJ.at :tliJ>n$ 0~1011.

MfitntE!tl.1ht tf , •• ~1 •. $1'.4. b·•l:41~- llf>til&J).fUI iU~· t~ tttC\tJ.,

$.ltd th$ pCU"i•n"i t>h3.ll'lt ia ~--~\U•»I MQ$t;t1t$ •.

J 416!$ J$i;I 3§9:4

~·•2·

a~''

lt·ct1tA M

I .•• ,.~_~) .s:a,.,,,1.1. U8i·Stl.

te•t·l•• II -i'·

. '.i -'''·:~,,-. .J94:,.o·e~

J~n.,,4fl~

P•~otnt tAe;nit llt .ltldlU H1meii1. ,, 0,.2a

0 .• 51 o.: .'C"J .:~<41

..,.1J1J< ... 1ili.m . .. V;t·""'":7

i .. t.$1 · :t:•J:>t.eni\ lk~it.1te11t 1a h:t$$$4it&·

1~1·i·~"' ''" . ~ ~-~·~. ··.T'·"~ y:r .... ,-..

-111t8 .... 11 .• ~

. -""" .111. . ·#'l ·. '~

·:le•tt•a .AA ~SHltt')l\ •... i'H!•·i

' 2,;t ·.

,J, •. I ,.,.· · .leflttlta 111;

· 1;a11at1 l!·t•l 3,.1

"··' 6 1 ·•·'·

.ioa1tis1.t.~1 '.StG:e:l. ~ .,__,.. __ --:- .t.."!"""!lJ._ ..... _!,(.ii · ......

' . . 5!&~·4.

•.···

't·l\"114. J.. ., ,.,.114'111.~

3

4

3 4

Oo.ncrete -·~]. I'$.f'5 ~ .

857

695 594 .'

519) 415

6'1

2550tt

ti!5659» Seotioll. :SJ

. TeU:$ion Steel Stress "<!',,,.._.., .7 'u

a.a955 19291 19$1.1

4. Fixed-En.tied ;\?e,§lVh BQl:.:-01eeg D!H~UI•

With l:'ef&l."anee to F1gut•$ ~a.

3000 psi e1an¢r$t~

Oomnraaaio:n Steel Stress

112933

16253 18840

n :iz lo, wDL + li1LL ~ 213, lbif/ft.

Beet ton A.A.:

p .::: o .. 011s;, p'' = 011,006, p'/p = o.506 As :.:: .2 •. :;7 tilq.1n11r, ! ''$ = 1.20 sq~ 1.n •.

S~oti on l31U

p = o.oo6t p• ~· o, 1'/p ttl: o As= l.20 sq, .. in .. ,.Jt..'s = o

Mo:aent~L and stresses in Seot1o~ AA and .BB baaed. on a uniform moment of itl~:rtia,

.. .,&,s in .fl:t;V'bl~m :;,, .

. MM lOOt!St661. ln.1:tt. Mi~ ~· SO),)Jlj. lJJ.1n •

s~oti~rt .·/AA . · ;:

As· :in · ~01\l•l.e:m l t b) l

$~ ()11~\ta i1' '

A.$. ill il:'~llle$ 1 ( ~C) 1

.· f~ ;:z;· 956

f$ • f$ ··~· $f)i<:;Q

t~ ·~ '9~6!~

Ooxufi~llti.ng. 1.lh4i l\on ... unttfl)r;ni mQm~111t» . . :l,:n~rtia at Se¢ttlo~li1U !Jj

·As l:l J?ritl'bl~m

~11. ~ .. 54~:9

i:ruil ~·· !1070 JJ>3r the . ~Q.njug~,t.~

l¢,AA. ~· .,.,.,,..,,,,,"" .. l}).~~­

lf,61.461 i~,tn. ~,, r., " ·, ",, -.

Stre$$•s tn S$.Q1ti().n M Al$ :\»: J?rt;\lbl~m l (

:tc sr 1'4£~ . p$;t .

$t!!?ei!t$e~ t~ $ee'ti~:fl. U

.ts :l.lit rr<Hl>·l~m 1 ( .~) f¢ .tiii' 171 psl

'.(t for" a more t$.$i~a11~(i s~pl~nf;:t>.lot!, ot a $~mt1~;Ji' tee®iqtui~ $(!),'e re~~~ettt;)fi!! ·ti).

For Section AA~ n ·~ 10 ein.\l p '/p .~ o. 506 , values of

pereen.'t ¢ha:nge: in st;re;:;;s are il'lte:rpolated from the

g:raphs of Figure$ 18 and 21.: For Seeti~n :3ll, ;n = ;LO

and p '/p :::: o, Pip:re li2 1$ thii:l proper graph to ttSfh

2

4

0 -2

3 4

OoJ1crete St res$

-Q1.5 ... J:ro .. s ... L}8,.S

Oo:nor~te .·Stress

... 32,J.

... :;8.4

"l 7·· ~--.

:3.6 6.o s.o

In Oomp:ir·ession

St.e2l St?fe~.f!,

55.4 95,4

1~6.S

As in ::P;robl<$m 3" eal)h ease of ereep. ~attSces a e'.h~,~g«J

in the mome:nt f):f' ine:ctia ~t: the 1;;~10 sectlone .and there-

b;r ea11aes a redistr1l:mt1,en Qf bending 1l'!omentli. :t!he

Obanges in st.:r~$S due ·to the :r€Hl1.~t:ribu.ttt)n of bending

moments ar~ ineluf$d, i.n the st:resa oaloulPJ.t1on13 belQW.

f4ome:tttiD· <}.:.f ll1$%"~1a

~nd ·~11e.

c -

. :t -

. ~~y.S?,;t"~'2!· ,..,~'f

.... 40.9:·

$iaat,i~~ AA M ~

. 46~,1~1· .. 4~1.·~•1.9\8

f<f;rccent Ol:i~~ge, tn .. i3$.t!dtn~. l~t11ell'.t·~.

. ~·~---~--·-- •"·+".("~'flo!"~~~-"-h~-.'·· ,' .:_•,

:~~it·t'li~n.t Olta:tl.~e ln llelJ:~li~~ l0lILt~t.~ Ii'!'))' _)!If._ ... _. 1:e;-, , .• _. ··~°l'fr .. --,--."'t?'i~ .... _c __ -~·. ·.:.w

. .

... ¢.).i?9

..,(),$0

In St::re$s~~ se~tt~n · ~JJ.,

1 .. ~.Q······.· .. n.·.·· .. ·.·.$ .. · :t .. ··.·~!}.· .• ·. · •.•.. s.:.•.~.•·. a ... · .. ·· ... ~.·.· .. · •. 1.· .. ·.·. t~mtlr~·~si.~ti $t~$1 j'f'itl"(r·•·- 0 _·:··~-y<ti·· .-· ··~

lS~~tl()• ~.1J f~1l$.l~:ti S1i~e'.l ~~~ .. W,~:iAl:lii~'ll' ..... ,J.i_,_.~.-4 J_, :·· il)li\ .,

~·.-. ... ~.

,·~.·· j£

., ~~:::~.~;; ~~;:1~ ·97;. . r:s11~1 · 1$00;J; 1,,. · as7ftii 1.1$~'•/.

', : .'~194J ' ' .. ;.': ·,! .·:,•

',,, ·, •,,.

. . ·~ .

\:· \;

: ;\ ;· .. :.

ii~!

'. I'?

:' ~ ~ ! ; ·, '; ·~ •.. ~ ·: · ..

· ... ···

I .'·' . ', ' '' :i:

·, . . : ' . ~ ', ' .·

' h

': _l ,,

~~'~1~li$ o.f 1'51it'$S.$ V$ri3llt:$ 0.t~eJI · a:ffe(:l~.~d ht V$.tla ...

t:tons l..lil ·tb.e ~eo~~1ri:tf ~.nd (!).£ the Th~ beam stz~s ttt!r#• !t!t~·.~1 q.ua.nt~tl~$ we:t$

e:noaen to t'epreae:nt t1pl.e'$.J.. pl'a~t.ifJ.!~l vsl'ttitHi».

Xa F:r~blern.s a:) ~(a) va.lues. $f t~ns:tle

i%), 'tih~ steel ltt~reased. lo p$:;ree~'ti 41~$ tttJ ~t~~P f!i>r o :.:; 4 wh11e th~ vs.lue1 .et}ro.11~~ssi:fl"e. .!$tt"E.il$S 1111 tll~ ~~~ ...

'1he .tn~lu$2\~n ¢>f

. .,, ..... """'"' m~1tnli:l:f:lt::riw .~t1;~et~4- th~s~ e'.ba;lng$a

ma:rl\.e dl:y. tu1ei! ( l'~Qbl~rott,t lfb)

an.a 2·h~1l:~1 ~1:;11~ ten.s.tl.~ stf.l(}1 .· ¢1:~.1:". !$.Jilptll))~:). ... ,.

me:tel;r :~$''t~ent whilEll' the o.~n:~:c~t~ stxi~i>1·S ~~~?f~~$ed ap"".:

pro·;&:t:mately- 4~} t ~. fbe Sltl*e.s$~al

in 'the ~o'mpre~e:ton ~t.eel:.

'b~tb pr<llhl,e!1'!$1J .• · l'lf~n ·'thott6b·

enrll:!m~ly

:pe.;i;~~~nt :\n E:!:npeJ::(i~~~eqi.

. w'ltlo.tn tlrt~a a;llol'w~hle vr(ll:"'.\t• .

So·iti.e . .,. (l) :t~c. l'J.Qt.$4

dattll)l:$d ti1eitr

:reoe1n:une11da1d:.on Qf th~ Q()~~

111rhich permits usin;~' twi~e e11nrn,uteta st:re13~i e~~pao1t;r b~se4

on ealoulated eliast1o $t:t~.tn.

f.or the beami:rl ill 3 lt1 th$' ditf'e:rGHH}ea 111

'the :mcnnents of inett1~ betl'reezi tlM~ 01.1,d-sedt.fl.onis ~tut the nttd·

ae¢t1():n o:f' ef;,.oh b~a;m '\iite:t~ t»a.k$ll :1..lilti!J oon$1dJSlrafb:1.on; ~~. eoia ...

pttt1ng tn$ inttial $la,j$t tc, m:omentis th.t-()1.:tg:hout tl:1e 'beam •.. l.t\

compari$o;n with O!i>filJ>U.t~1t\tq;n$ b1~vsed en; e:-n: .a,S$Urn.ed 'O{ilnstant

TAorrtel3,t of tnerti~, this r,a0re ~;itir;1;c~t mo1til!ttt a~Etl¥~i$ ~esulted

in slightly- h1['5her ¢oaputtd. m,t)'iJi~nt•lf1 ill th~ ~tltfer $Sot1on,m

Qf' the l'H"l;an.t a:n.d slighiil; l*)t'ife.:r ~~;,mpu:tei mGm~nts in, the less

sti.ff seet1on~ of the b.$ain• :tn '.l?rt,'.fbl$t\ts ·' ~a 4? tnel ten-

sile steel stres~es :tn the end. sei;t;ion$ (~4705 to 25659 and $4983 to i5906) e~~ee.d@d a.llOW"ablQ s·~ll?eas (~~l~,.ODO J;l$.i}

and 4 'When th~ V~?.tl"iatton bi motnfiJnt of tn~rtt~ vias t~'k:en iniHJ aec.u:rmnt. F~:ll an .@;,sisttmed C#'9:nste.nt moment 'Gf 1l).$:t1t1ia.1 th~ Qal;.,.

ottla:tied elast i.e. ten$.ile ~t.$el st:reasee ~re re 2'':3918 pa! ( l?r!l'ib•

l.em J) sJ'ild 2)9$6 p$i ( 1~o.tilem 4) •

It .1s also n0·ued that $tres.s ehangets du((; tQ ~reap

shown in f.ncam:ple$ l(a).; l(b),. 2(a.) ~.11.cl ~('b) a:re ent1relr in-

dependent ¢f mom;1e:l1t. l:l,"l theme l.oa.etea ,~~-ntileir~.:!!' os·r~,rns tlM11

moments remain c~nst~~t th:rou~hent 'the. ere.ep per:t.cni.

ln J1roblems 2 anlil 4 t.h~ Jj,jt;res$ <1'h$.n15eis tnoJ.:1.uled tb.e

e.f.:fects o.f morl!.en'b t'fhang~~ d:u:e to moW.t;l·nt :r~di,st:ribu:tion ais

well aw tne st:ce.ss Oha~~s due. to er~ep. J'($ir a p:t•()per

4 • . .t'or ,., ... i.·•1:1»!1"l·"'""'ii"'

$t,:cellt$ .o.f .

1:n~re.a$e

FrQm ~. r$vlelf, E>f tha :t1t.11.rr~:t;~~$:1 tba;r$ug)l. s.tu.d;y

mad$ ttn, the 1;Mittut$ $f $:t"~~P 'fill,~ te~tf;).rl!? atfeetl.n~ ~;r~.ep~

~h& equa:tit11'.JtJ devel(!);p$t)l a.~ti the g:cap:k\t$ l'!X'~~entell 1:1* ~h:\.$

pi~enli>ment>~ Qli th~ stl'$~s.e~ ln t~inf~t~~i QOJ;J~r~te 1ru1:1ml>~riil.

l:liustratGti problem.$ $h~~" tl~.t.~ ~an be UlifJe@., .

fh$ pt1:zi::v'4d.1; ef p$.;r~ ...

:meter$. 'lo:r tt1~ r~s$~:t'tllle:I?' 1l'.ttl!iirested 1:ia. ml'lre p.:t>eCl1$t valJle$

t~n can be t))ot.~tned f~om th~ g:r.e11~$, t:tte 41$t~l1ect 4ata ha$ .

hten ~el1eot~d ~d pltbl;l.$ha4 SeJ;i~r~t;e:t,.,., (9) f~H~1· 4ttts

~~velll'Jpel on fiitn l:i'~ .i>1g1t.~l .Qo.mput~r a:n<l th~ prr>gl"am

ts t:iaclu.d~d with the a~:h~.

D$V~.l~pn1$11t ilJlf 1'.ltla :it.s e;pplte·atle;):!t te tb.e lem.a has b:t"u;ught. ottt •e:it1;$1.ln lilt\~ts~ efte~t r~la't1e:fla*

to the graphs $.ntl in Saot;l(l)n, V

t:i:-$.~ed ;proble.:rn.Ja.,

.In su.m'maq the main i1()1\h'.tt1lfl m~y

Q.:r'~ep

(;)¢>:~le :;r.$t ~

Q,o~.st ant lH~n:~1l!11.g mQxn~nt ,

l'~· ~lllt~t t ~la. ltrQ,

an i:rurf:i:~~~tlle ln. tens1lll:l . stee1

. ~. All ~the1"

~tr~~~lt """'"''~""~

Sl:.ah3e~t~4 t~· en le:~~

rei~fort~d:

tlie b~am.

pe:r;~e~'

'b~~i11 w!Lth $. :t;n. 1~i$'1iiG4."lti1~t$

.w:ttb

1t1,tl.st e:l?mi~~tf.t

;J I

include« as a, pQ.til of il:luS'1Dtat,~d problem.s ; ~.nd 4.-, fur ...

ther study ot tbig · ltll.te jee1l may :cev~.$i1 ~a.Se$ ii<rh$'re t,b.e ~tre$s

changes due. t·o mom:ent :re,d;!st:lfi l)utJl.o~ alre ~ensil.tet"$bl.1 l~.:rge:r

·t.o thi$ thesis l:ifO'Lll.d ti~ $UClli ~ Stu~ b.tVOlving the etf'e~trt.s

of gtH1tnlii1try and loa.liil'l~.

stress Ohan.gas in :reiafo;r;~~\d Q1Sner~te meinb~rs due to sl'U"lnk""

ag.e. lll 1nant ease$ ·{})1a .sbrink;~ge etf«iH~ts ~.:t:l. S:tJ;esl;iieS: ni:ar ~e

mortil s.:Lg:nif1oant than th$ 4);t7$$p e.f:fe~·u$.... J.l.the'llg~ shrinltS!igQI

ef'feeta a.:i:~d creep ef'f~~ts oal!t not be. sep~;r;·.ateil, 1?1 tlle p~si­

()al st.ruotu:r:¢.t tt is pa~l))ribl~ to ae;pa:tatQ th~se two ~f':feets

tensive a·tu.d1 of i;;hr:tnk:e,:g1:t1 ant!'- ~hr1nJi:a.g~ ~.ffao'h·1$ '*7¢.nlld be a

s1gn1f1oatnt ailtdltio:nal t\l:ipi,t)• ·

Although the work of 't'.h1~J 'bheS:.t~ h1as been. delieloped

mi:n.di thlfl.t the w.Qrk e~ald bfJ prceft~a.bly · e:E:tand~Hf to eqver

pre stressed eonC:t'ete .f$1.truetU?'$~ •

l. ,' ' '

:Buti . .ltii:ng Qotta lequilt'emeilltt tc11 ·:ae.1nfor.e(i;ld. O~~lQ;rets A.<l~I• .au:tld!n; · Oed~·, ~im~rio.~,'A poD:o:rete ?nst1~.ut;~ Pllt}lieat1qn, 1956 ..

Ooru:n ... e·te . l'.¥d.el;lm~.tton, l?a:m:¢.hletJ, :Portla~(\t t1$jtl$Ut . As~t.1aia;.':" 't:ton.·~ i·':ne:f'le~tiel'll. ~:£.: Reinfor~eti: O~:n~r11rte Membe:rs, ~ 1941.•

'• .· l')a;v.i,$.fi . R,. i•••. Da'f1$~. Er~ E.t,·t .~nil !~railtot,1$! .~;,. S~, ,ffP.l~at1c Flow ot eonetete. trnde~ SUsta.1ned $t.re.$th n Ptoo~e€l:i:nge, JtS'fM;,··· Vol •. ~4., 1934,. pp ... :;3$!4 ... ;,86,. . .

4.. Fe1".'gnson.,. J?hi!t M • .,. Re · h~.i®is !l>ll .1'.flt;fr,:mat.a

. -~, .. ~ ... 1 iff·H .e7 and ~ililn.$ ; ' .tn•~ , ·.

ii.. Ge$u~~h :ff·.n.nS:·~ . nihrln.J(;~g~ anct '"reeJ; I'nt:lit.eiice .. on Dstle:~­'t ~tr>tt$,; ~1nwl M()~ez:r\i$ ~f Re:tnl,'.er~El!d. ~on~l".~tl'i 13eam$,tl A~a:.:t., V\;;l .•. 59 ... · l9e).i?.

7. :tat:~e, ,G .•. E,.. · · :~q . ~· !11d

... · ~e~:t 'X'o(rk ~

8 •. ri~~~Vef:lu:ml:4o •. '~;?e:t~~~t7 ot Oo~or~iH~ Cr$~p, 1~

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