,IU'!AL\"B!S CF A

R!LVOOCED Coti\~ 1mt\M IN ~OORSIO~i

by

Willlat?l Aiu.-and St~t I.1

in e;.;-,.,~idac..7 for the degree of

~t~r of Science

in

Ci Vil En.zilleezeing

~~ch 20, 1964 BJ.acksburg, Virginia

... .t...

I!.

3

LIST aF FIGURES AND ':£ABLE$

FIGURE TITLE PAGE

1-1 Examples of Combined Bending and Torsion 9

1-2 Failure Surface - - Pure Torsion ll

2-1 Example of Gasund's Dowel Action 20

3 .. 1 Photograph of Bea.ms Curing in the Forms 26 3-2 Photograph of Strain Gage Be:f'ore Wa."<ing 29

5-3 Photograph of' Strain Gage Af'ter Waxing 29 3-4 Photograph of General Testing .Arrangement :;o

;-5 Strain Gage :Loca:tions, Beam 1-1 35 3-6 Strain Gage Locations, Beam 1-2 :;6

3-7 Strain Gage Locations, Beam 1-3 3'T

3 .. s Strain Gage Loca:liions, Beam l-4 38

3-9 Strain Gage Locations, Beam 2-l ;9 3-10 Strain Gage Locations, Beam 2-2 4o

3-11 Strain Gage tocations, Bea.."l'l 2'."'3 ~·l

3-12 Strain Ga.gs Locations, Beam 2-h.. 42

3-13 Torque-Strain Flot, Beam. l·l 43

3-14 Torque-S·!~rain Plot, Beam l··l 44 3 .. 15 Torque~Strain Plot, Beam 1··2 li-5

3-16 Torque-Strain Plot, Beam 1-2 46

3 .. 1·7 Torque-Strain Plot, Beam 1-3 47 ;-18 Torque-Strain Plot, Beam 1-4 48

3-19 Torque-Strain Plot, Beam 2-l 49

i11U1Jin~

3··:2D

:HU

23 .. f!~~

3-23 ';(.?)1. J .:,.~

3 ... g5

3-26

;-~

, .. 2s

5-29 -; .... ~o .... ~ 3 ... 71

3~:)£!

3-3:;

,; ... ;;li , .. ,; ;-:;6

3-37 ;-;;8

, .. 39

1(..J~o .,, ?...!~l ...

h

'l1ITL7~

Ti)l'tfne""'St,rail.1 r~~i~, .. c, Beam ~~.~l.

'l~~rqtt(~ ... ~ltrs!n l"l~l't' ne~ 2;~f!

l1orq~~~~.sttrni:r1 Pl<lG, Beam 2~~t

Tc:t·q~tt~l't)St:i:·z;~in i~1,o~t ·' Be:mi 2~-:;

T-orque-Strs.in Plot., fj(,gm 2-1.1

Angle of P:rinci]/Ql Tensile Stred.n versue Torque, Bea.ms l-l a~d l-2

Angle of Frineipal Tensile Strain versus Torque, nearns l .. ~ antl l.J;.

Angle of PrineiIJaJ. TcnG!.le Strain ~rermie Torq,\ie, r~ams 2-1 and 2~~

Angle of l=':rincipal '.!1G>nBib' Strain verstw Torq.ue 11 Eea:ns Q ... ~ e.."'l.u 2-lf.

A:.r.gu]M D1aple.ee'l..mt}?l~i versus Torej.ue

Failm"e Cracks, Beam l·l

Failure C1"aek& ·' Df:am i .. 2

Failure Cr$;.l.c.}.:s_, Bosm 1-;

FailU!'e Cracks, Be~. 1~4

Fo.ilur~ Cracks, Beam ~-1.

Fa~.lure Ci~cka,, Beam 2·2

Failure Crackr;? ;B-?.?~r.t ~-5

T:ll-.1, ,._ II"' 't~ ' """' ··~ '·· 4'' Q.~.f.M,I. e \;l'S.C:xi.tl ; ci.;;;00 ~- -"!-

Photograph of Fc-"1lure Cracks, Beem l·l

Photograph of Fa.1.l.u-re Crat".k.~, Beam 1 .. 1

Photograph of Fnilut"e Crael~s, 'Be&"l 1-;

P1:1otogra.p'h of' Fa:il-moe Cracks, Beam l.J}

}~;\{~}~

~o

, .... ~ ;IJ,~

t.:::~.~ -":,.r;;,

~3

51:.

55

56

.':""•"'r· ;.;i l

58

;9 60

6J.

6~

6; 64-

65

66

6·7

63

l-0

69 60-

~·

F!GUFE

)-h2

3-43

3-1,4

3-45

TABLE

l

2

5

TIT:r..B

Photograph of Failure Cracks, Beam 2-2

Photograph of Failure Cracks, Bealll 2-2

Photograph of Faj.lure Cracks, Beam 2-~·

Photograph of Failure Cracks, Beam 2-4

Properties of Test Bemus

Comparison of Various Theoretical Predictions ·with Test Resul t.s

PAC~

70

70

7J.

7J.

PAGE

24

7'7

~1e autho1• would like to empress his appreciation to l~ofoss~l"

of the Civil Engineering Depart.men-:. of' Virgima

Polytechnic Institute tor bis leadership and guidance dvring t.t"lo

inveetigation a.."'id writing of thie thesis.

Ac!tna11loogeme11t ilil also made to

\-!hose help in the laooratol"";f wa-a inve.l®ble.

Appreciation i~ also iaxpre~~ed for the financial ass15tml©e

~i ve:a 'try t.he V. P. I , Engineering Illxperiment f:$tat:1on ·t~r.; pw."cll.Me t.hre

ooees@aey ~W.:r:•ifile, wi tb.out whie.?:t this thesis would not lwve been

}?OGSible.

:Fi:r..a.Uy the &uthor would like to f:~xpress his gl'a.ti tude i~ his

wife f.'or the tYJ~i.ug end proof reading of 'th® m..~u.scz'ipt.

I:!l th.e ~.at fci:ifty ;y~a?'!S the 'US£ of :;.'"einforeed conc~"et<~ :Z't;\;r•

fi:it.~:t1ct.u.r.s.l pw..~ses h~~ inm:·s..~~ed ·~hr,o.;:,ugi'! rese~.rch $Jld deve:to1:~nt t•J!

whe;·~ tt :'!JJ widel,,v used 'redey -r;ith c:ottl'!denee i.'l the meet eomplex

Gtrm:·tm"f~s. ~is ::a.pid. [~tlth ean be attributed to :i!np?'Oved. ~·~~}

high.er qt:w.11 ty ~inf.orcirtg stet~l$ anti t.he ad:ve.rJceraartt~~ wade in th~

t..ed'll'l:l<;t'Uf!S Of d.~$;1,gn ·

!J:1 ·t.~ design of cont.:b'.1t.....,us st.ruotu.t"c~ -the monclli ~'+lie mtnr.~ of

r~W.m,ce!i co:'lc~wte af'fo'.!?d..~ m;ri.:y adv~..nta.t,es e1.s well a.~ !1lme dis~A:w.l'!~A

-t.u~~ . Q!:\lfl mitj(lr disadVU'!l~ is that almost f'Neey c0J1cz:0ete mw;b-er !J

!Th:! nw,tter how simple, :lG sub,ject t-0 the aetiol3 of combined etl'efjSCs •

It is ofte~ ver; difficult if !t~t ~seibl® ·to ~~rze e $tructtil"e

aoS'.ll)lerol.;17 for every pos!Sli'bl~ loo1i?l/3 «>ntkit'ion mt-'lbjiaetil'.g i'& ~

~~':lbined. str.esaee. It ia ~n practice in 4eBiS1!i~ mero.oors ·c.u ·i~()l:>e f);f!.:j effect which l'lZ'O\"luees, in the desif1,fltn"af;j opinion,: !Q.egl.1gi't1lc

t$'tr.a£~es. 'lh:i.9 Nay or ~,r uot al'iff\VS resul.t in a safe atr·u.eture. A

l!.ltrese whidi may be l1esJ.1gi'ble ·when eonside>'.•e:l acting alone ~ not b<':

:ne{f).iei.ble 'When e<-lmbined w:1 t.n othe:r stresses, and thi(!;} f&ct. ~ nG·t

\11.l-wsy[i be evident to t..'iie de!llig!:Aer.,

In r.einfor~ed eoner.ete members designed to r.esiat bending the

problem of. tor.-s::ton ie usual.~ neglected by deeigneri;; in this OOtL."'ltr-y.

However, r.eeentl.y this p:r.oblem bu been gree:tly- ~mp!:!f.lsized in the

t.echt1ieal. li tere.ture, T'!le sub,jeet has a.ctu...,,lzy been inves·c.1~te\i peF ...

io.iically since 19.if~; bu·t the profession ~ meae no rec~ndationG

~!th t.ors:iQ.nal loads whicli mw introtluce-d in 1958, but its J~~ri~~

-eounterpm"t, the Ameriea..."l Concrete :tn:sti tute Building Code,~ n~e rii.O

m~ntion of it until 1963, 1;1..nd "'men &nly introduced the illf'~r€'??lce thiilit

eo~..$1dered by the designer, by :requiring that where sth"rups ere

;r-eq_td:r-ed they should be closed mad provided ~d th longi tu.\.linal cw.'lle:r

tA)r>~ioool actionp and tl:'m1sversely l.oad&l SJ."ohes develop torai~l

~t:r~~sses &t the supports (Fig. 1-1). The torsiiorel etreas iin t..hesc

~Ai~strallan Stand~J:><i Rule$ for the Use of !Joi"'mal Re:btf'oreed Concrete :in Buildingss 6~d Association of A'!.tStralia, Syduey~ 1958.

"'~~:tcan ton01"ete !ns~;itute Standa:l."d Building Cod1-e R.ec;uit·i:.;~ ri.ents f'oit' Reinforced ConcrErtie (A.CI 318 - 6,), Detroit, 1963.

9

Beam

Loaded Arch Girder RestreJ.ned Column

1-1 of Combi,ned Bending and T'Orsion

10

•:>f' tt~ bi~·mterial rt.Gtt.are of the J::'{:inforced et.m.crete ~ers ·; fJq};n ..

b~~r..iet.y of the cont;!?'ete &ld 1 ts complex reaction to combined

t~·t:r;'"eiS~es add.a f'urt..~r C('~lieo,;tiom;} n~rt; to ment:ton the 1$~"tpe Olf t'it!~

cro~I! S<"i1~t1 .. ~r.1..,

~.m{~~ w~li ht\e 'been do'oo along these line!l, ~Jut th.er.e ~~1~'.ln

tTt:li:lY' 1mar.tS'(Jerod. questions to be i..weBt.:tg~wd. !t is the intent ot

·Ck1i® thesis to reviei-1 em extend tl"'..e irt11e~ti@liltion of ·tors:tooa.l

resis~oe of reinforced concrete t-e~'TJ®, bot.b expel:"ime~nts.~r all\~

il'k".!.t~ytic~. Sper.d.fiotll~:. this thesis retll"e@ento ~ oonJcinl,l,"?t'tion tJl

~l..,vtech .. '1ie I:m»iti.tute :Lu 1959. A.\t.'ttoug..'l t.he pi~i!"!l.."U"J' ob,jectiv·e @f the

J;X~gre'lt~ is the px-aetS.cal r»ialysis of toi .. sion in eombin.'t.t.ion u~:tib

l.11..11~'.t:t..,g rr.1.d/or Bhenr ~ there iei ei. su.ff1eient ~ck of umdersw.nd.irt.?; 0£

p·il?'e tt1r8io~1 -00 W&"r"Sn't f'm-t.S:le:'t' inve~t1gr~t.ion. Thie thesis deW.s vd th

th~ ~~~bl.em ot vu~~ t~~$iou.

After W1 9.ll~;:lqe:a.J. study, ~~.gh.t l"ei.ni'oi"'ced concrete bs;;~,.m of'

.Oqi.m'.r.~ ~~:rosei ~e~tion# uning fO?.tt> c:lif:f'erf::nt spactngs of stirruf:$ l '!:1e!.."e

te®te<.1 :l.'l'l I»ll\"€1 torsion in thr::! l.abor~trJ:i:""J. The experw...ental Nglul t~

UC1~tS "'Om!'.are'l'l to ~a..\ytfi.ca.t ool'l.rt'.i~!W uoblg tcn.r t.".\eories of analysis ,

Sigtdfica!l1t differences wer.e noted.~ and ~ attempt uu fill9.de i'>() ex.rla.1n

t.fie 1~0.a.c~na ff)r the difference@. In tbe ·tents~ eeven of t.he mevloera

exhib1 ted a ve.:'r:y soodeli.1 bri ttlc f'rll?l.ctm"'e, e&.t"..h with a eingle lfu:"ge

ci-at~k , <t-thile the e:i~·tn developed m.w.eroua ct"aclta a."ld the appem:>ance

( a )

-o

Figure .l-2

'I'OP

1 ' -- J.

/ "

/I

" t "

( b )

Failure Surface --· Pure Torsion

x'iz-r:iii visible 1nd.ice:td.cn of 111:rG\in :'...::'. tt'Ut';lile era.eking of the conca:•c}'te

o;,-i ~ll faces e,t M angle of abcn.·rt 45° from the loll31 ·iWl.inal blSiil&i ~®.

®~: sp:trru.. (lrack 1.n-odi::1illi11s.tes, 'X·~xtimlss eu:-o~.md ·thre~IJ f'a.c~em ~ . .nd. :r."Gttll."'ns

oJ'J. ::Vt..Gclt 1<'.'Xl.t th:e follt'th :fa.c~; as eibai.m in Fig" 1-2, The 45° c:ra~*.fil !f£ra

15

lt'•~'.t'{1~e;th of i"~i?)i'o1•000 aonc1·0·~ was c:ar:t•:iA~d oi..'t by C, R. Ytn:iI:~--;,

W fl, ~'J:f.g;er t<J-.. nd C A. Ml.18;'bG<~l(e!l) ir~ 1922 at the Un:l~rere.ity o:f' Tc::~:·or.st.@.

~ffi.f::''!Y ·~:;ef.;ted ·tvGlve bea'l'~ (~f :t•eete.ngular crOSI$ seetioo. The tie?.Xw.:;i

~.~1elv;ded plsi!t ':cncz>q;:·t.(i1 c:cno.~ete r.oeinf'oreed 'l'fith lol:).gitudii1~1 hl'1:'.~t~

'ths:t :£1)j.J:-el ~~eirafo~ce..~r~t ~tnc"..t'~8~'ed the to~eque cs,:r~eity consid1c~~e::;.hly.

f\et•Jl Anderoen(l) "P.TeSei1t~~11. tbe reeul.ts of hia e~pe:i:·~.ntg; ~..r:1:t..~

ff!i€1tJtions t"'einf orced 'i.11 th ! inch sqtlare rods iz1 e~eh. cort1.t:n: ,1 !~1:!~..t·e

cr'C~~e sec:tione reinforced wi1:;h !I it.ieh eq~e f~r:ner !!'etls f"...nd 5/52 inch

~©tm~. 1;.:tes SpB;eed a:I:. e:tx 1.nehes ai1ril at three inchef~ 1 and beam .. q of

r,~q_~,1tl'.'"t'~z' (;l:t>OB~ sectiQn re~.l1.forced "1:1 th ~ inch ~qi:ae.re corner t'OOS ood

~/;'2 1nch !."Om1d for.ty .. f"ive degree spirals spaced at two ftnd. fl:>i~ ·(;.f.l,r<;%11

14

effective type of reinfor~nt.

In Andersen's second series of tests in pure torsion, which he

complerted in 1937~2 ) he teated t-wenty .. f'our specimens of rec~

cross section. ae used unreWoreed spectmene, specimens with wly

-,;a inch corner rode, and apecimens with -;/a inch oonier rod@ plus

f'orty•f'ive desree spira.l.8. W.s cor1clusions were that the modulus of

elasticity in shear of concrete ie e. d1.rect function of the modulus of

elasticity in compresaicmJ am that corner rods and rspiral re1nforce"'

ment do oot influence the elastic torsional rigidity of rec~

to torsion fail near the cen·ter of the longest side, due to high

tensile stresses :ln that region.. Be found that the square was the

rtOSt efficient rectangul.m' cross eeotion to resist toreion&l st.reases,

and that the addition oi corner rod& did U:ttle to raise the member 0 s

one fourth of one per cent ot the con~rete volt.i!DS could be relied upon

to increue the torsional atrength of concrete if' it we:re eeeurel,y

fastened t-o the comei" rods. Andersen also stated that the spiral

excess cf the ultimate tensile streragth of the concrete.

JI. i:,, Cowe.tl( 6' 1,8,.9 >10) has done a considerable amount of WOi'k

with reinforced eoncrete subject to pure torsion and c':Omb:lned bendii.11

am torsion. By using the general theory of tcreion developed by

15

M ~ Wl>"'2ttoml l?e~i~tm.'lee- of J.':llaUl eoncreti& c -C>< >!;'~ a b:-J'11e:t4'b:tli(;: fW.£<.ttion of' d/b

(2)

~~ ~ oo'.led &~~'nt due to s.t·iirrup '.tf?;illfox>~ilis

b 11 • vidtb of reiru:?orc!"l'Jg ~~

!\., >II:! Cl'OS~ seet1G-in a?>ett ,tJf one .Nu-

r .. .,, 111.i allowablEJ ateol st17es~

s !$ stiri"lt:p s:pacint~

~ :;::; a h.-roeroolie function of d'/be

con{:rete subject ta WI"i$ion Cem~n sssumee fW.l pla.stiei t'/ in tlie

seetio~. ~e tn= ret'er~ to ~~t.d, ( 16) mw ·bae shO'.m the toI•si~

b2 ( b ) 'M-.=-· d·::I' 8 -~9 2 ,

Since pw:'e tors:ton causes ecmpresa1~..re and teno11e etreoeeo 11ma.erieall;y

equal to the shear stress~ the ~unl sllear ~tretilG for eainc:rete 13

l:!.id:t:~d ·by :.l:ts maxiluum terwile strength. Rqw.lti011 (a) for the ecnt:ri-

bution cf l:'eiru•orcement to the ·torsio..'"lal ~t :.ta still valid. There ..

tore the ultimate torsional ~t for a rectangul.ai• :section ot• rein•

fo1:ced eonerete is given by the Gt.D of £q~tions (a) and (;}). For a

plain coom:'ete bG&::ll on~ Eq~t:1on ( 3) should be uood.

In 1957 a. c. Er~t(ll) oondu.cted a se:ries of expezdmentG to

rr

Ar = total area of' lonf,"1 tudinal steel plaeed a;ymmet<>~ica:L'cy .:i.1 "'iYi ti'.iirl th.e i~ririb.e:i:7 of the i•einforc~ ~"!.>::

f Y ~ yield str~ss of the tr~~V~'l"~e ties

ltv and l~ ~~ faet~s wtd.ch. when multiplied by b~ ~-00 'tu reap.~ctively_, :1:'1:z:p:i."~sent the leve:i~ &~ of the f()J!"i:,;(;!

in ter-ms of the :reini~Ol:'cir..g c~-e dim.ensions.

ll"!W.de • Sl11e:e ky b • repref!!lent!:l! th.e lever a.rm of the force dwelO!leti .it\

the ti~n a,,~.·ooe the top (er bott(l:11) of the beam~ 5.?).d 11_i t~ th~ l~'l.rer

that th.~ ti;.;~~ (kv + kh} 1rould seld,034 be much l.ta.l'ser ·Uw.l'.l 2.oD. If :tt

t-reit"e ~ t:11e..11 the lever '9.r:rs ct th~· forces in tlW transverse tiee would be

le.L·g~l.. 1:J:m.n the Mm0nllicns cf the membe:r ' ta:i:.er tests by s~{ 19)

tlte ~"!iluWl value ot (!ty. + kb) would be 2.00, letting eacll coe:l!'!ici~t

.;;.qual 1.00. llrnet's cor....puted values of (ky + ~h) 'U'Wr'.f ttom l.e, to

1 f'\t:. .. ;r;,.,.")\)

19

'thc.t t..'1\$ levei~ arm that dete:t"1llines the point, of' rotation of the f\.'):t."eefl

p&·esent in the trans·vers,., steel should lie within the member ..

Frofessor Hen$ Gesu:r.d(.12) of the UZ11versity of Kentucky ~ei

directed experimenwl investigatj.ons :L.1 !Hn"C; tol"aion a.nd hes al,lilo

l!'iade ms ~.nsly-tical study of "'~he problem ('?)Sund f'omd t.he f@i:l~m.Qe

mo~mnism ®h'CMn in Fig. i .. 1 to be valid) r.:md a~oo this me~hani~ w develo1;, a failure the~i'.'y @£1 ~;~>ll.ott.$:

Ttie conei?ete is &fj.Gumed to be cra.cl~ed &."id offers nt'> tensiJ.e

resists.nee to the torq.u.e, When the <X">ncrete c1"ackei 1 there ia a r~uis ...

tributio..~ of strea~e~ and the steel reinfo~cement t@.k'.es up tn~ load

&crO~:il ··the cre,ck. The resistance offered by tr..e rt.~inf'orcing ·cage ie

a.~numed "'oo be provided by four different actions: the tensile forces

in the ti<EG; the dowel a'1:tic:ra of the ties 2 the dowel action of 'the

lo:igitudinal steel1 and the b'endilli of the tiea !lc1~ose the hinge

{Pig. 2•1). If the c~as:tve ~e fOl<>mS oo top of the beW!l aoo s

is the l!l-"a.CUlg of tb.e ties) the number of eteel our.f'a.ce~ e~i-poood by the

fsilllI'e surface will be

2 dl + bl:. (6) s

l1her.e a1 and b1 at"e the depth end width of the :t"eiuf'o?"eing ~e. The

toi;..ia.l t~rque !"es:tsted by the t~a.nsverse ties in tension will 'be

m .11. f 2d1 b'il . A ,,, m h. 2 b11 ih A .1.""' ;;;': ... ~... 'I.?"'- - ......,,. •!> ~i.,.. :r,.r+ ~ ... = ""''" .,, . u .~1 f. ,.. ~ "" .Y·~ e 2 "" ,;, "' s s s yr,, (7)

where As is the area of one foou> of the stirrup and f Y"~ the yield st~es6

o:~ ·!;he ~teel .

The ~esis·tsnc~ due to dowel action of ·tne ties ia ~re ~ifficult

., I I I

"'1 l1 c;I 4.I

J' I I

Long S-curve

Figure 2-1

20

-; I Ji~{-Cl.I ·~---~ JI! ~--------'

Medium $-curve

·-b,--___...1

F = M 'Jt

dy'4

Median Tie

Short S-9urve

E.x:ample of Gesund • s Dowel Actio11

J,ength of the S is difficult to determine 1 since the cr&ck could cross

degrees~ and there a.re several stresse'Ci ties, the average length or the

s~cm .. ll'e would be ~ 1111 assuming tha·i the hi• formed on the top or

b()'tt.om fooe of the be-..... !Jhe hendilli resistance of the bm: ~be

~lng closer to the yield irane.nt eince the detormatiorJ.S are ~. If

·the beem eont.a:I.us enly oorner h.<\l"'s as longitudinal re:b1fo:rcement s tbe

Ttd=r\rt11~·~=6~~ l->41 4 s s

(8)

were ~ is the yield moment of the tie.

T'ne long! tudinal reinforcement e.lso provides torsioneJ. t•es~.et ...

e...!ilce by dowel action., ~is cu be handled in the same manner as the

dor<1el action of the ties. 1be length of the s-aurve in this case will

be one h8lf the spacing o:f the ties. Each long! tudinal bar is afi'eeted

action will be

where ~~, is the yield ~nt or the longitudinal bar, and r is the

perpemicula:t' ~istance fr"'..mi the bar to the a:d.s of i-otation ~

(9)

~e last component to be considered as resist:tns the applied

·torque is the bending of the ties a.bout the hinge (Fig. 2-1). 4lbe

i.r.ootai =

(10)

a~ As f;yt + 6 "1t ~ + t;: 2 ~ • "'+ M,,t • 2_1\+ bJ. (11)

ing torque ~ tbst both the longitudinal and transverse eteel must

yield before ultima:te f'ailm>e can take place. :tt would mee.m that there

e'!Ould be eond!tione of f'ailure '6here all the steel doeo not ylcld, as

tra.nDve"!l"Se steel. ia not f'ully developed;

2) only the trmssverse steel yields and the s't:t."'ensth of the

lo~.tudinal ate111l is not fu.\l.y dev-elopec1;

;) the eonerete first fails in sh9tn~ between the tiem; or

~\) the COllcrete first erttahea 1.~.A ccmip1•easion,

Suell failures wci.ald not be correctly evaluated by EqimtiOA'.IS ( 5)

and ( ll) . Whether or not t...1iese equations could be TiiOdif!ed t..o include

other com-11tiona of failure will require ~e study.

23

!II.. Er.ImmmN'l:rJ\ti W0!1K AND RE1~'iJ!}lS

A. E>;;perimenta.l. Woi~k .

~l:nex'0! were eigh'i:i beama teeted in tM .. s series, Each bee.m We$

fi. 3 ... 6\(• lo:ng1 m:th & ;ae -6° leng test t~ecticm in the middle' Jl-.J.l of the

beams hS!l t.he same croes secrt:lou ( 8" x er ) t m1d the: s~ omQtll'l't t;)f

longitudinal steel ( 4 Mo. 3 bru?s, oue be.r in ea.ell eor.n.eit). !flto

epaeirag of tra.nsverse reinforcing tiea was varied for ea~u v.a;ir of

beams; NQ. :; gitirrups were ple,eed at 2~t _, 4" and 6" c.pacing in each of'

three pm's ~ · no stirrups '"1el'e 'IASE.'d in a fourth ~r ( 1:7able l) ,

All the reinforeing 1!10.e AS*IM .. A - ;05 deformed hax>d ISf'sdi.1':

steel I\'fo. 3 'b-2.re, and woo obtained tram a local s.upply' eomp>.:>JlY. * ,,'4.

r~presentiative 6S!fiple of the reinforcing steel ims tested in ter.sion~

emd the yteld stt'"ength wu four.id to be app1,o:>:ime.tel.y 50,000 psi.

inie conerete used in these teete woo obtained from a local

ready~Et!.:r.: pla.nt** and contained the following quazrtities per cubic yard:

Portls.nd Cement (Type I) 522 lbs.

Se.:nd 1522 lbs.

No. 5'! orusb.ed stone 1508 lbs . (Maximum size 5/4'1 )

Water 374 lbs. (approx.}

t-Jat.er was added in tile mi>t~·truck, A relt.~:tively stiff m.1.x w-oo obt.sined

with a sloor.19 of less th.an 2" for the beams of group l, and ~ 31.1.liiap of

..._.......,. ..... y_~------------------~-·.-· ... '!> • "''l -•$$"*4

ti-Valley Steel Corpora:tion, Salein, Virginia.

**&acksburg Block and Supply C~ 1 Blacksbut>g, Virgirda.

TABLE l

Properties of 'l'est Beams

BFAM ~IE SP.A.CIMO AGE NJ.I 'mST CONCRETE l«JOOLUS OF (Inches) (Days} CYLINDER ELMmCITY

STRENGTH (PSI) (PSl)

1-l 2 58 6501 5<~5 x io6

i ... 2 4 !t.o 6,0l 5.,; x ir.P l-3 6 54 6301 5.;5 x io6

l .. 4 No Ties 51 6,0l 5.35 x 106

2-1 2 62 5320 J~.84 x io6

2 .. 2 4 55 5320 ,,..

4.84 x 10° ...

2 .. , 6 59 5320 4 . 84 >t lfl;;

2-4 No Ties 69 5,20 4.84 x 106

All reinforceme:nt consists of No. 3 bars.

All beams have 4 tit). 3 bare as longitudinal. reinforcement (one in ca.eh co:r.-ner) •

e,ve1"~gw f'o:r. t."aoh :pom". Xndi·ridwl test values vaeied +619 psi 1

·~640 p~:t. fl .. ~ the average. Table l gives the properties of each be~:wa

*Ba.JAwin-tim ... R&liilton Corpoi·ation, Eleotronics Di'itis:ion~ i.'1&:.ltlwm, *-~sachuootte.

SR-li. T9Pe FAB•25-l2 electrical resistance strain gages were

ueed to measure the strains in the reinforcing steel. These are wire

gages mounted in a bs.kel1 te bo<ly. Before each gage was mounted on a

bar, the bar deformations over a small area were removed using a hand

file~ and the area was polished using a fine emery e:loth. This

polished a.ree was only slightly larger than the gage itself, and the

change in bar diameter and cross sectional area was negligible (the

reduction in area was less than o. 7~). After the gage was mounted

and the cement had set for twenty-four hom. .. s, the lead-out wires were

soldered to the gage leads. The entire installation was then water-

proofed by brushing hot wa:{ on and around the gage until the gage, bar

and soldered connections were covered sufficiently to prevent any

moisture from penetrating a.nd causing a short circuit.. A minimlllll of'

bar area was vaxed. to reduce the effect of loss or bond streng'l~h.

Figures 3•2 a.nd ;-; show the gage before and after waxing.

SR-4 Type AR-2 45° strain rosettes were used to measure the

concrete strains simultaneously in three directions. The concrete

surface was rubbed smooth with a rubbing stone and dusted thoroughly

before the gages were applied. The rosettes ca..~ be seen near the

middle of the beams in the photogroapbs or the failure surfaces (eee

Fig. 3-38, for example).

The strain gages uaed had the :following ehara.cterist:Lcs:

Type Gage Resistance cement Fae tor (Ohms) Used

FAB-25-12 2.1; 120 Epoxy-150

AR-2 2.06 119.5 Epoxy ... 150

The location of t.he strain gages on the i·eintorcing steel ie

shown in Figures 3 ... 5 ·(;hrougb. 7-12. These figures are d:te.l-.<n so that

the right side of the f'igU!'e cc>rresponds to the movable head on the

t.est:Ulg me.chil'l.e . The :ma.chine head rotates clockwise when viewed from

the right end..

The beams were teated in a Tinius-Olsen gear .. dri vEm beS11'!

balance torsion :iilat~h.:tne with a rated capacity of (>O~OOO inch-pounds.

The maximum load applied dui"ing te£lti.r1g wae 76,000 !neh-pomds, l'"epre ..

senting s. tlienty-~even per" cent overload. The machi.:r~e ptill"fomed

satisfactorily during all the tests. Speed.al t?h.ueks were fabricated

to hold the beams in the ms.chine heads. These can be seen in Figure

,_4. ~i'he lJea.vn ends "rere placed i:t'l -the chuck~ (which were one fourth

inch lat>ger than the 'beam's cross aeetiontil dimensions), and ple.ste~

t::if PG.r.is was :packed a.round the etYle of the warn to pi·event th~: irreg-

ulszi ties of the chuck. fr1.')::n causing a tress concantre,tions.

The strain indicating equipment consiste(\ of two Baldwin Gage

Selectors w:tth twelve channels ea.ell, a."ld two Baldwin Strain Ind1,cators

(Model M). The strains iv-ez·c~ recorded to the nearest micro-inch per

inch.

.At the beginning of' ea.ch test the strain indicators wc~re

e.djusted usiri.g the potentiometers on the twelve channel gage selectors

to place the initial gage readings in the mid-range of t.he equipment.

A small initial torque of a.pprmdmately 100 inch .. po·.mds was applied,

initial gage readings were set and the torque then increased. in incre-

ments of approximately 5 » 000 inch .. pounds until fa:f.lure. Strain gage

AU G • 63

• AU G • 63

• AU G • 63

£9 • ~ n'1

31

readings ware recorded after each torque increment was applied. Af'ter

:ra.:t1w.~e 7 edd:itionaJ. torque was applied to develop the crack pattern

fully for visual study.

'ltlree persons assiete(i in conducting the tests. Tile ·testing

machine was operated manue.lly, ma.king it easier to WJ.intai11 the proper

loading increment. One person. operated the machine, and t.hen read and

reeorded the steel strains, 'l'he second person balanced the torque

indicator arm, and observed and mark.ed the crack patterns 0 Y1h1le the

third person read and recorded the concrete stra.ine. In ad.di ti,)n the

first and second persona also measured the rotation at each end of the

beam.

EL Results,

Beam 1-2 was the firet. specimen tested. The failure was

charact.eri~ed by numerous cracks forming continuousl,y throughout "the

test~ with no single crack predominating or widening as the torque

increased. Failure wa.e not sudden. As torque increased,, a level was

finally reached where the beam would no longer reeiet additional torque.

Eac.h of the other seven beams failed. -veey suddenly with little

W&"ning. Prior to fail.w:-e fev if atzy' large cracks had formed. A fey

ser:oncls before failure there was noticeable creep, a.coompanied by

audible popping. The failure mechaninm was sim.i.lar to that shown in

Figure 1-2. However, the comproes1ve hinge was not noticeable until

considerable t.wist was applied.

The result.s of the t.esta are given in Figures , .. 13 through 3 ... J~5.

each f igurt?. The steel stra.L"'lS increased ra.vi<Uy with increased twist

Where the strains extend be-yond. the limts of the gi.•aph, th.is is

indicated b".r an arrO".i, 11'he slo;pe of the a:r1·ow has no sign.if i<::ance.

Init.isJ. strains due t-o the dead weig:.1.t bend.L"lg were i:1ot reccn.:dcd and

are :rJ.ot i.."'lcluded on the plots of data.

each point. The principal tensile str;::iJ.n was computed by applying the

following equation involving these strains:

e :$: :rna.ximum tensile ~·train

ex and rz:Y = strains at :right ar..gles

eg :::e sh.earing st.rain computed from eg == 2 e450 - (ex+ ey)

'.fhere e450 :: 'the st:r.·t:dn !~5° f:t'Oill e~~ ~'1.d ey

T!~e a~gle of the principal tensile strain ( e ) was found using the

(12)

(l;)

'I'h<i m1g.te of the. p:i:incipal tensile strain versus the torque is

33

beam~ with the predominating e:rack shown as a heavy line and the cracks

of lesser. magnitude as broken lines ..

The total angular diepl.acement of' t.he test beams va.~ <inea.aured

and is plotted on Figure 3 .. 29. No data were obtained. for Beam i .. 2.

Data t"or Bee.ma 1-1, l•3; l .. 4 o.nll 2-2 \tere obtained from t.he im.licator

seale on the torsion machine whieh measured the rotation of one ~~

obtained by measuring the def.lectiona of horizontal extension arms

att~ched near tne ends Qf the beams.

During the t.est of Beam 2-2 :, it lffl.5 notieed t.he.t the pl.a.star

of Parie filler in the end chucks was crumbling and the rotation data

f'roni the machine indicat.or \i.ter<.~ considered invalid. Tb.erefore the

system of extension arms was used for the last three beame tested.

An analysis of the predicted rotation using St.-Venant.'e

\,. i principle ( 20 ) indicates that only the data for the last three bea.tr1s

can be eonaidered valid. Jl.Gtn:iming elastic action and neglecting the

ef£ect oi' the reint'orcing on the oodulus of elast1ci t;y and ·tne moment

of inertia of the section, St.-Vena.nt•s principle can be stated:

e :Cl tota,l angular displacement of. the beam

~. = appli.ed torq.ue \.1

t == lJe?...m lenf_lth

~ = a constant dependent upon b/e

'b c longer si.de dimension

c • shorter side d:!JnerlSion

G .:: Bhea:d.ng r,oo.ulvs ~ . __ J __ 2(l + µ)

E o modulus of elasticity

fa. ca Poisson's ratio

For these test beams:

M,. = applied torque "

la = 48a between extension arms

,l( I'll 0.17

G = 2,ot:-0,000 psi

ot' the beam up to tn.e e!•acking load. lt is also evident that th.ere

we.s considerable slip in the end chucks ae the plaster of Paris

yielded under the p~essures of the applied torque.

llllllllllllil!I 'I'op

11111111~1 H H H I . I Side

111111tf,IU1111 :Belt tom

H H H 1~1111.,11111 Side

~ - Hosette Gage

I I J~. I i 1 { ·j ,, " I Side

Bottom.

Side

~,- Rosette Gage

Figure ?-6 Strain Gage Locations, Beam 1-2

I [ I :: I ·j I

Side

I t· I I J I Side

~- Rosette Gage

----/?"'-------a.I

J_· ---4 ~~--

____ ,,"_ ---

Side

----!ta''-~--

I

Bot.t;om.

----161---......

•

Side

~ - Rose·tte Gl'.l.ge

3 Strain Gage Locations, Beam i .. l+

l 11111111:1 1111111

Top

1111111:1:11 ·l H H I Side

1111 i [ HJ 1111111 · I Bottom

H H f ,.11 1:1111.1111 I Side

~ - Rosette Gage

Figui•e .3-9 St.rain Gage I.ocat:.!.ons, ~wn 2-1

40

Top

Side

Bottom

Side

~ - Rosette Gage

Figure 3-10 Strain Gage Locations, Beam 2-2

ill

[ I I I -, J _, I I ·-------<

.I I:; I ·' I -7 l < l · -I

Side

\_j I f <.14 I I I

Bottom

I I f • I ~: I I I ~ "" Rosette Gage

..5-LL St.rain Locations_, Beam

~ I

/'9 II ----9ol

3

3

i-----17"------

... ~

Top

-- I

Side

Side

~ .. Hosette

Figure 3-1.~l Strain Gage I;ccations, Bt'"am

60

MAX

-r~. Tut '7ao. -~~~~~~-~~~~~~~~~~-=-:::::-:---::::;:;o.---="'-~

0 ·--/. t:1 l<> ~,<!? 1'"u NtoP, ~ - ~/·;. ..---! .e •

,;t;--- ./ 0 ·,.- • f 1.,'i:l[O

60 ," \ T,, tr.17

--..,.,£.,.--',-.-A

00 µ.. •rl ~'<1

I

{j r:J

H 40 i:.:: 11/.JrJ Gage No. S;ywbol •rl

C,)

~~' 1 -x-;::! -o'

ji /~' 2 :... - -a.-~ 3 - -o-

4 - -c:i-

~o Ill~ <:; ,, - ---7 - -1-

I I' I I I I I I I I I I O o 100 ~oc ~oo 400 -?<:>o

Strain in Micro-inches per Inch

Figure 3 Torque - Strain Plot1 Beam 1-1

tll Pl '" ~

I .d 0 s:: H

~ or!

(j) g. ~

~

eo

60

40

;:o

MAXL .---1\\-;~IO . ---~--;4."iO _,,--~.?6 - _....,..e:;_____,.- ~ -- ~y-/lo 1'34 '11:1. • I ·;--'i .,,~

/ • • .::l--"' - - \I~,,. • / ... . ·--=::--fiJ ~-"'" -~· ~· . r ~")~ / ,_...,._./

•-}IS-• 1 ·'--' ~~ 4 '• ---=-=-·-=--!! T::. '!!'~ To811 qi. l 7 " ·"" 1· r ' ._. .,/ / ,,..,,.. ,,

i ~I I /~ /,,, $. ·/ i1- i i ~/·,..,, I 'Q. .

l" .. = ,,. I :. j /./' . Jiii/ . . . ;_.s-&-11 = . I !II :~!fl iil my =

Gage No.

6 8 9

10 11

Symbol

- --11--·

.- --I.I'--- -~-

-·--=--12 - -ljJ-

o ·400 ~o~ 0 100 zoo. 300

Strain in Micro-inches per Inch

Figure 3-14 Torque - Strain Plot, Beam 1-1

60 /\11\X+~·-rrc=· '!!i;, - ::.. .<" .6 • - • - - - ;;>z D ===:_!":_.:========:.:::::==--='O:;;:::::==;:;:-~====::::!:::,.. ·~· .... ______ .

40 tQ Pl

<M .!i:l

I

ti ~ H

i:: I 111 / Gage No. Symbol "f1

~ l ~! 1 - -x-d 2 -'=" H - -A-

~ ~o I .. I V"

3 - -o-4 - -o-5 - ---6 - ----

0 0 JOO ROO 300 400 ~00

Strain in Micro-inches per Inch

Figure 3-15 Torque - Strain Plot, Beam 1-2

60

40 f;') p,

·..-! r"1

I ,;'.1 C) i::

H

I:: •rl

~ zy 1-1 g ~o

Ir 'I~ '-a.

i I ~

0 I t 0 100 ~00 aoo

Stre.in in Micro-inches per Inch

Gage No.

7 8 9

10 11 12

400

Figure 3- Torque "" Strain Plot, Beam l-2

Symbol

- --1--

- --11--

- --\n--

- -~-- -qi-

~00

en P; -rl .!4

I ..C! t.l s:::

H

;:: .,-I

<lJ ::1 C" f.., a

80

-:i. 10 i

. - .. '" cs t I ---ll~· . .........._ I ./ . . / • ~' . . I •-" __..~----

MAX.- •....-.....,_ ...

60

40

20

0

--• I ! i- - To/690

/ . i ./ _,...,-./ --------- 0 .---- • :_::::::.r· - ~ ""• s / /" .~:..---0~ ./------- ! ! / . /_/. ./ ....------"~ ""' .. """' \ < ./0 /~~·

i-~· VlT06i£_O

s I I

°"' ·.°"' ' ' /., ~-------\ : irff//. \ w1 -·

i .. y :V-0 10 ZO ao

Strain in Micro-inches per Inch

Gage No.

1 2 3 4 5 6 7 8

40

Figure 3-17 Torque - Strain Plot, Beam 1-3

Symbol

- -x-- --A--- --o--

~ - -c-·. -

-·--1--- --V\--

fQ

..,,.. -~

60

1"11'.X.

"1-0 , .. , p.,

•rl ~ ! r' ... ~t () ,:; H ,~

H •1-i

(J) ;;:$ U' ;.. ~ 20

0 0

----:~~:< / \ / i

b 4 ~ 0 I \ / I

r/ /_,/' /•+ I ) I . \ ! +I /.

..., '(/ 0 I /I /

1· i i 0/. /0 / l /

1· i I /. <> -'< 17 0

/ I I/ I . . . o / I/ I + "7. 0

/ I/ i / /'/

0 ~ t> 0

I I/ I ~-~___:---·

GI 1.~_:'D

L I I I I

0

10 '20 30 40

Strain in Micro-:i.nches per Inch

~

Gage No. .Symbol

l - --x--2 - -A-

/). :Z, - --o--.-'

4 - -o-

~

I I t I ?o 60

7 ) - Strain Plot, Beam 1-4

.t:~

(J)

60

40 w p.,

•Fl -~

I .c 0 Q H

~ .,..;

(.\) ,., ,... o• "" 20

0 -10() 0 leJO

St.rain in Micro-inches per Inch

fi~igure 3 Torque - Strain Plot, Beam 2~1

I>

Gas.e No,,

l 2 ~ ;'

4

' 6

Symbol

- -- x--- ·--A-·- -·-o-~ --o--.

20<:1

60

40 ) l, -i ~ I

;.~

~ .., H

k ·.-1

0 ::l o~

i ... t.:; l!tO

0

~?

MAX !sf~ it==- - r- ~o i Ir r..o l.s~- f

j//'j/~/$ ~&7151. Ul -

1 .· . /. -.I l ~{ -------=-. \

~

~ sZ. ~/ - I

~l \ i ~ /' 4 ~./.

. , \\ ~

~"\ \ ~~. ~

I/ lit 1\ / / iii "--..~ ~. - s.-e.

-100 0

Strain in Micro--inches per Inch 100

Figure 3-20 Torque - Strain Plot, Beam:2-l

7 8 9

10 11 1.2

No .

- -1--- Defective - --V'I--

- -qi-- --=--- -!(l-

,:;oo

60

lll p, ·ri :...: 40 ; .!:I () ~

H $:l ·rl

()) ~:3 d ~I a

20

0 -100

-9::>~- ---

• -<! ----------

\"" q~

- 3"0

.\ \ .\

v\ \

MAXI ,..,

a~ ,-_37 ,.. .. _ u-11 J. ">'-/":;~.-----. .

I I

I

/i/11~-/ . . . .., --...... I I • ', f 0 ' / <'>/ ~ '' - ...... ./ I I / "--.._~ /JI/ ~

.. . I f,11·

11· iii/

01

~ I Ql-1,ge :No. Symbol

\ ! .\ \ \\m

\ !~ Q' I

0

l 2 3 4 5 6

- --x--~ --A--

- --o--- --0--

·so Strain in Micro-inches per Inch

l''igure 5-21 Torque - Strain Plot,. Beam 2·~2

0 z t'- co ()\ CJ ,---....;; (\J

r""I r+ r·l

0 .,,.

0 <\(

I ~~--.--~----=- -===~-·--+==- 1 --t11 o ~ --·--lll--·--<11--·"-"'-· ~

{\J !

(\j

~J Ql LO ,..,

e•< .Ll t.~ +" ~=-1 r) ,..., i ... p,, Q) Cl:; r!

•rl f.t,). il.l ~lJ _!J "'~~· (J t!J ,~

"' '"r"ll • 0

j .. ; () -d ~?~

f:! ,,, :;:;; ·rl •\! )..,, +' i/2

60

!fl p, 'IO •Fl ~

I

~ ..,-!

~

°' l-o 0 E--t

20

0

r./I -2 ~ • mAX. ~__,..,..·\·-._....._a --- • • ' . .!_ . . <:? .~ 1~8 /;;, \ ..___ - ~v-~ --__,___ ~-

~ 6 0 \ \\ ~ / \ / . / .. ----------. ' . . ! ./\ -------------"" <: I I I/ "~ /. '?""' f /0 I I l;o " · . I.I X, i f /;· .

~\ 11! . . . . .. . .

4

. ~ ti. ~ ____-· -~

. I C> I I ...._V) .~

\ \\I r-------j [! ~I \ l ill I ~~(> \ ·:11/ J' ----r · l lf . ---------- /Tl \I/,/ .. ~ oo.~" _.,.

. --,.

I -::-

/ Gage No, Symbol.

\'"._J!

l - Defective 2 - -c..-;; - -o-.--'

l+ - -CJ--

5 - ----6 - ------7 - Defective I

8 - --II-

9 - -CJ>-

I I _ _j -?O n ?O 100 1';0

Strain in Micro-inches per Inch

Figure 3 Torque - Stra:tn Plot, Beam 2-3

I© p..

•rl .!>!

; ;:::

l!,,) >:'l t4

~ -r-l

Q) :::; cl i...,

60

MAX.-

~o

0 E-i 20

0

- 4a • 40 ·- l\

0 ~~ ."" I <>

\ 0

----- 11 '7

• ------ i- ---- • /--- .

/+.

I

~

)(

Ai ~/ i +

\ <J

0 .' ~,\\ \ I 1 l

1 ;1 •. " .. • I

'I-

Gage No. Symbol

1 ~ -x-2 - -A-J - -o-h - -c:J-

<'.l 100 ~cc

Strain :.!.n Micro-ini::hes per Inch

Figure 3 Torque - Strain Plot, Beam 2-4

70

60

s:l ·@ 50

~ rn v .-! ,,,.,, ~ 40

E~ .-! ~. 'F! ('.) I'.'.: 30 .,-,

~ i'.i-! 0 () th zo s:: '• <'~

10

0

0

'>')

--¥ Beam 1=1 Rosette 1 - -- x -Rosette: 2 - - o -

Beam 1-2 Hosette 1 - -o-Rose·tte 2 .. - ru -Hosette 3 - -- - --

I ~--- .. ,

-------- ~·i C'l -='~--f M 1-e

__.......--- 0 =- I MAX $EA

, ;:::::>" --·~ :- ''-_ I

' / ,, ___ , - 0 • --- . ' ·-·----.,~·-----a ·-o '+- ./ • ____ ' ~

10 l.O

Figure j / of~

30

:!.n

\ : 'f- I \ .,,._, ·-ic-.-- I

I I

MAX._1 .e, E P./VI /- I

4-o ~o 60 70

•rensile Stn:d.n <rerfiHHJ 'l'o:rque

Beams 1=1 and 1··2

70

60

Ci ""'.I 50 <'!;! ;:..,; +:' en tJ r-! ,,.., ~ 40 .... v 8 r-! ~

<rl u >:l lO ,,.., H

i:,\.;

(;.~

0 di

..-! zo b!J n ·d!

10

0 0

Bea'll 1-'.) Rosette l - -x-

2 - --c--

Bea:m l~·~· Hosette 1 - -- o --Rosette 2 - -- tv --

Rosette :3

K I

----

' x ·--x --·~'r 0 --·-- Q ::::0~-!.- -e; : ><t--·-- I ·-() ·- ()--·-"' () ·----Clf--0

){ I

I I i·-- <=>--.! -o'::::::j I I f'1A><

/'-rv ,_. N--·~ /·-rv-: i BEAM/-:?. ~ ~ ~ I /\ / ~ / I MAX J!,EP..M 1-4 . I

\

\

10 zo JO 40 ;o 60' 70

'I'orque:· in Inch=kips

Figure 3 of Pr:l.nctpal 'l'ensile Strain versus 1l'orque

Bt~v .. ms and 1-~

7P

60

i:.1 •d 50 ~1}

~ lf.l

v r-1 •.-! !IJ 40 Cl v 8 rl ti.'! P,1

•.-! \.) Q 30 Ti

k ~ 0

• ii)

'"iii 20

·~

/0

0

0

')

Beam 2-1 Rosette l - -- x --HoGette 2 - -- a --

Beam 2-c: Rosette l - -- rv -Hosette 2 - - --

I -i -"j. I

I

• - ~ ------ •-.._ ----- • : I ll --- - \.I I ·-.........__. <" ~-- -·-"' -:J;.'(j -.:::::::..-0, _ ~o "'-· , , ----1--...'V~.. O?'-<. -- .. ~"-'--·- : : MA'>< 2-1 ·--------- ·-._. -....::::::::: ·7 "' ' : E>.E,>.M --------------.

10 '20 so 40

Torque in

~o

MA..X BEAM u__:

I

I

60

Figure ~J-27 Angle of Principal 'I'ensile Strain versus Torque

Beams 2-1 and ;2~·2

70

70

60

\'.1 .,~ ~o ~n i.,, +' w. tJ rl y~

w 4-0 ~ ~)

8

30 ~ri

H p~'

\,"~

0 ti r·I 20 t1f)

9 ~.

10

0 0

-.........__'1, --.._____

·-~ --· -- "'---; ·---- 1\1

Beam 2~3 Hosett.e 1 - -- x --Hosette 2 .. -- o --

Beam 2~4. Rosette l - -- "'--

~' ~·-o.i-·

)( ---·--N-; .......... ·-)<•-· ""-.. ----" ' ~ ' ---- M>.X: ~: ~EA.MI '-...::! j2· 4 I : -1 ' I I MA"><

!~<!.-3. t-

10 ao .30 40 rn 60 70

Torque

Figure ') Angle of Principal 'feneile Strain versus Torque

Beams 2-3 and 2-4

-II'-

-4'-

-0-

·-c--v-·-

tr-0 ~-(?, C'.-2

0

0

Oc

O'E

Of,

Ol

i-·3 0 ··i

1Cl t:.: m

~·-1 ;::! () :::J' I

;>;;' f,•·

!-(j r11

' ' I \

' \ \

\

',

\

\

\ \

'.

' \ \

,, \

' ' '

\ \

'

\ ', \

'

\ y

\

' ., ' \

\ \

\

\

\ \

\

\ \

\ I \

\

\

\

\

I \

' '

\

' '

\ \

\ \

' \ \

\ \

Side

\ \

\ \

\ \

\

Bottom

Si.de

'

3- Failure Cracks,; Beam 1-- 1

.... , \

\ \

\

,,.,.., o •.

Top

Side

Bott.om

Side

Figure 3-31 Failure Cracks, Beam 1-2

I

SI

St de

Figure 3- Pai:)..ure Cracks, Bem:rt 1

I~ .·.I

I ·:s; . I Side

.s: .· Bott.om

Side

) ... Failure Cracks,, Beam

Bottom

Slde

F:tgure )-- Failure Cracks 1 Beam 2-1

' ,

Top

\ I

Side

qottoru

Slde

Failw:e Cracks, Beam 2-~.?

' '

[

Figure 3-

'.I'op

,,.,,,.--- ....... \ ,, I I \

Side

Side

~I

\,J I ' I' \ I

' ........... , ' '

\

Failure Cracks, Beam ;::-5

6'7 ~ I

I.~ Side

k~.----------,-------"-7 Bottom

Side

68

JUL 63

Figure 3 ... 38 Fai ure Cr acks, Be

JUL 63

Figure 3-39 Failure r acks, Beam 1-1

£9 • 9 nv

£9 9 nv

70

AUG • 63

Failure Cracks, Beam 2·2

AUG • 63

Figure 3-43 Failure Cracks, Be 2-2

£9 !> mt

AUG • 63

1

72

lV. DISCUSSION

Although the ~J objective of the program of which this

thesis ie a part is the practical analysis of a. reinforced ~o~rete

beam subjected to combined bending and torsion, the purpose of this

thesis wee to renew and enel'll:l; bot.lit experimente.U.v and ene.!ytioally,

the invest:i.ga:tion of' the torsional resistance only. In preVious

theoretieaJ.. and experimental. work dealing With reinforced concrete

beams in torsion two conf'lieting assumptions have been presented. One,

that at the ultimate torque onq the steel resists the tensile stresses,

with both the steel and concrete resisting the compressive stresses; arld

'tWo, that both the concrete, in a plastic state, and the steel resist

both t!le tensile and comprese1ve stresses induced. For this thesis

eight l.>eatliS were tested a..'lld the data evaluated to dete111Td.ne if there

would be definite agre~ut with either assumption.

Seven of the eight beams tested failed sUddenly with little

or no warning. Only two (1-l, 2•1) of the seven beams shoved evidenee

of cracking before the ultimate torque was reached. On these beams

the cracks were bai.rline craclts, and although with increased torque

the length ot the cracks increase~,. the widths did not appear to

increase until failure. At ultimate load &'lld failure in the other

five bee.ms, the concrete seemed to era.ck along one failure zone instan-

taneously 1 leaving only the steel :reinforeinl to :resist the applied.

torque. :tn no ease was the steel able to resist wq additional torque,

nor to hold the maximum torque; and the beam failed.

73

center in the o·the:r· beams. ~t.1.e f'a:tlu:r:e su:i:'f'aee of five of the o·thel:"

seven beams was sim:U&· to that i•e;poi··t.ed p1~eviousl.y by Covan ( "(), . ' ( )

l~rnst\'ll) and Gesund"J.2 (Fie;. 1-2L with a l!lp:'lral teJ:l.FJion c:r.•a.cili

eases see:raed to be a. h;y'brid :t"ailure ·' W'i th the concY.'ete te:nd:lng t~:.i :tW.l

in tension at an angle gx•eater tha..'"l 1~5° 1 rather than at !~5° as eAJieX'"'

able i:a :Seam 1-1~ which is heavily reinforced (ties e.t 2" spacin,~),

with its relatively short failure zone. Ernst(ll) predicted a f'W.lure

o:f this eort for heavily reinforeed bee.ms.

Beam 2•1 (ties at ~r spacing).• which vas theoretically stronger

than Beam 2-2 (ties e.t 411 spacing) becau.~e of closer tie spacing, failed

near.one end at a lower ultimate torque. This was probably due to the

laek of extra reinforcement near the end where the chuck induced local

stresses which were probably very high. Thie was the only bemn that did

Although these tests were designed as pure torsion tests; the?e

was some dead load bending momen·t pt"eeent. At failure tile n10ment/tm:que

ratio was abou-t. o.olt. Apparent4' th.is had little if any ef'f'ect; st

least it did not induce the compression binge to form on the top L~

each test. Four of the beams had a compreaaive hinge form on a side

(l-3, 1-4, 2~1 1 2·3) and two on the bottom (2·2, E-4). There was r..o

distinguishable OO!.lp:ress:tve h.tnge on the other °t'lfO beams (1 ... 1_. l..,2).

In nor.ma.lly re1nt·orced bemna subjected to combined bending and torsion,

t,_ne bending moment al.ways ca.uses the hinge to form on the top compreo-

T.ue steel strains in these beams remained sma.11 (up t..o 50 .. ,;wo meJ:>o-inchee par inch) until a reletively large torque ca.used the first

crack to appear. In no case was there a. general ;yieldit1g of the rein-

toreement. On the contrary, there were only a few isolated C"ases where

yield strains at the ultimate torque. Both ~s were near a predom ...

ill.-"m~ cl"ack. It is very likeJ.y that .some of the strains read from.

where ~;ield st:r.•ains were recorded were in B-""'B.t!l 2-2, on one top longi ..

tud.inal bar and on one stirrup, and in Beam 2·3, on one stirrup.. ~te

yield strains were not i .. ecorded until a.f.ter the ultimate torque was

The only beam that. did 11ot e.xhibi t a sudden type f'ailtu-e was

'{6

l.?! thin th<: range of' eiq,-iervoontal 01"rort• arul in.di1;.a.te onl.y that the

etra.i:ns m:e very srraJ.l,

i'ose·<;te :!) on Benm 1.J.~ showed u. tendency for the princi1w..l

st!'ain to :i.wUeate i:rw:r1eased lcmgj,tudinaJ. tenoion nt n. torq.u.e of

a.bout l!i inch .. ld;ps. This c0t.tl.(l have been clue to a f'aJ.se start of. a.

compress:ton h1.nge on the oppos~:l:;e fr:i,ce"

As a. m~a.ns: of con11iari~cm: Tabl.e ?- was prepru~eo. giv~.ng th.c

thoo:retica1 ul:t:br.ate i;c'!:eq,_,ue$ <;>.(:ifdpu.ted by methods suggeated; by three

invcst:igai..'.{):i!'S 1 (l),nd tl1e ·!:;&is·~ vmJ:i,w ult:i.m.~<:.e and cra~king i;orq.u.e. In

E!'rist~s ulti:mste atr.®r:igth fo:i:m.i.Ua the factor ;k1,r y it.a,) was ta\!:,en at

2 . O .. ar3 ar1igg~!eited tin Cha10 ... te1" !.'.'\: o $mSt ~ s eqi~t.ion for the bea,,'!JS ~4:l th

no ·t.::te&i :!,$ ·r;he sa.Tt:i.e as t,he elast.:lc eqti.<:l.tion suggested by Cowa..~.' but

:m1•r!st ~:tsen the ul:t.imat.e tensile str{".ngth of the concrete in~te&l of the

allowable elastic srtrengtb. The :part of' Ccr.:an~s ultimate t..orque

fomm.f.!.a p~·edict:tn13 the toi·que cavricd by the reinforcing steel elQoo

:ta also g:tven (I!~q.uat:lon 2).

'fi1e eat,:l!nated c11 ncldng torque for ·the test. beams was obtained

by aas•~mtir.ig that t.he eon~rc·te ct•ac~kell -when the pr:l:ncipa.l. t.cnaile

strains reaeh.ed a value of 8.J?l:JJ~oxirr:ately 100 m.tcro-inchcs per inch.

The cracki11.$ torque repo:t>tt:;~J. was the Gllk"lllest torque at which e.n;r of

th~ :wr:tnc:lpe.1 't.e:neile 3tt•ai:ru3 reached n value of 100 :micro ... in~hes per

inch in tensiono

It 1;1as expee:t~d tf'l.v;t t.he maY.:iunrm Gt.rain would occur on the

oenterl:tnse,. An e~raluation of ·the strain data shOW!$d a. random dis-

t:dbuti~n for -t.he location Qf a.na~1.mum strain 011 those beams where

;LABI.11 ,.~

Ccm't1arinon or• 'J~iOU$ '.i1'heoi:·e.ti~cJ. l~"adic·t1ons \;1.J.t.:.h 2ea·i; Hesu:Lts

(All 1lalu.es in Inch..,li:ipo.)

1}~3llifi f} r.!J.~ ~,..,,__,. !_~,. \"""'"·"''"'"·:.t· ,:;-~,,.~~ t<·:l •'>(l") ¥ • ..:~·~~ ,.- ....... ""°{,.J

1 .. 1 (ft' l 1-2 (4'1 )

1-:5 (61•)

1-4 {llc t:tez:.)

2-l (::;;::•)

<'} -"' (.4"'' ) ~:.,;.;

~ 3 (f:tt) ~- '.:.il

2-4 {!@o t.:1CEt$)

CDi~i:.!PB :r:;r.._t\STIC ~~tfilMIJ',~* ·~· . P c~, D(i~ ...,.,'!~· ~J

·72.0

5'7°2 5":; ~ --~ ..

9.3

68g6

53.8 48 :::i e.\..t

8.6

!i'REDlCTIC:SS C~J.<(J!.'N'l'S CGM1~'S tm£E\!lt\TE gr. W.Jl'lMAT'E: :-6'0m.{m:.AH·

Eq.(2

82.5 168.6

I 41.2 I 127.:5

I 27,5 I 113.6 ...... .,. 86.1

C2.5 1 55 .. - .J.

41.2 113.8

ft7~5 l-JO.l

-·--- 72.6

*USillG 1963 A.Cl Buildir.e; Code 'V;l.lue~

EmlSl''S aESm~:o·s ULTIMATE UIEll{L~_;

Fen~L1'Lt"\** 1~'011'.t<i:JL.'l*~

5 ~1~l ll}

66.o+ ii:~~~;·:) ~ i.

... 6 fl 0 ., •. I lO'.j.(i

65.o 'fO.O

5}.7 ..,_._....,.,

66.o+ I 2.l()ol

I 66.o+ I lO~.O

66 .. o I

70,.0

45.4

TEG'T R.E:CUL'l'$ 1;smt1A'll.ID (J:i.M;fiW\1 'l"Ut.~UZ

54.Ji.

55.0 49 (\ . ::; 47 ... 6

5'7·7 44~6

49.J.;.

50)f

I

I

UL&lltlfi~

'lU;::t~UE

···c ,.,.o 58.9 66.(s

55.0 -

64,.7

6'7 .·7

~t .. 8

5.4.1

With 6'tir·rups: .t~ n 5 -ff'e ; .f s : 18,v..i'.:J psi,; f 1 e gt;-0\,\P 1 ~ 6y0l :psi.; Z' e grou;f1 2 = 5:120 1~s1

With.ou:'~ trtirrrms: :P = 1.1 ~ - max e *'*i't ult = .. oe t' e ; fO' yield = 50,000 p:d . .; ky + }3n = i2~0

V:etw:ic eq,uation modified fol' less than minim't.Un longitudinal steel. see text, p .. ·rit

-3 ~

cnmput!Hl fol: t.he 100 filic.~'0-.. ir.tch t1er :.tncti at;;.•.r1ii:lii en ·tbe ~s ofl the

1.:eu'ucc~.u~s.

An e)tmdn!liotion oi' ~~'U.bl~\ 2 shows sr.;;vora.l interestir~ thl.:ttr.fl:

l) There is qui t.e a 1.Jide range in th€ pt•edi~ted utt:Un!i:iie

cr~pa.i~iti.ea ±'or every beam.

2) Meet. p-cedlcted -.U.timate capa.citiea are great;.cr thim. the

e.ctu.~l, '~:,;:;:·L "·E,.:;__,_;~Eil; -whic.~h ~JS 'l'JU\t t-'he pr.-etiir;·tion equation.9 -i;fcn.'.ld.

h~ve be~n dfl.\1gBrot.1s to use f:'or ·t."»ie desisn ot theoo bQamB.

3) l~xcept for the belUS 11itb no tics (l-4~ 2 .. 4), Cawan'i

l#Orldng cnpaci ties do not prcwide &l t'!l!ie~oote sate·b'y far.:·tor esa:tnst

the "i.<est fail.tu.·*~ ''orO.ll..1es Qf these btJ.aJD3·

'*) '£be use of tieu i!lcreased ·~e ui·~imate cai•ci 1-y of '(;be t.~st

b:::!a.1~ a.bout t.we:irty per cent .•

5) '!'h~ i'J:timai~e capa.ci·~ies ot' tne· teat 'beanw ie::-tceed~ ·tne

CJ?&.uld.ng ev.:pa.ci ties by e w:lde 11a.;\~e, from seven ~r cent ·to i:i:tty .. o!'le

l)tllf.' Cf,Ult ..

6) Cowan9 s oltizuate <mpa.ci.ty for th·e l"eim'o1'ci11g ~rt.eel alOtl!:.l

givctt a. reu:mable p1~ed:lc1.;ion fol." the t11Jst l.)CG.."11.S~ if it :ts ~S&'i.liill.x1 t~..at

the cont::ibutio.n or the con\'.!1•et.e is less as the quant.i t:y of ~ ti@a

1fl increas£:d •

:tlm four longltud1:llal bare used m~mt tM: miniiilunl 1-eqw.rementa of

Ex•n.!.\it.'s Eqt!3.ti011 (4) only ir.1 ·t;j,"Wse beams (1 .. :;, 2 .. ;) with the stir.tupe

spaced et s:b:; inch.es. .4.c:cm·dtng to J.u•nat, in test ~.:ms J. .. 1, l ... 2, 2-1

79

Md. a, .. 2 the longi:tudina.1 stt.'el wottld have yield.f;'d v causing tl1c beams

to f'a.il t.ei'orc the clo$er EiptAeed. st.trrups could develop jl'i.e.lti 6$~:-~&tms.

lt tJ:.i.e ai·ee. of longi tud.inal re:bn'oroement is leas than tl'ia't r.(..>q1ured by

Equai:.:ton i-4-, the torque ca)t'll!tc1'ty can be computed by assuming -tb.at the

.mood.mum t'or-ee thail ca:t1 be d~ve!Qped in "Gile stirrups is pr-o}IQ;w·ci~

(Equation i'+) to tlw &~?a of the lcngi tw:tinal rcini'o1•cement, pro~.tled.,

~hei•erort'! Beauis i .. 1~ i ... a, 2-1 &"ld 2 ... 2 would c.®:rr"<J the same uJ:titrAte

torque or ti.6.o inch ... kiii>~.

Oilly llemn z ... 2 rrei.rl ;yield strains recordett in th<.1 longitudi1".!Sl

s.tcel.~ W.tho~ ·cne)."e could have been local yiel.din-.g: that was no·t

picked il'!? 'by th.e gages, Beam 1 ... 1.,,. with less t.ba.n the u.in:tmum lo~ ...

tv..dinal. 1rtf>)el f\>1• stirrups spaced at r.vo inch.es; had a subst~ltiai.1.y

higher W.tima:te torque ~ any llthe:r beam ·testedp which would support

Gesund'e or Cowan•e predictions rather than .Ernst's.

A further atudy of the test data showa that the pi•esence of

$tittupu inc1·eases the toJ.~que ee.pa~ity i but t.ha't 't;he concrerw ts al.SO

eff'e~ti;;e 1:n resist.tng t.he torque a'a ultimate ecttxlitiona .• since thi;;

beQrJ:W wi'th higher strength cona.t'e·t.e failed genera.Uy st td.{#l.er torque

ca.paeitiee,. This e.ction ot• tlle concrete was Pal"tic~ul&·ly evideJ;Jt: in

t,h~ bettmG tl".at :failed r.mddenly and did not develop visi'bl"~ eracl~

uni;il tht.~ ultimate torque wu renci1ed.

~ne t.ht,,>orieli presented :i.n Ch~pter ll ~.?.oncerning reinforcei.\

concret.e belC!W.$ subjectt.>d to torsion assume an :ideal t's.ilur(> am·:e&ee ot'

lJ,5-::~ cracks on three i'acel.l t:Uld a compresai'V'e hinge on the fo\.ll."th ±'ace,

*l"'hie is based on the fat.~t that 'the 1+;0 plane if! tlle pri:rmiP0l :plane

"f1~ --

H.1

~l q C!'.)l·JCLUE.J:Ol.;1~S

In ge~ral the era.cl.;: patte-rns and failure surfaces th~t

oc~urred in these tests were similar to those previously reported 'by

Cowan('?)~ Ernst(ll)" Gesund(l2) and Shflm(l9). Five of the 'be&ns had s

failure surfa<!e simil.ar to tl'la.t shown in Figure 1-2. The i.."WO bemns.

(l-l, 2-l) ·with the stirrups spaced at 2" had a hybrid shear-tensior.'l

type of failure as predicted by Errust(ll). The eighth beam (l-2);. with

stirr~ spaced a.t 4n .~ as well as Bem l·l, bad a unique cJrnck po.tt,,ez•n

in t-ha,t apparently no compressive hinge fol'lne(l. After the ultimate

torque was rea.<:lled in ~'t'Gm i ... 2~ adi.litional torque was applied 1.n

hopes at forcing a noticeable hinge to form, but the beam developed

~ e~cks in the s~~'lirel. :pat.tern instead. This beam was the yoUtlgeSt

beam ·test.ed ( t'orty days as camps.red to the others at e.bout fiftY·f;iX

days)') and the concrete seemed to crack easily and uniformly, transf'er·

ring the applied torque to the steel. The other seven beams, as des ...

cribed !n Chapter IV, failed suddenly, and in some casea did not. develop

visib.le cracks until the ult:il11ltte torque was reached.

A comparison or theoretical ~,red1eticms with the test. results

{Table 2) indicates that a111peren·t~v no sin.~le formula. ie valid for

all the test beams,; howeve1,0, Ernst• s equation a.a U.mi ted by ·the

l®gi:ttrlinal ateel areE\ is r.easonably a~curate. C0tta.n • s elastic

eqtie:tions ;yield resnlt!'1 that P...re too cloae to the nltimate t~r.~oo~ for

the mf:.mbero to be oons,.(lererl a~f.e., TM.a suhett.-mt1a.tes e. s:L'llJ:J..lar eon ...

eluei">n Drevio,.tsly reached. b~r Shem< l9) , Howe,rer ,~ t.his is o~ly true of

·those bemr.s that contained. s·tirrups. The 'lw..ams ·~t had no etirru,~a

had. an ultimate torque about five t:i.mas Cowan' ts ia•edicted elastic

tol~que; whieb WO't!ld U.ldicate that Ci thGr the allowable st~eSS used WU

too ccm.Gervative, or that Ca.mu ts equation J1i8¥ not be valid, or tbat

he did not g:tve credit to dat1el aetion of tllc longitudinal steel.

Qesuud' e ultima:te torque equation was relatively accura'te ;i.n

predic.-ting tile carJB.ei ty tor the beams W1 th stirrups spaced at 6" • But

the beams with stin .. ups at 2•r and 4" had an ult1mte torque cal.eulated

f'rom bo"'im Ccwa?J. • s ar.d Ocsund' s equations much greater than the ·tea·t.

ultimate torque. This may be expla.1ued by the hybrid shear-temsion

type fa:U.ure observed in Beams l•l mad 2-2 instea.U of a tensile type

failure on 'fhich "the equat:b,ms are based.

There was no significant iuf'ormation pined as to the bend:lng

moment induced in a beam by the action or stresses in the etif'rups.

It was hoped that the bending moment eoUld be measured by measurine

the atrair..s in the lo11gi.t.udinaJ. steel, but ·the differences be·twe~n

~ne;1 tudir.al strain• on opposite faces of the beams wei•e too inaon ...

siste~t to develop a usable ec;.ua.tion. In none of the tefJts of beams

vith stil·rupe did the c~ack pattern develop over ·the 61,ea wn~e the

st:eain sagoa were pl.act-.."-'J so tha't u&e.tul data could be obtained,.

l. And~"l"Se?J., Paul. 11E:qierimeute with Concre·te in rl'Ol"S1on, u ~ansaetions of the }\merican Sopieti of ,Civil ~~eer~P "il'olwne 100 it 1935 ii :op. 9li'9 .. 983 ·

2. rulde1rsen, Pat:J., 1'Rec"'Ga.ngular Concrete Sections Unde:t• Torsion.,~ ~o~~ie_C?i' the American Concttete Inst:i.t~, Volu:oo 34, 1938,1 pp. l-lL

; , })Oa;;>,; I" B. ° Combined Bending and Torsion in Reir.iforc.1cd Coneretr~ Beams. 11 ~'hes is for the degree of Master oi" Sd(;;':lce .• Virginia 1\':Jlytcc:hnic Institute, 1962.

4. B.t•etljlc'!:;.~: Borifh 0 Diseui:t.sion of a Paper by !!em"'J J. C<1i.Yi,'J.J:1/1

i?roeeedint)! o:f the American Concrete Institute, Volume 56~ i959-1/)t;o,. PP· '1;a9 ... 14oo. · · -

r .. ~. ~.,,/h<:tv ,,,,.., . t• '."> ''l~'l "'·'m'·'te K.·ii--.... ¢1.,. "-f' "-"r::.r-+"°~"'"1'''' 1~·1• f'!""'l" """°'t"" r ..ou'"'-... ~.l,c .. .,,,;,a.,., ':;Jr; ~.'·.-; ~l--·~•J..,,g1_6l-.f.I 1.,,1~.!.."<,;;' ...... ~-·), v- 4)-~.,-~~~ ... ~~ .. ~"'·-~~·;..,.~..-~~ --

Beams wi·th ar.ii vitbout Tran$verse Reinforcing 1n Combined Beril:lr.Jg :mrl To:t'i:d.on" 11 Thea is t cy1.: the degr.ee <:Jf Maste1• o±" Sc:tence, University of Kentucky, 1962 ..

6. Cowan.~ f£. J. 11 iJeed.g:n of Beams Sul)ject to Torsion Rel:a.ted t;o the N<1w A.11r;;1;r.aJ.ia:n Code, 11 Proceedill.:1S cf t.,'l;.e i'Jr.ericsn Concrete Imltitute, Volume' 56, i§59:-i.'§OO, pp. 591.:bir.-

7. Cowan, H. J, "An Elastic Theory for the Toraiona.l St-cengt.h of Rect.;i;i.ngular Reinto:t•ced Concrete BeQl!!S," M~i.~ine Cl~ Concret! .l!esea.r~,,. Volume 2, Number 1~, July, l95<i;"i?'i). :;-8.

8. CowRn, H, ,y. 0 The Strength of Plain, Reinforced and. Pre$·!Jresse1.t Concrete UtW.er the Action of Cor.ibined fl'\~~ecses, with Pa.rtieul.ar Reference to Combined Bending and 'X-orsion fjf H•1c·ts.r.;zular Sectiol'l!:$ /' ~ zine of Concrete Hese11rc:!!_, Vol'!1l'l1E' 51 lfumber 14, December, 19531 pp. 5 •

9. Cowan, H. J. 0 Toreion ot a Reetangula.'r EJ.a.stic :tsot.rop:tc ~am. Rein.forced w·ith Recta.."lgulru." I!f:1lices of Anoth\:!r ~~1.ter .. iaJ., 0 AJmlied Sg,1~.ntific Research, Volume 3, 1951•19531 l'P, ;i~.!t:§li'.8 • .

10. Cowan, E. .J. a.n.d P.rmatl"Oilg, S. ''Exper:L.""le~ts on the Strength of Re1-'l'lf'orced ~~ld. PJ.9estreaeed concrete Deams and of.• Cont~t·et<"; .. :zncasec1 Steel Joists !n Co-!llb1ned Bend:tr.tg ar.d !orsion.,:' wine or~pon~~e Research, Volume 6, Humber J.9, ~b.1•cn, 195;:;i 2 pp, ;t··;~O.

ll. Ernst, G. c. 'fUltime:te 'I'<>rsione.l Properties of Rectangular' Reinf01•ced Concrete Beams," fr.oeeed~ o~ -~~ A.1ller:i.c!!! Con~rete Institute, Volume 54, 1957=195S, pp. ;41 ... 35fj ..

l.2. Gesund, Rans. ('Ultimate Strength of Reinforced Conm"ete Bea.ms in Combined Torsion11 P,.ending a'!l<i Shear,." Unpublished paper of Professor Gesund, University of Kentuel"..y1 1960~

13. Gra.ssam.~ N. ~. 3. "Ex-perime:nts on Conorete Un<ler Combined Beu.ding ~ Torsion, 0 Ppoce~i§S of the Inst~tut:lon fjf ~iv:l.l Ene!nee1"s, Volume 5, Number 21 March, 195(), pp. iJ9 .. 1~5~

14. Lessig, N. ff. uneterm:tnation of the Load•Bearing Capacity of Reinforced concrete Elements with Rectangular Cross Section Subjeeted to Flexure with Torsion," ~o~~d*!gs, £2mrrete c.nd Reinf..2.t2!.~J!9ncrete .J;nstit:,t!t.e, Volume ; , ~ Moscow, 1959. ~"lSl.ated from Russian by ).1a.?'garet Coroin, Foreign Literature Study Number -,·n,..)

15.. ?k9L"1all, W. T. 11The Torsional Resistance of Plastic Ma:teriale with Special Reference to Concrete, ° Concrete ~ Constructional; E£ne~riSI, Volume '9, Numoe:r 4; ~, pp. B:;-SS.

16. I~adai, Ar.Pad. Theolt ef' Flow· and il'aeture or Solids, Volume l, Second' Ed-ii.on, McGraw ... llill Boolt Com:pe,ny, 1950, w. );,.90 ... 5u.

17. Pandit, G, s. 11u"lt1mate Strength of Reinforced Concrete in '.i:o:rsion," g~n~ ~.£.O!!cre~.J July...September, 1962.

18. Schuette, F. J. •tultiiiate Strength of Square Concrete Beams with Tra.nsveree Reinforcing in COll.lbined Bending and Torsion. 11 Thesis for the degree of Ma&ter of Scient!e 1

Unive~sity of Kentuck-y, 1.962.

19. Sh.am, E. Q,. •rulti.mate St.rength Theory of TorGion Applied to Re:t.ntoreed Coneret..e :seams.If Thesis for the degt"ee of Master of Science: Umveraity of Kentucky, l96o.

2<). Timcshenko, Stephen. §!r~ ,O,f Material!, Part ll, Third Edition, 'D. Van Noetrand Company, 1956, pp. 237 -25!1, ~

21.. Young, c. R •. , SGger, w. L. and Hughes; c. ,A.. 11Tors:t.01ml Strength of Rectangular Sectiono of concrete 0 Plain snd Reinforced,, 11 The American Architect and the Architectura.1. !evi~:; Volmue 124, Ninb...~ 2425, August 11 19231 pp. 12'!..:126.

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Willim1i Aurm'ltl Stuar·~ !l

A'.BSTRA.CT

Ei&bt reinf'or{!OO. concrete beams 'ltf,.th t"'wo different comere-i:-..e

strengths a.nrl tour different. sehemes or re!nt'orc1r1g were teated tc1

f.'a;ll .. '1.U"e in pvl'e to-rsion. SR-.11. st:re..:tn ~es were uaed to measure the

steel strains at selEwted r;oints., while stra~..n rosettt?.fl wm~e used to

IDenaure the 1'!.oncrete strains at the sm:·fa.ce of the test bean~.

A r.eviow of ar.al.'ltica.1. and e~r:Wental studies in the lit~r ..

a:tu:re is prescm.t.ed, and bri.efl.y cU£.cussed. The test reaulta were then

compmted w:t.t'b thE;! tbeoretii:'!al pre-dictions developed by C0'\1an, JZl:'nst

and Gesu.ntl.

In general the fai.lure surfa.ee of the ·test. beams agreed with

those previously predicted. The t~st nltime:te torq'l.ie va~ied co:ns1d.er-

s.'bJ.y frotr. the predicted ultimate torques or Cmran and Ge1.mn,1.. Ernstis

basic equation uas modifie...4. to compensate for le..ss than th~ mnim1:rm

longit1Jdinal reint'orcement in four of tbe beams. With this 1J10d:tf:lca-

tion Ernst.•s theoret.ica,l predict.ions were very close to the test results.

A study of' tbe test res'l:u ts also shmis that one of th~ :p;i.~csented theories is valid for one particular case~ but that none of the theories

is valid for t.~ei-y ease. consequently :J there is a nead fo:t• further

research in this subject in order to develop a more exact tn.eoi-y for

use in design.