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,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 [email protected]'.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~
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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,..)
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Unive~sity of Kentuck-y, 1.962.
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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.