M e m b e r A m e ric a n C o n c r e te In s titu te
S Y N O P S I S
T he results of th e tests of 44 beams of reinforced concrete w ith stirrups welded to th e longitudinal reinforcem ent are presented. T he beam s were designed in such a way as to produce failure by diagonal tension. Vari
ables including th e size an d inclination of th e stirrups, type of concrete and ratio of longitudinal reinforcem ent are studied. A comparison of the stren g th of welded stirrups w ith th a t of loose stirrups, as reported from form er tests on web reinforcem ent, is attem pted.
1 — IN T R O D U C T IO N
T h e use of w elded stirru p s as w eb rein fo rcem en t for concrete beam s, in s te a d of th e c o n v en tio n al loose U -stirru p s, should p ro v id e im p ro v ed co n stru c tio n , since i t gives p erfect c o n tin u ity to th e reinforcing steel e m b e d d ed in con crete a n d fa c ilita te s a c c u ra te steel-settin g . I ts p ra c tic a l use in u n its fo rm ed b y lo n g itu d in a l b a rs a n d stirru p s w elded to g e th e r h as been su g g ested as econom ically possible. J
T w o m ain q u estio n s arise in connection w ith th e p ro b lem : (a) W ill b eam s c o n ta in in g stirru p s rig id ly connected to th e m ain re in fo rcem en t d evelop hig h er sh e arin g stre n g th s th a n th o se u sing com m on loose stirru p s?
(b) C a n th e h ig h er th e o re tic a l efficiency of c e rta in inclined stirru p s be u tilized th ro u g h th e use of w elded u n its?
A fa v o ra b le a n sw er to th e se qu estio n s is im p o rta n t, since th e econom ical use of w elded stirru p s w ill d e p en d to a g re a t e x te n t u p o n th e a d v a n ta g e s to be secured b y th e use of such w eb reinforcem ent.
* S u b m itte d to th e I n s t i t u t e M a rc h 31, 1945.
f G r a d u a te F ello w in C iv il E n g in e e rin g , U n iv e rs ity of Illin o is.
J D . M . M c C a in , “ W e ld ed S h e a r R e in fo rc in g fo r C o n c re te B e a m s ,” C iv il E n g in eerin g , J u ly , 1939.
(141)
P ro b ab le a d v a n ta g e s in s tre n g th due to w elded s tirru p s lie in im proved anchorage of th e stirru p s, p re v e n tio n of slip p in g of in clin ed stirru p s along th e m ain bars a n d im p ro v ed anch o rag e of th e m a in bars.
T h e a d v a n ta g e to be secured b y th e use of c e rta in in clin ed stirru p s is in d icated th eo retically , as show n b y th e follow ing s u m m a ry of th e usual analysis of web stresses. In such analysis, it is co m m o n to assu m e th a t th e ac tio n of a reinforced concrete beam m ay be lik en ed to t h a t of a tru ss in w hich th e to p chord is form ed b y th e com pression zone of th e concrete, th e b o tto m chord b y th e lo n g itu d in al re in fo rcem en t, th e te n sio n web m em bers b y th e stirru p s, an d th e com pression w eb m em b ers b y p o rtio n s of th e concrete web of th e beam . T h e a ssu m p tio n is also m ad e t h a t th e com pression web m em bers are inclined a t 45 deg. to th e axis of th e beam . T h e analysis shows t h a t th e s tre n g th of s tirru p s h a v in g a giv en volum e of m e ta l p er u n it volum e of concrete d ep en d s on th e in c lin a tio n of th e stirru p s w ith resp ect to th e axis of th e b eam .* Also a cco rd in g to the analysis, v e rtical stirru p s a n d stirru p s a t 45 deg. are e q u a lly strong.
T h e m axim um stre n g th is given b y stirru p s in clin ed a t 67.5 deg. If the s tre n g th of th e b eam w ith v e rtic a l s tirru p s or s tirru p s in clin ed a t 45 deg. is ta k e n as u n ity , th e an aly sis show s for s tirru p s in clin ed a t 67.5 deg. a b eam s tre n g th a b o u t 20 p e r ce n t g re ate r. T h e in c lin a tio n to be given to th e w elded stirru p s should be th e one t h a t gives th e m axim um stre n g th .
T his p a p e r co n tain s th e resu lts of th e te s ts of 44 b e a m s of reinforced concrete, loaded a t th e th ird p o in ts w ith a sp an of 8 ft., a n d designed to produce d iagonal ten sio n failure. T h e te s ts w ere in te n d e d to give results show ing th e s tre n g th of w elded stirru p s an d to p o in t o u t w h ich is the b e st in clin atio n , w ith resp ect to th e axis of th e b eam , to be given to those stirru p s. T h e follow ing v ariab les w ere in tro d u c e d in th e in v e stig a tio n :
(a) ty p e of concrete, (b) ra tio of w eb rein fo rc em en t, (c) in c lin a tio n of th e stirru p s, a n d (d) ra tio of m a in re in fo rcem en t, w h ich w as v a rie d in a su p p le m e n ta ry series of te s ts on four beam s. T h ese fo u r b eam s are tre a te d se p a ra te ly th ro u g h o u t th is p a p e r a n d are re ferred to as Series la . A com parison is also m ade w ith fo rm er te s ts on loose stirru p s.
N o ta tio n
T h e follow ing n o ta tio n is used, p a rtic u la rly in th e h e a d in g s of T ables 1 to 3:
f c — average compressive strength of 6 by 12-in. concrete control cylinders (3 for each beam)
fy = yield point stress of web reinforcem ent A ,
V ~ , , — steel ratio of longitudinal tension reinforcem ent
B u lle tin N a Y e e .’ In v e s tig a tio n of W eb S tresses in R e in fo rc e d C o n c re te B e a m s ," U n iv e rs ity of Illinois.
_ A , _ .
r ab ~ ra ^10 we^ rein f°rcen^ent, where A v is the area of the two single stirrups
k
a t a cross-section, a is th e stirrup spacing, measured norm al to th e direction of th e stirrup, and b is the w idth of the beam
= ratio of the distance from the neutral axis to th e top of the beam, to the effective depth d
= (sin a + cos a) sin a = 1.0 when a = 45 deg. or 90 deg., and 1.20 when a
= 67)4 deg.
a = the inclination of the stirrups to the axis of th e beam Pu = u ltim ate load on beam
Fc = vertical shear (one-half the load on the beam) a t which diagonal cracks were first noted
V u = vertical shear (one-half th e ultim ate load) a t ultim ate load
V v = vertical shear (one-half of the load on the beam) a t which yield point stress was reached in web reinforcement
_ V ° u ■ ■
Vc ~ bjd ~ s*learing stress a t which diagonal cracks were first noted.
. •
= rr~, = shearing stress a t ultim ate load bjd
V v , ■
Vy — ^ = shearing stress corresponding to vertical shear, V v Vu calc = shearing stress vu, calculated from Eq. 8.
vy calc = shearing stress vy, calculated from Eq. 6.
2 — DESCRIPTION O F TESTS
R e la tiv e ly few of th e specim ens heretofore te s te d to d eterm in e th e s tr e n g th of w eb rein fo rc em en t in beam s of n o rm al dim ensions h av e h a d a defin ite diag o n al ten sio n failure. I n itia l failure b y b o n d has p lay ed a n im p o rta n t p a r t in th e b e h av io r of th o se beam s, resu ltin g in a failure d u e to th e sim u ltan eo u s effect of b o n d a n d d iagonal tension. M a n y of th e m h a v e failed in ten sio n , p o in tin g o u t th e necessity of using a fairly larg e a m o u n t of ten sio n steel, w hen th e web rein fo rcem en t is of som e im p o rta n c e , in o rd er to av o id th is k in d of failure.
T h e specim ens used in th is in v e stig a tio n w ere designed to produce a tr u e d iag o n al te n sio n or shearing failure an d to avoid as fa r as possible all o th e r influences. C are w as ta k e n to e lim in ate all v ariab les except th o se being stu d ied . T h e beam s w ere all 10 ft. long, 5.5 in. wide, and 21 in. deep. F o r ty of th e b ea m s w ere reinforced lo n g itu d in ally w ith fo u r 1-in. sq u a re b a rs p laced in tw o layers, w ith an effective d e p th of 18.25 in.
T h e o th e r fo u r b eam s w ere reinforced w ith tw o 1-in. sq u a re bars, w ith an effective d e p th of 19.5 in.
T h re e series of b ea m s w ere te s te d : tw o groups, each of tw e n ty beam s, w ere id e n tic a l ex cep t for th e ty p e of concrete. In Series 1, M ix N o. 1 (1:3.34:5.0, b y w eight, w ith a w a te r-cem en t ra tio of 0.75 b y w eight) w as
T A B L E 1— O U T L IN E O F B E A M TESTS
No Stirrups 2 '8 "
-( c
= ■■" ' n- n T
I
— 3 > Î
hI'-O"]. 8'-0" J . O
-Verticol Stirrups | , M 6 /j
S±2 - r
g>
-5" Gage Lines on both fa c e s
Fig. 1— G eneral dimensions, details o f reinforcement and location o f strain gage lines of beams.
Fig. 2— V ie w o f the reinforcement of beams, (a to p ) U n it w ith vertical stirrups, (b center) U n it with 45 deg. stirrups, (c b ottom ) Com plete reinforcement o f a beam with 67.5 cleg, stir
rups.
in clin a tio n ; D , 45 deg. in clin a tio n . T h e fra c tio n re p re se n ts th e d ia m e te r of th e s tirru p s used. F o r exam ple.
1-N in d ic a te s: Series 1, b eam w ith o u t w eb rein fo rcem en t.
2-D Y i in d ic a te s: Series 2, 3^-in. s tirru p s a t 45 deg. in clin atio n . 1-1 3 /8 in d ic a te s: Series 1, 3 /8 -in . stirru p s a t 67.5 deg. in c lin atio n . la -V Y± in d ic a te s: Series la , Y - i n . v e rtic a l stirru p s.
T h e p h en o m e n a of th e te s ts show ed th e follow ing g eneral p ic tu re : U p to a lo ad of a b o u t 50,000 lb., co rresp o n d in g to a co m p u te d sh e a rin g stress of 300 psi, no diag o n al cracks w ere d eveloped. T h e stresses in th e stirru p s w ere v e ry sm all, th e d iag o n al te n sio n b ein g ta k e n u p b y th e concrete, w hich w as still uncrack ed .
t o r th e b e am s w ith o u t w eb rein fo rcem en t, a fte r th e d iagonal cracks s ta rte d , a m ain crack developed on each side of th e beam . T h ese cracks, w hich w ere inclined a t a b o u t 45 deg. to th e h o rizo n tal, s ta rte d in th e low er h a lf of th e b ea m a n d progressed u p w ard as th e ra te of deflection of th e b ea m increased. A t u ltim a te load, th e u p p e r p o rtio n of th e beam sh e ared off in th e com pression zone.
F o r th e beam s w ith web reinforcem ent, th e d iag o n al cracks s ta r te d in th e low er h alf of th e b eam as before. As th e load increased, th ese crack s ex te n d e d u p w ard , while new cracks t h a t s ta rte d a t th e b o tto m of th e beam s ex ten d ed v e rtica lly for a while, th e n tu rn e d a t a b o u t 45 deg. a n d ra n a p p ro x im a te ly parallel to th e o th e r d iagonal cracks. T h e n u m b e r of cracks developed an d th e d istan ce b etw een th e m was, on th e average, d e p e n d e n t on th e ra tio of w eb reinforcem ent. A t u ltim a te load, th e b eam s w ith Y - i n . stirru p s h a d a m ain diagonal crack on each side, a t each end, w ith rela tiv e ly few cracks p arallel to th ese. I n th e beam s w ith 3 /8 -in . stirru p s, th e diag o n al cracks w ere m ore n u m ero u s a n d th e d ista n c e b etw een th e m w as, on th e average, sm aller. All beam s w ith 3^-in. s tirru p s failed in com pression w hen th e stress in th e w eb re
in fo rcem en t h a d b a re ly reach ed th e yield p o in t, so t h a t th e d iagonal cracks w ere n o t fu lly d e v elo p ed ; how ever, a t u ltim a te load, th e n u m b e r of crack s w as generally g re a te r th a n for th e o th e r beam s.
I n general, alm o st all of th e d iag o n al cracks were well s ta rte d a t th e lo a d a t w hich th e w eb rein fo rc em en t w as stressed to th e yield p o in t.
In c re a sin g th e lo ad m erely opened up th ese cracks, ex ten d in g th e m u p w a rd u n til th e u ltim a te load w as reached b y shearin g off th e u p p e r p o rtio n of th e beam .
All th e b eam s failed b y d iag o n al te n sio n except th o se w ith Y~in.
stirru p s , in w hich failu re w as d u e to cru sh in g of th e concrete in com pres
sion, a n d th o se of Series l a w ith 3 /8 -in . stirru p s, in w hich failure w as d u e to th e sim u ltan eo u s effect of ten sio n in th e m ain steel a n d diagonal te n sio n . F o r all beam s t h a t failed b y d iagonal tension, failure occurred in a b o u t th e sam e w ay. One of th e tw o m ain cracks a t each side of th e specim en, w hich h a d ex ten d ed o v er a b o u t 7 /8 of th e d e p th , s ta rte d to open u p ra p id ly a t m id h e ig h t as th e stirru p s yielded, developing a t b o th ends u n til th e re m a in in g u n c ra c k e d concrete a t th e to p of th e diagonal c ra ck failed b y sh ear, causing a sliding of th e tw o p a rts of th e b eam along th e crack.
3— SERIES 1 A N D 2, A N A L Y S IS O F TESTS Experim ental d a ta
T a b le 2 gives p ro p e rtie s of th e m a te ria ls u sed in th e beam s, to g e th e r w ith th e p rin c ip a l re su lts of th e te sts. F ig. 3 gives average curves
show-TABLE 2—RESULTSAND ANALYSISOFTESTS (Two companion beams ofeachtype shown)
S train in Web R einforcem ent
Fig. 3— A v e ra g e 'shear-strain' curves fo r the web reinforcement. Series 1 and 2. a. K in stirrups, (b) % in. stirrups, (c) V2 in. stirrups.
150 JO U R N A L OF Tl—IE A M E R IC A N C O N C R E il. h u i n u i u iNuvemoer 1^3 in g th e re la tio n b e tw e e n v e rtic a l sh e a r a n d s tra in in th e w eb rein fo rce
m e n t for ea ch ty p e of b e am s te s te d . I n th e r ig h t h a n d m a rg in of each figure, th e sh e a rin g stre ss c o rresp o n d in g to th e v e rtic a l s h e a r (co m p u ted as ex p lain ed la te r) is show n. T h e p lo tte d p o in ts in th o se figures re p re se n t a c tu a l v a lu es fro m th e te s ts . I n p lo ttin g th e se cu rv es, a n d in an aly zin g th e re su lts, only a b o u t h a lf of th e m ea su re d s tra in s (th o se w h ich reach ed th e y ield p o in t first) w ere considered. T h e se w ere g en erally lo c a te d on gage lines across or n e a r th e p rin c ip a l d ia g o n a l cracks.
Position o f the neutral axis
F ro m th e flexural stra in s d e te rm in e d a t m id sp a n , th e p o sitio n of th e n e u tra l axis w as co m p u te d fo r ea ch lo ad in g . T h e v a lu es so d e term in ed , a n d co rresponding to th e la s t five in c re m e n ts of lo a d in g a t w hich s tra in m ea su re m e n ts w ere ta k e n , w ere a v e rag ed a n d are giv en in T a b le 2 for each beam . As can be seen from th is ta b le , k ran g es fro m 0.507 to 0.578 w ith a g eneral av erag e of 0.549. T h is figure co rresp o n d s p ra c tic a lly to th e v alu e of k co m p u te d for th e crack ed sectio n w ith n = 10. F ro m n — 10, th e v a lu e of j = 0.833 w as d e te rm in e d fo r th e cracked section a n d th is c o n s ta n t v alu e w as used to co m p u te th e sh e a rin g stresses for th e beam s of th e se tw o series.
I t is realized t h a t th is v a lu e of j m a y n o t be th e a c tu a l v alue a t th e o u te r th ird s of th e beam s, b u t i t is in a g re e m e n t w ith th e u su al p ra ctice in d e te rm in in g th e sh earin g stresses. F u rth e rm o re , it is n o t believed t h a t th e lever a rm a t th e o u te r th ird s is v e ry d ifferen t from t h a t a t th e c e n te r w hen d iag o n al crack s h a v e developed.
A v e ra g e shear a t the y ie ld p o in t stress in the web reinforcement
T h e v e rtic a l sh e a r a t w hich th e s tra in in th e w eb rein fo rcem en t was e q u a l to th e kno w n y ield p o in t s tra in for th e steel w as d e term in ed for each gage line. T h e sm a lle st e ig h t sh ears so selected w ere averaged.
T h is av erag e is a r b itra rily defined as th e “ a v erag e sh e a r a t w hich th e stre ss in th e w eb re in fo rce m en t re a c h e d th e y ield p o in t” a n d will be called V v. T h is sh e a r a n d th e co rresp o n d in g sh e a rin g stre ss h a v e been selected as th e basic v alu es for th e an aly sis of th e te s ts . T h is selection a n d th e co n sid eratio n of t h a t sh e a rin g stre ss as th e b asic v a lu e on w hich a perm issible stress sh o u ld b e b a se d is ju s t one crite rio n for t h a t p u r
pose. A lth o u g h all of th e b e am s t h a t failed b y d iag o n a l te n sio n to o k m ore lo ad th a n n ecessa ry to stre ss th e s tirru p s to th e y ield p o in t, th is e x tra lo a d w as ta k e n a t th e expense of a n excessive in c rease in w id th of th e d iag o n al cracks. I t re m ain s to b e d e te rm in e d w h e th e r a sh e a r V v su sta in e d in d efin itely w ould n o t p ro d u c e failu re of th e b eam .
M e th o d o f a n a lyzin g results
I n T a b le 2, th e an aly sis of th e re su lts of th e te s ts fo r th e se tw o series is p re se n te d . C olum ns 1 to 10 of th is ta b le give v a rio u s q u a n titie s t h a t
£ i n p si Fig. 4 — Relation between v„— Krf„ and concrete strength.
h a v e been d e term in ed b y th e te sts. T h e y are self-explanatory or hav e been explained previously. In colum n 11 are given th e values o f :
vy - K r f y
K r f v is th e th e o re tic a l shearing stress a t th e yield p o in t s tre n g th of th e w eb rein fo rcem en t w hen th e an alogy w ith a tru s s is m ade in th e analysis of th e a c tio n of stirru p s in reinforced concrete beam s.*
T h e valu es of vy — K r f y h a v e been p lo tte d in Fig. 4 ag ain st th e concrete s tre n g th as d e te rm in e d from s ta n d a rd te s t cylinders. T h ro u g h th e p o in ts so d e term in ed , a s tra ig h t line h as been passed, for w hich th e e q u a tio n is
vy — K r f y = 200 -f- 0 .0 4 /'c (in psi)
T h e sh earin g stress a t w hich th e stirru p s are stressed to th e yield p o in t, as defined previously, is th u s rep resen te d fairly well, for th ese series of te s ts b y th e follow ing eq u a tio n :
V y = K r f y + 0.04/'„ + 200 (in p s i) ...(1) T h e sh earin g stresses d e term in e d b y th is form ula are show n in C o lu m n 12 of T a b le 2 an d are d e n o te d as “vy calc.” C olum n 13 of th e sam e ta b le gives th e ra tio b etw een th e sh earin g stresses calcu lated b y fo rm u la (1) a n d th o se derived from th e ex p erim en tal v alue V y.
T h e c a lcu lated a n d e x p erim en tal values of vv agree m ore closely th a n could generally be exp ected for te s ts on reinforced concrete beam s. W hen in d iv id u a l beam s are considered, th e ra tio betw een calcu lated an d ex
p e rim e n ta l valu es ranges b etw een 0.914 a n d 1.027. T h e general average for th e beam s w ith stirru p s is 0.976, for th o se w ith % - i n . stirru p s,
*See, fo r e x a m p le, B u lle tin 166, U n iv e rs ity of Illin o is, loc. cit.
fc in p s i
Fig. 5— Relation between v„— v„ and concrete strength.
0.983, a n d for th o se w ith J^-in. stirru p s, 0.998. T h e g en eral average for all th e specim ens is 0.986.
C olum n 10 of T a b le 2 show s th e difference b e tw een th e e x p erim en tal
V
sh earin g stresses, — , a t u ltim a te load a n d a t V v. T h is difference, bjd
Vu — Vy, h as been p lo tte d a g a in st con crete s tre n g th as abscissas in Fig.
5. A s tra ig h t line h as been p assed th ro u g h th e p o in ts so d eterm in ed . T h e e q u a tio n of th is s tra ig h t line is :
VU ~ Vy = O.OG/'c su b s titu tin g vy as given in e q u a tio n (1):
vu = K r f y + 0.10f ' c + 2 0 0 ... (2) W ith th is form ula, th e sh ea rin g stresses a t u ltim a te lo ad h a v e b een com p u te d for each beam , are given in C o lu m n 14 of T a b le 2 a n d are d e n o te d as vu calc. C o lu m n 15 gives th e ra tio b e tw e e n th e sh earin g stresses c o m p u ted b y fo rm u la (2) a n d th o se d e te rm in e d exp erim en tally , t h e a g reem en t b etw ee n c a lc u lated a n d e x p e rim e n ta l v a lu es is again
\ e r y good, w ith ra tio s t h a t ran g e from 0.830 to 1.086 fo r in d iv id u al beam s. T h e general av erag e for th e b eam s w ith 34-in. s tirru p s is 0.998, for those w ith % -in. s tirru p s, 0.983, a n d th e gen eral a v erag e for all th e beam s t h a t failed b y diag o n al ten sio n , 0.99.
T h e m easu red deflections a t th e c e n te r of th e b eam s w ere co m p ared w ith th e th e o re tic a l deflections c o m p u te d b y M a n e y ’s e q u a tio n .* F o r a beam load ed a t th e th ird p o in ts, th is e q u a tio n is
n 23 I2 , , . D = (c + t)
_______ 216 d ’
A . S ^ aM y T ? c h n ^ a l ^ P a p e r s " ^ u T p .1 « " W H ? “ ^ D efiec tio n in R e in fo rc e d C o n c re te B e a m s ," P ro c .
5 4 0
o
f-C 3 0
• 2 0 v>
.? 10
B e a m w i t h
'
6 3 2
5 2 6
4 2 1 i
CL
3 I6 £
«ftm 2II i
in I0 5 IJZ
tn 0
^0.00! "¿jn
S t i r r u p s
S tra in in Web R e in fo rc e m e n t
Fig. 6 — A ve ra g e "shear-strain” curves for the web reinforcement. Series 1 a.
in w hich I is th e span, d th e effective d e p th , c th e s tra in a t th e to p fiber, a n d t th e av erag e s tra in in th e ten sio n steel.
T h e m easu red deflections a t u ltim a te load w ere in c e rtain cases as m uch a s 40 p e r ce n t g re a te r th a n th e co m puted. A t th e lo ad corresponding to
T h e m easu red deflections a t u ltim a te load w ere in c e rtain cases as m uch a s 40 p e r ce n t g re a te r th a n th e co m puted. A t th e lo ad corresponding to