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8/9/2019 Round-robin analysis of the RILEM TC 162-TDF - 2.pdf
http://slidepdf.com/reader/full/round-robin-analysis-of-the-rilem-tc-162-tdf-2pdf 1/10
Ma ter i a l s and S t ruc tu res /
M a t d r & u x e t C o n s tr u c t io n s ,
Vol . 36 , Nove mb er 2003 , pp 621-630
~ RILE M TC 162-TDF: Tes t and des ign methods for s tee l f ibre re inforced concrete
R o u n d r o b i n a n a l y s i s o f t h e R I L E M T C 1 6 2 T D F b e a m
b e n d i n g t e s t : P a r t 2 A p p r o x i m a t i o n o f 8 f r o m t h e C M O D
r e s p o n s e
Prepared by B.I.G. Barr~and M .K. Lee ~, w ith contributionsfrom E.J . de Place Hansen3,
4 5 2 5 3 4
D. D up ont , E . Erdem , S . Schaer laekens , B . Schnt itgen , H. S tang and L. Vandew al le
1) Cardiff University, W ales
2) Belgian Research Institute, Belgium
3) T echnical University o f Denma rk, Denm ark
4) Kathol ieke Universiteit Leuven, B elgium
5) Ruhr-University of Bochum , Germany
ABSTRACT
A round robin test progrmmne was carr ied out using the
beam-bending test recommended by RILEM TC 162-TDF [1] .
The test programme included both plain and steal fibre
reinforced concrete (SFRC) beams. A detailed analysis was
carried out to investigate the influence of different test
configurations on me asurements o f the crack mo uth opening
displacement (CMOD). Linear elastic fracture mechanics
(LEI?]VI) and non-linear fracture mechanics (N LF M )m eth od s
were uti l ised to investigate the problem analytically. From the
analytical studies carried out, i t is proposed that the CMOD
should not be m easured at a d istance m ore than 5mm from the
bot tom f ibre o f the beam. A larger d is tance than 5m m
will
cause
the deviat ion between the mea sured CMO D and the mac CM OD
to reach an unacceptable level. A simple rigid body model has
been proposed to relate the CMOD to the mid-span deflection.
The N LF M analysis and exper imemtal esul ts w ere compared for
both the plain and SFRC beam resul t s and i t was tbund that
resul ts based on the basis of CM OD can be compared to those
based on deflections, tbr practical purposes, using the simple
rigid bod y m odel. T he experimeaatal results strongly suggest that
the r ig id body model could be effect ively appl ied for al l the
types o f materials tested in the round robin test programm e. In
addit ion i t was fotmd that the conversion from CMOD to the
equivalent m id-span deflection, 8~, revealed g ood agreem ent
between the load-average mid-span deflection (P-6) curve and
the load-equiva lent mid-spa n deflection (P-8r curve especially
~br the SFR C specimens. It is proposed th at the load-CM OD (P-
CM OD ) curve be used to calculate the proposed R ILE M design
parameters (as opposed to the P-6 curve) via the use o f a
correction factor determined using the sim ple rigid bo Jy m odel.
Un prog ram me d '~ sa is comqx~ra t~ en tre la t~ora to ires a d t~
r&dis 'O pou r &rd uer he te s t de txm Oe sournise & la f l ex ion , cornm e
prescrt~p a r T C
1 6 2 - TD F d e l a R J L E M
Des'pouttes
e n b ~ t o n n o r m a l
e t e n b ~ o n d e i c e s m d t a l l ~ ( B f ;~ D s o n t p ri se s" e n co ra p te d a m l e
p r o g r a m m e d ' e ss a & U n e & d e d ~ t aL ll & a d t~ e f f ~ r p o u r
e x a m i n e r l 'i qf lu e ~ ce d e d ~ r e n t e s m d t ho d es ' d 'e s s a i d e m e s u r e s d e
l 'c m ~ e m a 'e d b r i j ~ e d e j ~ s w e ( O O F ). L a m d ~ l n i q u e d e j P a c a z e
l in d a ir e e t n o n - l ~ a i r e a d t~ u t i li s & ~ e x a m i n e r le p r o b l & n e d ' u n e
f o4 :o n a n a , ~ e . S u r b a s e d e s
~tudes'
ar~lytiques, on p.o_pa~e qu e
l 'O O F n e so i t ~ m ~ v a r & a u n e d i st a n c e d e p l u s d e 5 m m d e l a b a~ e
de la tmu tre . A u-de ld d e ce t te dA tcau:e de 5 ram, la d~&~ mce en tre
l ' 0 0 F m e s u r d e e t l ' O O F v d ri ta b le a tt e i n t u n n ; n e au i n a c~ * p ta b le . U n
m t ~ l e s im p l e, b a sd a z tr d e ~ c ~ p s s o ii de s , e s t p r ol n~ s d u m r e s t b n e r
l ' O O F h l a l e x io n a u c e n t re d e l a t n m t re . L ' a ~ a l ~ e d e l a m ~ . m i q u e
d e f i z z: t w i n n o n - l i n & ~ a ~ t ~ c o m p a r & w e e c l e s r & ~ d tc a s
e x ' t ~ i m o g a w c t x m r le a" p o u t r e s e ~ b d t on n o r m a l e t e n B F M : E n
p r a t iq u e , o n a o o m t a t d q u e h es r & u l t a t s b a s' ds ~ l ' 6 K ) F p e t c v o t t ~ e
com par & arv~c lea ' r&~ qa ts ba~& a~ r la dO qex ior~ s i on u t il i se un
mo dd le bcz~d u r des co rps ,solides. Lea"r ~ t a t s a ~ gg b re nt c l a ir e m e n t
~ e he modk le , t v . z~ , s~ tr & " c~)q :~ .wl ide , , l~ ' i s' se ~ tre c~ p l~ d a~ tr
tom" le s ma t&iau x te s t& ~ Ie p ro gra m m e dez ' ewsais' compara(i [J.
E n p lm ; la o )rrverskA n en tre I 'O ( )F e t la d (e flex ion &iu ivahen le au
c e n t r e d e t b p o u t r e ( 6 ~) a r & ~ l d u n e g r a n d e c o ~ ) r m i t d e v f f e l e
d i a g r ~ ' n m e d e la o r c e e n l ~ .c c k) n d e l a & f l e x i o n m o y e n n e ~ c e n t r e
de la po~Ire ca he d iagra mm e d e la Jbrc e en Jbnc t ion de la d~ ,1exkm
& l u iv a le n te a u c e m r e d e la l x m t r e . C e z, i e s t c ~ , m c m t l e c a s l x n e c
B F M O n ~ e q u e le d ia ~a m m e d e
kL/bme
e n J b n c t io n d e
l 'O O F (conO'a~avxen t au d ic~ ,ramme crvec la br ce e n Jbnc tion d e la
dd f lex ion ) so i t u t il i sd lgn a 9 le ca lc~ l de~ pam mk tres de
d i m e m i o n n e m e n t , p r o p a ~ d p a r l a R I L E 3 / [ P o u r c e f lt ir e , o n p ca t t
u ti li se r u n j & c t ~ r d e c o rr e ct km , d d t e r m ~ p a r l e m ~ xt O le , b a s' d a , r
des" cor ps ~91ktes.
1359-5997/03
9 R I L E M 6 2 1
8/9/2019 Round-robin analysis of the RILEM TC 162-TDF - 2.pdf
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T C 1 6 2 - T D F
1 . I N T R O D U C T I O N
This paper g ives an accou nt of a d e ta i led invest iga tion of
the relationship between the mid-span deflection, 8, and the
crack mouth opening displacement , CM OD , obta ined in a
round robin test pro gramm e executed by f ive test laboratories.
The roun d robin test programm e was carr ied ou t using the test
eonfi=marationproposed by RIL EM T C 162-TDF [1] .
Further details of the round robin test progrm~me are
reported in a companion paper and will not be repeated here
[2,3]. Essentially, the rou nd robin test programme was executed
in two phases with two concrete strengths considered. The
concrete grades considered were C25/30, termed as normal
strength concrete (NSC) and C70/85, termed as high strength
concrete (HSC ), Th e letter C indicates a characteristic
compressive strength and th e numerals indicate the compressive
strengths in N/mm2 mea sured from cylinders and cubes
respectively. Three types of fibres and fibre dosages were
employed during the round robin programme i e l?'~arnix65/60
BN ( tbr the N SC beams wi th 25kg/m 3 and 75kg/m 3 of f ibres) ,
Dm mix 80/60 BN (for the NS C beams wi th 50kg/m 3 of f ibres)
and Dramix 80 /60 BP ( fo r the HS C beams wi th 25k~ m 3 o f
fibres). Plain concrete beam s w ere also included as part o f the
test programme to play the role o f control spec imens and w ere a
me ans o f investigating the strengths, l imitations and sensitivity
of the proposed test metho d, as the y do n ot contain the -variations
introduced by fibre distribution an d orientation.
F ive t e s t i ng l abora to r i e s we re i nvo lved i n t he beam
bend ing round rob in p rogram me , w i th Ca rd i f f Un ive r s i t y
(UWC) as the task co-ordina tor , The labora tor ies a re ( in
alphabetical order):
C S T C - Belg i an B u i ld ing Resea rch Ins t i t u t e
D T U - T e c h n i ca l U n i v e r si t y o f D e n m a r k
K U L - K a t h o l i e k e U n i v e r s it e it L e u v e n
R U B ~ R u h r - U n i v e r s it y o f B o c h u m
UW C - Ca rd i f f Un ive r s i t y ( ta sk co -o rd ina to r )
The above ac ronyms fo r e ach pa r t i c ipa t i ng l abora to ry
wi l l be used henee fb r th .
2 . R E S E A R C H S I G N I F I C A N C E
The t e s t me tho d advanced by t he RILE M TC 162-TDF i s
a no t ched beam unde r t h ree -po in t bend ing which i s shown
schem at ica l ly in Fig. 1 , From the exper ience ga ined from the
round rob in t e s t p rogramme , i t was conc luded t ha t t he t e s t
wou ld be m ore s t ab le i f execu t ed unde r CMO D con t ro l
par t icular ly for spec imens wi th a high st rength matr lx~ I t i s
an t ic ipat ed tha t t he CM OD wou ld be m easured a t a c e r ta in
d i s tance away t i om the bo t t om sur face o f t he beam. K ni t~
edges (or other s imi lar devices) could be used to ho ld the c l ip
gauge (or LVDT in some cases) in place , The kni tb~edges
wo uld have a certain thickness , do, ~ i l lt lstrated in Fig. 1.
The measured CM OD w ould have a sys t ema t ic ~ r ~e la ti ve
to t he t rue CM OD ~ wi th t h e dev i a t ion p ropor t iona l wi th
respect to s om e fianction of the thictmess do, It is thus
essent ia l to l im i t the thickness of the kni i~-edge to a cer ta in
range to avoid unac ceptably la rge devia t ions.
The t e s t me thod i s e iose ly r e l a t ed t o t he c r -e me thod
proposed by t he R1LEM TC-TD F [4 ] . The t e s t me thod
requi res tha t measurem ents be made o f the mid-spa n
def lec t ions on bo th sides o f the beam, 8i an d 62, in addi t ion
I
I -
t
L P'~ L
d
$ . L__~
2
E a 4
L
CMOD H
Fig. I - Schem atic illustration show ing knife-edges (or
similar devices)having a certain thickness, do.
t o t he CMO D. S t r ic t l y speak ing , the RILE M TC-TD F des ign
parameters should be ca lcula ted f rom the load-average mid-
span de f lec t ion, P-5, curves. How ever , there ma y be cer ta in
t imes where this i s not possible . Ei ther , one or both
de f l ect i on measurem ent s may be t b tmd to be e r roneous and
th i s wou ld e l im ina t e t he poss ib i l i t y o f c a r ry ing ou t
ca lcula t ions based o n the P-6 curves. Another scenar io wh ich
may ar ise , i s tha t some researchers may desi re to carry out
t he t e st s i nd i rec t l y by measur ing on ly CM OD .
Add i t ional ly , the possibi l i ty of obta ining a l l necessary design
pa rame te r s f rom the l oad-CMOD (P-CMOD) curves wou ld
appeal to the prac t i s ing engineer as i t reduces se t -up t ime an d
cos t. A mode l r e l a t ing t he CM OD measurement s t o t he mid-
span de f l ec t ions wou ld pave t he w ay to such a poss ib i li ty .
The i dea o f us ing t he P-CMOD curve i s no t a new one .
Gopa l a ra tnam
e t a L
[5] ibund tha t CMOD are less prone to
errors than beam def lec t ion measurements and postula ted
tha t tbe P-CMOD curve could be a possible a l te rna t ive
measure o f t oughness . Al so , Gopa l a ramam e t a l [6]
demonst ra ted tha t there was vi~aal ly a l inear re la t ionship
be tween mid- span de f l ec ti on r e sponse and CM OD and
suggested tha t the two could be re la ted, for a l l prac t ica l
purposes, using a simple re la t ion. Barr e t a L [7] concluded
tha t t he P-CM OD curve would be a goo d bas is fo r use i n t he
measurement o f t oughness and ca r ry ing ou t t e s ts on no t ched
beams via CMOD control leads to a s table post -peak
re sponse. M ore r ecen t ly , Nava lu rka r e t a L [8] proposed tha t
t he P-CM OD curve be used , i n s t ead o f t he P-6 cu rve , t o
evalua te the f rac ture energy o f concre te .
3 . C O R R E L A T I O N B E T W E E N C M O D s A T
D I F F E R E N T D E P T H do
3 . 1 S c o p e o f s tu d y
Thi s sec t i on g ive s an accoun t o f a s t udy ' t o i nves t i ga t e
the e r~ r i n t roduced by us ing d i f i ' er en t kn i fe -edges (o r
s imi l a r dev i ce s t o measure / con t ro l CMOD) wi th d i f f e ren t
6 2 2
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Ma ter i a l s and S t ruc tu res /
M a t d r i a u x e t C o n s t r u c t i o n s ,
V o l . 3 6 , N o v e m b e r 2 0 0 3
th i ckness , do . In a beam t es t , t he CMOD i s t yp i ca l l y
m e a s u r e d a t a f i x e d d i s ta n c e f r o m t h e b o t t o m o f t h e b e a m .
I t i s u n u s u a l t o m e a s u r e C M O D d i r e c t l y f r o m t h e b o t t o m
f ib r e o f t h e b e a m , T h i s i n t ro d u c e s a f i x e d a m o u n t o f
s y s t e m a t i c e r r o r i n t o t h e m e a s u r e d o r n o m i n a l v a l u e o f
C M O D ( d e f in e d b y C M O D i n F i g. 2 ) r e la t iv e t o t h e a c tu a l
v a l u e a t t h e b e a m s u r f a c e
Le .
C M O D ~ i n F i g . 2 , w i t h t h e
magn i tude o f t he e r ro r i ncreas ing wi th an i ncrease i n t he
th i ckness o f t he kn i f e -edge (do ).
' CMO D" ;
Fig. 2 - Il lustration of difference between CM OD and C MO D
Two ana ly t i ca l methods were used t o i nves t i ga t e t h i s
i ssue; Ferrei ra
et al .
[9] used l inear elast ic f racture me chan ics
(LEFM) whereas S t ang [10 ] u sed non- l i near f r ac tu re
me chan i cs (NL FM) . E r ro rs i n t roduced i n t e rms o f t he
d i f f e rence i n the CM OD mea su red us ing a cer ta i n kn i f e -edge
th i ckness , do, wi th t ha t o f t he CM OD mea su red exac t l y a t t he
b o t t o m o f th e b e a m w a s s tu d ie d . T h e " r e a l " C M O D a an d t h e
nominal va lue , CMOD, as ob t a ined exper imen ta l l y a re
related as fo l lows:
C M O D ~ = c r ( 1 )
where c~ i s t he t heo re t i ca l co r rec t i on t hc to r fo r conver t i ng
t h e m e a s u r e d v a l u e t o th e " r e a l " C M O D .
3 . 2 P r o p o s a l f o r a li m i t t o do
From the s t udy us ing LEFM [9 ] , i t was found t ha t fo r a
kn i f e -edge th i ckness o f 10 mm , t he e r ro r vari es f rom va lues o f
20% to 8% as a func t i on o f mid , span defl ec ti on . B y assuming
a m ax im um er ror o f 5%, t he s t udy showed t ha t a kn i f e -edge
should not have a th ickness o f mo re than 2 .5 mm . For an error
level of approxim ately 10%, the s tudy suggests that a kni fe-
edge should not have a th ickness greater than 5 ram.
The a nalysis using NL FM [10] showed simi lar result s wi th
the error increasing w ith increasing knife -edg e thickness. By
consider ing kni fe-edge th icknesses of 2 m m and 5 m m, several
correct ion factors were obtained for var ious mid-span
def lect ions. The resul t s are shown in Table t . I t can be
observed that af ter a cer tain m id-span de f lect ion, the correct ion
facto r approaches a con stant value.
In essence , bo th ana ly t i ca l s t ud i es showed t ha t t he e r ro r
i n t roduced i ncreases wi th an i ncrease i n t he he igh t o f t he
kn i f e -edge used and t ha t t he co r rec t i on t hc to r , c r var i es as
t he t es t p rog resses . How ever , t he change i n t he v~ ue o f c~
i s i n s ign i f i can t whe n t he t es t i s i n t he pos t - c racked phase . I t
i s p roposed t ha t t he kn i f e -edge (o r s imi l a r dev i ces )
t h ic k n e s s i s l im i t e d to a m a x i m u m v a l u e o f 5 m m , T h i s
th i ckness i s no t t oo smal l t hereby m ak in g i t re l a t i ve ly easy
t o f a b r i c a te w i t h re a s o n a b l e a c c u r a c y a n d w o u l d n o t b e t o o
cum bersom e to hand l e , espec i a l l y fo r p rac t i s i ng eng ineer s .
4 . C O R R E L A T I O N B E T W E E N A N D
C M O D
4 . 1 M e t h o d o l o g y
This section gives an a ccou nt of an investigation to determine
whether there is a simple correlation between the mid-span
deflection, 15, and the crack mouth opening displacement,
CM OD . This i s to enable calculat ions of the RILE M TC 162-
TD F design parameters f rom P-C MO D ctawes to be considered
as a parallel or al temative m etho d of analysis.
In the post-cracked phase, using r ig id body ~ n a t i r i t can
be show n tha t theoretically the average mid -span deflection, ~5,
can be related to CMOD. The only var iable that one may
encounter wi l l be the value ol d , , which i s def ined in Fig . .
The equat i on r e l a t ing ~5 o CM OD i s as fo l l ows :
~5 L 1
C M O D 4 H ( 2)
where , L - span a t wh ich t he beam i s t es t ed mad H = the
apparen t dep th abou t wh ich t he beam ro t a t es
Equ at ion (2) im pl ies that the rat io o f ~3/CMOD is a
cons t an t g iven by L/4 , mu l t ip l i ed by t he va lue 1 /H. The o n ly
p rob l em l e f t is t o eva lua t e t he va lue o f H . The po in t a t wh ich
the beam ro t a tes can be assum ed (as a f i r s t app rox imat ion ) t o
be a t t he veu ,, top su r face o f the beam . In t he case o f t he
round rob in t es ts , H i s t hen es t imat ed t o be t 50+ .~ . In t he
fo l l owing sec ti ons , t he nom inal va lues o f CM OD
i .e ,
C M O D i n F i g . 2 ) w i ll b e u s e d f o r th e a v ~ y s e s a n d s h o u l d n o t
b e c o n f u s e d w i t h C M O D a.
In t he f i rs t phase o f t he round rob in p rog ramm e, f il e t est
l abs u sed d i f lb ren t va lues o f do i n t he wo rk c ~ r o~ t. Thus ,
t he CMOD resu l t s fo r t he p l a in no rmal s t r eng~ concre t e
beams , C25 /30 , and no rmal sgeng th concrer w i~ a f i b re
c o n t e n t o f 5 0 k g / m ~ was ob t a ined v i a . t he ~ e o f d i f f e ren t
va lues o f do . Th i s g ives t es t da t a t o i nves t i ga t e t he
assumpt ions impl ied in Equat ion (2) , Table 2 tabulates the
va lues o f do adop t ed by t he var i ous t es ti ng l a l~ du r ing t he
f i rs t phase o f t he round rob in l es t p rog ramm e.
By p lo t t i ng ~5 /CMOD agains t l / I t , a ~ by ca r ry ing ou t
l i near r eg ress ion w i th t he es t imat ed r eg ress ion l ine pass ing
th rough t he o r i g in , one can eva t~uate t he a cc ~a ey o f t he
Tab le 1 - Correction factors at various mid-span deflections (from non-linear fracture mechanics analysis)
Correction factor, er prescribed mid-span deflection
do (mm ) 0A 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
2 0 .9 73 3 0 . 9 8 3 7 0 . 9 8 5 2 0 . 9 8 5 8 0 . 9 8 6 2 0 ~ 9 8 6 4 0 . 9 8 6 5 0 : 9 8 6 5 0 . 9 8 6 6 0 .9 86 6
5 0 .9 3 57 0 1 9 6 0 3 0 . 9 6 3 9 0 , 9 6 5 3 0 . 9 6 6 1 0 . 9 6 6 6 0 .% 68 0 . 9 6 7 0 0 . 9 6 7 1 0 .9 6 72
Note: Figures in bold indicate,flaeprescr ibed mid-span def lect ion in mi l l im eters . . . .
6 2 3
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T C 1 6 2 - T D F
T a b l e 2 - L o c a t i o n o f C M O D m e a s u r e m e n t
d e v i c e s a t t h e v a r i o u s t e s t l a b s d u r i n g t h e f i rs t
p h a s e o f t h e r o u n d r o b i n t e st p r o g r a m m e )
Testing lab Thickness do mm )
CSTC* -4
DTU 7.5
KUL 14
RUB 0
UWC 5
Note: * CSTC measured CM OD above the bottom fibre on
both sides of the beam, ? RUB m easured the CMO D on
both sides of the beam, at the bottom corners
m od e l . A s lope in c lose agreem ent wi th the theore t ica l va lue
of L /4 would im ply tha t the m ode l i s app l icab le . Hence , the
ra t io o f 6 /CMO D was ca lcu la ted a t d i st inc t m id-span
deflect ions , 6, and these values are plot ted agains t 1/H. This
w a s d o n e f o r b o t h t h e p l a in a n d S F R C b e a m s . E x a m p l e s o f
these p lo t s a re shown la te r in F igs . 7 and 9 . )
S t r i c t ly speak ing , the m ode l i s on ly app l icab le fo r t e s t
re su l t s in the pos t -c racked phase . The ac tua l re la t ionsh ip
b e t w e e n 6 a n d C M O D i s a t h n c t io n o f th e c r a c k l e n g th , t h u s
i m p l y i n g d i f f e r e n t 6 / C M O D r a t i o s i n th e p r e - c r a c k e d a n d
pos t -c r acked pha ses a s i l lus t ra ted in F igs . 3 and 4 . F or
p r a c t i ca l r e a so n s , a c o n s t a n t v a l u e o f 6 / C M O D i s h i g h l y
des i rab le . The pre -c racked phase i s re la t ive ly shor t in
com par i so n wi th the p os t -c racke d phase , wh ich . jus t if i e s the
u s e o f a c o n s t a n t 6 / C M O D r a ti o . T h e a s s u m p t io n o f a
c o n s t a n t 6 / C M O D r a t i o s h o u l d n o t h a v e a s i g n i f i c a n t l y
a d v e r s e e f f e c t , e s p e c i a l l y f o r S F R C b e a m s a s t h e y a r e
te s ted we l l in to the pos t -c racked phase .
A p a r t f r o m u s i n g t h e e x p e r i m e n t a l r e s u l t s f o r t h e
v e r i f i c a ti o n o f E q u a t io n 2 ) , a c o m p u t e r p r o g r a m d e v e l o p e d
a t th e T e c h n i c a l U n i v e r s i t y o f D e n m a r k , D T U , w a s u s e d i n
a d d i t i o n t o t h e s i m p l e p r o p o s e d m o d e l . T h e c o m p u t e r
p r o g r a m t o g e t h e r w i t h a d e s c r i p ti o n o f t h e a n a l y s is i s g i v e n
i n r e f e r e n c e [ 1 0 ]. T h e c o m p u t e r p r o g r a m w a s e x e c m e d w i t h
f i x e d m a t e r i a l a n d g e o m e t r i c a l p a r a m e t e r s b u t w i t h
d i f f e r e n t v a l u e s o f d o. S i m i l a rl y , t h e p r o c e d u r e o f o b t a i n i n g
6 / C M O D a t v a r io u s m i d - s p a n d e f l e c t i o n p o i n t s a n d p l o t ti n g
~ / C M O D a g a i n s t 1 / H w a s c a r r i e d o u t u s i n g t h e c o m p u t e d
resu l t s ob ta ine d f rom the p rogram .
4 . 2 V a l i d i t y o f m o d e l
I f we a s sum e the c rack l eng th i s s ign i f i can t ly l a rge in the
no tched th ree -po in t bend t e s t , the m ode l sugges t s tha t the
r a t io o f 6 / C M O D i s c o n s t an t i n t h e p o s t - p e a k r e g io n .
E q u a t i o n 2 ) a l so i m p l i e s t h a t th e r a t i o o f th e a r e a u n d e r P - 6
t o t h e a r e a u n d e r P - C M O D i s a c o n s ta n t . T o t e s t t h e v a l i d i ty
o f s u c h a m o d e l , p lo t s o f 6 / C M O D a g a i n st m i d s p a n
def lec t ions we re ca r r i ed ou t . Addi t iona l ly , p lo t s o f the ra t io
o f t h e a r e a u n d e r t h e P - 6 c u r v e s a g a i n s t t h e a r e a u n d e r t h e
P - C M O D c u r v e s w e r e a l s o p r e p a r e d .
F igs . 5 and 6 sho w p lo t s o f 6 /CMO D against m id-span
deflect ion, 8, tbr pla in and SFR C b eam s respect ively. I t is seen
tha t the ra t io o f 6 /CMO D approaches a p la teau a f t e r a ce r~ in
valu e of mid-span deflect ion. In general , the constant pla tean
of 6 /CMOD occupies m os t o f the pos t -peak reg ion and th i s
sugges ts tha t the a s sum pt ions beh ind the p rop osed m o de l a re
va l id e spec ia lly wi th in the pos t -peak reg ion .
E
0 , 8
t~r
0
0 . 6
C
~ , 0 . 4
E
o.2
0 005 0~ 0 . 15
CNOO r a m }
0
o 2 0 .4 0 . 6 0 . 8 1
C r a c k m o u t h o p e n i n g di s p la c e m e n t , C M O D m m )
Fig. 3 - Typical plot of 6 against CMO D tbr a plain concrete
beam with the inset figure magnifying the initial portion of the
curve.
0.3
25 /
0 2
0,15
_ =
~ o.~ 0.2 o y
e
O
O
m
>
< 0 / , . ... , , .........~.
0 0.5 1 1.5 2 2.5 3
C r a c k m o u t h o p e n in g d i s p la c e m e n t , C M O D r a m )
Fig. 4 - Typical plot of 6 against CMOD for a SFRC beam with
the inset figure magn i~ing the initial portion of the curve.
F i g s . 5 m a d 6 w e r e p l o t t e d u s in g t h e a v e r a g e v a l u e s f r o m
al l the t e s t ing l abs . Thus , i t i s no t un expe c ted tha t the p lo t s
f o r t h e N S C a n d H S C b e a m s d i v e r g e f r o m e a c h o t h e r . T h i s
i s b e c a u s e i n t h e f i r s t p h a se o f t h e r o u n d r o b i n p r o g r a m ,
te s t ing l abs used ve ry d i f fe ren t va lues o f do- On the o the r
h a n d , i n t h e s e c o n d p h a s e o f t h e r o u n d r o b i n , al l b u t o n e
u s e d a d ,, o f 5 m m w i t h t h e e x c e p t i o n o f R U B , w h o u s e d
3 r a m , b u t t h e d i t ~ r e n c e i s n o t s e e n to b e s i g n if i c an t .
T h e d i v e r g e n c e t b r t h e S F R C b e a m s i n F i g . 6 i s n o t a s
s ign i f i can t a s s een in the p la in conc re te beam s . Th is i s a s
e x p e c t e d , s in c e t h e S F R C b e a m s w e r e t e s t e d w e l l in t o t h e
p o s t - p e a k r e g i o n w i t h r e l a t i v e l y m u c h l a r g e r m i d - s p a n
def lec t ions in com pax ison wi th the p la in conc re te beam s .
6 2 4
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Ma ter i a l s and S t ruc tu res /
M a t O r i a u x e t C o n s t r u c t i o n s
Vol . 36 , Nov em ber _00~
1.2
0,8
i
0
0 , 6
0 . 4 -
0.2 84
13
0
[ ]
0 0 1 3 0 0
O C25/30 (NSC)
[]
C70t85 (HSC)
i J
0.1 02 03 0 .4 0 .5 0 .6 0 .7 0 .8
A v e r a g e m i d . p a n d e f l e c ti o n
( r a m )
F i g . 5 - T h e a v e r a g e r a t i o o f 8 / C M O D f o r p l a in c o n c r e t e
b e a m s f o r a s e r i es o f a v e r a g e m i d - s p a n d e f l e c t io n p o i n t s .
1.00
O
o
oo
0.96
0.92
[ ]
0.88 9
0 . 8 4 .
0'.80
&
x
A
I
x A
O
C25t30(25)
[ ] C25130(50)
a
c25/3o(75)
x c 7 0 / s s ( 2 s )
[ ]
i
0.5 1 1.5 2 2.5 3 3.5
A v e r a g e m i d - s p a n d e f l e c t i o n ( r a m )
F i g . 6 - T h e a v e r a g e r a ti o o f 6 / C M O D f o r S F R C b e a m s f o r a
s e r i e s o f a v e r a g e m i d - s p a n d e f l e c t i o n p o i n t s .
4 3 R e s u l t s a n d d i s c u s s i o n
Fig . 7 shows a p lo t o f t /C M O D agains t 1/H fo r a mid - span
def lect ion of 0 .4 m m for the no rmal s t rength plain concrete
beams. The plot uses b oth the experhqaental resul ts and the
resul t s computed using the aforesaid computer program from
DTU. This type of p lot wa s obtained for several mid-spa n
def lect ions and the gradients were obtained by m eans o f l inear
regression analysis . Al though f ive labs were involved in the
round rob in p rog ramme fo r t he beam-bend ing t es t, on ly fou r
experimental data sets is available for analysis tbr the plain
NSC spec imens because a l l t he beams i n one o f t he l abs
und erw ent catastrophic failure.
F ig . 8 shows t he g rad i en t s ob t a ined t b r a l l average mid -
span def l ec t i ons cons idered . Bo th t he exper imen ta l r esu l t s
and t he ana ly t i ca l r esu l t s a r e qu i t e c l o se t o t he p red i c t ed
v a l u e o f L / 4
i .e .
125 for th is s tudy. In general , i t can be
0.8
0.6
0.4
E
y 133.13x
9 Exper imen ta l IS /CMOD at ,5 = 0 .4 ram
0 . 2 [ ] S t a n g
- - I J n e a r (E xp e r im e n t a l ~ V C M OD a t ~s = 0 . 4 m m )
- - Linea r (Slang)
0.0060 0.0062 0.00 64 0.0066 0.0068 0.0070
I l H
F i g . 7 - R a t i o o f 6 / C M O D a g a i n s t
1 / H
f o r p l a in N S C b e a m s at
a n a v e r a g e m i d - s p a n d e f l ec t i o n o f 0 4 m m
180
3 :
9
120.
W
C~ iO0.
0
0
80 ~
60.
0
40
2 0 .
O :
in
o D
[ ] [ ]
0 [ ] E l Q O 13
o o o <> o 0 o <>
O Experimental gradients
[ ] S t a n g
- -- - Rigid bod y mod el
0.2 0.4 0.6 0.8 1 1.2
A v e r a g e m i d - s p a n d e f l e c t io n ( m m )
Fig . 8 - Gm di~ ts obta ine d a t s e ve ra l po in t s of m id- spa n de f le c t ion
l~om expe rimen ta l and analUdcal results for pla in NS C beam s.
conclude d t ha t the exp er imen ta l and an a ly t i ca l r esu l t s a re i n
c l o se a g r e e m e n t w i t h t h e p r o p o s e d m o d e l .
F ig . 9 Shows a p lo t o f 5 /CMO D against I /H fo r 8 = 2 rnm
for the norm al s t rength SFRC be ams w i th a f ibre content of 50
k g / m3. Fig. 10 show s the gradients tbr a l l ~is considered. O nce
again , there i s close agreem ent between the exper imental and
anal)~cal results.
4 4 R a n g e o f a p p l ic a b i l it y
In the analysis repo rted in Section 4.3, consideration wa s
o n l y g i v e n t o t w o r ~ e s o f c on cr et e ~ s ; t h e p la i n N S C
beam and t he NSC beam w i th a t ib re con t en t o f 50 k /m . A
similar ana lysis cannot be carried ou t f iar beam resul ts f rom the
second phase of the round robin test s because dur ing th is
phase, al l but one laboratory used a d , , of 5mm. Thus a
di f ferent approach was adopted to investigate whether the
proposed mo dal appl ies to al l o f the concrete types considered
6 2 5
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T C 1 6 2 - T D F
r~
o
0,8
0.6
0,4
= 130.45x
O E x p e r i m e n t a l 8 / C M O D a t ~ , = 2 r a m
0 2 [ ] S t an g
- - L i n e a r ( E x p e r i m e n t a l 5 / C M O D a t 5 = 2 m m )
- - - L i near (S tang)
0 .0060 0 .0062 0 .0064 0 .0066 0 .0068 0 ,0070
I I H
F i g . 9 - R a t i o o f S / C M O D a g a i n st 1 / H f o r S F R C C 2 5 / 3 0
5 0 k g / m 3 o f f i b r e s ) a t a n a v e r a g e m i d - s p a n d e f l e c t i o n o f
2 . 0 m m .
2 0 0
160
E
m 120
0
r 40
[ ]
O
[ ]
~' g 6 w 9 [ ]
_rot__
,~ Exper ime n ta l g rad ien ts
[ ] S tang
- - R ig id body mod e l
0 1 2 3
Average mid-span d e f l e c t i o n ( r a m )
F i g . 1 0 - G r a d i e n t s o b t a i n e d a t s e v e r a l p o i n t s o f m i d - sp m a
d e f l e c t i o n s f r o m e x p e r i m e n t a l a n d a n a l y t i c a l r e su l t s f o r C 2 5 / 3 0
5 0 k g / m 3 o f f i b r e s ) b e a m s .
in the round robin programm e. W ith the conf i rm at ion that the
p roposed model wo rks wel l wi th t he p la in concre te beams an d
SFRC beams f rom the f ir s t phase , th i s conc lus ion can be use d
in the analysis of the secon d phase beam resul ts . T his t ime, the
CM OD at vari ous va lues o f 5 was ob ta ined.
Fig . 11 shows a p lot of CM OD against 5 for the p lain
concrete beams. I t can be observed that both sets of CMO D
increases in a l inear manner in relation to 5. Also, the results
tbr NSC beams are qui te close to HSC beams. Again , the
divergence can b e exp lained because o f the d i f ference in do
adopted in the f i rs t and second phase. The plain concrete
beam s wo uld be expected to be m ore sensit ive to d i f ferences in
do due to the fact that the test s had relat ively low values of
mid-span deflections.
F ig . 12 shows a p lo t o f CM OD agains t 5 fo r the SF RC
beams. O nce again , the CMO D increases l inear ly wi th 5 . This
t ime, t he ag reemen t be tween a l l t he SFRC beam types i s
0 8
~ " 0 . 6
t~
O
0 4
0 2
8
O
KI
r
O
r
l
O
= o C 2 5 / 3 0 ( N S C )
Q
* a C 7 0 / 8 5 ( H S C )
0 0 .2 0 .4 0 .6 0 .8
A v e r a g e m i d - s p a n d e f le c t io n m m )
Fig . l 1 - R e l a t io n s h i p b e t w e e n C M O D a n d 6 f o r p l a i n c o n c r e t e
b e a m s .
3 ,5
A 2 5
E
2
O
t~ 1 ,5
0 5
g
I
J
i t
S
x C 7 0 / 8 5 ( 2 5 )
9 C 2 5 / 3 0 ( 7 5 )
. C 2 5 / 3 0 ( 5 0 )
9 C 2 & ' 3 0 ( 2 5 )
0 0 5 1 1 5 2 2 5 3 3 5
A v e r a g e m i d - s p a n d e fl e c ti o n r a m )
F i g . 1 2 - R e l a t i o n s h i p b e t w e e n C M O D a n d ~i f o r S F R C b e a m s.
ex t r emely good . The da t a po in t s seem to l i e on t op o f one
another . D ivergence betw een the f i rs t mad second phase resul ts
i s not obvious because the SFR C be am tests were loaded wel l
in to the post -peak region mad thus w ould not be as sensit ive to
di f ferences in do as that o bserved for the p lain concrete beams.
Us ing t he p l a in NSC beams as a benchmark f rom the
previous s tud y (refer to Section 4.3), Fig. 11 indicates tha t the
relat ionship found betwee n 5-CM OD wou ld also apply for the
plain HSC tbavugh logical deduct ion (as the data poims for
both types of concrete are qui te close wi th each other in
Fig . 11) . Simi lm ly , i t can be deduce d f rom Fig . 12 that the
relat ionship between CMOD and 8 not only holds for the
C25/30 w i th 50kg/m3 o f t ibres, but also applies to all the fibre
concrete typ es considered.
Figs, 11 and 12 we re plot ted using the average values
calculated using all the results available from the round robin
t es t p rog ramme. The r a t i o o f 5 /CMOD was compi l ed a long
wi th coeff icients of var iat ion to determine whether any
6 6
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Ma t er i a ls an d S t ruc t u res / M at iaux e t Cons t ruc t ions , Vol . 36 , No vem ber 2003
C o n c r e t e
g r a d e
T a b l e 3 - T a b u l a t e d r a t i o s o f 5 / C M O D a t v a r i o u s p r e s c r ib e d a v e r a g e m i d - s p a n d e f l e c t i o n s a n d t h e i r
c o e f f i c ie n t s o f v a r i a t i o n f o r t h e p l a i n c o B c r e t e b e a m s
a t i o o f S / C M O D a t p r es c r ib e d a v e r a g e m i d - s p a n d e f le c t i o n s 6 = --~-.-~
0 . 1 0 . 15 0. 2 0 . 2 5 0 . 3 0 . 35 0. 4
0.874 0.822 0.792 0.776 0.765 0.766 0.765
C25/30
( 9.0 0) ( 6 . 4 3 ) ( 5 . 0 4 ) ( 4 . 0 8 ) ( 4 . 3 4 ) ( 3 . 7 8 ) ( 3 . 8 0 )
0.987 0.902 0.867 0.859 0.849 0.840 0.832
C70/85
(8 ,6 6) ( 8 . 2 0 ) ( 6 , 8 8 ) ( 5 . 0 6 ) (5 .8 2) ( 4 . 7 1 ) ( 3 . 9 9 )
Note: Figures in bold are the prescribed average mid-span deflections,
8,
in millime tres
Figures in brackets are the coefficients of variat ion, V (%)
0 .5
0.750
(6.37)
r
0.820
.... 3.50 )
0.6
0.757
(4.34)
0.816
(2.83)
0 .7
0.760
(4.33)
0.818
(2.32)
T a b l e 4 - T a b u l a t e d r a t i o s o f 5 / C M O D a t v a r i o u s p r e s c r i b e d a v e r a g e m i d - s p a n d e f l e c t i o n s a n d t h e ir
c o e f f ic i e n t s o f v a r i a t i o n f o r t h e S F R C b e a m s
C o n c r e t e
g r a d e
F i b r e
d o s a g e i n
kg / m3)
C25/30 (25)
C25/30 (50)
C25/30 (75)
C70/85 (25)
Note:
R a t i o o f 6 / C M O D a t p r e s c r ib e d a v e r a g e m i d - s p a n d e f l e c ti o n s
0 .25 0 .5
0.917 0.856 0.833
( 9 .5 8 ) ( 6 . 0 0 ) ( 5 . 2 5 )
0.898 0.854 0.840
( 7 .1 5 ) ( 5 . 6 2 ) ( 5. 44 )
0.969 0.894 0.865
(4.29) (3 .8 7) (3~41)
0.923 0.870 0.850
( 10 .8 ) ( 7 . 1 6 ) ( 5 . 3 9 )
8 = - g .
0.75 1.0 1.25 1.5 1.75 2.0
0 . 8 2 2 0 . 8 2 1
.828
(5.60)
0.835
(5.31)
0.851
0.824
(5:43)
0.832
5 . 3 5 )
0.843
0.823
(5.05)
0.830
(5.34)
0.837
(4.67)
0.830
(5.68)
0.833
(4:36)
0.832
5 . 1 o )
0.829
2 .5 3 ,0
0.822 0.823
(4.17) (4.t2)
0.833 0.836
(4 .8 2 ) (4 .8 8 ) ,,
0.828 0.824
2 . 3 2 ) g . 6 0 )
0.824 0.822
O.28) i @ 0)
3,03)
0.840
(4A7)
(3,03)
0.835
(3,93)
3,09)
0.831
(3.63)
(2.87)
0.829
(3.65)
2.85)
0,828
(3.55)
Figures in bold are the prescribed average m id-span deflections, 8, in millimetres
Figures in bracke ts axe the coefficients of variation, V (%)
deviat ions f rom l ineari ty wa s lost due to averaging. Tables 3
and 4 show t he r a t io o f 8 / CM OD at vari ous average mi d - span
def l ec t i ons t b r t he p l a i n concre t e and SFRC beams
respectively. It is clear that the coeffic ients of variation, V , for
al l the values c onsidered are below 10% . Ft tr thermore, the
ma jor i ty of the var iat ions are wi th in 5%, This su ggests that the
linear relationship w ould no t solely be applicable to the
averaged values, but w ould be appl icable g lobal ly ( for the test
d imensions use d in the round robin test programm e).
5 . I N T E R C H A N G I N G B E T W E E N 5 A N D
C M O D
T o s t u d y t h e i m p a c t o f i n t e r c h a n g i n g b e t w e e n ~ a n d
C M O D u s i n g t h e p r o p o s e d s i m p l e m o d e l , t h e r o tm d r o b i n
b e a m r e s u l t s w e r e r e - a n a l y s e d b y c o n v e r t i n g C M O D v a l u e s
i n t o equ i va l en t ~ va l ues . Th i s was ach i eved us i ng t he
express i on :
L 1
5 e . . . . . C M O D ( 3)
4 H
where , 6e = equ i va l en t mi d - span def l ec t i on , L = span a t
wh i ch t he beam i s t es t ed , H = t he apparen t dep t h abou t
w h i c h t h e b e a m r o t a t e s a n d C M O D = n o m i n a l ( o r
m e a s u r e d ) C M O D .
T h e P - 6 c u r v e s f o r t h e b e a m s p e c i m e n s w e r e c o n v e r t ed
t o P-8~ cu rves u s i ng E quat i on (3 ) . Typ i ca l cu rves fo r a p l a i n
concre t e beam are show n i n F i gs . 13 and 14 .
B y c o n v e r t i n g t h e C M O D v a l u e s to e q u i v a l e n t m i d - s p a n
def l ec t i on va l ues (6~), t he a g reem en t be t wee n t he P-6 and
P-CMOD curves (F i g . 13 ) i s i mproved (F i g . 14 ) . Th i s i s
espec i a l l y t rue where t he pos t -peak r eg i me i s concerned .
The d i f f e r ence be t ween t he t wo c u rves i n t he i n i t ia l par t o f
t he cu rves i s ac t ua l l y i ncreased , bu t t h i s i ncrease i s no t seen
as subs t an t ia l . The app l i ca t i on o f Equat i on (3 )h as t he e f f ec t
o f s h i f t in g th e P - C M O D t o w a r d s t h e l e f t i n a l i n e a r f a s h i o n
because t he absc i s sa va l ues a r e r educed by a t ~c t o r . Th i s
f ac t o r i s a func t i on o f the appa ren t dep t h abou t wh i ch t he
b e a m r o t a te s , H .
F i gs . 15 and 16 show t he i m pact o f t he conver s i on f rom
t h e P - C M O D c u r v e t o t h e P - 6 ~ c u r v e f o r a ty p i c a l S F R C
beam. As t he SFRC spec i mens a re t es t ed fo r a r e l a t i ve l y
14
lO
P - c u r v e
0 0.1 0.2 0.3 0.4 0.5
8 o r C M O D m m )
0.6
Fig. 13 - P-8 and P-CM OD curves for a typical plain NSC beam.
6 2 7
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T C 1 6 2 - T D F
14
12
10
8
v
0 ~ r T I
0,1 0 ,2 0 ,3 0 ,4 0 ,5 0 ,6
5 o r ~ e ( r a m )
F i g . 1 4 - ,, P - 8 a n d P - 8 ~ c u r v e s f o r a t y p i c a l p l a i n N S C b e a m .
l o n g t i m e a s c o m p a r e d t o t h e p l a i n b e a m s , t h e a g r e e m e n t
be t ween t he P -6 and P -8~ curves i s g r ea t e r . S i mi l a r l y , t he
d i f f e r e n c e b e t w e e n t h e t w o c u r v e s i n t h e i n i ti a l p a r t o f th e
curves i s i nc r eased bu t no t s i gn i f i can t l y so .
T o h a v e a q u a n t i ta t i v e e v a l u a t i o n o f t h e i m p a c t o f th e
c o n v e r s i o n , l o a d s a t p r e s c r i b e d m i d - s p a n d e f l e c t i o n s w e r e
o b t a i n e d f r o m b o t h t h e P - 6 a n d P - 8 ~ c u r v e s . T h i s w a s
c a r r i e d o u t t b r b o t h t h e p l a i n a n d S F R C s p e c i m e n s . T h e
d i f I ~ re n c e b e t w e e n t h e l o a d s o b t a i n e d f i ' o m t h e t w o c u r v e s
i s q u a n t i fi e d u s i n g t h e e x p r e s s i o n :
Pee
-
P~
A D = ~ • ( 4 )
w h e r e , A D = d i f f e r e n c e b e t w e e n t h e l o a d o b t a i n e d f r o m t h e
P - 8 a n d P - 8 ~ c u r v e s a t a p a r t i c u l a r p r e s c r i b e d m i d - s p a n
d e f l e c ti o n , P ~ = t h e l o a d e v a l u a t e d f r o m t h e P -8 ~ c u r v e a n d
P ~ = t h e l o a d e v a l u a t e d f i o m th e P - 8 c u r v e
T ab l es 5a and 5b show t he r e su l ts ca l cu l a t ed fo r t he p l a i n
T a b l e 5 a
-
C o m p a r i s o n b e t w e e n t h e l o a d s o b t a i n e d
f r o m t h e P - 8 a n d P - Be c u r v e s a t v a r i o u s m i d - s p a n
d e f l ec t i o n s f o r th e p l a i n N S C b e a m s
C n l n g e
0.1 0.15 0.2
P-8 5.38 3.30 2.19
V( ) ( IL6 ) (20.0) (23.4 )
P'Se 4.86 3.14 2,17
V ( ) ( 1 5 . 0 ) ( 1 8 . 9 ) ( 2 1 .9 )
AD( ) 9 .56 4,96 0.5 63
L o a d a t p r e s c r i b e d m i d - s p a n d e f l e ct i o n s k N )
0.25 0.3 0.4
1 .5 6 1 . 1 4 0 . 6 6 2
[ ( 2 L 5 ) ( 2 9 ) ( 3 3 . 2 )
1 .6 0 1 . 1 7 0 . 7 0 9
( 2 4 . 1 ) ( 2 8 . 3 ) ( 3 0 .6 )
2.49 3.22 7.16
Note: Figures in bold denote the prescribed mid-span
deflections in millimetres (mm)
Figures in brackets are the coefficients o f variation, V( )
N S C a n d H S C b e a m s r e s p e c t i v e l y . T h e d i f f e r e n c e , A D , i s
gen eral ly wi thin 10 indicat ing a sati sf~actory agr eem ent
be t w een t he P -8 and P -8~ c t u 'ves fo r t he m i d , span de f l ec t i ons
cons i de red .
T a b l e s 6 a t o 6 d s h o w t h e r e su l t s o b t a i n e d f o r t h e S F R C
b e a m s . T h e a g r e e m e n t b e t w e e n t h e P - 6 a n d P - 6 ~ c u r v e s i s
14
12
10
z 8
4
2
(1
8 r C M O O
ram)
P -8 cu rve
~ - P - C M O D c u r ve
0 5 1 1,5 2 2,5 3
8 o r C M O D r a m )
Fig. 15 - P-6 and P-CM OD curves for a C25/30 with 25kg/m 3
of f ibres (Dramix 65/60 B N) SFRC beam.
14
12
10
Z 8
m
o 6
0
8
or 6, rm~)
- - P-8 curve
P-8e curve
0.5 ~1 1,5 2 2,5
8 o r ~ r a m )
Fig. 16 - Typical P-6 and P-Be curves for a C25/30 with
25kg/m 3 of f ibres (D ramix 65/60 BN ) SFRC beam.
T a b l e 5 b - C o m p a r i s o n b e t w e e n t h e lo a d s o b t a i n e d fr o m
t h e P - 8 a n d P - Be c u r v e s a t v a r i o u s m i d - s p a n d e f l e c t i o n s
f o r t h e p l ai n H S C b e a m s
C u r v e L o a d a t p r e s c r ib e d m i d - s p a n d e f l e ct i o n s k N )
0.1 0.15
0.2 0.25
0.3 0.4
P-8 5.39 2.67 1.68 1.13 0.8 01 0. 47 0
........V ( ) ( 1 73 ) ( 1 7 , 9 ) ( 2 1 . 2 ) ( 2 L6 ) ( 2 4 . 3 ) ( 3Z 8 )
P-B e 4 .1 7 2 .3 5 1 . 5 4 0 . 9 8 5 0 , 7 4 5 [ 0 . 4 4 1
V ( ) ( 1 8 . 1 ) ( 2 1 . 3 ) ( 2 & 5 ) ( 2 6 .9 1 ) ( 2 4 . 9 ) [ ( 34 .9 )
AD ( ) 22.6 1i.9 8.18 13.2 6.95 6.24
Note: Figures in bold denote the preseribed mid-spa n deflections
in millimetres (ram)
Figures in brackets are the coefficients o f variation, V( )
e x c e l l e n t f o r t h e m i d - s p a n d e f l e c t i o n s c o n s i d e r e d w i t h t h e
d i f f e re n c e , A D , b e l o w 5 .
I n a d d i t i o n , t h e v a r i a t i o n s a t t h e p r e s c r i b e d m i d , s p a n
d e f l e c t i o n s h a v e b e e n p l o t t e d m a d a c o m p a r i s o n m a d e
b e t w e e n t h o s e o b t a i n e d f r o m t h e P - 8 c u r v e a n d f r o m t h e P -
8~ c u r v e , F i g s. 1 7 a n d 1 8 s h o w s u c h a c o m p a r i s o n i b r t h e
p l a i n c o n c r e t e s p e c i m e n s . S i m i l a r l y , F i g s . 1 9 a n d 2 0 s h o w
6 8
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M at e r i a l s and S t ruc t u r es
/
MatOriaux et Constructions V o l . 3 6 , N o v e m b e r 2 0 0 3
t h e c o m p a r i s o n f o r l h e S F R C b e a m s . I n g e n e r a l t h e _ o - [
v a r ia t io n d o e s n o t c h a n g e s i g n i fi c a n t ly w h e n t h e P - 8 c u r v e
i s t rans fo rmed in to the P-8~ cu rve us ing the p roposed
s i m pl e r i g i d bo dy mo de l . T h i s i s t r ue fo r bo t h t he p l a i n and __g
S F R C bea ms . ' ~ 30
T a b l e 6 a - C o m p a r i s o n b e t w e e n t h e l o a ds o b t a in e d f r o m
t h e P - 8 a n d P -B e c u r v e s a t v a r i o u s m i d - s p a n d e f l e c t i o n s f o r
t h e C 2 5 / 3 0 ( w i t h 2 5 k g / m 3 o f D r a m i x 6 5 /6 0 B N f i br e s )
S F R C b e a m s
Cu rve L oa d a t p r e sc r i bed r e d - span de f lec t ions (kN)
0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0
P-~ 7.75 7.54 7.31 6.98 6.60 6.33
V ( % ) ~ 2 4 . 3 )..... (26 ~7 ) (27,6) (28.2) (2 8~ 8) (29 .6)
P-6e 7.85 7.65 7.39 7.00 6,61 6.31
V(% ) (25.3) (27.9) (28.5) (28.9) (29.5) (30.1)
AD(%) 1.29 1.49 t .07 0 . 4 1 3 0.219 O. 197
Note: Figures in bo ld denote the prescribed mid-span deflections
in mill imetres (mm )
Figures in brackets are the coefficients of variation, V(%)
T a b l e 6 b - C o m p a r i s o n b e t w e e n t h e l o ad s o b t a in e d f r o m
t h e P - ~ a n d P -B e c u r v e s a t v a r i o u s m i d - s p a n d e f l e c t i o n s
f o r t h e C 2 5 / 3 0 ( w i t h 5 0 k g / m 3 o f D r a m i x 8 0 /6 0 B N f i b r e s )
S F R C b e a m s
F
L oad a t p r e sc r i bed m i d- span de f l ec ti ons (kN)
Cu rv e [ 0.5 1.0 1,5 2 . 0 2 . 5 3,0
P-8 16.5 18.0 t8.2 17.8 17.0 16.6
V(% ) (22.5) (21.9) (22:2) (22.9) (22,9) (22.8)
P-~30 16.6 18.0 18.2 17.7 16.8 16.4
V(%) (22.2) (21.9) (22.3) (23.2) (23.1) (23 .t)
AD(%) 0 .5 8 1 0.299 0.147 0.764 0.967 1.20
Note: Figures in bold denote the prescribed mid-span deflections
in mill imetres (m m)
Figures in brackets are the coefficients & varia tion , V(%)
T a b l e 6 c - C o m p a r i s o n b e t w e e n t h e lo a d s o b t a i n e d f r o m
t h e P - 8 a n d P -~ e c u r v e s a t v a r i o u s m i d - s p a n d e f le c t io n s
f o r t h e C 2 5 / 3 0 ( w i t h 7 5 k g / m ~ o f D r a m i x 6 5 /6 0 B N f i b re s )
S F R C b e a m s
Curve
P-5
Y( )
P-8~
v( )
L oad a t p r e s c r i bed m i d- span de f l ec ti ons k N )
0.5 1.0 1.5 2.0 2 .5 3.0
20.0 20.4 19.8 18.9 18.0 17.3
[(1 4,9 ) (16.0) (16,9) ( 6 ,8) (15.5) (15.( ) )
/9.9 20.1 19.5 18.4 17.6 16.7
(14 .7) (14 .9) (16 .2) ........... 15 .5) (13 .9) (14 .6)
AD(% ) 0.127 1.39 1.88 2.47 2.07 3.18
Note: Figures in bo ld denote the prescribed mid-span deflections
in millimetres 0rim)
Figures in brackets are the coefficients o f variation, V (%)
T a b l e 6 d - C o m p a r i s o n b e t w e e n t h e l o a ds o b t a i n e d f r o m
t h e
P - ~ a n d P - ~ e c u r v e s a t v a r i o u s m i d - s p a n d e f l e c t io n s
f o r t h e C 7 0 / 8 5 ( w i t h 2 5 k g / m 3 o f D r a m i x 8 0 / 6 0 B P f i b r e s )
S F R C b e a m s
Curve
P-8
V(%)
P-6~
V(%)
AD(%)
L oad a t p r e sc r i bed m i d- span de f l ec t ions , kN)
0.5 1 .0 1 .5 2 .0 2 .5 3 .0
13.2 t5.6 16.8 17.3 17,2 16.7
( 3 0 . I ) . .. . ( 3 0 .7 ) ( 3 ] , 4 ) ( 3 ] . 5 ) ( 3 1 . 2 ) ( 3 1 .9 ) .....
13.2 15.5 16.5 17.0 16.9 16.5
(25.5) (25.5) (26.7) (26.8) (26.4) (26.1)
0.171 1.10 1.84 1.86 1.96 1.43
Note: Figures in bo ld denote the prescribed m id-span deflections
in millimetres (ram)
Figures in brackets are the coefficients &v ariatio n, V (%)
i
O
c
O
O
20 84
~0
O
O
l>
O
<>
D
O
n n
o
C25/30 CNSC)
I l l C70 /85 (HSC)
o .~ 0 .2 0 ,3 04
A v e r a g e m i d - s p a n d e f l e c ti o n , 6 ( r a m )
0.5
Fig. 17 - Variation of measured load at prescribed average
mid-span d eflections, 8 (variation obtained from P-8 c urves)
for the plain concrete beams.
| J
ii
E
3
20 84
r
. ~ I 0
o
G
O
O
r NSC)
O C70/85 HSC)
O
r O
O O
O
O
D
OA 0 ,2 0 ,3 0 .4 0 ,5
E q u i v a l e n t m i d . p a n d e f l ec t i o n , 5 e ( m m t
Fig. 18 -V ari ati on o f measured load at equivalent mid-
span deflections, 8~ (variation ob tained from P- ~ curves)
tbr the plain concrete beams.
40
>
30
E
15 2 0
O
i 1 0
0
X X X X X X
0
O O D D ~ G
A A
A t~
0 c25/30(25)
Dc25~30(50)
X C 7 0 J ~ 5 2 5 ~
0 5 1 1 5 2 Z 5 3 3 5
A v e r a g e m i d - s p a n d e f l e c t i o n , 8 ( ra m )
Fig. 19 - Variation o f measured load at prescribed average
mid-span d eflections, 6 (variation obtained fi'om P-8
curves) for the SFRC beam s,
6 2 9
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T C 1 6 2 - T D F
40
~ 2o
@
G
O O 0 O
X X X X
X
0 El 0
r l I 0 I : l
A A
A
A
o
C 2 5 / 3 0 (2 5 )
a C 2 5 / 3 0 (5 O)
C 2 5 / 3 0 (7 5 )
X C 7 0 / 8 5 (2 5 )
0 0 . 5 1 t 5 2 2 . 5 3 3 . 5
E q u i v a l e n t m i d - s p a n d e f l e c t i o n , c b ( r a m }
Fig. 20 - Variation o f measured load at equivalent mid-span
deflections, tS~ (variation o btained from P-8r cu rves) f or the
SFRC beams.
In sum mary, i t is found that the s imple r igid bod y model is
appropria te lbr convert ing the P-C MO D curve into a P-6, curve.
The P -8~ curve i s found to agree we l l wi th the expe r im enta l ly
obtained P-6 curve, especia l ly for the SFR C beams.
It is therefore suggested that the P-CM OD curve be used to
obtain des ign parameters ra ther than the P-6 curve. ~ s wil l
faci li ta te the possibil ity of measuring CM OD alone to obtain the
parameters us ing a s imple rela tionship betw een 6 and CM OD as
given by Equat ion (3) and thus save set-up t ime and cos t . This
s imple re la t ionship should be u sed to o btain a c orrect ion factor
to in te rchange be tween 6 and CM OD ( tb r exam ple fo r a kn it~
edg e thickness, do = 5 mm , the ratio of~SJCMOD = 0.8065). Thi s
method can be used to e i ther evaluate toughness or res idual
s trengths us ing the P-C MO D curve.
6 C O N C L U S I O N S
The s tudy repor ted he re shows tha t the re would b e a ce r ta in
am ou nt o f e r ro r wi th rega rd to d i f fe ren t va lues o f d ,, be ing used
wh en condu c t ing a beam -bending t es t. Th e am oun t o f e r ro r
introduced in general increases as d, , increases . From the
analyt ical analysis , fo r an error level o f 10%, a knife- edge of
5ram w ould be accep tab le . Bea r ing in m ind the va r ia tion due
to the dis tr ibut ion of f ibres a knife edge thickness o f 5mm is
thus recom m ended . Th i s va lue was chosen a s a com prom ise
be tween accuracy o f m easurem ent and prac t ica l cons ide ra t ions
(especial ly for pract is ing engineers) ,
The accuracy of a p roposed m ode l , based on r ig id body
kinematics, to obtain a relationship between 6 and CMOD was
investigated. Analysis was carried out asing both experimental
and analytical results. In both plain and SFRC beams, it was
found tha t there was c lose agreem ent be tween the m ode l and the
experimental and analyt ical resul ts . Another s tudy conducted
showed that this re la t ionship between 6 and CMOD was
independen t of the f ibre contents and the concrete s t rengtl~s
considered, Fu rthermore, P-CMO D curves were converted into
P-8~. curves us ing the s imple r igid b od y model . T his was found
to be in c lose agreement with the experim ental ly measured P-6
curve e s~e ia l ly fo r the S F RC beam s .
The resul ts show that toughness can be evaluated from ei ther
the P-6 o r P-C M OD curves . W ith the support o f both a~,lalyt ical
and experimental evidence, i t is recommended that the P-
CM OD curve be used f or evaluat ing the toughness or residual
s trengths of SFR C. H owev er, the re lat ionship betwee n P-6 and
P-C M OD needs to be. es tabl ished tbr may given ge ometry.
A C K N O W L E D G E M E N T S
The work repor ted in th i s pape r fo rm s pa r t o f the Br i t e -
E u r a m pro jec t Tes t and Des ign Methods fo r S tee l F ib re
Re inforced Concre te , con t rac t no . BRP R-CT98-0813 . The
partners in the project are : N.V.Bekaert S .A. (Belgi tm~ co-
ord ina tor ) , Cen t re S c ien t i f ique e t Techn ique de l a Cons t ruc t ion
(Be lg ium ) , Ka tho l ieke Unive rs i t e i t Leuven (Be lg ium ) ,
Technica l Unive rs ity o f Dem n ark (Denm ark) , Ba l fou r Bea t ty
Ra i l L td (Grea t Br it ain ), Unive rs i ty o f W ales Card i f f (Grea t
Bri ta in) , Fert ig-Decken-Uqqion CanbH (Germany), Ruhr-
Unive rs i ty -Bochum (Germ any) , Technica l Unive rs i ty o f
Braunschweig (G erm any) , F CC Cons t rucc ion S . A. (S pa in) ,
Univers i ta t Polytrc nica de Catal tmya (Spain).
R E F E R E N C E S
[l] RILEM TC 162-TDF, RILEM TC 162-TD F: Test and design
methods for stee l fibre reinforced concrete. Bending test,
Mater S t ruc t 33 (2000) 3-5.
[2] Barr, B.1.G. and Lee, M.K., 'Definition o f Round Robin
Test. Preparation o f Specimens. Execution and Evaluation of
Round Robin Testing' , Report t?om Test and Design
Methods for Steel Fibre Reinforced Concrete, EU Contract -
BRPR - CT98 - 8t3 (2001) 105 p.
[31 B arr , B.I.G , Lee, M.K ., de P lace Hansen, EJ. , D upora, D.,
Erdem , E., Schaerlaekens, S., Schn fitgen, S., Stang , H. and
Vandewalle, L. , 'Round robin analysis of the RILEM TC 162-
TDF beam-b ending test: Part t - Test method evaluation',
?dater Struct 36 (2 63) (2003) 609-620,
[4] R1LEMTC 162-TDF, RILE M TC 162-TDF. Test and design
methods for steel fibre reinforcedconcrete. 6-~ design method.
Recommendations', Mater Struct 33 (226) (2000) 75-81.
[5] Gopalaratnam, V.S., Sh ah , S.P . , Ba tson , G.B., Criswell,
M.E., Ramakrishnan, V. and Wecharatana, M., 'Fr~,'ture
toughness of fiber reinforced concrete' , A C I M a t e r i a l s
J o u r n a l 88(4) (1991) 339-353.
[6] Gop alaratnam, V.S ., Gettu, R., Carmona, S. and Jamet, D.,
'Characterization of the toughness of fiber reinforced
concretes using the load-CMOD response' , in Proceedings
FRAMCOS-2, edited by Wittmann, F.H., AED1FICATIO
Publishers, Freiburg (1995) 769-782.
[7] Barr, B., Gettu, R., AI-Oraimi, S.K .A. and B ryars , L.S. ,
'Toughness m easurement - the need to think again' , C e m e n t
and Concre te Compos i tes 18 (1996) 281-297.
[8] Navalurkar, R.K., Hsu, T.T.C., Kim, S.K. and Wecharatana,
M., 'True fi~cture energy of concrete' , A C I M a t e r i a l s
J o u r n a l 96(2) (1999) 213-227.
[9] Ferreira,L., G ettu, R. and Bittencourt, T. , 'Study of Crack
Propagation in the Specimen Recomm ended by RILEM TC
162-TDF Ba sed on Linear Elastic Fracture Mect-~aics', Report
from Test and Design Meth ods for S teel Fibr e Reinforced
Concrete, EEl Co nt rac t-B R PR - C '1 98- 813 (2001) 9p.
[10] Stang, H., 'Analysis o f 3-point bending test ' , Report from
Test and Design Methods tbr Steel Fibre Reinforced
Concrete, EU Contract - B R P R - CT98 --- 813 (2001) 7p.
6 3 0