Round-robin analysis of the RILEM TC 162-TDF - 2.pdf

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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



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




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



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




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



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



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



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



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



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

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