The International Journal of Cement Composites and Lightweight Concrete, Volume 10, Number 3 August 1988

Model l ing of fibre pull-out f rom a cement matr ix

Youjiang Wang*, Victor C. LF and Stanley Backer*

Synopsis Theoretical analyses of fibre pull-out from a matnx reported in the hterature are bnefly rewewed The effects of Poisson's ratio, elastic-frictional bond strengths, and bond strength vanat=on with shppage d=stance on the pull-out relation are discussed. A theoretical model motivated by observations of fibre surface abraston ~s developed to predict the pull-out force versus d~splacement relationsh=p. The model takes into considerat=on the vanat~on of the frict=onal fibre-matrix bond strength wtth fibre shppage d=stance. Good agreement is ach=eved between model pred0cted pull-out behav=our and experimental pull-out curves for nylon and polypropylene fibres For these synthet=c fibres, the bond strength increases wtth the shppage d=stance dunng the process of pull-out. The model also pred=cts reasonably well the pull-out behav~our of steel fibres for which the bond strength decreases w=th the shppage dtstance.

Keywords Mathematical model, fibre remforced concrete, bonding strength, portland cements, nylon fibres, polypropylene fibres, metal fibres, pull-out tests, fibre cement compos=tes, f=bre--matnx bond, bond stress, shear strength, frictional bond.

INTRODUCTION F=bre pull-out tests are often used to study the fibre- matnx bond behavtour m fibre reinforced cement compo- sites (FRC), a factor strongly mfluenc=ng the properttes of the compos=tes. The pull-out test 0s also =mportant by itself as it simulates the fibre bridging-pull-out phenom- enon dunng the fracture process of FRC In relating the pull-out test results with the fibre-matnx bond character- ~sttcs, numerous models have been developed and many of them have been revtewed [1,2] A uniform shear bond strength between the fibre and the matnx as often assumed =n FRC models [3, 4] and the bond strength from pull-out tests is frequently reported on terms of the average value over the embedded fibre surface area [1, 5, 6] Obviously, the uniform shear stress does not necessanly represent the actual stress state at the fibre-matnx interface.

When a load is apphed to pull a fibre out from a cement matnx, triaxial stress state generally ex=sts at the fibre-matnx interface, pr0manly due to the rad=al contrac- t=on of the fibre The effect of fibre radial contraction (Po~sson's effect) during pull-out was d~scussed by Kelly and Zweben [7], Pinchm [8], and Baggott and Gandh= [9] Their analyses show theoretically that for certain types of

* Department of Mechamcal Engmeenng and t Department of Cwd Engineenng, Massachusetts Institute of Technology. Cambndge, MA 02139

Received 25 January 1988 Accepted 20 Aprd 1988

(~ Longman Group UK Ltd 1988

0262-5075/88/10301143/$02.00

fibres (e.g. polypropylene), the pull-out process will revolve unstable debondmg and concrete reinforced w~th these fibres ahgned in the matrix w011 not exhibit multiple cracking due to their unfavourable Potsson's ratios However, the experimental results of Baggott and Gan- dh~ [9] indicated that slight mJsalJgnment m the specimen and the aspent~es of the fibre surface could offset the Polsson's effect Multiple cracking of FRC reinforced with ahgned polypropylene fibres was observed tn their d~rect tenston test, which was not expected accordmg to the theones for perfect fibre ahgnment and smooth fibre surface S~nce shght m~sahgnment ~n pull-out test ~s unavoidable and no unstable load drop has been observed even ~n the authors' pull-out tests of nylon and polypropylene fibres which have relatwely h~gh Polss- on's ratios [10], ~t follows that in general the Po~sson's effect can be neglected in fibre pull-out analysis and that a one dimensional model ts usually adequate to descnbe the fibre pull-out test.

Lawrence [11] and Gopalaratnam and Shah [12] have obtained one dimensional solutions to the fibre pull-out problem in which gradual debondmg of the elastic bond was considered (Figure 1) In both models, a perfect elastic bond between the fibre and the matnx is first assumed and the elastic shear stress field ~s obtained Debondmg takes place when the maximum shear stress reaches the elastic bond strength, "rs. The shear stress ~n the debonded regaon ~s determined by the frict0onal bond strength, "r,. Both % and -r, are assumed to be constant during the pull-out process. Because of the inadequately defined geometry in Lawrence's model whach uses some parameters to lump the effect of geometry and matenal properties, the use of th~s model

143

Modelhng of hbre pull-out from a cement matnx Wang, L~ and Backer

S M a t r i x

F i b e r

.h'...'h'/I// L Elastic Fnchona l Bond Bond

J

Figure 1 Schematic illustration of a partially debonded fibre during pull-out

l l t embec lment length ('~/2)=0.51n fiber cliemeter =O.O1 ~n ~,~:600 psi, 7,~= 275 psi

n

6

0

Q.

j:atast rop t c /~ ~//clebond, ng ~[w)

i t fmctlonol F ~ , , , ~ p u l l-out

~ ~ --- expt, N Q Q m G r l ",, ~ and Shoh

" " , . . . _ . .____~pu l l-out mode=

0 0 0.1 0.2 0.3 0.4 0.5 0.8 0.7

dlsplocement , (~ (m)

Figure 2 Comparison of the pull-out load displacement behaviour predicted by Gopalaratnam and Shah's model [12] with experimental results by Naaman and Shah [13] (11b = 4.45N, 1 in. = 25.4mm, I psi = 0.0069MPa; After Gopalaratnam and Shah [12])

is very hmlted [1 ] The model by Gopalaratnum and Shah has been used to predict the result of steel fibre pull-out test [12] (Rgure 2)

It has been shown [11, 12] that both % and .r, are important parameters to characterise the debondmg process of fibre pull-out and to determine the conditions for unstable debondmg However, the effect of the elastic bond strength, %, on the overall pull-out response ,s not always significant During debondmg, the shear stress =ntenstty in the elastically bonded region decays approximately exponentially with the distance from the debondmg point, wtth a maximum stress equal to % The length of the fibre segment strongly influenced by this elastic stress field (the length over which the shear stress ~s not near zero) is dependent on, among other quantities, the rat,o between fibre modulus (El) and the matrix modulus (Em), and the fibre cross-sectional area For low modulus fibres, or even high modulus fibres with high aspect ratio (length to diameter ratio), the elastic stress field has relatively httle effect on the pull-out

toad-d~splacement relation except at the very early stage of debondlng Figure 3 shows a calculation example In which the Ioad-d~splacement curves for both % = 2~', and % = 0 are compared for cases with El/Era = 10 but w~th different fibre lengths, based on the theoretical relations derived by Gopalaratnam and Shah [12[ As shown by this specific example, there is no noticeable effect of the elastic bond strength (%) on the pull-out Ioad-d,s- placement relations for embedded length L1 = 20mm, and some, but not significant effect for L1 = 5ram From Figure 3(a), it appears that the inclusion of the elastic shear stress field m the calculation would result ~n an initially less stiff P-6 response than for % = 0 This could be caused by neglecting the matrix deformation ~n the calculation of fibre pull-out displacement [12] Presuma- bly the matnx deformation is more Iocallsed and larger near the exit end of the fibre for the case of bonded fibre, and more uniform along the fibre length for the case of purely fnctlonal resistance If the P-6 curves in Rgure 3(a) were corrected for the matrix deformation, the discrepancy in the stiffness of these curves noted earher could be expected to disappear It can be shown that for lower El/Era rabo or higher fibre aspect ratio, the % effect ~s even less Notice further that the d~splacement before complete debondmg corresponds only to a minute fraction of the total pull-out displacement. After com- plete debond~ng, the elastic bond strength has no consequent effect Therefore ~n most cases the pull-out analysis can be slmphfied by neglecting the elastic stress field

L ,, I IL,

Z v

C~

3

;20

15

10

(a) L1=5 m m /

Ts~O

i

0 0.5

8(

6C

40

20

(b) LI=20 mm

. !

0 5 10 15

Figure 3

DISPLACEMENT 6 (~m)

Effect of the elastic bond strength % on the theoretical load displacement response before complete debonding: (a) L1 = 5mm, (b) L1 = 20mm (~, = 2MPa, fibre diameter dt = 0.5mm, Ef = 270GPa, Em = 27GPa, matrix shear modulus Gm = 11.3GPa, matrix cross-sectional area Am = 200mm 2)

144

Modelhng of hbre pull-out from a cement matr,x Wang, Li and Backer

The fnct~onal bond strength ('r,) ~n general vanes with the slippage distance (s) between the fibre and the matnx during the process of pull-out. The effect of the variation of "r, with s has been discussed by the authors [10] In steel fibre pull-out, ~, often decreases with s as a result of the breakdown of the cement at the fibre-matnx ~nterface due to the stiffness and hardness of the steel fibre The decrease in -r, dunng steel fibre pull-out ~s responsible for the significant d~screpancy between the experimental curve and theoretical result by Gopalarat- nam and Shah using a constant .r,, as seen nn the descending part In Figure 2 In synthetic fibre pull-out test w~th nylon and polypropyelene fibres, ~, was found to increase w~th s due to fibre surface abrasion and accumulation of wear debnes [10] Clearly, it is haghly 0naccurate to simply use a constant fnctnonal bond strength to descnbe the fibre pull-out process after complete debondmg of the elastic bond In this paper, a theoretical model ~s developed whuch takes into con- s=derat=on the vanat~on of the frictional bond strength during pull-out The pnmary objective of this study ~s to understand the behav=our of a fibre =n an FRC member during fracture W~th additional cons=derat0ons of fibre onentat=on and fibre d~stnbut~on, the results of th~s study can contnbute to the predict=on of FRC constututwe relations

THEORETICAL MODELLING Basic assumptions Among the many configurations often used for the pull-out test, the pull-out of a fibre embedded across a matnx crack, dlustrated m Figure 4, 0s of partncular ~nterest because of ~ts s0mflanty to the fibre bndgmg s~tuat~on 0n FRC fracture process To predict the pull-out load vs crack separation relation for th0s 'two-sided pull-out' problem, a few s~mphfymg assumptions are first made. The fibre ~s assumed to be hnear elastic and sufficmently strong to enable complete pull-out without rupture The contnbut~on of matnx deformation to the crack separation 0s neglected Based on the above d~scussJon, both the Po~sson's effect and the elastuc bond strength are neglected ~n th~s analys~s The destmc- twe feature of th~s model hes ~n ~ts assumption that the fibre-matnx fnct~onal bond strength (denoted as "r after- wards) for an infinitesimal fibre segment ~s a funct;on of

Flber. .~ L1

Matrix

L 2

Matrix

" " - ' - I ~ P

- " ~ 8

Figure 4 Schematic illustration of a pull-out spemmen configuration

BEFORE LOADING:

2' 1 x

L 1

DURING PULL-OUT: i

s(o) . j s(x) j ~ . .~ 4

(X=O) (x=L i)

Figure 5 Illustration of durect fibre pull-out problem

/J - " ~ P1 2

nts slippage distance (s) wroth respect to the matrix, described by a "r-s law Using such a "r-s relationship for the fractional bond strength instead of a constant value has been shown necessary in the prevnous section

The direct pull-out problem Before solving the two-sided pull-out problem, the direct pull-out (one-sided) of an embedded fibre from a cement matrix, shown in Figure 5. needs to be considered The x coordinate system in the figure is w~th respect to the undeformed fibre geometry (before loading) and ~s associated with the material partncles For example, during the process of pull-out, x always equals 0 for Point 1. the embedded end. and equals L1 for Point 2, the loading end

The slippage dtstance, s(x). wuth respect to the matrix of an infinitesimal fibre segment at x is equal to the sum of the slippage dJstance of the fibre embedded end, s(O), plus the elastic elongation of the fibre segment located between 0 and x

s(x) = s(0) + fXe(x) dx (1) o

Note that s(0) = 0 before complete debonding, and s(0) vanes from 0 to L~ during subsequent pull-out The axial strain of the fibre, e(x), can simply be calculated from the fibre axtal load P(x)'

4 ~(x) = P(x) (2)

11" d f 2 Ef

and P(x) is given from the equilibrium condition:

~'X-dx) P(x) = Po + df [1 + e(x)l dx (3) o

where Po 0s a constant representing the fibre end anchorage effect when the fibre end sltps (Po = 0 before

145

Modelling of hbre pull-out from a cement matrix Wang, LI and Backer

Pmax

I ~ L1 _~

I - -;I P1

• (s)

P1

~ 61 ,L 1

P2 p - '?-JI P2

~max ~ ÷ u n l

L2

oading

| I t I I I I I

k 2 6 2

Figure 6 Illustration of calculation procedures for the two- sided pull-out problem

~ 6 P Pmax'

P = P1 = P2 6 : 6] + 6 2

L 1

6

complete debonding) Po usually ~s very small, and can be estimated from the expenmental load vs crack separa- tion curves. The shear stress, -~(x), between the fibre and the matrix is determined by the slippage distance, s(x), given by the ~'-s law as "r(s), for fibre sections reside the matrix:

,r(x) = ~(s(x)) for x+s(0) < L1 (inssde matrix) (4a) "r(x) = 0 for x+s(0) /> L1 (outsldematrlx) (4b)

These equations, equation (1) to equateon (4), are, in general, coupled nonlinear questions, but can be solved by numencal methods w~thout ~terat~on

At each loading stage, the load and the correspond- mg d~splacement can be obtained after solwng these equations from

P1 = P(x = L1) (5)

61 = S(X = L-l) (6)

The two-sided pull-out problem For a fibre embeddedma matnx across a crack plane, as that shown ~n Figure 4, the pull-out of the short and the

longer fibre segments are first analysed separately, and then these load and d~splacement responses are combined to obtain the total load-displacement relation, as illustrated in Figure 6 Detads of this procedure are described in this section

The fibre in the matrix is not directly loaded by external forces Instead, the load =n the fibre is trans- ferred from the fibre-matrix interface. Apparently, the longer fibre segment (L2) would provide a higher pull-out force than the shorter segment (L1) if they were pulled out independently For this reason, the shorter fibre segment wdl be pulled out from the matrix as though it were directly pulled out by an external force, and the Ioad-d~splacement relation P~-81 derived ~n the previous section apphes to th~s process Due to equ~hbnum constraint, the pulling forces for both segments must be equal, l e P~ = P2 The maximum pulling force experi- enced by the longer fibre segment dunng the two-sided pull-out wdl be the same as that for the shorter fibre segment

Before reaching the maximum load Pm~×, the longer fibre segment (L2)is monotonically loaded with pulling force provided by the shorter fibre segment (L1), and the

146

Modelling of hbre pull-out from a cement matnx Wang, L/ and Backer

longer segment deforms and shps ~n response to the load according to the Ioad-d~splacement relattonship derived for the one-sided pull-out problem (after replacing subscript '1' with '2' tn equations 1-6)

The axial force, the axml strata, and the shppage d0stance states at P = Pmax provide the ~ntt~al cond0t~ons for the unloading process Refernng to F~gure 7a, the ax0al force, strata, and shppage d~stance as functions of coordinate x at the instant of P = Pmax are designated as Prn(X), era(x) and s~(x), respecttvely After reaching the max0mum load Pmax, the load ~n the shorter segment wdl drop, and so will the load in the longer segment (P2) Since the longer segment stdl has the potential for h~gher loads and the load actually decreases, the segment can no longer be pulled out and ~ts embedded end wdl remain where It was when the load reached Pmax Dunng unloading, part of the elastic elongation of the longer fibre segment ~s recovered and the fibre tends to 'shnnk back' into the matnx, and th~s ~s resisted by the f ibre- matnx tnterfactal fnct~onal stress For a gtven unloading mstant, a parameter h =s defined to be the x-coordinate at which the shear stress reverses tts d~rect~on as shown ~n F~gure 7b The fibre sect=on w=th x ~< h =s not affected by unloading For x <~ h, the axial force, stratn, and shppage dtstance assoc0ated wtth Pm~× are mamtamed at Pro(x), Era(X), and Sm(X), ~e

(a) AT 1°2 = Pmax I I P, =

"c Ix)

I

J f P~,,x

I l._ x

L 2

(b) DURING UNLOADING

x=h,,%

, q "c(x)

2(x)

0

p- P=

P2

. x h L 2

Figure 7 Schematic diagram for determination of load displacement relation at Pma, and during unloading for the longer fibre segment (L2)

P(x) = Pm(x) (x ~< h) (7)

e(x) = era(x) (x ~< h) (8)

s(x) = sm(x) (x ~ h) (9)

Beyond h (i.e for x > h), the load is found from equd~bnum cond~ttons as

P(x)=Pm(h)-~X'r(x)'~df[1 +e(x)]dx (x>h) (10)

and the slippage ~tstance at x ~s given by the shppage distance when P = Pmax (sin(x)) plus the sliding dtstance due to fibre elast0c shnnkage: /x

S(X) = Sin(X) + [em(h) - e(x)J dx (x>h) I l l ) h

S~mdar to equation (4), the shear stress d~stnbut~on Js gtven by

"r(x) = "r (s(x)) for x + s (0) < L2 (reside matnx) (12a)

"r(x) = 0 for x + s (0)/> L2 (outside matrix) 112b)

and the relatoonshlp between the axial force (P(x)) and the axial strata (((x)) Is the same as equation (2) Once these equations are solved, the pull-out load and the d~sptacement corresponding to the instant ~n the unload- mg process can be obtained from

P2 = P(x = L2) (13)

52 = s(x = L2) (14)

By decreasing h from L2 (no unloading when P2 = Pmax) untd P2 = O, the P2-52 relationship for the unloading process can be obtained

To calculate the complete Ioad-d~splacement (crack separation) relationship, first the P1-51 curve for the shorter fibre segment ~s computed, from which the maximum load Pma× Is identified Then the Ioadmg and unloading parts of the P2-52 curve for the longer fibre segment are calculated Fmally, by adding the displacements of both segments together (5 = 51 + 52) for the corresponding load (P = P1 = P2), the complete load vs crack separation curve, P-5, ~s obtained Thts procedure has been dlustrated schematacally tn F~gure 6 Note that the function 5(P) ~s multwalued, therefore thts operation must be performed on the branches before and after P reaches Pmax separately

Theoretical predictions Theoretical load vs crack separation curves for the pull-out tests were calculated based on the model The "r(s) functional relationship used Jn the model can be expressed =n any convement form, such as hnear, multdmear, p~ecew~se constant, or polynomial In the calculations, the fnctlonal bond strength (-~) was assumed to be a quadratic function of the shppage dtstance (s).

"ds) = ao + al s + a2 s 2 (15)

where ao, al, and a2 are constants, determ=ned empm- cally such that the theoretical curves are reasonably close to the expenmental results

147

Modelhng of hbre pull-out from a cement matnx Wang, L~ and Backer

The theoretical predictions for nylon and poly- propylene fibre pull-out are shown in F0gure 8 and the expenmental results are gwen ~n F~gure 9 In these tests, the specimen configuration shown ~n Ftgure 4 was used, with df = 0.508mm, L~ = 50mm, and L2 = 70mm, for both nylon and polypropylene Cement paste matnx was used and the specimen age at test was three days Details of the expenments have been reported else- where [10]. Figure 10 shows the theoretical prediction for steel fibre pull-out together with the expenmental result reported by Naaman and Shah [13] The difference between the predicted and the expenmental curves ~n the ascending part in F~gure 10 is presumably due to the inclusion of deformations other than fibre elongation (e g testing machine and fixture deformations) in the

expenmental curve, for a fibre elongation of 0 25mm (displacement near peak load in expenmental curve) would correspond to a uniform fibre strata of 2%, requmng an axial load over 800N ~n the fibre if the steel fibre could stdl remain hnear elastic

By comparing the expenmental load vs crack separ- aUon curves w~th the theoretical results, ~t can be seen that the proposed model is ~ndeed able to predict closely the pull-out test result. However, it should be pointed out that the ' r -s relations determined empmcally are unhkely to be fully independent of geometry and loading condF t~Qns An alternatwe expression based on the theory for fibre on cement fnctlon and wear might be more desmrable to represent the -r-s relationship

It ~s interesting to note from these results the strong

L4" F-- L-- ,~6

v

0

15

I0

5 ..J

0 20 40 0

s (~)

Polypropylene

20 40 60

CRACK SEPARATION 6 (mm)

Figure 8 Theoretical predictions for nylon and polypropylene fibre pull-out. Figure on left shows the ~-s relations used. (dr = 0 .508mm, Ef = 1GPa, LI = 50mm, L2 = 70mm, Po = 0)

Ale 12o ,b, i i 5 I0

0 , 0 , 0 20 40 60 80 0 20 40 60 80

CRACK SEPARATION a (mm)

Figure 9 Exper, mental results for the fibre pull-out tests: (a) Nylon fibre, (b) Polypropylene fibre

2 C}.

0

30 , LI p

20

TM

_.J

0

0 5 I0

s (mm)

I " I

0 5 10

DISPLACEMENT ~ (mm)

15

Figure 10 Comparison of model prediction (solid line) with experimental result (broken line) by Naaman and Shah [13] for steel fibre pull-out test. Figure on left shows the ~-s relation used. (dr = 0 .254mm, Ef = 209GPa, L1 = 12.7mm, Po = 2N)

148

Modelling of hbre puff-out from a cement matnx Wang, LI and Backer

~nfluence of the ~'-s relation on fibre pull-out response In the case when -~ increases w~th s due to fibre surface abrasion as seen ~n synthetic fibres hke nylon and polypropylene, the pull-out load could remain at relatwely hgh level over a w~de range of crack separation, and the final crack separateon could be greater than the embedded length of the shorter fibre segment. These contnbute to an ~ncrease ~n energy absorption dunng pull-out and could have ~mportant ~mphcattons on the crack resistance of FRC [14]

CONCLUSIONS Ftbre pull-out Js often modelled with an elastic bond strength and a fncttonal bond strength for the fibre- matnx interface However, the fnct~onal bond strength generally vanes w~th the fibre slippage distance relatwe to the matnx In th~s study, a theoret=cal model for the pull-out test has been developed whtch takes =nto account the bond strength vanat=on dunng pull-out Good predictions of the expenmental load vs crack separat=on relat=ons are obta=ned for pull-out tests w=th nylon, potypropylene as well as steel fibres

ACKNOWLEDGEMENTS The authors would hke to acknowledge the support of the Sh~mmzu Construction Company, Ltd and the Pro- gram of System Engmeenng for Large Structures at National Science Foundation

REFERENCES 1 Bartos, P, 'Review paper Bond in fibre reinforced

cements and concretes', The International Journal of Cement Composites, Vol 3, No 3, August 1981. pp 159-77

2 Gopalaratnam, V S and Shah, S P, 'Fadure mechan~s~sm and fracture of fibre re~nforced con- crete', ACI Special PubhcatJon SP-t05 Fibre Rein- forced Concrete Properties and Applications, Amencan Concrete Institute, 1987, pp 1-25

3 Aveston, J, Cooper, G A and Kelly, A, 'Single and multiple fracture', ~n The Properties of Fibre Composites, Conference Proceedings, National Physlca Laboratory, IPC Science and Technology Press Ltd, November 1971, pp 15-26

4 Wang, Y, 'Mechanics of fibre reinforced concrete', S M Thes~s, Massachusetts Institute of Tech- nology, 1985

5 Mmdess, S and Young, J F, 'Concrete', Pnnt~ce- Hall, Inc, Englewood Chffs, N J, 1981. pp 633

6 Wang, Y. Backer, S and D, V C, 'An experimental study of synthetic fibre reinforced cementmous composites'. Journal of Materials Science, Vol 22, No 12, December 1987, pp 4281-91

7 Kelly, A and Zweben, C, 'Polsson contraction in aligned fibre composites showing pull-out', Journal of Materials Science, Vol 11, No 3, March 1976, pp 582-7

8 Pmchm, D J, 'Potsson contraction effect =n aligned fibre compos0tles', Journal of Materials Science, Vol 11, No 8, August 1976, pp 1578-81

9 Baggott, R and Gandhi, D, 'Multiple cracking in aligned polypropylene fibre reinforced cement composites', Journal of Materials Science, Vol 16, No 1, January 1981, pp 65-74

10 Wang, Y, D, V C and Backer, S, 'Analysis of synthetic fibre pull-out from a cement matnx', ~n Bonding in Cementltlous Composites, Edited by S Mmdess and S P Shah, Material Research Society Sympos0um Proceedings, Vol 114, Matenals Research Society, P~ttsburgh, 1988, pp 159-65

11 Lawrence, P, 'Some theoretical considerations of fibre pull-out from an elastmc matnx', Journal of Matenals Sctence, Vol 7. No 1, January 1972, pp 10-6

12 Gopalaratnam, V S and Shah, S P, 'Tens=le fracture of steel fibre reinforced concrete', ASCE Journal of Engmeenng Mechanics, Vol 113. No 5, May 1987, pp 635-52

13 Naaman, A E and Shah, S P, 'Pull-out mechanism in steel fibre re~nforced concrete', Journal of the Structural Dwlslon, Amencan Society of Civil Engi- neers, Vol 102, No ST8, August 1976. pp 1537- 48

14 D, V C, Wang, Y and Backer, S, 'Effect of fibre-matnx bond strength on the crack resistance of synthetic fibre re~nforced cementrtlous composl- t~es', in' Bonding ~n Cementtt~ous Compos0tes, Edited by S Mmdess and S P Shah, Matenal Research Society Symposium Proceedings, Vol 114, Matenals Research Society, P~ttsbough, 1988. pp 167-73

149