University of Nigeria Creep of Polypropylene...in Civil Engineering Department of the University of...

University of Nigeria Research Publications

MBAEZUE, J.I.

Aut

hor

PG/M.Sc/Ph.D/81/1257

Title

Tensile Creep of Polypropylene Fibre Reinforced

Concrete Laminate

Facu

lty

Engineering

Dep

artm

ent

Civil Engineering

Dat

e 1985

Sign

atur

e

1

TITLE PAGE

TENSILE CREEP OF POLYPROPYLENE

FIBRE REINFORCED CONCRETE LAMINATE.

Engr. J. I . Mbaezue,

PG/M.Sc/Ph.D/81/1257

Submi t ted i n F u l f i l m e n t of t h e Requirement

f o r t h e Rward o f Doctor o f Ph i losophy

i n C i v i l Eng ineer ing Department o f t h e U n i v e r s i t y o f N i g e r i a ,

Nsukka.

D E D I C A T I O N -------------------

i I I

ACKNOWLEDGEMENTS

My spec ia l g r a t i t u d e goes t o Pro f . R.M.Madu (my Superv iso r ) ,

Dr. N.Egbuniwe and Dr.A. S t r ze l czyk , f o r t h e i r ass i s t ance and guidance

i n niy p resen t work.

S ince c reep observa t ions r e q u i r e a l o t of pat ience, my thanks go t o

t h e l a b o r a t o r y s t a f f and i n p a r t i c u l a r Messt-s. S-Nweke and Ndukwe f o r t h e i r

ass i s t ance i n t h i s regard. I am a1 so g r a t a f u l - to P ro f s .A~iiazigo, ~dukwe, ' ~n j ' i ?

N d i l i , Okoye, Ezediinma,Ogbu-Kalu, Eze i lo , Dr.Sebestian,Dr.Onyegegbu, 3

Dr.Anyanwu,Dr.Aniuc heaz i , for t h e i r encouragement and moral suppor t i n t h i s

p r o j e c t . The f i n a l p roduc t i on o f t h e t h e s i s cou ld n o t have been p o s s i b l e b u t f a F

t h e e x c e l l e n t and ded ica ted i n p u t o f Mr.E.O.Ekeziem who typed and he lped

i n a r r a n g i n g and p roo f - r ead ing t h e whole t e x t .

On t h e domestic f r o n t , t i l e au tho r i s e t e r n a l l y ve ry g r a t e f u l t o h i s

dad, Nze M.O.P.Mbaezue f o r g i v i r ~ y him educaticul; t o t h e T r a d i t i o n a l Ru le r o f

h i s town, Eze J.M.Udoji, f o r h i s suppor t ani! ~'nc-oi~ragement and conferment

o f I c h i e t i t l e w h i l s t s t i l l a s tudent , and t o t-he f d n ~ i l y o f t h e Adigwes,

e s p e c i a l l y unc le Frank, f o r t h e i r suppor t , encourage~nent and adv ice; t o my

grand paren ts , t h e Odid ikas i n p a r t i c u l a r , fro111 where I main ly d e r i v e my

s t r e n g t h and success.

To a l l my f r i ends , r e l a t i o n s , w e l l w ishers and s t a f f o f Jimbaz (MA) 1;P;d

t h e a u t h o r i s a l s o g r a t e f u l and i n p a r t i c u l a r Messrs.Imoh,Ogwu-Chinuwa,

Chikwendu,Onwuka, Okpara, White,Ogui ke,Nwosisi ,Enma Michael , Obi ,An iagoh,

Drs.Eze,Obasi,Cele,Anowi and C h i e f Dr.G.Mbanuyo and Rev.Frs.Ughanze and

Ekwunife who took c a r e o f b o t h t h e domestic arid o f f i c i a l ma t t e r s w h i l s t I waf;

engaged i n t h e p resen t work.

F i n a l l y , though n o t t h e l e a s t , I g i v e g l o r y t o t h e Most High f o r

g i v i n g me t h e l i f e t o ach ieve t h i s . - - - - .. - . - -

TABLE OF CONTENTS

CHAPTER ONE -

CHAF -

2 .

2 .

2.

TWO -

T i t l e Page

D e d i c a t i o n

Acknowledge~nents L i s t o f Tables L i s t o f F i g u r e s

N o t a t i o n s

A b s t r a c t

INTRODUCTION

Background

O b j e c t i v e o f t h e Study

Importance and Na tu re o f t h e Problem

Approdch t o t h e Study

LITERATURE REVIEW .. . . F i b r e s . . . . Vegetable o r N a t u r a l F i b r e s . . S y n t h e t i c o r A r t i f i c i a l F i b r e s . . I n o r g a n i c F i b r e s . . F i b r e R e i n f o r c e d Composites . . General P r o p e r t i e s o f F i b r e Re in fo rced Composites

Geometry o f A1 igned F i b r e s . . F i b r e M a t r i x Mechanics . . F u r t h e r Composite C h a r a c t e r i s t i c s . .

x i i f i

x v i

Table Table of Contents - &

Compactibi l i t y . . . . Strength . . . . Shear Resistance . . . . Fatigue Resistance . . Toughness and Impact .. . . Durabi 1 i t y . . . . Polypropylene F i b r e Reinforced Cement, Concrete and o ther Matr ices . . Creep (Time-Dependent Deformation) . . Creep Study by Model . . . .

CHAPTER THREE: DtFINlTION OF THE PROBLEM AND METHODOLOGY

Methodology . . . . L

CHAPTER FOUR: RHEOLOGICAL MODEL ANALYSIS, IHEOKY AND - EXPERIMENTS . . . .

4.1 Kheological Model . . . . 4. I (a) Rheological Analys is . . . . 4.2 So lu t ion f o r Constant a ( S i m p l i f i e d Method)

4.3 Second Method o f So lu t ion f o r a . . 4.4 Th i rd Method o f So lu t ion f o r a by Newton Raphson

4.5 Experiments . . . . 126

4.5.1 Test Apparatus and Ma te r ia l s . . 126

1.. 4.5. l(a) D i r e c t Tension Creep Equipment- Descr ip t ion 126

4.5.l(b) Dimensions o f Apparatus . . 128

4.5.2 Experimental Procedure . . . . 129

4.5.3 Preparat ion o f Test Specimens f o r D i r e c t Tension Test . . . . - 131

T -

v i

Table - Table o f Contents - Page

4.54 F lexura l Tests . . . . 131 \

4.5.4(a) F lexura l Test Creep Equipment: Descr ip t ion 131

4.5.4(b) Preparat ion o f Test Specimens f o r the F lexura l Test . . . . 134

4.5.5 Procedure f o r F lexu ra l Tests 135

CHAPTER FIVE: PREDICTION OF EMPIRICAL CREEP EQUATION FROM EXPERIMENT AND RHEOLOGICAL THEORY . . 142

5.1 Ca lcu la t i on o f Constant a . ' 142

5.2 P red i c t i on o f Empir ica l Equation f o r Tens i le Creep from Experiment and Rheological Ana lys is 145

CHAPTER S I X : RESULTS, DISCUSSION AND LIMITATIONS - . . 6.1 Rheological Model o f Composite Behaviour

6.2. Determinat ion o f Constants . . 6.3 P red i c t i on o f Empir ica l Equation f o r Tens i l e

Creep o f Composite . . . . \. 6.4 Graphical P lo t s , . . .

6.4(a) . D u r a b i l i t y . . . . 6.5 L i m i t a t i o n s . . . . 6.6 P r a c t i c a l App l i ca t ions . . . . CHAPTER SEVEN: CONTENTS, CONCLUSIONS AND RtCOMMENUATIONS

FOR FURTHER WORK

7.1 Conclusions . . . . 184

7.2 Recommendations f o r f u r t h e r Work . . 186

BIBLIOGRAPHY . . . , 187

Computer Analys is . . . . 202

Computer P r i n tou ts - Graphs . . 203-212

p la tes , Photographs . . . . 213

Increased load on the Mod i f ied Tecquipment Creep 214 Machine

I n i t i a l load on the Mod i f ied Tecquipmenl: Creep 215 Machine

Fron t view o f Creep R ig tquipment w i t h lodd ing 216

- .- - --. Increased load ing on Creep Rig Equipment -.-.-_- 217 ----...

v i i L I S T OF TABLES

Table

1 Tens i le Strength and Tens i le Modulus o f E l a s t i c i t y o f Untreated Bamboo . Compressive Strengths and Compressive Moduli o f E l a s t i c i t y o f Seasoned un t rea ted bamboo.

Basic Engineerinn Proper t ies o f Some Vegetable F ib res

Weight Change o f Polypropylene a f t e r 30 days exposure t o var ious chemicals showinq res is tance.

Cha rac te r i s t i c Proper t ies (Mechanical ,Thermal , E l e c t r i c a l ) of Polypropylene

Dependence o f Impact Strength o f Polypropylene on Molecular weight (Me1 t Index)

L i s t o f Polypropylene Producers and Trade Names

Basic Engineerina Charac te r i s t i cs o f Tensar Geogrids . Spec i f i ca t i ons of Tensar SS1

Spec i f i ca t i ons o f Tensar SS2

Spec i f i ca t i ons o f Tensar SS3

P r o o ~ r t i a c o f Various Inoraanic F ib res . Hexaaonal Array Stress Concentrat ion Factors

E fcec t o f Method o f Compaction on the F lexu ra l St rength o f Steel F ib re Reinforced Mortar 66

Toughness Index Resul tr for DENCLL T e s t Specimens 74

(polypropy'\ene Fibre concrete)

Touqhness Index Resul ts f o r Compact Conlpression Tes t specimens (Steel F ib re , 30.. Notch)

v i i i

List of Tables I Table

f 17 1 18

Toughness Index Results for GRC Material

Durabil i ty of Carbon Fibre-reinforced cement uniaxial load

Properties of Carbon Fibre-reinforced cement composites

Summary of some Rheo logical Models for concrete

Creep Results for Flexural Tests - Experi~ner~tal Constant Load, o = 14,3 j g

0

Creep Results fo r Flexural Tests Constant Load, oo = 13.4 kg

Creep Results fo r Flexural Tests Constant Load o0 = 12.9 kg

Creep Results fo r Flexural Tests Constant Load, u = 12.35 kg

0

Creep Results for Flexural Tests Constant Load uo = 11.8 kg

Creep Results for ~ l e x u r a j Tests - Predicted and Experimental for uo - 14.3 kg

Creep Results for Flexural Tests - Predicted and Experinlental for II = 13.4 k g

U

Creep Results for Flexural Tests - Predicted and Experimental for u = 12.9 kg

0

Creep Results fo r Flexural Tests - Predicted and Experimental fo r oo = 12.35 kg

Creep Results for Flexural Tests - Predicted and Experimental fo r o0 = 11.8 kg

Load-Strain Relation for t = 600 hrs



Load-Strain Relation fo r t = 1000 hrs.

F i g u r e - 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

i x

LIST OF FIGURES --

S t r e s s - S t r a i n r e l a t i o n s h i p o f j u t e f i b r e s

V a r i a t i o n s i n Weight d u r i n q w e t t i n g and d r y i n g o f Akwara

Tensar Manu fac tu r ing Process

U n i a x i a ' l l y Ur ien%ated G r i d s o f Netlon L t d . , England

B i a x i a l l y O r i e n t a t e d G r i d s

T y p i c a l Creep curves f o r Tensar SR2 a t 2 0 ' ~ (Po lypropy lene)

T y p i c a l T e n s i l e Tes t K e s u l t s f o r Tensar SS2 Geogrids L o n g i t u d i n a l D i r e c t i o n (Po lyp ropy lene)

'Typical T e n s i l e Tes t R e s u l t s f o r Tensar SS2 (Geogr ids) t r a n s v e r s e d i r e c t i o n ) (Po lyp ropy lene)

Creep l e s t R i g

Kecovery compl idnce e r ( t - t ) i i acli!ir~st. load a

( t i m e of ' loading ( 9 . 3 ~ 1 0 ~ sec) l o r v a r i o u s t i m e ( s e c ) o f o r i e n t e d po lyp ropy lene f i b r e

O r i e n t e d Po' lypropylene c o r ~ d i t i o r ~ e d success ive creep and recovery

Creep Compliance S t ress o f Po lyp ropy lene F i b r e

Creep compliance ( s t r e s ~ ) ~ f o r Po lyp ropy lene

Recovery compl iance ( s t r e s s ) 2 f o r po lyp ropy lene

Creep compl iance-s t ress po lyp ropy lene f . ibre

( a ) Creep & Recovery Response t o s t e p l o a d i n g programnle

(b) Creep Response t o a % - s t e p lond- in<j prograllllne

Log ( s h i f t f a c t o r ) as a Funct ion o f r e c i p r o c a l temperature f o r po lyp ropy lene f i b r e

Non- l i near Isochronous s t r e s s - s t r a i n curves f o r c reep and recovery o f po lyp ropy lene fi h r e

Page - 10

13

17

27

28

29

30

3 1

34

35

36

38

39

40

4 1

43

4 3

4 7

48

X

L i s t o f F igu res

F i g u r e

19

20(a)

20 (b )

2 1

22

23

24

2 5

26

2 7

28

2 9

30

3 1

32

33

34

35

36

37

I s o ~ h r o n o u s comp l iance /s t ress curves creep a t 100 C of i s o t r o p i c polymer

T y p i c a l F i b r e embedded i n m a t r i x w i t h t e n s i l e f o r c e a p p l i e d

T y p i c a l S t ress d i s t r i b u t i o n f o r a c t i o n i n 20 (a )

Composite C i r c u l a r C y l i n d r i c a l C e l l

Lemon Shaped F i b r e

F i b r e w i t h Lemon-shaped ends

T e n s i l e S t r e s s - s t r a i n cu rves f o r GRC Composites

F l e x u r a l l o a d - d e f l e c t i o n c h a r a c t e r i s t i c s o f GRC lam ina tes and asbestos

V a r i a t i o n o f s t e a d y - s t a t e c reep w i t h s t r e s s

Steady-sta. te Creep as a f u n c t i o n o f T e r ~ s i ' l e S t r e s s f o r Cu/W Composite

T y p i c a l T e n s i l e S t r e s s - l o a d / e x t e n s i o n curves

T y p i c a l cu rves showing t e n s i l e s t r e s s - l o a d extensometer e x t e n s i o n

T y p i c a l cu rves showing t e n s i l e s t r e s s - l o a d i t o t a l e x t e n s i o n

Schematic d iagram o f f a b r i c and t e s t specimen

F l e x u r a l response cu rves f o r wa te r c u r e d specinlens

F l e x u r a l response cu rves a t 7 months a f t e r c u r i n g i n t h r e e d i f f e r e n t environments

Load d e f l e c t i o n c u r v e f o r Po lypropy lene f i b r e cement composi te

An I d e a l i s e d Composite

Forces a c t i n g on t h e f i b r e

Creep behav iou r o f 1aminat .e~ exposed i n s i d e t h e l a b o r a t o r y and under n a t u r a l wea the r ing

Page -

4 9

54

54

57

6 2

64

68

6 9

70

7 1

84

85

86

89

90

91

93

99

100

105

L i s t o f F i g u r e s

F i g u r e s

38 Creep behav iou r o f nor~r la l ( 1 ) under-cured 142) and p a r a f f i n cu red ( 3 ) l amina tes under n a t u r a l weather ing

A t y p i c d l composi te C e l l o f M a t r i x and f i b r e s p laced between 2 c a r t e s i a n axes x and y

The Rheo log ica l Model

M o d i f i e d Tecquipmeat

Tes t Specimen f o r D i r e c t T e n s i l e Tes t

Creep R i g f o r F l e x u r e

T y p i c a l Po lypropy lene Re in fo rced Co w r e t e Laminate t e s t e d i n F l e x u r e

V a r i a t i o n o f Extens ion w i t h T i n ~ e f o r Var ious Constant l oads d u r i n q D i r e c t T e n s i l e Tes ts

V a r i a t i o n o f S t r a i n w i t h Time f o r v ~ r i o u s cons tan t loads f o r F l e x u r a l T e n s i l e Tes ts

R e l a t i o n between oo and a

R e l a t i o n between oo and t_

R e l a t i o n between oo and E~

P r e d i c t e d and Exper imental Curves o f T e n s i l e Creep o f Po lypropy lene F i b r e R e i n f o r c e d Concrete Laminate f o r uo = 14.3 kg.

P r e d i c t e d and Exper imenta l Curves o f T e n s i l e Creep o f po lyp ropy lene f i b r e r e i n f o r c e d c o n c r e t e l a m i n a t e f o r U, = 13.4 kg.

P r e d i c t e d and Experimental Curves o f T e n s i l e Creep o f Po lypropy lene F i b r e R e i n f o r c e d Concrete Laminate f o r u = 12.9 kg.

O

Page -

x i i Table o f Figures -

Figures - Page L.

54 Predicted and Experimental curves o f Tens i le Creep o f polypropylene f i b r e re in fo rced concrete 165 lannnate f o r co = 12.35 kg.

55 Predicted and experimentdl curves o f Tensi le Creep o f polypropylene f i b r e re in fo rced concrete 166 laminate f o r go = 11 ,0 kg.

56 Va r ia t i on o f Creep C o e f f i c i e n t w i t h t ime f o r d i f f e r e n t values of loads

174

5 7 Va r ia t i on o f Creep Con~pliance wi th t ime f o r d i f f e r e n t l e v e l s o f loads oo 175

58 Var ia t i on o f load w i t h creep f o r var ious times (Isochronous curves)

I76

x i i i

P = Force on boundary per u n i t volume - sur face fo rce ( i n vector form)

t = t ime

7 = shearing stresses i n appropr iate d i r e c t i o n ; the Xy'[xz"yz f i r s t subscr ip t i nd i ca t i r rq the a x i s t o which the

f o r c e i s perpendicular and the second the d i r e c t i o n t o which the shedring s t ress i s p a r a l l e l .

n = s t ress tensor - u = a r i t hme t i c mean o f the normal stresses. u ,o and np and

c a l l e d the pressure i n t he f l u i d denoted 6y negat ive s ign -p.

u = c o e f f i c i e n t o f v i scos i t y . L

u X

- f r i c t i o n a l t e r m o f s t ress components

0 X

= normal s t ress i n t he d i r e c t i o n of f i b r e i n x - d i r e c t i o n

'y = normal s t ress i n the y - d i r e c t i o n - u

X = average normal s t ress i n the matrix in x - d i r e c t i o n

"T = fo rce exerted by the ma t r i x or1 the F ib re t o the r . ight o f p o i n t (x,o)

t o t a l fo rce on f i b r e

t o t a l average s t ress ' i n the x - d i r e c t i o n

distance o f f i b r e from o r i g i n i e , y = H (between 2 p a r a l l e l f i b r e s )

v e l o c i t y o f f i b r e i n the x - d i r e c t i o n

v e l o c i t y o f the f i b r e i n t h e y - d i r e c t i o n

stream func t i on

non-dimensional quant i t y

l eng th o f f i b r e

Notations

gap (distance) between f ib re ends

substantive acceleration

body force ( in vector form)

velocity vector

the three orthogonal components of the vector f i e l d W

the three orthogonal components of the vector f i e ld F

f lu id density

p,,py,p2 = three orthogonal coniponents o f vector f i e ld P

g = vector acceleration due t o gravity

P = f lu id pressure

( t ) = elongation per unit length of specimen a t time t or creep a t time t .

U ( T ) = time-dependent uniaxial s t r e s s

J ( t ) = creep con~pliance

I ec( t ,oo) = s t ra in response

I e = creep recovery r e c = creep response

C. Vf = volume fraction of f ib res

r f

= radius of f ib res

2R = centre t o centre spacing

xv Notat ions

.I = a constant depending on geometry o f array o f f i b r e s

Px = t e n s i l e e x i a l fo rce i n x -d i rec t ion

icl and uz = dashpot and spr ing constdnts respect ive ly

€1, EZ = s t r a i n i n systems I and 2 respect ive ly

EI, = s t r a i n i n spr ing f o r m d i f i e d Maxwell model i n System 1

E ~ s = S t r a i n in spr ing f o r system 2

E I ~ = s t r a i n i n dashpot f o r modi f ied Maxwell Model i n System 1

,az = st ress i n systems 1 dnd 2 respect ive ly

L = t o t a l s t r a i n

u = t o t a l s t ress ( load)

!A = dashpot constant f o r Maxwell system

a = constant

r = constant

= spr ing constant (Modulus o f E l a s t i c i t y ) i n system 1

M2 = spr ing constant (Modulus o f E l a s t i c i t y i n system 2 )

uld = dashpot constant i n system 1

Em = < t r a i n a t t ime t = i n f i n i t y

6 = s t r a i n a t t ime t = o. 0

OO = constant s t ress ( load)

x v i

For f i b re - re in fo rced composite elenients, nlathen~atical formulat ion!

t o subs tant ia te experimental work i n i nves t i ga t i ons i n t o time-dependent

behaviour o f organic f i b r e re in fo rced concrete elements a re rare. Most

o f t h e i nves t i ga t i ons have been main ly expwimenta l . I n t h i s thesis ,

an empir ica l equat ion i s obtained t o p r e d i c t t he t e n s i l e creep o f

polypropylene f i b r e re in fo rced concrete lan~ ina te , making use o f

rheo log ica l and experimental r e s u l t s . The v a 1 i d i . t ~ o f t he proposed

model i s tes ted by a comparison between the pred ic ted s t ra in - t ime

curves and experimental data. The type o f rheo log ica l model used was

determined by making some p re l im ina ry assumptions and cons ider ing the

laminate ds a two dimensional composite o f m a t r i x re in fo rced by a

regu lar a r rdy o f s i m i l a r f i b r e s and loaded i n tension i n a d i r e c t i o n

p a r a l l e l t o t h e f i b r e d i r e c t i o n . By app ly ing bdsic equations governing

motion and neg lec t ing the body forces, an e q ~ ~ a t i o n o f t he fo rm

was obtained.

Without so l v ing the above equation, t he terms

a2+ 2

+ ) + 2 and Li(E2 + s x 2 ) cha rac te r i se

respec t i ve l y viscous and e l a s t i c behaviours i n d i c a t i v e o f v i s c o - e l a s t i c i l

and hence a rheology i s modelled t o s u i t t h i s . The model i s made up o f

l i n e a r spr ings and dashpot w i t h a non-return valve.

x v i i

The model i s analysed using a constant , t e n s i l e load and

appropr ia te dynamic and s t a t i c equations t o ob ta in a response o f form

Experimental i n v e s t i g a t i o n o f t he laminate f o r the determinat ion o f

t he cors tants was c a r r i e d o u t by

(a) Fabr ica t ion o f polypropylene re in fo rced concrete lamidates.

(b) App l i ca t i on o f u n i a x i a l t e n s i l e load i n the d i r e c t i o n

o f the f i b r e s using d i r e c t and f l e x u r a l methods w i t h

the appropr iate creep equipments fabr ica ted.

( c ) P l o t t i n g o f s t ra in - t ime r e l a t i o n s h i p w i t h steady constant

l oad over a long per iod OF no t l e s s than

s i x months.

Using numerical techniques f o r the determinat ion o f constants and

curves o f best f i t t o r e l a t e them, empi r ica l fornlulas were obtained

and hence an equation t o p r e d i c t the t e n s i l e creep o f polypropylene

f i b r e re in fo rced concrete laminate given by

i s derived.

Hence i t i s concluded that,

a new Rheological Model i s

discovered for investigating the tensi le creep of

polypropylene f ibre reinforced concrete laminate;

the creep response to the said model in ( i ) i s

given by equation ( l ( a ) )

within a certain range of loads, the tensi le creep of

polypropylene f ibre reinforced concrete laminate can

be predicted without actually performing experiments

using equation (I (b))

the said n ~ i n i m u n l load in ( i i ) above i s 60.09% of the

ultimate.

CHAPTER ONE -

INTRODUCTION

1 .1 Background: The use o f organic f i b res as a r e i n f o r c i n g element

i n const ruc t ion indus t ry has been on the increase i n recent years.

The need f o r such i s more predominant i n places l i k e East Asia,

India, A f r ica , Mexico, Egypt, Libya, o i l producing, t r o p i c a l and

developing countr ies. I n these areas, the p o t e n t i a l s f o r a v a i l a b i l i t y

o f such f i b r e s are many. These organic f i b r e s comprise vegetable and

synthet ic . Examples o f these a re bamboo, saw-dust, coco-nut husk,

akwara, j u te , s i sa l , c o i r , polypropylene, nylon, polyethylene and saran.

Inorganic f i b res , comprising mainly steel , asbestos, glass and

ceramic are being used extensively i n developed count r ies l i k e Europe and

U.S.A., where the po ten t ia l s o f a v a i l a b i l i t y are q u i t e high. The

disadvantages i n the use o f inorganfe f ib res ape

many ranging from the hea l th hazard caused by asbestos t o

d i f f i c u l t i e s i n handl ing dur ing mixing o f s tee l and glass f i b r e s i n

format ion o f composites.

Polymer f i b r e s l i k e polypropylene are by-products o f crude o i l .

O i l producing countr ies have great po ten t ia l s i n the production o f

t h i s type o f f i b r e . A s tep i n t h i s d i r e c t i o n has already been taken

i n N iger ia i n the s e t t i n g up o f Petrochemical Indus t r i es f o r such

production i n Warri and Kaduna. The polymer f i b r e s come i n d i f f e r e n t

- forms ranging from mono-filaments, f i b r i l l a t e d , t o mesh network.

The method o f conversion i n t o d i f f e r e n t forms and s i ze depends on the

? I , . ,

I 2

objectives of a par t icular moulding compa . ,. ~"

Cement denotes any kind of organic o r ' inorganic adhesive.

In construction, i t generally implies a binder or matrix fo r aggregates

o r f ibrous materials t o produce various concretes, which otherwise may

be known as COMPOSITES. A composite formed by polypropylene f i b r e and

cement o r concrete may be regarded a s a composite of duct i le f i b re in

a b r i t t l e matrix. Certain properties of composites a re insensit ive t o

t h e i r microstructure, while other properties a re sensit ive. ~ e n s i ' t ~

and heat capacity, fo r example, depend only on the volume fract ion

of the phases, whilst thermal, e lec t r ica l conductivity, Young's

modulus and strength a re sensit ive t o the geometry and orientation of

the discrete and continuous phases.

Many composites may be described a s tiomogeneous and isotropic

materials in which par t ic les of a second hornogeneous isotrnpic phase a re

dispersed. Assuming a uniform volume concentration, i t may be

possible in principle t o calculate the micros t ruc ture-sens i t i ve

properties of the composite from the inherent properties of the

composing elements. A considerable e f fo r t has been devoted in

recent years t o the determination of overall properties of

inhomogeneous composite materials. A greater part of t h i s e f fo r t

has been concentrated upon l inear e l a s t i c isotropic materials. The

basis of the investigation en t a i l s calculation of the e l a s t i c f i e l d

due to single inclusion o r inhomogeneity i n an in f in i t e matrix of

dif ferent e l a s t i c property. One such fundamentdl property i s

' C R E E P 1 a s applied t o non-1 inear e l a s t i c components of the composites.

Another important proper ty i s t he o v e r a l l . e l a s t i c moduli

of t h i s inhomogeneous system i n terms o f t h e i n d i v i d u a l moduli

and r e l a t i v e volumes o f t he phases. Other p rope r t i es a r e bond,

shrinkage, d u r a b i l i t y . These p rope r t i es present respect ive problems.

1.2 Object ives of t h i s Study: Theoret ica l p red i c t i ons o f deformation

and f l o w o f composite ma te r i a l s from a knowledge o f phase p rope r t i es

and geometry a re genera l l y r a r e and p r o h i b i t i v e l y d i f f i c u l t owing t o

the complexi ty o f t h e d i f f e r e n t equat ions and boundary cond i t i ons t h a t

represent t he problem. Most o f t h e work so f a r done i n t h i s area has

been experimental. Pure a n a l y t i c a l approach i s necessary f o r t h e

fo l low ing: -

To f i n d the most s u i t a b l e f i b r e f o r r e i n f o r c i n g concrete f rom

a rev iew o f t he 1 i t e r a t u r e .

To study the long term behaviour o f a composite formed w i t h

the f i b r e polypropylene.

To f i n d a s u i t a b l e rheo log ica l model t o p r e d i c t t h e

deformation r e l a t i o n s h i p .

To de r i ve a response equat ion f o r t he behaviour i n (c) .

To f i n d an empi r ica l equat ion f o r p r e d i c t i n g t h e creep o f t h e

composite w i thout performing experiments.

To determine o the r empi r ica l r e l a t i o n s h i p and experimental

values f o r creep c o e f f i c i e n t and compliance f o r such composite.

To determine s u i t a b l e curves t o f i t t h e experimental data.

R s lab i s one o f t he s implest s t r u c t u r a l elements. On account

o f t h i s , i t i s used i n t he i nves t i ga t i on .

1.3 Importance and Nature o f the Problem: The problems i n mix ing two

phases t o form an inhomogeneity o f composite mater ia l a re many.

Important p roper t ies o f these composites are - (a) D u r a b i l i t y

(b) Bond

(c) Strength

Durabi l i t y e n t a i l s time-dependent s tudies and hence creep. Creep

i n v e s t i g a t i o n may e i t h e r be long term o r shor t term. Associated w i t h

creep i s shrinkage. The advantages o f f i b r e re in forced concrete l i k e

increased toughness and impact res is tance cannot be over-looked.

The nature of creep problem i s such t h a t a l o t o f pat ience i s

requ i red i n ca r ry ing ou t long-term observat ions. Concrete i s genera l ly

known t o be b r i t t l e mater ia l and weak i n tension. Tensi le r e s u l t s

o f concrete alone are no t very encouraging. Reinforcement the re fo re

o f concrete w i t h a d u c t i l e mater ia l l i k e polymer f i b r e may improve t h e

t e n s i l e proper t ies o f the composite formed. It i s the re fo re necessary

t o i nves t iga te the creep o f such composites when subjected t o t e n s i l e

1 oad.

1.4. ~ p p r o a c h %a t he Study: I n order t o determine the type o f

Rheological Model t o be used, an ana ly t i ca l approach i s employed by

i d e a l i s i n g a composite formed o f f l u i d mater ia l re in forced w i t h a

regu la r a r ray o f f i b r e s and loaded i n tension. Under c e r t a i n

assumptions, equations f o r two dimensional creeping f l o w o f a non- l inear

f l u i d are appl ied. This procedure reduces t h e boundary value problem

for the Navier equation to a relatively simple equation by using some

constitutive equations and classical theories of e l a s t i c i ty .

.. The type of model having been chosen, i t s response when a steady i

t ens i le load is applied is analysed by applying basic dynamic and s t a t i c

equations and making use of Heaviside Step-by-step Integration fo r

boundary analysis.

Experiments are performed to determine creep behaviour of the

composite. These involve fabrication of polypropylene reinforced

concrete slab i n form of a laminate and the use of adequate r igs fo r

Flexural and Direct Tensile t e s t s applying dead loads. Creep curves

a re plotted for these results.

The resul ts of these experiments enable the constants i n the

general creep response equation to be determined making use of numerical

techniques f o r the solution of ensuing mathematical problem.

Approximation theories are applied i n relevaht areas making particular

use of Least Squares technique. I

Empirical equations are formed relat ing these constants. By

some mathematical manipulations, a final equation i s obtained t o predict

the tensi le creep of the composite. The range of appl icabi l i ty of

loads i s determined.

Creep curves are plotted which include predicted and experimental

results. Creep coefficient and compliance are determined. Computer

analysis and print-outs are mde i n relevant areas for comparative

results.

CHAPTER TWO

LITERATURI: REVIEU -

2.1 FIBRES: The study o f f i b re - re in fo rced elements has been on t h e

increase i n recent years. F ibres nmy be c l a s s i f i e d i n t o two main

groups, namely -

( i ) Organic, which comprises (a) Vegetable o r na tu ra l f i b r e s

(b) Synthet ic o r a r t i f i c i a l f i b r e s

and

( i i ) Inorganic f i b r e .

Zonsveld [ I581 c l a s s i f i e d organic and inorgan ic f i b r e s respec t i ve l y as

LOW-Modulus and High Modulus f i b r e s .

2. l (a) - Vegetdble o r Natural f ib res : These comprise f i b r e s such as s i s a l ,

coco-nut husk, j u te , bamboo, akwara, ~ p l a n t a i n bark, sugdr-cane bagasse.

They have been inves t iga ted by authors l ike A z i z e t a1 C131, Cook e t a1

[27], Das Gupta e t a1 C321, N i lson E1051, S w i f t el, d l [243], Uzomaka C1511,

Cox C311 and Ramaswamy e t a1 [I 13,lO,13,26 ].l'hey yenerd1 l y concluded t h a t

( i ) Vegetable f i b r e s have low nlodulus o f E l a s t i c i t y

( i i ) There i s inherent problem o f poor bond and d u r a b i l i t y

( i i i ) There a re some i n h i b i t i o n s i n t h e i r use as re inforcement

t o cement o r concrete caused by t h e i r c e l l u l o s e and l i g n i n

contents.

Basic engineering c h a r a c t e r i s t i c s o f one o f t he vegetable f i b r e s

c a l l e d bamboo a re summarised i n t ab les 1 and 2 by Cox [31]. The

accuracy o f r e s u l t s g iven f o r t he t e n s i l e and compressive modulus o f

e l a s t i c i t y seems exaggerated. The f i g u r e s given should be used w i t h

Source of Information

E.F.Smith and K.L. Saucier

B.J.Menti Zinger & R.P.Flourde

A.Purusho- tham

Hazine

V.A. Purugganam et a1

TABLE 1

Tensile Strength and Tensile Modulus of Elasticitz

of Untreated Bamboo by Cox 131

Location

Clemson S.C. U.S.A.

Jackson, Miss. USA

Villanova, Pa USA

India

Germany

Japan

Manilla Phillipines

Specie

A.Gigantia & Phyllos- tachys

A.Tecta

Unknown

Unknown

Unknown

bladake of the Shiudeasa Genus

Probably BBmbu8a Spinosa

Moisture

Green & seasoned

Green

Seasoned

Unknown

Unknown

Unknown

Seasoned

Average Tensile Strength Kg/cm'

2400

667

1269

2300

1986

2400

1950

Average Tensile Modulus. of Elasticity Kg/cma

195,721

130,771

125,147

145,536

179,283

Unknown

167,500

TABLE 2

Compressive Strengths and Compressive Moduli of

Elasticity of Seasoned Untreated Bamboo;Short Time Static Test

by Cox [311

Source of Information

J.C.Espinosa

S . Mehra, et a1

H.E.Glenn

V.A. Purugganaa

Location

Manila Philippines

India

Glemson S.C.USA

Manilla Philippines

Specie or Origin of

B. Spinosa

B. Balcooa

Probably Phyllosto- chys

Bambuscides

Probably B. Spinosa

Moisture Content

Unknown

Unknown

Unknown

Unknown

AVERAGE

. . .

Average lompressive Strength Kg/cmP

verage ompressive

of lasticity Kg/cmP

Unknown

Unknown

150,900

114,300

132,600

caut ion, since the species o f bamboo are r a t h e r too many. Change i n

c l i m a t i c condi t ions, harvest ing periods and v a r i a b i l i t y i n t e s t methods a f f e c t

the engineering proper t ies . Glenn C491 repor ted i n the Clenlson study,

d u r a b i l i t y o f bamboo as a reinforcement t o concrete t h a t a coa t i ng o f

aspha l t emulsion on ban~boc~ increases the bond and loaa c a r r y i n g capac i ty o f

concrete.

One of the main problems i n the use o f bamboo as r e i n f o r c i n g f i b r e

i s i t s undefined geometrical s ize which a f fec ts the aspect r a t i o defined

as the r a t i o of l eng th d i v ided by l e a s t cross-sect ional dimension. Since

bamboo has t o be s p l i t t o ge t a t t h i s dimension, i t fo l l ows t h a t t h e

performance of bamboo w i l l depend on the d e x t e r i t y o f the f a b r i c a t o r .

bbnsur e t a1 C931 who attempted a cha rac te r i sa t i on o f Ju te f i b r e

obta ined values f o r s p e c i f i c g r a v i t y as 1.02, t e n s i l e e l a s t i c modulus

as 9.59 N/mm2 and a s t ress -s t ra in curve as.s l~own i n f i g . 1. Though these

r e s u l t s compare favourably w i t h those o f Majumdar [92] and Castro [158],

f o r S i sa l f i b r e , Mansur f a i l e d t o consider t h e i r E las to -p las t i c behaviour

and d u r a b i l i t y w i t h regards t o environmer~tal f a c l m s and hence the r e s u l t s

obtained may be t r e a t e d w i t h caut ion . One main problem h i g h l i g h t e d

by Castro e t a1 on s i s a l i s t h a t o f standardisa%ion o f s izes of

vegetable f i b r e s w i t h p a r t i c u l a r reference t o volume f r a c t i o n de f ined as

the percentage content o f f i b r e i n the o v e r a l l volume of composite.

Table 3 shows a summary and comparative c h a r t o f basic engineering

c h a r a c t e r i s t i c s of some vegetable f i b r e s . One obvious f a c t from t h i s

t a b l e i s the di f ferences i n r e s u l t s obtained by d i f f e r e n t i nves t i ga to rs .

hi^ My again be due, as i n t h e case o f bamboo, t o species, c l i n ~ t i c

cond i t ions of growth, method o f f a b r i c a t i o n and tes ts .

1 STRAIN (

50

4 0

- nl

30- E \ 2 - V) V) lu 2 0 - 0:

5,

Fig. I. Stress - $ t ra in relotionship of jute fibres

..

A

- A

/IX

xi

,/ 0

SAMPLE I

a SAMPLE 2

" 0" X SAMPLE 5

after Mansur [93]

SPECIFIC GRAVITY 1.02

MODULUS OF ELASTICITY a 9.59 ~jrnrn'

ULTIMATE TENSILE STRENGTH 850 ~ / m g

--L 0 2 4 6 8 10

TABLE 3 -

Basic Engineering Properties of Some Vzqot~b1c Fibres by

Different Invest igators -- -.--. . -

Type o f Vegetable Fibre

S i s a l

Jute

Akwara

Sisal

I l lV88t ig~ to l8

Majumdar[92]

Mansur 1931

Uzomaka [I511

Castro et a1 [231

-

-...-.

C r i t i c a l Volulne fract ion f o r crmcnt composite Vf(%)

0.6

0.75

-

7

---- --

Speci f ic Gravity

1.5

1.02

0.96

-

Young's Modulus KN/mm2

-

9 .59

20.0

2 1

----+------A

I

Tensi le Strength N/nun2

0.80

5 0 . 0

- I

55.2

Another vegetable f i b r e found predominantly i n N iger ia and some

t r o p i c a l countr ies isNAkwara': b o t a n i c a l l y known as piassava f i b r e . It

has been invest iga ted by Uzomaka [I511 who observed the fo l lowing: -

( i ) t h a t i t i s a vascular bundle, cons is t ing o f a sheath o f f i b r e s

surrounding annular l a y e r known as primary phloem and w i t h i n which

there are two metazylcms. The sheath i s made up o f numerous

f i b r e c e l l s .

( i i ) The f i b r e geometry i s variable, from c i r c u l a r , rectangular t o

e l l i p t i c a l . It tapers along the length w i t h diameter vary ing

from 1 .Omm t o 4.0m. On the average it i s about 1 . 5 ~ long,

w i th a spec i f i c g r a v i t y o f 0.06.

( i i i ) On dimensional s t a b i l i t y , i t i s s tab le i n water and appears

durable i n cement mat r ix environment. Var iat ions i n weight

dur ing wet t ing and dry ing as shown i n f i g . 2.

( i v ) The s t ress-s t ra in r e l a t i o n i s l i n e a r , the mater ia l f a i l i n g i n

tension by b r i t t l e f rac tu re .

( v ) The tangent modulus i s o f the order of 2 KN/mm2.

The charac te r i s t i cs ofW~kwara"as determined by Uzomaka were

obtained f o r the brownish zone on ly and n o t f o r the whole leng th and

hence it may be sa id t h a t the r e s u l t s are n o t comprehensive. To

determine the overa l l propert ies, the whole leng th should have been

tested. The durat ion of h i s observation was no t speci f ied, and hence

more def ined long term exposure and charac ter isa t ion i s required.

Whi ls t the t e n s i l e modulus o f E l a s t i c i t y obta ned by Castro C231

f o r S isa l f i b r e (21 N/mn2) compares favourably w i t t h a t o f 'hkwara"

(20 N/mm2), cas t ro concluded t h a t - i

TM- Days

Var~otion in weight of okworo d u r i n ~ cyclvli o f wetting otld drying F i g ' af ter Uzomaka L I S I J

( i ) Vegetable f i b r e s have s i g n i f i c a n t p rope r t i es t h a t make them

e l i g i b l e as p o t e n t i a l r~einforcement f o r cementious matr ices.

( i i ) Tens i le s t rengths of upto 80 psi(0.552 N/mm2) and e l a s t i c

6 ~nodu l i of upto 3x10 p s i (21 N/nm12) a re obta inable.

( i i i ) Lengths o f f i b r e s upto 75mm and volume f r a c t i o n s o f up to 11%

can be mixed w i th Por t lqnd Cement mortar ma t r i x .

( i v ) E l s t o - p l a s t i c behaviour i n f l e x u r e and m u l t i p l e c rack ing can

be achieved f o r volume f r a c t i o n of f i b r e s above seven percent.

I n general most of the authors agree on the problem of bond,

d u r a b i l i t y and h a n d l e a b i l i t y dur ing mix ing i n t he format ion o f composites.

These areas need f u r t h e r i nves t i ga t i on . Furthermore, most of t he work so f a r

done on vegetable f i b r e s i s ~ n a i n l y experimental. C lass ica l t heo r ies on

t r a c t i o n s w i t h i n the compos i t c a re no t many enough t o v a l i d a t e these

experiments and hence the r e s u l t s so far obtained may be used w i t h caut ion.

Test methods have n o t been star~dardised nor Codcs o f Prac t ice . Values f o r

p roper t ies such as Izod Impact s t rength, Hardness shore, thermal expansion

and conduc t i v i t y have no t y e t been un iversa l l y repor ted. Creep behaviour

o f vegetable f i b r e s under load has n o t been very much inves t iga ted .

2. I (b) - SYNTHETIC OK ARTIFICIAL. FIBRES: These comprise f i b r e s such as

polypropylene, polyethylene, and ny lon a1 1 chemical ly known as polymers.

They have been inves t iga ted by authors l i k e N i c h o l l s [ I041 and Frank C461.

Polypropylene and Polyethylene a r e p a r a f f i n chains several hundred carbon

atoms long. On account o f t h e i r h igh molecular syauwtry, they a r e non-

p o l a r and the re fo re i nso lub le i n p o l a r so lvents such as water.

Polypropylene i s a c r y s t a l l i n e thermoplast ic p o l y o l e f i n r e s i n [46 ,3,511.

being the f i r s t member o f t he growing f a m i l y o f syn the t i c i s o t a c t i c

polymers t o achieve i n d u s t r i a l importance [46]. Polypropylene can be

produced d i r e c t l y along w i t h polyethylene by p y r o l y t i c conversion o r

thermal c rack ing o f selected hydro carbon feed stocks. The feed stocks

range from ethane t o crude o i l . Chemically, polypropylene i s represented

by t h e formula

H H H H H

I I I I I C -- C - C -- C -- C

I I I I I H H -C-H H H - C - H H

I I H H

Under c e r t a i n cond i t ions i t can be polymerised t o g i ve a l l t he methyl

groups on the same s ide o f t h e main carbon chain r e s u l t i n g i n a

' s y n d i o t a c t i c ' polymer o r w i t h a random arrangement producing an

' a t a c t i c polymer. Polypropylene f i b r e i s manufactured by m e l t i n g t h e

Polymer and squeezing t h e m e l t through very f i n e holes, keeping a tens ion

on t h e thread thus produced u n t i l i t sets. This i s the spinning method

o f product ion.

Proper t ies o f Polypropylene: On account o f i t s c r y s t a l l i n e character ,

polypropylene, l i k e o ther c r y s t a l l i n e po lyo le f i ns , i s so lub le on ly a t

e levated temperatures C841.

Polyolef ins are genera l ly b i o l o g i c a l l y r e s i s t a n t t o chemicals,

micro organisms and termites, hence i t can be used fo r subsoil construction

works. Table 4 shows a quanti tat ive conlpilation of polypropylene w

resistance to various chemicals. Fuming sulphuric and n i t r i c acid and

other oxidising agents however attack polypropylene slowly. The weight

change i n Table 4 i s due to the sum of e f fec t s of swelling and dissolution.

The character is t ic properties of Polypropylene involving mechanical,

thermal and electr ical properties a re given in Table 5 as complied by Frank

C461, and the dependence of impact strength of polypropylene on molecular

weight (melt index) i s shown in Table 6. Frank's resu l t f o r Ultimate

Elongation as shown in Table 5 however appears too exaggerated.

Polypropylene goes by different trade names and these as well a s

t h e i r producers are shown in Table 7. In U . K . the trade name as

produced by Netlon Ltd. i s Tensar. Tensar Geogrids are used in the

present investigation.

Tensar geogrids are high tens i le strenyth polymer grids developed

specif ical ly for reinforcing so i l s and cementious ~matrices. rhey a re

manufactured from specially selected polyolefins t o produce consistent

and chemically iner t engineering materials, sur table f o r long-term

reinforcement.

Tensar Geogrids a re manufactured a s shown in f i g . 3 by stretching

punched sheets of a selected polymer to align the long chain molecules.

This molecular alignment provides the grid w i t h i t s high tens i le strength.

The s i ze , shape, pitch and pattern of the holes i n the polymer sheet

before stretching a re a1 1 closely controlled, since they have a

s ignif icant influence on the hape of physical properties of the i; result ing Tensdr grid.

I I

Figure 3 Tensar nlanufactur trig process by courteriy ::f Netion Ltd, Blackburn England.

'TABLE' 4

weight change' a f Polypropylehe'after ' 30 days exposure

to'variouf 'chemltals' showing resistance as compiled

Chemical

Sulphuric Acid 98%

Nitric Acid, fuming

Sodium hypochlorite 20%

Gasoline

Benzene

Xyl ene

Methylene chloride

Carbon tetrachloride

Turpentine

Transformer oil

by Frank [46]

Weight change (%)

19

E l e c t r i c a l )

o f Polypropylene as compiled by Frank [46] . . ;, . . :

8 -

Density (gm/cm3 )

Re t rac t i ve Index nD

Tens i le Y ie ld Strength kg/cm2

Tens i le Y i e l d Elongat ion ( Z )

Tens i le Modulus (kg/cm2)

U l t imate Elongat ion (%)

S t i f f n e s s i n f l e x u r e (kg/cm2)

F lexura l modulus (kg/cm2)

I zod Impact Strength,Notched

2 3 ' ~ (crn.kg/cm)

Hardness Shore

-5 0 Thermal Expension x 10 / C

Deformation under load (%)

(50°c/24 hrs. )

Thermal Conduct iv i ty x l o m 4 -1 -2 (Cal sec cm ; 1°c, cni -1

S p e c i f i c Heat ( c a l / g / ' ~ )

Mold Shrinkage ( Y )

D i e l e c t r i c constant (60 cycles-50mc)

D i e l e c t r i c s t rength KV/mm

Volume R e s i s t i v i t y (ohm.cm)

Surface R e s i s t i v i t y (ohrn.cm-')

D i ss ipa t i on ~ a c t o r ( 6 0 cyc les - - 100 mc)

Fig. 4 and 5 show respec t i ve l y U n i a x i a l l y and B i a x i a l l y o r i en ted

g r i ds . Typical dimensions and spec i f i ca t ions of Tensar Geogrids a re

given i n Tables 9, 10 and I 1 and basic engineering c h a r a c t e r i s t i c s a r e

given i n Table 8. Net lon L td ' ,s resu l i o n a t break o f 300%

appears r a t h e r too exergerated., '

McGown e t a 1 [94] i n : a symposiu na ture and Proper t ies

o f Tensar Geogrids h i g h l i g h t e d the use o f t h i s polymer i n embankment

and o the r cons t ruc t i on i n d u s t r i e s and t h e i r f a i r l y good d u r a b i l i t y as

evidenced from creep curves.shown i n f i g . 6 , They however, d i d no t

i nves t i ga te the long term behaviour o f the f i b r e when used as a

composite mater ia l .

Typical t e n s i l e t e s t r e s u l t s a r e shown g r a p h i c a l l y i n f i g s . 7 and 8 ,

The b lack co lou ra t i on g ives the polymer a p ro tec t i on f rom u l t r a v i o l e t

1 i g h t degrada Lion. The creep curves fo r polypropylene tensar g r i d s

dep ic t v i sco -e las t i c behaviour. A thorough examination o f t he l o n g term

creep p rope r t i es o f Tensar Geogrids has been undertaken i n t h e U n i v e r s i t y

o f Strathc lyde, Glasgow and i n the l abo ra to r i es o f Net lon Ltd., Blackburn,

England. F i ve temperature c o n t r o l l e d l abo ra to r i es have been s e t up t o

study the e f f e c t s o f sustained constant loading i n c o n t r o l l e d

environments a t temperatures o f ~ O C , I ~ O C , 2 0 ' ~ and upto 40'~. These

t e s t s a t Net lon were on Tensar SR2 g r i ds us ing 3 r i b s wide x 8 bars

l ong and those a t t he Un ive rs i t y o f S t ra thc lyde were on 15 r i b s wide and

5 bars long. The creep t e s t s a t Net lon a re conducted e s s e n t i a l l y f o r

mon i to r ing the v a r i a t i o n i n creep performance o f product ion mater ia ls ,

w h i l s t those performed a t the Un ive rs i t y a re t o generate design ,

Dependence o f 1111pac t Stren@h o f Polypropy l ene on --- - -I-__.- -.--

Molecular weight (Me l t index) . Conlpiled Fron~ Frank - ----- C461, , ' . . .,

- -

M e l t Index, f l o w r a t e 9/10 n ~ i n

23o0c/2. 16 kg)

I zod Impact s t rength, notched

12 9.8 2.2 2.2 I I

Tens i l e Impact S t reng th

( type S ) (cm kg/cm2) 43

Z Z

TABLE 7

List of Polypropylene Producers and Trade Names of Polypropylene

Country

Rustria

France

Germany

I ta ly

Japan

U.K.

U.S.A.

Company

Petrochemie Schwechat A.G. (Danubia)

Naphthachimie S.P., Societe Normande de Matieres Plastique

Badische A n i l i n - & Soda Fabrik A.G.

Chemische Werke Huels A.G. Farwerke Hoechst A.G.

Montecatini Edison S.p.A.

Chisso Corp Nitsubishi Petro chemical Ind Mitsui Chemical Ind. Co.Ltd.

Sumitome Chemical Co. Ltd.

Imperial Chemical Industries L

Shell Chemical Co. Ltd.

Netlon Limited

Alamo Polymer Corp.

Avisum Corp.

Trade Name

Dapl en

Napryl Pry1 ene

Luparen Novol en KR 1300

Vestolen PP Hostalen PP

Moplen Mople fan Mera k l on

Chisso Polypro Nobl en Noblen

Noblen

Propathane Ul stron Nufil

Carlona P

Tensar

Marlex PP

Avisum PP 01 efane 01 ef 1 o Olemer Dl eform

TABLE R

Basic Engineering Characteristics of Tensar g e o w --

by courtesy of Netlon Ltd., England.

Property

Vicat Softening Point (OC)

Shore Hardness (0)

Tensile Strength (MN/rn2)

Elongation a t f a i l u r e (%)

Abrasion Resistance (Mg/lOOO cycles)

Test Method Result --

148

TABLE 9

Spec i f i ca t i ons o f Tensar SS1 [94]

R o l l Dimensions

Length 50m

Width

Physical Proper t ies I Q u a l i t y Control o f Gr ids Proper t ies

4m

Approx. w t .

Colour 1 Black I I ? i / m )

44 Kg.

TABLE 10

Spec i f i ca t i ons o f Tensar SS2 C941

R o l l Dimensions

Length

Width

Approx.dia.

Approx. w t .

- -

50m

4m

0.5m

68 kg.

Physical Proper t ies o f Gr ids

0 .3Kg/m2

Colour I Black

Qua1 i t y Control Proper t ies

I I

Trans- verse / 31.51 11.0

KN/m S t ra in : X

Longi t u - d ina l 17.5 12.0

TABLE 1 1

Specifications of Tensar SS3 [941

Roll Dimensions Physical Properties of Grids

Length

Width

Approx .Dia

Approx.wt.

Wt.

Col our

0.25 Kg/m2

Black

Qua1 i ty Control Properties

Strain

Trans- / 21.4 1 10.6 verse

1 7

LONGITUDINAL RIB

E \

I

I TRANSVERSE BAR FIGURE 4

UNlAXlALLY ORIENTATED GRIDS ( ~ i g ~ are manufactured by stretching punched sheet in one direction u caretully controlled conditions by courtesy of Netton Ltd.,Engl

- +Longitudmal r ib

/ :mnsverse rib

Figure 5 :

Biaxially orientated grids (Figure 5 ) are ~roduced by stretching unkxially or~entated grids, again under carefully controlled conditions, in the d~rection normal to the un~axial orientation-rensmssz by courtesy of Netton Lt d., ~ n ~ l o n d .

load K N /m+

29

26

2'4

14 9

10-

- 9

1 100 200 300 LOO 500 GOO 700 800 900 loo0 t~me(hrs)

Figure 6 Typical creep curves for 'Tensart SR2 at 20°C

( Polypropy\ene) after Net (on .

8

G

It

5/ -

3

2

1 sample, - SIZQ %bars long x 3

r ibs wide 4

Figs. 9 ( a ) and 9(b) show creep t e s t r i g s used i n the U n i v e r s i t y

o f Strathc lyde and Netlon. The creep curves obtained exper imental ly as sh,

f i g . 6 a re non-1 inear.

Attempts have been made by authors 14ke Ward e t a1 C152J a t

es tab l i sh ing theo re t i ca l represcntat ions o f non- l inear isothermal

behaviour o f v isco-e las t ic polymer l i k e polypropylene. To t h i s end,

he proposed a m u l t i p l e i n t e g r a l representat ion f o r non- l inear s t ress-

s t r a i n behaviour i n the form

where i t i s assumed wi thout lois o f genera l i t y t h a t t h e Kernels

J,, . . .. . . . J N are synletric func t ions o f t h e i r arguments. An in t roduc t ion

o f t h i s representat ion can be obtained by consider ing a given load ing

programme n ( r ) as a super-posit ion o f f i n i t e s i m a l load ing steps. The

integrand o f the f i r s t term i n equation 1.1 can be ' i n te rp re ted , as

representing the i nd i v idua l and independent c o n t r i b u t i o n o f t he loading.

step do(^,,) t o the f i n a l elongation. The integrand o f the second term,

on the o-ther hand, may be regarded a s represent ing the j o i n t c o n t r i b u t i o n

o f t he load ing steps d a ( ~ ~ ) and da(r,) t o t h e f i n a l e longat ion.

From equation 1 .I i t may be seen t h a t the time dependent

.on t h e loading programme a(~)=0. T<O; u(T)= u = constant, r > o is 0

given by t h e expression,

Ward and Onat then obtained creep compliances from experiments

conducted under f i v e d i f f e r e n t l oad l e v e l s as shown i n f i g . 10 on a

l o g scale. From the p l o t o f fig.10, the non- l inear na ture o f polypropylene

mater ia l i s s t r i k i n g l y v i s i b l e . The curves a r e parabo l ic and i t

i s o n l y f o r small t imes and low load i n t e n s i t i e s t h a t t h e ma te r ia l can

be considered as l i n e a r . Another i n d i c a t i o n o f n o n - l i n e a r i t y o f t he

ma te r ia l i s obtained by comparing creep and recovery curves as i n f ig .11.

BY t h e l i n e a r theory, creep and recovery curves i n f i g . 11 should

co inc ide f o r a g iven i n t e n s i t y o f load. AS t h e f i g .11 show;, t h i s

i s n o t so except f o r low load i n t e n s i t i e s . I t may be noted t h a t

instantaneous o r sho r t t ime recovery i s l a r g e r than t h e i n i t i a l creep

response, inc reas ing w i t h increase i n app l i ed load.

I n t h e mathematical representat ion o f t he observed non- l inear

behaviour, i t i s assumed t h a t the e longat ion a t a g iven t ime i s a

non- l inear f u n c t i o n of t he s t ress h i s t o r y t o which t h e f i l a m e n t has

been subjected and t h a t experimental r e s u l t s can be represented w i t h

reasonable accuracy by approximating t h i s f u n c t i o n by the sum of a

l i n e a r and t h i r d order he red i ta ry funct ion5. The 1 oading programmes

employed i n the study by Ward prov ided subs tant ia l in fo rmat ion f o r t he

cons t ruc t i on o f the Kernels de f i n ing the two funct ion.

Fig.9: Creep Tes t Hig by Courtesy of Netlon l t d

Recovery e r ( t-t, i g against load db ( t ~ m e of loading (9.3 * lo'seC) for various times (sec) of oriented polyprop~lene - ' f i b re after Ward st-a1 [I53 I -

F ~ g u r e 11 O r i p n t ~ d po\ypropylcne, C ondi tioned succssslve Creep a n d recovery, a f ter Word e t a l I1531

I

A few years l i t e r ,

behaviour of selected po

Ward and Wolfe 11531 investigated the creep

lypropylene monofilament under uniaxial loading.

Multistage loading programes were used to confirm the consistency of the

mathematical representation proposed ea r l i e r by Ward and Onat,[152]. A

f ibre whose non-linear representation contains only l inear and third order

rnultiple integrals was chosen for experin~entat i~n because a l l the terms

can be expl i c i t l y evaluated by a n~oderate progratalne. The total f

el ongation was predicted for a more complex superposition 'loading

programme by measuring only the creep, recovery and siniple superposition

of identical loads. The f ib re used in the investigation has a diameter

5 of 5.56 x 10-~cn\, a viscosity average n~olecular weight of 1.5 x 10 and

an optical birefrigence of 0.037. Creep observations were performed

a t ~ l o r ~ s t a r ~ t contro'l 1 ed temperature of 2 1 " ~ and rela civc humidity of 60%

under cyclic load ing . Creep and recovery curves in the for.111 -of

e c ( t , U , ) / U ~ aga ins t time and r,(t-t,. co against t 'n~e were plotted.

From these resul ts curves were constructed o f e ( t , o < ! ) / ~ and C 0

e r ( t - t , , o j/i a g a i n s t a c ' fur various times. The curves are 0 0 0 0

shown in f ig s . 12 , 1 3 , 14 and 15.

Figs. 12 and 13 are predominantly l inear in shape and th i s i s also

verified by the curves in f ig s . 14 and 15. The additional

compliance curves ar.e also parabolic to a s i n ~ i l a r degree confirming

tha t the creep, recovery a n d simple superposition behaviour i s

described adequately by equation 1 . 1 . This c o n t i w s that the

representdtion of non-linear visco-elastic bchaviour proposed i s

consistent when applied t o complex two stage loading experiments.

0' 5 0 100 - Stress dyne/cd(ldj 150 Figure 12

c compllonce stress of pdypropy[~he ' O f W and wolfe [!53 j

sec

F ~ g u r e 13 Creep compliance 4stressP for polyproP!4ene after Ward and Wolfe L1531

F~gure 14 2 Recovery compliance Ostren ) for polypropylene

after Ward and Wolfe [l53]

sec

d I I 2.5 5.a

(Str~ss) dyne/cd(ld) Figure 15 Creep complmnce-stress polyprop ylene aftei Ward and Wolf e [ 153 1

J 42

I n con t ras t w i t h i s o t h e ma1 inves t i ga t i ons o f polypropylene by

Ward e t a l , Morgan e t a1 [ I 0 1 1 studied creep, recovery and l oad super-

where the creep t e s t i s de f ned by the load ing programme i U(T) = 0, T<O; ( T I = oO = constant; T > 0 ..... (1.4)

as shown i n f i g . 16(a).

pos i t i ons o f same f i b r e a t

They found, consider ing mul t . ip le

l i n e a r v i s c o - e l a s t i c behaviodr,

where e ( t ) i s t h e e longat iod per u n i t l eng th o f t he specimen a t

t ime t; U(T) i s t h e t ime d pendent u n i a x i a l s t ress and J ( t ) i s t he

creep compliance. Equation (1.2) may be i n t e r p r e t e d as meaning t h a t I s t r a i n i s a l i n e a r cont inuo s he red i ta ry func t i ona l o f t he s t ress i

vary ing temperature range o f 2 8 ' ~ - 60'~.

i n t e g r a l representa t ion f o r t h e non-

t h a t experimental r e s u l t s can be

h i s t o r y as enunciated i n

s t r a i n response ec ( t , oo)

i n t e g r a t i o n o f equat ion 1.2

Boltzman's superpos i t ion p r i n c i p l e . The

i n t he creep t e s t was obtained by d i r e c t

g i v i n g

represented w i t h reasonable

s t ress terms. They a l s o shoved

representa t ion i s s h i f t e d by

accuracy by f i r s t , second and t h i r d order

t h a t each o f the Kernels i n t h i s

t h e same f a c t o r w i t h change i n temperature.

The load ing programmes emplo ed i n t h e i r ana lys i s a re shown i n f i g s . t: 16(a) and 16(b) w i t h the c o r esponding response. They expressed the

response o f a l i n e a r s o l i d i the i n t e g r a l form n

i i

Stress t

Loading

I

- Y , time ( t ) '

Figure 16:

c$

Loading programme

- ( a ) ' Creep a Recovery response to step loading programme

m ( b )

I I

I time I I I

( b l Creep response to a 2 - step loading programme after Ward et a1 0533 '

Thus f o r a l inear so l id , the creep compliance is independent

of a. and i s given by J ( t ) . A creep and recovery t e s t i s defined by

the 1 oading programme

The recovery response i s given a s

e ( t a,, $1 i s the elongation measured fo r t > t, in the creep and I recovery t e s t and ec ( t , a,) i s the creep or elongation which would be

obtained i n the creep t e s t a s shown i n f i g . 16(a) f o r a l inear sol id

........ er = J ( t - t l ) a, (1.6)

For a two-step loading programne shown in f i g . 16(b)

a=O, T < O ; a =a ; O<r<tl ; a = uo, T ....... o > tl (1.7)

and corresponding additional creep response i s given by

1 ec = e ( t , a,, t l ) - e c ( t , a. where

1 e ( t ,uo , t , ) i s the elongation measured a f t e r the application of the

second s tep of loading and hence for a l inear sol id ,

1 ec = J ( t - t , ) o0 ........ (1.8)

The results of Morgan e t a1 can therefore be shown t o be

adequately represented by the f i r s t three terms in the expansion of

45

equation (\.la) given by Ward et a1 C1521, thus

2 3

for creep ec(t,ao) = J1(t)ao + J2(t,t)ao + J3(t,t,t)oo ...... (1.9) The recovery test, already defined, yields by intergration for t>t,

2

2 J2(t,t-t,) + J2(t-t,, t-t,) oo +

The recovery response is obtained by subtracting (1.7) frorn(1.6) giving

er(t-tl,oo) = J1(t-tl)ao - J2(t-tl,t-tl)oi +

2J,(t,t-t1)ao2 + J3 (t-t1 ,t-tlt-tl)uo3 +

The two-step loading response is given by

These equations therefore suggest that creep, recovery and superposition

w i l l r e q u i r e t h e same s h i f t f a c t o r s t o be app l i ed t o t h e load ing t imes t,

as t o the response t imes t and t-t,.

It may the re fo re be s a i d t h a t t he behaviour o f i s o t r o p i c polymers

a t d i f f e r e n t temperatures can be pred ic ted us ing the same s h i f t f a c t o r

f o r each Kernel which suggests t h a t t he m u l t i p l e i n t e g r a l representa t ion

has physical s i g n i f i c a n c e and t h a t t h e Kernels a re associated w i t h the

same molecular processes.

Fol lowing f u r t h e r dev610pments, Foot e t a1 [45] i nves t i ga ted t h e

non-1 i n e a r behaviour o f another c r y s t a l 1 i n e unor iented polymer, polyethy lene

te rephtha la te . The i r i n v e s t i g a t i o n f o r creep recovery and load

superpos i t ion was i n t he temperature range o f 70 - 1 0 0 ~ ~ . The r e s u l t s

suggest t h a t i n t h e temperature range considered, t he t ime/temperature

equivalence can be represented by a constant a c t i v a t i o n - energy theory,

t he a c t i v a t i o n energy being independent o f s t ress . F ig . 17 shows s h i f t

fac tors p l o t t e d as a f u n c t i o n o f t he rec ip roca l temperature. W i th in

reasonable accuracy, these r e s u l t s f i t a constant a c t i v a t i o n energy of

143 K cal/mole.

The non-1 i n e a r i t y i n behaviour f o r creep and recovery i s shown i n

f i g . 18. This shows the departure from l i n e a r i t y o f the creep data

a t longer t imes. For recovery, t h i s departure occurs a t sho r te r t imes.

I n o rde r t o consider t he data i n terms o f a m u l t i p l e i n t e g r a l type o f

r e l a t i o n s h i p , isochronous compl iance/st ress curves were p l o t t e d as shown

i n f i g . 19. The form o f t h i s r e l a t i o n s h i p suggests t h a t t h e data can

be adequately represented us ing o n l y t he f i r s t two Kernels i n t h e

representa t ion . Values of these Kernels may be determined f o r

d i f f e r e n t t imes f rom these data.

./ 3 300 sec

-creep --- recovery

04 0.5 0.6 07 08 09 10 1.1 1.2 1.3 11, 1.5 1.6 1.7 stress * lo-' dyne cm"

Figure 18 Non-Itnear isochronous stress-stram curves for creep 8 recovery of Polypropylene fibres after Ward and Wolle [ 1531

2.2.(i

i n h.is

Compos

A f a i r l y new organic f i b r e c a l l e d PRO-49 (Kevlar) has been

developed i n U.S.A. They cons i s t o f chains o f aromatic r i n g s l i n k e d by

non- f lex ib le groups such as CO-NH. They have r e l a t i v e l y h igh s p e c i f i c

modulus and st rength. The i r use i s meanwhile l i m i t e d .

2.1 ( c ) INORGANIC FIBRES:- Commonly used inorganic f i b r e s are graphi te,

asbestos, s i l ica,g lass,steel , and tungsten.[28,29,30,33,34]. T h e i r

engineer ing proper t ies are l i s t e d ou t i n Table 12.

Compared w i t h organic f ib res , t h e i r Modul i o f E l a s t i c i t y a re

r e l a t i v e l y much higher. [351. Other advantages o f these f i b r e s i n c l u d e

h igh chemical resistance, h igh temperature res is tance and a v a i l a b i l i t y

i n var ious forms (mat, tape and loose f i b r e ) .

2.2 FIBRE REINFORCED COMPOSITES: This may be def ined as t h e marr iage

of f i b r e and a b ind ing element u s u a l l y known as matrix,[21,25,27]. This

ma t r i x may be b r i t t l e l i k e cement, su lphur o r d u c t i l e l i k e bitumen,resins

The f i b r e s may e i t h e r be organic o r inorganic.

I n cons ider ing the composite o f organic f i b r e s 1 i k e polypropylene

and a b r i t t l e ma t r i x l i k e cement, d i f f e r e n t t heo r ies have been

proposed by some authors w i t h regards t o f i b r e ma t r i x mechanics.

) 1 KellyComposites:731

i n v e s t i g a t i o n o f M ic ros t ruc tu ra l Parameters o f A l igned Fibrous

i t e found t h a t some proper t ies o f composite a re independent o f

absolute s i ze o f t h e f i b r e w h i l s t o thers which invo lve i n t e r a c t i o n

between two components depend on t h e rad ius o f t he f i b r e and i nvo l ve

proper t ies l i k e compac t i b i l i t y , s t rength, shear res is tance, f a t i g u e

TABLE 12

- M a t e r i a l

PROPERTIES OF VARIOUS INORGANIC FIBRES

as compiled by N i c h o l l s [ I 0 4 1

Graphi te

Asbestos

Drawn S i l k a

Boron Glass

Carbon Steel

Mol ybdonum

Tungsten

-

T e n s i l e Strength6 p s i x 10

Young ' S Modulus

ps i x 10 6

S p e c i f i c G r a v i t y

Typical diameter (m)

Me1 t i n g Temp.

(OC)

resistance, Toughness and Impact Resistance, D u r a b i l i t y , Creep and

Shrinkage. Other p rope r t i es o f a f i b r o u s composite such as thermoedastic

( i n c l u d i n g the e l a s t i c moduli and thermal expansion c o e f f i c i e n t s )

m g n e t i c and e l e c t r i c a l proper t ies, depend on the r e l a t i v e

concentrat ions (volume f r a c t i o n s ) o f t h e two phases b u t a r e

independent o f the absolute s i z e o f t h e f i b r e s (except i n so f a r as t h e

p rope r t i es o f t h e f i b r e s themselves depend upon i t s s i ze ) . They a r e

a l s o governed by the p rope r t i es o f the cons t i tuents and by t h e i r

geometrical arrangement only. The mechanical p rope r t i es o f a f i b r e

re in fo rced composite ma te r i a l depend very s t rong ly on the f a b r i c a t i o n

method used. The usual var iab les associated w i t h t h e f i b r e s namely

aspect r a t i o , o r i e n t a t i o n and d i s t r i b u t i o n a f f e c t these p rope r t i es i n a

way which can be pred ic ted from theory.

I n f i b r e cement composites, add i t i ona l f a c t o r s such as p o r o s i t y

and degree o f penet ra t ion o f reinforcement ( p a r t i c u l a r l y i f i t i s i n

the form o f a bundle) by t h e cement p lay r o l e s i n the development o f

p roper t ies . The e f f e c t s o f these proper t ies are examined below.

2.2( ia) Geometry o f a l i gned f i b r e s : - K e l l y s ta ted t h a t f o r any

a r ray o f f i b r e s

1 Vf = aa(r f where

K

Vf i s the volume f r a c t i o n o f f i b r e s

rf i s the rad ius o f f i b r e s

2R i s the cent re o f spacing and

a ' i s a constant depending on the geometry o f t h e ar ray .

Kelly's theories may be defficient due to the following

assumptions made:-

(i) perfect bonding

(i i ) regular array of fibres

In practice regular arrangement of reinforcing fibres is difficult

to achieves [139,141] except where the manufacturers have made it so

like in the case of polypropylene mesh fibric.

2.2(ib) Fibre-Matrix Mechanics: The mechanics of Fibre-matrix

stress transfer is as shown in figs 20,and 20b C60,61,72,73,741.

Nicholls [I041 considered a rigid cylindrical fibre in a matrix

subjected to a tensile axial force Px. By equating axial forces and

neglecting adhesion force at fibre end, he obtained

where .Px is the fibre tensile force at a distance x from

fibre end,

rf is the radius of-fibre

a is the bond or interfacial X

Typical variations of fibre stress and shear stress T when both

fibre and matrix deform elastically are shown in fig. 20b.. T is maximum

at fibre end and minimum at the centre and vice versa for a . In

Figure 20. a: Typical f ibre ombedded in

matrix with tensile fo rce

Figure 20. b: Typical stress distribution

tor action in 20 (a ).

d i f f e r e n t coioposi tes, t h e l a r g e r t h e r a t i o o f shear ~ ~ o d u l u s o f t h e

111aLrix t o t h e Young's ~uodulus , t h e more r a p i d l y t h e 1.ibt-e s t r e s s

inc reases w i t h d is tar rce fro111 f i b r e end, [ /Y,~O,~J! : / .

I t lriay the re fo re he s t d t e t l t h l ; s l ~ c a r stre!;; w i l l a c t r ~ i l l l y be

concen t ra tc t l ql1 the f i b r e end -in pt.oport;-ion t o l.l~(? t l - i fTerence i n

e l a s t i c ~lrotl~r l i o f t h e f . i h rc at-~d III~I L r i x , r r i : l i t i v c . ~ :;~tly t o t h o scluare

r o o t o f thr: 111.irl ~IIIUIII I-a J i u s oS c u r v ~ ~ t u r . t > (11' i.tlt. P i l l r e . 'Ftle III~ L r i x n e a r

t h e end t h c r e r o r e l j e y i n s l o f a i t a t l o w e r :jti-e.;5 l ~ v r l s i l lart. ( ;ha t o f

t h e u'l t inrate s t r e n g t h o f . the c o ~ ~ ~ p o s i t e .

When ma- lx ix and f i b r e e l o n g a t e e las l : i ca l l y , (.he Young's Modulus

o f a con t i r i uous f i b r e conlposi l:e .is a p p t - o x i ~ ~ r a t c l y g i v e n by N i c h o l l s 1 lO4,l3l I

II js the Yourrqs Modulus, V -i: Ltic vcllrrriw f r ; ~ c l . i ~ l t ~ dnd t i le s u l ~ s c r i p t s c,f

Spencer e t a1 [ I 2 4 1 analysed e x t e n s i v e l y t h e i n t e r f a c i a l t r a c t i o n s

i n f i b r e - r e i n f o r c e d co~npos i tes as shown i n f.ig.21 t,o d e t e w i n e t h e o r e t i c a l l y

t h e f o r c e s a t t h e i n t e r f a c e between a f i h r e drld 111r11.rix i n a f i h r e r e i n f o r c e d

coinposi te near t h e end o f d discont. inuous I ' i h ~ c or ill t h e neighbourhood

o f a break i n a f r a c t x r e d f i b r e . They ass~~alct f ths I: f i h r e and III~ t r i x

behaved as l i r ~ e a r e l a s t i c :;ol i d s imd c h a t t l ~ e r o -it; jn.rl(.:cl: b o ~ d iug,

hence der.ivetl equa t ions f o r r a d i a l allt i a x i a l d i ~ p l ~ r c e r ~ ~ i ~ r ~ t s a s

where t h e s t r e s s Fur~c l io r r $1) i s chosen suc:h I;hi11

11 [, . - - ("":',,, - 'I ? I ( , )

I:) ;I % I.

arid o , c,,, CJ, are t h e s t resses r e s p e c t i v e l y I t r a i l t a ~ ~ g e n t i a l rf

and a x i a l d i r e c t i o n s u s i n g p o l a r c o o r d i n a t e s . .lr.,.~ i s the :;hear s t r e s s ,

rf t h e r a d i u s o f f i b r e ; v t h e p o i s s o n ' s r a t i o , L' Lhc~ w x l ~ ~ l ~ ~ of E l a s t i c i t y ;

u and w a r e t h e d i s p l a c e ~ ~ i e n t s i n t h e r a d i a l and a x i a l d i r e c t i o n s

r e s p e c t i v e l y .

Fig. 21: Composite circular cylindrical cell. Tractions in a fibre embedded in

cylindrical matrix.

Most o f t he so lu t i ons o f t he roo ts o f t he above equat ion (1.14) a re

t r i v i a l . The n o n - t r i v i a l s o l u t i o n can be app l ied t o the composite.

Since complex roo ts occur i n conjugate pa i r s , on l y t he p o s i t i v e ones

may be app l i cab le f o r purposes o f t h e i nves t i ga t i on .

Spencer's above method o f approach i n analys ing t r a c t i o n s i n

f i b r e - m a t r i x composite, i s pu re l y t heo re t i ca l . The r e s u l t s may be

acceptable i f t h e n o n - t r i v i a l so lu t i ons obtained g ive reasonable r e s u l t s

when app l i ed exper imenta l ly t o t he composite. A probabl i s t i c theory

f o r t he s t reng th o f shor t f i b r e composites was proposed by Fuduka e t a1

C471 i n which the s t reng th o f u n i d i r e c t i o n a l sho r t f i b r e composites was

predic ted. They assume t h a t t h e f a i l u r e o f t h e composite occurred due

t o t h e i n a b i l i t y o f the sho r t f i b r e s b r i d g i n g a c r i t i c a l zone t o c a r r y

the load. The s t ress concentrat ions on t h e f i b r e s b r i d i g i n g a f i b r e end

gap are evaluated as a func t i on o f t he number o f f i b r e ends forming t h e

gap. The s izes o f t he gaps were then pred ic ted f rom a p r o b a b l i s t i c

approach. The sho r t f i b r e composite s t reng th was then est imated f rom t h e

gap s i z e and corresponding s t ress concent ra t ion f a c t o r . By tak ing i n t o

account t he p r o b a b i l i t i e s o f f i nd ing f i b r e gaps w i t h the number o f

adjacent ending f i b r e s K , greater o r equal t o KO, the u l t i m a t e s t reng th

o f t he composite, ocu was determined as

c I 1 N P j 'cu = 'fu j;K 7

5 J t 0 '

where Vf = f i b r e volume f r a c t i o n o f composite

N = number o f f i b r e s

P = p r o b a b i l i t y f u n c t i o n

a = u l t i m t e f i b r e s t rength f u

K = s t ress concentrat ion f a c t o r

Numerical r e s u l t s obtained f o r a hexagonal a r ray o f f i b r e s i s shown

i n tab1.e 13.

Table 13: Hexagonal Array Stress concentrat ion f a c t o r s by Hedgepeth e t a1 [58]

Number o f discont inuous f i b r e , r

Maximum s t ress concentrat ion f a c t o r , K

I t may be po in ted o u t t h a t Fuduka e t a1 on ly adopted the

concept o f a ' c r i t i c a l zone' i n t h e i r ana lys is and hence the r e s u l t s obta ined

are u n i v e r s a l l y acceptable. Furthermore, q u a n t i t a t i v e comparisons o f

theory w i t h experiment was d i f f i c u l t because o f l ack o f in fo rmat ion on

f i b r e l eng th and o r i e n t a t i o n d i s t r i b u t i o n s , as we l l as f i b r e c r i t i c a l

l eng th i n c r i t i c a l sho r t f i b r e composite systems.

The above probabl i s t i c theory had e a r l i e r been analysed by

Zweben e t a1 C1591 who proposed a s t a t i s t i c a l theory o f ma te r i a l

s t reng th w i t h a p p l i c a t i o n t o composite mater ia ls . They considered

ma te r ia l s t o be imperfect heteregeneous cont inua composed o f d i s c r e t e

volume elements whose c h a r a c t e r i s t i c s a r e r e l a t e d t o ma te r i a l s t r u c t u r e

and imperfect ions. The s t rength o f the elements was assumed t o be a

s t a t i s t i c a l q u a n t i t y and as t h e ma te r ia l s were loaded, elements f r a c t u r e d

randomly throughout the body causing l o c a l i s e d s t ress concentrat ions.

The accumulation o f such breaks resu l ted i n o v e r a l l f a i l u r e . By

r e l a t i n g s t rength t o mater ia l s t ruc ture , Zweben's theory attempted t o

b r i dge the gap between the microscopic and continuous approaches t o

f r a c t u r e mechanics. The theory was then app l i ed t o composite ma te r i a l s

re in fo rced w i t h whiskers and continuous f i b r e s .

Comparisons w i t h experimental data showed f a i r l y good agreement,

w i t h t h e r e s u l t s f o r Whisker - re in fo rced composites appearing t o

p rov ide a good p r e d i c t i o n o f s t rength and an explanat ion t o the

d i s p a r i t y between t h e s t rength o f i n d i v i d u a l whiskers and the s t reng th

o f composite made from them.

It may be observed t h a t t h e approach t o ma te r i a l s t reng th

proposed by the above authors makes i t poss ib le t o r e l a t e m a t e r i a l s t reng th

t o s t r u c t u r a l imperfect ions. This then f a c i l i t a t e s t h e importance o f

thevar ious types o f inhomogeneities observed i n ma te r i a l s and the'

theory proposed can be used as t h e basis f o r s t a t i s t i c a l approach t o

designing w i t h imper fec t ion - s e n s i t i v e mater ia ls . The above theory

however has n o t conf irmed whether t he c r i t e r i a a r e app l i cab le t o l a r g e

volumes of ma te r i a l s which are more representa t ive o f r e a l s t ruc tures .

Furthermore t h e cumulat ive weakening-load concentrat ion ana lys i s was

app l i ed t o two dimensional composites which gave a p red i c ted f a i l u r e

s t ress t h a t i s n o t q u i t e as good as the f r a c t u r e propagation c r i t e r i o n

w i t h i t s at tendant cumbersome a p p l i c a t i o n which may r e q u i r e f u r t h e r

study.

Burgel e t a1 [22] considered another method o f ana lys ing s t ress and

s t r a i n i n t he ma t r i x o f a f l a t f i b r e - r e i n f o r c e d mater ia l w i t h i n the

e l a s t i c region. To t h i s end, a s i n g l e long e l a s t i c f i b r e as shown i n

f i g . 22 was considered as shown embedded i n and bonded t o a two-

dimensional matr ix . The assumed f i b r e l eng th i s 21,with D(x) as t h e w i d t h a t

p o i n t w i t h coordinates [x,fD(x)]. The problem i s reduced t o t h e s o l u t i o n

o f an i n t e g r a l equat ion f o r t h e f i c t i v e f o r c e f ( x ) ; t h e r e s u l t i n g s t ress

and s t r a i n i n t he m a t r i x and m a t r i x - f i b r e i n t e r f a c e shear s t ress.

Fig.22. Lemon Shaped. fibre after Burgel [221

The i n t e g r a l equat ion obta ined by Burgel e t a1 f rom the

above ana lys i s was o f t he form

where

cll (x,y) i s t he t o t a l s t r a i n caused by the app l i ed

s t ress and f i c t i v e fo rce f ( x ) .

0 0 011 O = 0, a,, = 012 = 0 are the homogeneous s t ress

app l i ed t o the ma t r i x a t i n f i n i t y .

The geometry o f t he f i b r e and the mater ia l constants en ter i n t o

the i n t e g r a l equat ion on ly through the c h a r a c t e r i s t i c f u n c t i o n K ( s ) .

Equation ( 1 . l 8 ) was solved numer ica l l y f o r a rec tangu lar f i b r e and a

f i b r e w i t h lemon-shaped ends and the r e s u l t s agreed f a i r l y s a t i s f a c t o r i l y

w i t h experiments.

A general curve t h a t can be used t o est imate the i n t e r f a c i a l

shear s t ress reasonably f a r away f rom the end reg ion i s given by t h e

above authors i n f i g . 22,

0 06 1.0 a 11

Figure 23: Fibre with lemon-shaped m d s fictwe force f vs relative distance a / \ from f ibre end for different values of K at the centre of the f ibre by Burgel 121 I

It may be noted t h a t t he i n t e g r a l equat ion i n (1.18) can be

used t o c a l c u l a t e t r i v i a l l y t h e f i b r e shape i f f i s s u i t a b l y given.

2.2(i i Fur ther Composite Charac te r i s t i cs

2 .2( i fa ) Compactibi l i t y :- Aggregates i n f ib re-concre te c rea te compaction

d i f f i c u l t i e s ,[45,42,431. Swamy [ I371 recommended t h a t i t i s advantageous t o

increase t h e f i b r e s i z e as the aggregate s i z e increases. For a

g iven aggregate volume, the re i s a c r i t i c a l f i b r e content beyond

which compact ib i l i t y i s g r e a t l y reduced.

Compac t i b i l i t y i s a l s o a f f e c t e d by bo th t h e l eng th and diameter

o f f i b r e [134]. The VB t ime genera l l y increases w i t h increas ing l eng th

f o r a given f i b r e volume and diameter w h i l s t there i s a decrease i n

c o m p a c t i b i l i t y as the diameter decreases f o r a g iven l eng th and volume

o f f i b r e s [134,39J.The d i f f i c u l t i e s o f compaction associated w i t h f i b r e

concrete mixes a r e n o t r e f l e c t e d by the compaction f a c t o r as s ta ted by

Edginton [34]. I t was a l s o discovered t h a t more energy i s requ i red t o

compact f i b r e concrete than convent ional concrete,[66,67]. Mould v i b r a t i o n f o r

compaction i s p re fe rab le t o i n t e r n a l v i b r a t i o n s ince i n t e r n a l v i b r a t i o n

can cause agglomeration o r al ignment o f t he f i b r e s and i n some cases

b a l l i n g . Table 1 4 shows the e f f e c t o f method o f compaction on t h e

f l e x u r a l s t reng th o f s t e e l f i b r e re in fo rced motar [130]. The r e s u l t s

o f Table V ib ra t i on a r e r e l a t i v e l y much h ighe r than those o f needle

v i b r a t i o n [135,136,1371.

2 .2 ( ia i b ) Strength:- F ib re- re in fo rced mat r ices can f a i l i n t h e f o l l o w i n g

modes under d i r e c t t e n s i l e stresses ,[7,8,9,ll,I2].

Effect of Method (of Compaction on the F lexura l

Strength o f Steel, F i b r e Reinforced Mortar [I301

4 : 1.0 1 I O . Sand Volume: 50%

-i bre Volume

%

p,lexural Strength Rat io

Needlei V ibra t ion Table V ib ra t i on I

( i

( i i )

( -i i i )

I I 6 7

M a t r i x f a i l s , t h e n f l b r e s F a i l i n t e n s i o n o r bond a t a

l o w e r conlpos i t e s t r e s s l e v e l . M a t r i x r a i l s , , t hen f i b r e s f a i l i n 1:ension a t a h i g h e r

coi i iposi te s t r e s s l e v e l . M a t r i x f a i l s , thcrr f i b t ~ s f d il i n horill )I, a h i g h e r

co~ripos ilx s t r e ~ s s 1 eve I . F i g . 24 i s a r e p r e s e n t a t i v e %tr f~ ! is . -s t , ra in i:l.ir.vt:s i n d i r c c t ter is iot i

f o r Glass f i b r e K e i r ~ f o r c e d Coili l~osi Le (GIK) l a i u i r r a t c ~ a s comp i!ed l ~ y

Swalny [140].

The curves show d i s t i n c t l i n e a r , p o s t - c r a c k i n g I&IJV-iour.

A t h i g h e r s t r a i n s , t h e r e i s a p r o g r e s s i v e r e d u c t i o n i n the s t i f f n e s s

o f t h e composil:e clue t o cracking,l.36,h!; ,6!3].

Therc p e r t o l)r a d-is!:.inc:L rii f'Ferr?llcc! i n I.I>c .( Ii!xura'l

composi te showrd ;I loore b r i t t l e f < ~ i l u w I I I I ? i t . Vibre or

Fig. 24: Tensile stress-strain curves fa GRC compositer [Ref 1403

Fig. 27, Steady-state Creep as a function of Tensile Stress far Cu/W Composite after Street 11263

o r l i m i t i n g f i b r e volume f r a c t i o n i s 6-8% beyond which the re i s

l i t t l e improvement i n t e n s i l e s t rength because o f increased p o r o s i t y

i n t h e composite. Ai r -cured specimens have h igher s t rengths than water-

cured specimens. The increase i n t e n s i l e s t rength i n s t e e l f i b r e

composite i s l i m i t e d t o t h e order o f 30-50% f o r 2-3% f i b r e volume f r a c t i o n .

Compressive Strength:- The presence o f f i b r e s holds t h e composite together

even a f t e r complete f a i l u r e and increases the compressive st rength. The

compressive s t rength i s d r a s t i c a l l y a f f e c t e d by presence o f voids. For

b r i t t l e matr ices, compressive f a i l u r e r e s u l t s i n t e n s i l e s p l i t t i n g on t h e

maximum p r i n c i p a l s t ress plane. Therefore i t i s recommended t h a t f i b r e s

be a l i gned i n the d i r e c t i o n o f maximum p r i n c i p a l s t ress o r normal, f o r t h e

d i r e c t i o n o f maximum compressive s t ress r a t h e r than p a r a l l e l t o i t as f o r

d u c t i l e matr ices. Compressive f a i l u r e o f a specimen a l s o r e s u l t s

from the l a t e r a l i n s t a b i l i t y o f t he columnar elements separated by

tens ion cracks,[20]. Since poor bond reduces s t reng th and increases

res is tance t o v ib ra t i on , the combined use o f organic and inorgan ic I

f i b r e s (eg. polypropylene and s tee l ) i n se lected propor t ions w i l l

op t im ise bo th the s t a t i c and dynamic s t reng th properties,[761,77,781.

2 .2 ( i i c ) ' Shear Resistance: L i t t l e i n v e s t i g a t i o n has been c a r r i e d o u t

i n t h i s area. However, Batson e t a1 [I81 found t h a t bo th t o r s i o n a l and

shear res is tance a re increased by t h e i n t r o d u c t i o n o f f i b r e s i n

concrete. The type of f i b r e and volume f r a c t i o n a l s o a f f e c t shear Cl81.

More d e t a i l e d work has t o be done on laminar and t ransverse shear

s t resses of composites.

73

2 .2 ( i i d ) Fa t igue Resistance: Fat igue s t rength increases w i t h increase i n

volume f r a c t i o n o f f i b r e s [17]. Batson,[17,18,19,62] discovered t h a t f a t i g u e

6 s t reng th o f 90% o f t he f i r s t crack s t reng th a t 2 x 10 cyc les i s a t t a i n a b l e

w i t h 2.3% s t e e l f i b r e volume f r a c t i o n f o r non-reversal t ype o f loading.

With g lass f i b r e composites subjected t o c y c l i c load ing i n tension, t h e

permanent deformation increases w i t h peak s t r a i n reducing the composite

s t i f f ness . Br iggs [20] discovered t h a t h igh s t rength and h igh modulus

f i b r e s such as carbon gave f a t i g u e st rengths up t o 70 t o 80 Nmm2 when used

i n composites. These st resses a re much h igher than normal unre in forced

cement matr ix .

2 .2 ( i i e ) Toughness and Impact: Ba r r e t a1 [16,69,75,142] i nves t i ga ted the

toughness index t o measure t h e energy absorpt ion o f f i b r e r e i n f o r c e d

concrete. By toughness index i s meant t h e measure o f amount o f energy -

r equ i red t o de f l ec t the f i b r e concrete beam used i n t h e modulus o f rup tu re

t e s t by a g iven amount compared t o t h e energy requ i red t o b r i n g the f i b r e

beam t o Bhe p o i n t o f f i r s t crack [16]. The f i b r e s used [ I 6 1 f o r t h e

toughness index t e s t were polypropylene, s tee l and GRC mate r ia l . The index

r e s u l t s f o r polypropylene are as i n Table 15 f o r DENCEL t e s t specimens. The

toughness index r e s u l t s for compact compression t e s t specimens f o r s t e e l

f i b r e i s as shown i n Table 16 and t h a t f o r GRC ma te r i a l i n Table 17 [16].

Barr e t a1 drew t h e fo l low ing conclusions on t h e above - ( i ) Toughness index va r ies from 0.25 f o r p l a i n concrete upto a

maximum o f u n i t y f o r very tough f i b r e re in fo rced concrete ma te r ia l .

( i i ) S i m i l a r index values are obta ined when s i m i l a r s i ze o f

notches i s in t roduced i n polypropylene specimens.

TABLE 15

Toughness Index Results f b r DENCEL Test

Specimens (Polypropylene F i b r e Concrete) -

F i b r e Con t e n t

( % I Average

Notch Depth (mm)

Toughness Index

Results

Average Toughness

Index

TABLE 16 -

Toughness Index Results f o r Compact Compression

Test Specimens (Stee l F i b r e , 30 mm Notch) -

F i b r e Content

(%I Toughness

Index Results

Average Toughness

Index

( i i i ) The index values f o r polypropylene was f rom 0.32 t o 0.41, f o r s tee l

0.34 t o 0.84; and f o r GRC f rom 0.91 t o 0.93. These values r e f l e c t

numer ica l l y i n t h e shape o f l oad d e f l e c t i o n curves.

Ba r r [I61 recommended f u r t h e r study on 'bond' specimens. B a r r e t a1 1151

a l s o i nves t i ga ted the f r a c t u r e toughness o f polypropylene f i b r e concrete

and obta ined r e s u l t s almost s i m i l a r t o those o f Ba r r and L i u [161.

Hannant C561 reviewed the developments i n f i b r e concrete techno1 ogy w i t h

specia l emphasis on toughness and impact res is tance. Swamy e t a1 [I421

i nves t i ga ted impact res is tance o f s tee l f i b r e r e i n f o r c e d 1 i gh twe igh t

concrete and concluded t h a t -

( i ) A low impact res is tance t o f i r s t c rack ing does no t necessa r i l y

i n d i c a t e low impact res is tance t o failure,[127,128,129,130].

( i i ) With p l a i n l i g h t w e i g h t aggregate concrete, t he re appears

t o be a l i m i t t o concrete s t rength beyond which impact

res is tance decreases. A s t rength o f 45 N/mrnz appeared

t o be such a l i m i t . I n the presence o f f i b r e s , such a

l i m i t d i d n o t appear t o e x i s t .

( i i i ) With a f i b r e volume o f I%, subs tan t i a l increases i n impact

s t rength and energy absorpt ion can be achieved over those o f

p l a i n concrete.

( i v ) A h igh f i b r e l eng th and aspect r a t i o w i t h sur face deformations

which enable extensive debonding o f the f i b r e s appear t o be I

essent ia l c h a r a c t e r i s t i c s f o r h igh impact res is tance.

TABLE 17

Toughness Index Resul ts f o r GRC M a t e r i a l

Specimen S ize

Toughness I n d e ~ Results

Notch ilepth/Specimen Uid th I ~ o u ~ h n e i s

Index

0.91

0.93

a g =0.4

0.926

0.946

a g =0.5

0.900

0.927

a =0.6

0 .907

0 .929

2 . 2 ( i i f ) D u r a b i l i t y : Long term behaviour o f f ib re - re in fo rced matr ices have

n o t been very exhaust ive ly invest igated. Steel f i b r e composites have n o t

go t much d u r a b i l i t y problem except t h e r u s t i n g o f t h e s tee l i t s e l f if

exposed [135,144]. P l a i n and f i b r e r e i n f o r c e d normal and l i g h t w e i g h t aggregate

concrete exposed t o var ious aggressive environments i n an uncracked

cond i t i on over a pe r iod o f th ree years by Edginton and Hannant [341 have

shown no s i g n i f i c a n t d e t e r i o r a t i o n a p a r t f rom sur face s t r a i n i n g . Steel

f i b r e concrete a l s o shows e x c e l l e n t f reeze - thaw res is tance. A f t e r 300

f a s t cyc les o f f r eez ing and thawing, t h e d u r a b i l i t y f a c t o r f o r 1.67 f i b r e

volume va r ied from 73 t o 85%,[71].

A l i e t a1 [4,5], summarized t h e i r i n v e s t i g a t i o n i n t o t h e s t reng th and

du rab i l i t y o f carbon- f ib re r e i n f o r c e d cement composites as shown i n -

Table 18. From t a b l e 18 i t i s c l e a r l y seen t h a t a t ambient temperature,

bo th a i r and water-cured samples r e t a i n t h e i r i n i t i a l s t reng th over a

pe r iod o f one year. I n t h i s respect, t he carbon f i b r e s a r e b e t t e r than

commercially b o r o s i l i c a t e g lass (e.g. E-glass) f i b r e s . When t h e carbon

f i b r e composite was s to red cont inuously i n water a t 5 0 ' ~ a b e t t e r

idea o f the d u r a b i l i t y was obtained. There was s l i g h t d e t e r i o r a t i o n

i n s t reng th over a per iod o f one year, though t h e r e s u l t s were more

super io r than those o f cement composites made w i t h E-glass f i b res , t h e

s t reng th of which reduces t o ma t r i x s t rength when subjected t o t h e iame

storage.

The mechanical p rope r t i es o f carbon-f i b r e cement composite are' 1 i s t e d

i n Table 18,[4,201. From t h i s , i t i s seen t h a t t h e s t reng th and s t i f f n e s s

Matrix

Ordinary Portland Cement

Ditto

Ditto

Ditto

Ditto

TABLE j$

Durability of Carbon-fibre-reinforced cement

Uniaxial Load by Ali 141

Type of

Fibre

:arbon Fibres

Jniaxial

landom mat

tandom mat

'E' glass fibre

Volume

Fraction

1 - 4gm in tensile

!nE zone

3% uniformly dispersed

4.8% uniformly dispersed

Modulus of Rupture (mN/mZ

7

days

48.1

-

-

-

32.5

air cure6

- 180

days

water cured

- 365

days

:ured in water

o f t he composite i s g r e a t l y improved. Using the mixture law, an at tempt

was made t o c a l c u l a t e t h e e l a s t i c modulus and t e n s i l e s t reng th o f carbon-

f i b r e cement composite tak ing i n t o account the o r i e n t a t i o n e f f i c i e n c y

f a c t o r proposed by Krenchel [79]. There i s a reasonable agreement

between ca l cu la ted and experimental values as shown i n Table 19.

2 . 2 ( i i i ) Polypropylene F ib re Reinforced Cement,Concrete and o ther Matr ices:

I n recent years i nves t i ga t i ons have been c a r r i e d o u t i n polypropylene

f i b r e re in fo rced cement o r concrete,[50,53,54,55,56]. One o f such

i n v e s t i g a t o r s i s Dave e t a1 [32] who showed exper imenta l ly t h a t t h i n

cement sect ions re in fo rced w i t h chopped polypropylene f i b r e s can sus ta in

use fu l s t resses i n th ree p o i n t bending beyond an i n i t i a l crack st ress.

The h ighes t pos t crack f l e x u r a l s t resses a r e obta ined f o r t h e h ighes t f i b r e

concentrat ion and the f i n e s t longest f i b r e s which can be s a t i s f a c t o r i l y

incorporated i n t he cement. The re in fo rced samples were o f d isc - type

w i t h chopped f i laments incorporated t h e r e i n by a mixing, dewatering and

pressing technique,[891.

Wh i l s t t he comparative values o f pos t crack s t ress achieved by

Dave's i n v e s t i g a t i o n a r e r e l a t i v e l y low ( l e s s than 20 MN/m2) compared

w i t h Hanfiant's C561, they i n d i c a t e t h a t t h e h ighes t s t resses can be

achieved f o r

(a ) maximum f i b r e concentrat ion cons is ten t w i t h homogeneous

d i s t r i b u t i o n o f f i b r e s and w i t h no s i g n i f i c a n t weakening

o f t h e mat r ix .

I Matrix

Ordinary

Portland Cement



Tppe of Fibre

Bigh moduls

Staple Fibre (Em)

High Strength Staple Fibre

(Hs)

Bigh Modulus Random Mat

TABLE 19

Properties of Carbon Fibre-reinforced

Cement Composites by A l i 141

Form of Fibre

Continuous

Uni-direc- t iona l alignment

Continuous Uni-direct- ional

Chopped Fibre Mat Random i n a Plane

E las t i c Modulus (GNDlastic Modulus (GN/mZ

Tensile Strength (MN/rn2) -

volume f rac t ion experi-

mental

'T -

calcu- la ted

experimental

I Bod Impact Strength (Wu mm/m

( b ) f i n e s t , lorrgesl f lbres which can be s C l t i s f a c t . o r i l y

incorpovated.

( c ) l l i g l ~ e s t f i b rc? Young's modulus.

The f i r s t two c o r ~ c l u s i o n s wou'ld bc expected f ' i . ~ 1.11(! i :r~:at.~iirnl o f

m u l t i p l e c r a c k i n g g iven by Aveston e t a l I ] . I n t h i s , !.he r e i n f o r c e m e n t

o f an o the rw ise b r i t t l e i n a t r i x by a l - i g i e d f - i b r e s o f h i g h e r t e n s i l e

s t r e n g t h than t h e m a t r i x has been cons idered. The va lue for t h e

1 l i m i t i n g c r a c k s e p a r a t i o n (between x ' and Z x by t h i s method) . i s

where Vlll :. I I f r c i o ~ ~ of i i i .- II I'

s l r c w y i h o f i ~ ~ a t u i x

r - rddiu; o r c y l i n t I r i c < l f i b r e

T = fibre bond s t r e s s w i t h c e ~ r ~ e n t

the experimental study carried out by Dave et a1 did not however detect

the influence of any fibre property other than concentration on the

pre-crack matrix properties and therefore it may be justified to define

the role of fibres as load-bearing once the matrix has attained and

exceeded a failure strain.

The behaviour of polypropylene reinforced cement in both tension

and flexure was investigated by Baggot [ I41 who obtained results for

direct tensile tests.

The loading rate was lm/min.

Figs.39 and 40 show typical stress - load/specimen extension and stress-load extensometer extension for 0.35 matrices reinforced with

fabrillated filaments.

Fig.41 shows curves for stress-load/extension for varying length

of fibres. It can be seen that there are load drops at first cracks

which suppressed when sufficiently long fibres greater than 6mm are used.

- ---

Figure 28:

Typ~cal tensile st ress - load/ extensioh curves for 0.35 matrices reinforced with 5 .0% monofilament (type 130/261 after Baggot [14]

- 3.0% 6 mm

0.1 EXrENSOMETER EXTFNSION (MM)

Figure 29:

a Typical curves showmg tensile stress-load extensometer cxtenslon for for 0.35 matrices reinforced with f ~ b n l l a t e d filaments by Baggot [ 14 ]

1 2 3 4 5 EXTENSION (mml

FIGURE M): Typical curves showing tensile stress. load/total extension for 0.35 matrices reinforced with fibrillated filaments by Baggot [14]

Figs. 28, 29', 30 as p l o t t e d by Baggot show t h a t polypropylene

cement composite e x h i b i t s a s t ress s t r a i n curve w i t h a r i s i n g non- l inear

reg ion and f i n e , c l o s e l y spaced m u l t i p l e c rack ing dur ing t e n s i l e and

f l e x u r a l deformation, when the f i b r e s a r e discont inuous won0 and

f i b r i l l a t e d f i laments. F ine c l o s e l y spaced m u l t i p l e c rack ing was

produced dur ing f l e x u r a l deformation o f l abo ra to ry scale specimens w i t h

lengest f i b r i l l a t e d f i l amen ts a t maximum concentrat ion. S i m i l a r

mu1 t i p l e c rack ing a1 so occurred dur ing t e n s i l e deformation b u t was accom-

panied by major v i s i b l e cracks. Hannant [55] i n h i s i n v e s t i g a t i o n o f

f i b r e cement and f i b r e concretes discovered t h a t t h e o r e t i c a l l y a ma te r i a l

w i t h a t e n s i l e s t ress s t r a i n curve having a ho r i zon ta l non-e las t ic reg ion

and no s t ress drops, should produce a r i s i n g f l e x u r a l .stress s t r a i n curve

w i t h a r e i n f o r c i n g r a t i o o f 2.4. Baggot's r e s u l t s demonstrate t h i s

phenomenon q u a l i t a t i v e l y , al though n o t q u a n t i t a t i v e l y . They a l s o show

t h a t a c e r t a i n amount o f s t ress drop i n a t e n s i l e curve can be converted

i n t o a r i s i n g s t ress i n the f l e x u r a l mode, which i s a l s o p red i c ted

by same Hnnant's theory. A1 though Baggot discovered t h a t increased f i b r e

l eng th and concentrat ion l e d t o improved s t reng th i n tension and

f lexure , t he upper l i m i t o f t h i s va lue was n o t establ ished. Fyrthermore

f i b r i l l a t e d discont inuous polypropylene f i l amen ts prov ided b e t t e r

re inforcement than monof i laments because of t he more f ib re-mat r ix bond.

Authors 1 i k e Wal t o n and Majundar, [go J Mai e t a1 , [ 89 ]

Baggot [141 have a l s o t r i e d t o achieve a r e i n f o r c i n g

performance w i t h bo th discont inuous and continuous monofi lament po l y -

propylene i n cement based matr ices w i t h l i m i t e d success. They obta ined

f l e x u r a l load d e f l e c t i o n curves w i t h a r i s i n g post-crack reg ion w i t h

discont inuous reinforcement and s i g n i f i c a n t s t ress drops associated w i t h

crack development occur r ing dur ing the non- l inear deformation.

Gardiner and C u r r i e [48] i nves t i ga ted pu re l y t h e f l e x u r a l behaviour

o f composite cement sheets us ing woven polypropylene mesh f a b r i c s .

The type o f f a b r i c used i s as shown i n f i g . 3 1 w i t h f i x e d spacing i n

w a f t and w e f t d i rec t i ons . This can be compared w i t h Tensar geogrids

e a r l i e r described and w i t h polypropylene monof i laments and f i b r i l l a t e d

f i l a m e n t s %

The mode o f t e s t i n g by Gardiner was by f o u r p o i n t bending,

as i n fig.$! w i t h the component exposed t o d i f f e r e n t environments.

, Flexu ra l response curves f o r water cured environment i s shown i n f ig .32 ,

Flexura l response curves a t 7 months a f t e r c u r i n g i n wet,

d r y and r o o f environments are shown i n f ig.33.

Gardiner e t a1 concluded t h a t cement composite w i t h polypropylene

f i b r e f a b r i c has super io r f l e x u r a l s t rength than t h a t us ing open networks

of f i b r i l l a t e d f i b r e s . The use o f f a b r i c makes handl ing eas ie r , g ives

f i n e cracks on the tension faces a t a spacing equal t o t h e warp spacing

and p r a c t i c a l l y i n v i s i b l e t o the naked eye. I f proper c u r i n g i s n o t

a t t a i n e d and l a r g e shear fo rces a re present, then delaminat ion may

occur w i t h i n t h e composite.

There was no determinat ion i n f l e x u r a l s t reng th a f t e r 7 months of wate

cured specimens, Also there was no d e t e r o r i a t i o n i n s t rength when t h e

composite i s p roper ly cured f o r 48 hours and then subjected t o normal

warp tex

w e f t tcrx

Crns

dlrction ( reinforcing

222 ( f ibri l lated )

direction )

61 Fobrlc structure

f I

I

1 5 _ o m Test speclmen and rig dimensions

Figure 31 .: Schemat~c d~agram of fabric and test specimen

after ~ a r d i n e r [48 I

I crossheqd rate 0 05em/md

- . - a - 7 m o n t h ~ - - - - - - 2 8 days 7 days

W/C 0 .5 mlx l c / k environment 20' water

Flexural. response curves for water cured specimens taftent~~Gardin.r and Currie [ 4 8 I

spmmen th~cknoss lOmm

120-

100 - <

..

01 x

. w e t ----- - root ~nv~ronment for 7 months

wA 0.5 mlx l c / l s

0 1 2 3 5 5 6 crosshead movement h m

F~gute 33 Flexural response cwrves a+ 7nonths after curing ,, thren d ~ f l ~ r e n t pnvlronmento

Qardlner and CUrrlQ 148 1

weathering environment f o r up t o 7 months.

As a comparison t o Gardiner 's inves t iga t ion , Fathuhi [37] examined

the toughness o f polypropylene re in fo rced t h i n s labs supported on f o u r

s ides under f l e x u r a l o r impact loading. Continuous load-centra l

d e f l e c t i o n curves were obtained as shown i n fig.134.

The curves show a r i s i n g l i n e a r reg ion w i t h a drop i n l oad a t

t he f i r s t crack. The t e s t s ind ica ted t h a t t he h ighest f l e x u r a l

s t rength and toughness resu l ted when slabs were re in fo rced w i t h

polypropylene network w i t h the slabs e x h i b i t i n g m u l t i p l e c rack ing u n l i k e

those re in fo rced w i t h sho r t s tee l f i b r e s [67] which f a i l e d main ly by a

s ing le crack. The t a i l o f the curve i n f ig.34 demonstrates the

gradual t r a n s f e r o f s t ress from the mortar ma t r i x t o the network

reinforcement. Di f ferences i n the i n i t i a l slopes o f the curves may be

mainly due t o d i f fe rences i n slab thickness.

Attempts have been made.by Netlon Ltd. a t s tandardis ing t e s t

methods and design p r i n c i p l e s f o r Tensar (pp) geogrids as reinforcement

i n s o i l s . They were used i n embankments h o r i z o n t a l l y t o c u t t he shear

plane thus prov id ing s tab le foundation o f a bank o f optimum height .

The inc lus ion o f polymer g r i d s i n t o the shoulders o f t he

embankments i s very usefu l as i t al lows compaction equipment t o operate

I I

.

r'

- c . -- - ----c4

I +---- -----_----- I I

I I II 2 I 5 I I I

Z 5 k I 'r I 8 Central deflection - (mm)

Figure 34 L ood dsfloction curve for Polypropylene f ibre cement

Composi te by F a t t u h i [37]

[67] reported i n Japan the success o f us ing tensar i n embankments f o r

res is tance o f earthquakes.

Murray e t a1 demonstrated the techn ica l and economic success

o f r e p a i r i n g s l i p f a i l u r e s by rep lac ing the o r i g i n a l s o i l i n predeter-

mined compacted layers, us ing Tensar polymer g r ids . The g r i d s n o t on l y

s t i f f e n up and r e i n f o r c e the base, b u t they a l so provide the necessary

t e n s i l e f o r c e t o prevent a deep step f a i l u r e . Net lon ~ o . L t d . formulated

a design c r i t e r i a f o r reinforcement Sta t ing t h a t

Factor o f sa fe ty FOS = Restoring Moment D i s t u r b i n g Moment

If the ca l cu la ted f a c t o r o f sa fe ty i s l e s s than the requ i red f a c t o r

o f safety, t he r e s t o r i n g moment can be increased by the i nc lus ion o f

reinforcement i n the form o f Geogrids l a i d i n ho r i zon ta l l a y e r s i n the

one o f weakness, t o c u t the shear plane, thus prov id ing a s tab le

foundat ion t o a bank o f optimum height . The requ i red g r i p length, L,

i s g iven by

L = -&- x FOS

Tensar g r i d s a p p l i c a t i o n was a l s o analysed i n a p p l i c a t i o n t o

Embankment const ruc t ion w i t h foundation mattresses by Net lon Co. Ltd-

FOS = f a c t o r o f sa fe ty

E f f e c t i v e Mattress w id th i s given by

28 = FOS p4'

I n Reinforced s o i l r e t a i n i n g wal ls , Tensar r e i n f o r c i n g gr ids ,

arranged i n l aye rs d i s t r i b u t e d throughout the depth o f f i l l , can be

employed t o r e s i s t the ho r i zon ta l pressures r e s u l t i n g f rom t h e mass o f

f i l l and any surcharge app l ied t o the f i l l , whether ho r i zon ta l o r v e r t i c a l .

Considering a granular f i l l mater ia l o f known dens i ty and shear

res is tance angle, subjected t o a surcharge, a t any depth (h) f o r a v e r t i c a l

wall,. and ignor ing the e f f e c t s o f f r i c t i o n between s o i l and

wa l l , Net lon Co.Ltd. a t t a i n s f o r design s t a b i l i t y

FOS = 74- a(ah t q ) d ....... ( 1 .24

where

TI i s t he permiss ib le t e n s i l e s t rength o f the Tensar gr id ,

K = 1- s i n 0 a 1 t s i n 0

h = depth o f tensar reinforcement

q = surcharge

d = d is tance between 2 p a r a l l e l tensar reinforcements.

2 . 2 ( i v ) Creep (Time-Dependent - U e l o r ~ i ~ a l i o n ) :

I n t c n s i o n , t h e a d t l i t i o n o f f - i b res t o i na t r i ces , I-c?tluces c r e e p and

shr inkage. I ldg ington C341 f o u n d t h a t t h e use ul' f i b r e s i n i l i a t r i c e s has

n e g l i g i b l e c f f c ! c l on c r e e p bl. c o n c r e t e i n c o i i ~ p r e s s ~ i o r ~ . 1-1; has been found

by U r i g g s [;20] t h a t carbon F i b r e s reduce r l e x l r ~ x l crc?ep d e f l e c t i o n by a

-Fat-tor o f 6 a t 2% . f i b r e volume f r a c t i o n . A t 9% volume f r a c t i o n , t h e

c r e e p r e s i s t a n c e inc reases by 40 t.ilues.

S t r e e t [ I261 i n v e s t i g a t e d steady-creep o f f i b r e r e i n f o r c e d

r n a t e r i a l s and c a r r i e d o u t t t l c o r e t i c a l ~~ red~ Ic1 : io r rs s u p p o r t e d by exper iments .

1 t Nils f ound t h a t a l s d 1 vo lu l l s frac: t i o n o r 1: i b re , V t,he r o l e o f

m a t r i x anr riot be iynoret l . Thu ill-eis !;el\i I i;iv i l,y o f t:lie co~aymsil;e

.is dependerrt on i ff , yc aitd l i ; (whorr! and ,- ai.c. ro:,pect.ivc.ly C f ,

s t r e s s e s and s t r a i n s iri corl~pos i t e ) .

S t r e e t [ I 2 6 1 developc?d t l i c o r i e s f o r si.a;lily-f;l:i:tc> c r e e p b e h a v i o u r o f

f i b r e con~po!;ites and corllparetj t l lese w.iI:It exl~cl ' i l~lr!r~l:s. l i o t h c o n t i n u o u s

and disco111;inuous f . i b res were cons idered. IFnr ... i : ! ~ ~ ~ l . i r ~ u o u ~ . . ~ ~ I ; ibrc Conpos i tes ,

S t r e e t o b t a i n e d an express ion f o r c o ~ ~ q ~ o s i t e c r e e p s t r e n g t h I > ~ as

where ac = Composite creep s t rength

E = Steady s t a t e creep r a t e i n composite C

E = Steady s t a t e creep s t r a i n i n mat r ix m

afO, tf0, n and amo ' 'rno , m a re known constants f o r a g iven

temperature. The above expression was obtalned under the f o l l o w i n g

assumptions and procedure:-

( i ) App l i ca t i on o f a constant load t o the composite thereby causing

s t ress i n f i b r e and mat r i x which a re ca l cu la ted by assuming equal

s t r a i n and having a knowledge o f s t ress /s t ra in curves.

(ii) D i f f e r e n t i a l creep i s n u l l i f i e d by the emergence o f steady-creep when

e q u i l i b r i u m i n phases i s a t t a i n e d hence

( i i i ) A power-law r e l a t i o n between steady s t a t e creep r a t e E and s t ress a

found t h e o r e t i c a l l y and emper ical ly f o r many mater ia ls was used thus

if ' I n af = afo [--I and

" fo

'm 0 = a C-I

'"'3 'rno

( i v ) The normal r e l a t i o n which p a r t i t i o n s the composite s t ress oc between

the components according t o t h e i r volume f r a c t i o n s was used thus

This expression shows i m p l i c i t l y t h a t there i s no i n t e r a c t i o n

between phases and may be considered v a l i d f o r a l l values o f t he composite

creep s t rength oc, which may be def ined as the st ress t o produce a s p e c i f i e d

creep r a t e o f the composite.

If the above assumptions are v a l i d f o r r e a l behaviour, t h e s t r a i g h t -

l i n e nature o f above equation suggests t h a t i t should provide a very good

est imate o f creep s t rength experimental ly. Su rp r i s ing l y , t h e above

theory has no t been va l ida ted experimental ly.

It would appear t h a t the on ly w e l l documented creep-rate data f o r

cont inuous f i b r e composites showing an apparent steady s t a t e are those o f

Copper matrix/Tungsten f i b r e by Mcdonels [88]. To compare theory

w i t h experiment these data were combined w i t h those f o r Tungsten w i res

o f H a r r i s and E l l i son . The pred ic ted composite behaviour i s given f o r

a minimum volume f i b r e f r a c t i o n Vf = 0.10 and a maximum Vf of 0.74 a t a

temperature o f 816'~. As much as theory agrees f a i r l y w i t h experiment

i n t h i s case, more intermediate values o f Vf cou ld have been explored t o

conf i rm the behaviour.

Fol lowing S t ree t ' s inves t iga t ion , F e r r i s [38] c a r r i e d o u t a

r igorous mathematical ana lys is o f a r e g u l a r l y re in fo rced mqte r ia l i n the f o r

o f a two-dimensional composite subjected t o un i fo rm tensiop as shown i n f i g s

35 and 36. H is theory as shown by Schl icht ing[119] involves t h e no t ion o f

mater ia l where t h e arrangement o f t h e f i b r e s i s so chosen as t o g i ve an

arrangement o f r e g u l a r l y repeat ing conf igura t ions ,as i s the case o f polypro-

pylene f i b r e mesh (Netlon) e a r l i e r described. This repeat ing u n i t s having

been chosen, the problems o f so lv ing the biharmonic equation w i t h complex

Fig. 35.. An ldealised Composite after Ferris [38]

Fig. 36. Forces acting on the Fibre after Ferris U81

101

boundary conditions are solved to determine the resulting flow in the

matrix and stresses produced in the fibres. He assumed an idealiaed

composite with equations for two-dimensional creeping flow of non-1 inear

fluid. By applying some basic equations of motion and constitutive

equations, Ferris arrived at the bi-harmonic equation of the type

where

$ is the stream function such that Bwm" a* u = 3 ; \ I= --

a~ ax mMm

U and V being velocity components in x and y cartesean coordinates.

Equation 1 .ela is obtained on the assumption of Newtonian flow. It is also assumed that the flow is so slow and steady that time function

in the above equation is absent. The solution of the equation is by

combined analytical and numerical techniques with appropriate boundary

conditions. Typical results obtained by using finite sine transforms

and truncating a number of times gave results of the form

X 4D IA(P) (sech 0-1) - cr = 4c1 + LX

0.1

where zx is the average normal stress in the x direction, L is the distance between end of fibres,

i s t h e ~ o e f f i c i e n t o f v i s c o s i t y ;

A(p) , L ( p ) a r e f u n c t i o n s o f p t o be deter i i t i r~cd by t h e boundary c o n d i t i o n s

p 11 al; y = o and y - 1 . arid O i s . I. I-G

'The r e s u l t s ob ta ined show tl~cit urider c e f t a i n corid-i t<ioris a reversed

f l o w regime i s s e t up i lnd hence t h e nraxiiiiur~l si:rc:s';os i n the f i b r e s do r io t

occur a t t h e i r ni i t lpo' ir i ts. T11e I - i i r i i t a t i on o f F e t ~ r i c , ' a n a l y s i s a r e . tha t

( d ) t h e n w t e r ~ d l niusl. be r e i n f o r c e d i n a r e g u l a r way

(b ) t h e e f f e c t o f d i s c o n t i n u i t y i s n o t cons ide red

( c ) no exper i r l iental v a l .ids-Lion o f t h e t h e o r y was c a r r i e t i o u t .

( d ) t h e t i m e Kleriierit was o n l i t l e d because o l the assurr~pt ion n~iide

above Newtorriarr f l o w .

Invesl ; ic jat ing F u r t h e r i n t o I ' e ry i s theore t . i c - i l ar~; i . lys is, M i l e i k o

19611 t l c r i v e d !;irn]~'lc'r slhcdr t l i c o ~ ~ y by b u i 'Id i r q l ip 2 ::hear rr~odcl. T h i s

shear imode'l f w I i.l:a tcxl i:hc s o l ~ r t iorl uf t i I . i I ~ u b ' l e i r ~ i t t h e r e s u l t s

o b t a i n e d showed b e t t e r iigrrcriicnl. w i 111 1-crt-is c f i l i :u l i l t io~ i r . . M i l e i k o

r e p e a t i n i l e1enlerr.L: i s sub jec ted t c ~ pure tensior t a s ?tiown i n fi51.'(5 .

Fro111 t h e above ( f i y . 3 5 ) d i i igrani i t i s assirrr~ed t;hat AUCC

undergoes s l lear w l r - i l s t t h e r e s t o f t h e elenrent: .is -icjriu'retl, hence t h e

nallie 'S imple Shear Model ' . Other a Cterirpts a t forr i iula t i i i y equa t i o r i ol- c r e r l ~ o f coiriposi t.e

were i~iade by Ross and Lornran 11 17.1 who exyresst?d c:rec!p as a f u n c t i o n o f

t i m e i r l t h e f o r m

r -- t . i . . .> 1~ .r>..... ..*.... ,..,..+" +,.

be determined from experimental r e s u l t s . This expression i s hyperbo l ic

1 i n na ture and has C = ?; which i s t he l i m i t i n g value o f creep when t

tends t o i n f i n i t y . As a f u r t h e r improvement, t he Uni ted States Bureau

o f Reclamation expressed creep o f concrete C as

C = F ( K ) l o g e ( t + l ) where

K i s t h e age a t which t h e l o a d i s appl ied,

F(K) a f u n c t i o n represent ing t h e r a t e o f creep deformation w i t h t ime

and t i s t h e t ime under load i n days. The above equat ion g ives curves

o f exponential na ture and app l i cab le f o r per iods under ' load up t o f i v e

years. The main problem i n fo rmula t ing equat ions f o r creep o f concrete

i s t h a t i n p ropo r t i on ing concrete i t i s n o t poss ib le t o change one f a c t o r

w i thou t a l t e r i n g the other . Possib le f a c t o r s a r e f o r example water-cement

r a t i o , w o r k a b i l i t y , mix p ropor t ions . As s ta ted by N e v i l l e [ I 021 f o r

unre in forced concrete, i t i s on l y t he cement paste t h a t undergoes creep,

t he r o l e of aggregate being t h a t o f r e s t r a i n t and hence creep may be

regarded as a f r a c t i o n o f vo lumetr ic content o f cement paste i n concrete

w i t h a non- l inear r e l a t i o n . On t h i s bas is there fore , N e v i l l e proposed

t h a t t he creep o f concrete C can be represented by t h e equat ion

g i s t h e volumetr ic content o f aggregate

u , t h e unhydrated cement

C~ , the creep o f neat cement paste o f t he same q u a l i t y as

used i n concrete and

where

Ir - - Poisson's r a t i o

Ea - - modulus o f E las t

E = modulus o f last c i t y o f concrete i -

"a - Poisson's r a t i o I f aggregate.

When concrete i s t he re fo re r i i n f o r c e d , a d i f f e r e n t form o f creep

equat ion i s obtained. Equations

i n t he case o f organic f i b r e

t o t h i s e f f e c t a r e r a r e e s p e c i a l l y

reinforcement.

J a i n e t a1 [68,6] i n v

o f g lass f i b r e - r e i n f o r c e d

was c a r r i e d o u t f o r a pe r iod

increase i n percentage s t r a i

than 1% o f maximum creep. T

For r e l a t i v e l y low s t

has a considerable e f f e c t on

curves show an exponential

hours, t h e creep curves o f

slow and un i fo rm increase i

deformation a t 1000 hours m

c l i m a t i c change f rom w in te r

r a t e s o f deformation are d i

Ja ins ' i nves t i ga t i ons took

temperature and e f f e c t s o f

s t i g a t e d exper imenta l ly t he creep behaviour

o l y e s t e r laminates. The observat ion which

o f 1500 hours was used i n determin ing t h e

f o r n ine types o f laminates which was l e s s

e creep curves a re as shown i n f igs.37 and 38.

t i c load ing t h e curves showed t h a t weathering

the creep r a t e and deformation. The

ependence o f s t r a i n on time. Up t o 1000

11 t h e laminates a re almost l i n e a r w i t h a

s t r a i n w i t h time. The sudden r i s e i n t h e

y be due t o r i s e i n temperature due t o

t o summer. The curves a l s o show t h a t t h e

f e r e n t f o r d i f f e r e n t types o f laminates.

ognisance o f v a r i a t i o n s i n humidi ty ,

a i n and sunshine.

Fig. 37: Creep ba bborotory and rtms: normal(1 and type bn

Fig. 38: Cmrp behl paraffin wed(:

viour of laminator exposed bmi& the ldrr natural wwthwlng at ~ O M N / ~ nder-cured (2)pamffin cured (3) rte (6) C145J

wr of normal (I) under-cumd (2) and ominam under natural weafhen'np [64rg

I 106

: A number o f i nves t i ga to rs c a r r i e d o u t t h e

the use of models. Prominent amongst

these are Nev i l l e , D i l g e r an Brooks [ I021 who a p a r t f rom desc r ib ing

o the r models as l i s t e d i n Ta l e 20, s ta ted t h a t f o r a q u a l i t a t i v e

s iml l la t ion o f rheo log i ca l be av iou r o f concrete, a model should have the

f o l l o w i n g c h a r a c t e r i s t i c s - f

( i ) i n i t i a l s e t on J p p l i c a t i o n o f load, p a r t l y constant

and pa r t1 y s t r e s-dependent ;

i i ) instantaneous e a s t i c deformation; i i i ) delayed e l a s t i c deformation

i v ) a creep proper t 4 dependent on s t ress and t ime ;

v ) a perlnanent ( i r ecoverable) time-dependent set, r predominating a stresses near ing t h e u l t imate ;

( v i i ) delayed e l a s t i c

The i d e a l i s e d deformations usled i n b u i l d i n g up the r e a l behaviour a r e

e l a s t i c , v iscous o r p l a s t i c a d a re represented by a spr ing, a dashpot b and a f r i c t i o n element respec i v e l y . The bodies w i t h these i dea l l i n e a r It p rope r t i es a r e r e f e r r e d t o as Hookean s o l i d , a Newtonian l i q u i d and

S t . Venant body respec t i ve l y 1041. There a re two bas ic models [ I041

known as a Ke lv in (Vo ig t ) mod 1 and a Maxwell model shown i n f i g s . 39

and,40'respectivel y.

Name o f Model

Burgers

Summary o f some Rheolog.ca1

by Nev i l 'e ,

Ross

F l ugge

Table 20

Model f o r Concrete as discussed

D i l g e r and Brooks [ I 0 2 1

Cowan

Freudenthal and

R o l l

Powers

Gopalakrishnan, N e v i l l e & Ghal i

Tor ro ja and Paez

B juggren

G luck l i ch

I s h a i

- -

Type o f Combination o r equivalence

Ser es o f combination o f Ke lv in and Maxwell I Kel i n w i t h a sp r ing i n se r i es i Kel i n i n se r i es w i t h 2 dashpots, one o f whi h has a constant v i s c o s i t y and t h e 0 t h r vary ing. i

c a l l y s i m i l a r t o Flugge's w i t h t i o n a l p a r a l l e l dashpot

Set o f b r i t t l e spr ings i n se r i es w i t h 2 Kel i n model s;one o f which has a dashpot w i t / non-return va lve r e s u l t i n g i n non- rec very o f seme o f t ime-deformation.

one Maxwell ; a l l i n The Maxwell has a l i n e a r spr ing

w i t h respect

Sorpt ion element as aga ins t po t i n Ke lv in .

A cdmbination of Powers and Cowan

~ e s d r v o i r o f viscous f l u i d connected t o ashpots w i t h spr ings as i n Maxwells. 1

spring, wedge,claw,closed w i t h te lescop ic

F r i c t i o n e d spr ings i n ser ies w i t h quasi-Kelvin.

and a l i n e a r sp r i ng -

t o Ke lv in model i s given

Some elaborate rheo l

such as spr ings moving i n

u n i d i r e c t i o n a l dashpots w i t t

C501, Sorpt ion elements

and sets o f spr ings g i v i n g

these add i t i ona l elements r

e l a s t i c i t y f o r which springs

1 - e t t l t l ) where

)g ica l models use o the r mechanical devices

dishpots (Tor ra ja and Paez's model) [150],

non-return valves (Gluchl i c h ' s model )

(Gopalakrishnan, N e v i l l e and Gha l i ' s model)[521,

r a y under d i f f e r e n t l oad l e v e l s . The use o f

cognises a departure from simple v isco-

and dashpots are s u f f i c i e n t , The response

The response t o a Maxwell mo e l i s g iven by 1

x = extension o f

J = spr ing compl

P = appl ied load

t = t ime and t,

spr ing

ance

7 r e t a r d a t i o n time.

r e f e r r i n g t o Maxwell and eld din respect ive ly . The response t o a

Pt xm = PJ + - Vrn

The subscr ip ts m and k

i n Burgers Model i s g

1 .

combination o f Maxwell and

C1021 by

iven . Ke lv in model as

N i c h o l l s [I041 describes o ther

Je f f rey , Schwedoff and

ranging from v i sco -e las t i c

Determination o f ConStants:

ensuing equations a r i s i n g

mathematical procedure i nvo lv ing

46,57 and some c l a s s i c a l

models such as Bingham, l e t h e r i c h ,

PaynJ:ing-Thomson t o charac ter ise behaviours

t o an e l a s t i c .

The determfnat ion o f constants i n

f rom the models may be achieved by some

Approximations o f func t ions [1,2,24,

theor ies o f e l a s t i c i t y 87, 1551.

I 109

CHAP ER THREE -+- DEFINITION OF T ~ E PROBLEM AND METHODOLOGY

The fo rego ing l i t e r a t u e reveals t h a t vegetable f i b r e s a r e T suscept ib le t o decay, inadeq a t e as regards bond and have v a r i a b l e

engineer ing proper t ies depen i n g on species, c l i m t i c growth and method

o f preparat ion.

f Asbestos f i b res , in-as much as they have proved themselves i

usefu l i n t he manufacture o f c e i l i n g and p a r t i t i o n boards, r o o f i n g sheets

and pipes, have presented t h as h e a l t h hazards. A.

Glass f i b r e s a re and w i t h p r o h i b i t i v e cos ts

a r e spa r ing l y used.

Steel f i b r e s , apar t f r o b t h e problem o f segregat ion dur ing mix ing

a re equa l ly r e l a t i v e l y For sho r t s tee l f i b r e s , t he bond

problem e x i s t s w i t h the a t t e n a n t d i f f i c u l t i e s caused by d i s c o n t i n u i t i e s . P Kevlar, PRD 49 and PRD 9, desp i te i t s h igh s p e c i f i c modulus and

s t reng th a re r e l a t i v e l y new a d hence n o t f u l l y tested.

Polypropylene f i b r e s ha e good chemical res is tance, a re

unaf fec ted by moisture, have i g h t e n s i l e s t rength and e longat ion a t

break w i t h low s p e c i f i c g rav i y. They a re e a s i l y handled dur ing mixing. I Polymers s u i t a b l e f o r t he pro u c t i o n o f polypropylene f i b r e s a r e d by-product o f petrochemical dus t r y which are r e l a t i v e l y low-cost

polymers and a r e a v a i l a b l e o i l producing countr ies. The product ion o f

t he f i b r e s i s i n several ranging f rom cont inuous c y l i n d r i c a l . . f i laments which can be t o s p e c i f i e d lengths c a l l e d mono-filaments

o r as f i l m s and tapes y be f i b r i l l a t e d t o form f i n e f i b r i l s of

r e c t a n g u l a r c r o s s - s e c t i o n i r i t h e fo r in of rensar geo-q r ids . E n g i n e e r i n g

p r o p e r t i e s o f t h i s f i h r e ' l i k e t ( ? i ~ ! i i I e strt ! t ig.th? YOUIKJ'S ~~roc lu lus can be

v a r i e d d u r i n g product io r r . Hence, po lyp ropy lore Fibr.es l e n d

therr~selves rriore addp tab le f o r use, i is a r e i r i l c ~ r ~ i n g e l e i w n t t.o coricrel:e

o r c;es~ent t h r ~ r i any o t h e r t ype o f f i b r e . I l i o y urlso have v i x o - e l a s t i c p r o p e r t i

I n cons i i le r . ing 11o1,ypropy 1 w e , t h e i~resh t:ypt? i l: Irrore atlvaii taqeous

tharr e i t h e r t h e nionol i ' la i~~enl ; o r f i b r i l l a t e d s t i o ~ t f i l n i ~ i ~ r i t s s:ince t h e

two i d t t e r do n o t lprovide adequate bond. 111 a d d i t i o n , t h e p roh le~r i s

encountered i n m ix i r i g o t h e r t ypes o f f - i b r e s w i t h c o n c r c t e nniriely

segrega t ion , l ) i ~ l l i n g and inho i~ iogenei ty a r e v i r t u a l l y abser i t .

1-hey a r e trrore d u r d b l e ut ider t l i f l e re r r l : env-irori~~ie!li l-al c .or i t l i t ior~r ; arrd do

n o t p r e s e n t any F i r e h a z ~ ~ r d s . O t i i ~ c c o i ~ n l ( 1 1 it::. i~i l i?vcrr!: 11 i r a b i l i t y o f

ve ry l ow f a i l u r e st . ra in i ind weakircss i n i . e ~ r s i i ~ i i , i i. Irocoi~~c?; r leci?ssary t o

i n v e s t - i g a t e the d u r a b i l i t y o f isarr- iaqe o f l.li,i!, I r i - i t - t ' l c l r i a t r i x wif.h a

t h a t s tudy.

Ward [ I 5 3 1 and H i l l e t : a l [601 i n t h e i r n ~ u l t i p l e i n t e g r a l

a n a l y s i s of po lyp ropy lene f i b r e a lone , showed t h a t t h e f i b r e e x h i b i t s

a v i s c o - e l a s t ~ c behav iou r and the c r e e p b e l ~ a v i o u ~ " c,lrr be d e s ~ r i b e d i n

t h e furn l

l h i s equai . ior~ i s o n l y l o r t he f i l i r e o lotre anil n o t wller~ uscd as a

conlposi t e i ~ r a t e r i a l . I'heorel: i ca l a t t c ~ u p l : ~ b y Spencc?~- 1, 124 1 i ~ r i r l F e r r i s 1381

i nves t . i ga te r e s p e c t i v e l y t h e t r a c t i o n s and c r e e p b e h a v i c ~ u r d l t h i n

s7sbs i n for111 o f f i b r e - m a t r i x la i l l ina te gave i n s t r i n s i c r e s u l t s alust

o f wh ich a r e t r i v i a l . F e r r i s ' r e s u l t s may be c o n s i d e r e d c o n s e r v a t i v e

s i n c e t h e c o n v e c t i v e p a r t o f a c c e l e r a t i o t ~ i n h i s e q ~ ~ a l - i o u was assuoled

ze ro and hcnce t h e tiillc! c o l ~ ~ l i o r ~ e n t was absent i n t h e ger leral b i -ha rmon ic

:1v I a i l ,la.t = ...... . - Y - & ' , Y f i iax ;,y

( i i ) C u n s t i t ~ ~ t i v e equa t ions f o r f l i r i t i i~k~ l : , . ' i x ,

( i i i ) C o n t i n u i t y equa t ion

and choos ing a stream f'uncl.ion such t h a t

then i 1; may bc shown t h a t

:;elution o f which w.il 'I g i v e r:i.rcsser,, I I I I I ~ I ~ , I I, i~ue Col- the

proposed coir iposite.

tho f o l l o w i n g I b e observed.

( a ) The composi te l a w i n a t e analysecl lx+llilvi.s a!; d

v i s c o - e l a s t i c n l a t e r i a l evidericetl b y the e las t3c:

and v i scous cor~lponents o f t h e s a i d equa t ion i n

Cerliis o f parameter w - i l h i n 1:he c o ~ ~ ~ p o ~ i t u ~ t s

( b ) Thc f 'unct ion :I, i s ,I s t r e s s funct ic jr~ w l ~ i c l ~ i n t u r n

M i s t h e m a t e r i a l c o n s t a n t HI.

Modulus o f E l a s t i c i t y .

(c) The t i n ~ e r l e ~ r ~ e n l i s no l o n g e r r c r o as i n

F e r r i s r q u a t r o ~ ~ .

( d ) F o r i h e pred ic l3or r o f f, t r a i n - t i ~ w r c l l. ion,

e i t h e r a c l a s s i c a l s o l u t i o n LIT t h e a l~ov t ' eq~ra l : ion

(3.11) lnay be I I I ~ I ~ ~ o r a K h e o l o g i c ~ i l Model ~ i r r d l y s i s

used. l h e l a t t e r ~ ~ ~ e t h o d i s adopted i n t h e

l ~ r e s c n t work.

i n predic t i r tc l t h e (.:reep beh i l v iou r o f concret.e, N e v i l l e , lli l g e r p, ~~~~k~

1. 1 07 11 i o n F i t-~~ied t h t c;onc.retc! e x h i b i 1.s v i :c .o-das t i c ~nove~nerrt under

load, obtir- ined elitpi r im l f o r ~ ~ r u l i ~ s p r t d i c t.iny C I W ! ~ h111. d i d nol.

inve!;I;-iyatc the c reep behav iou r o f po lyr~~e~.- re . in i t ) r ,cet l cor~c.ret,c. The

e x p o n e n t i a l exp ress ion o b t i l i t l ~ ? t l b y abovr? a ~ ~ l : l i o r ~ ; i o r t l ~ r c r e e p o f

u n r e i n f o r c c d c o n c r e t e y i v e n b y

has n o t shown ve ry good agreerwnt w i t h experis~t?nt.$l dat.a, r i o r can

c r e e p cu rves be p r e d i c t e d w i t h t h e above e q u a t i o n w i t h o u t a c t u a l l y

p e r f o r ~ n i n y exper i i r ten ts . , The e r r o r i r r t h e above equal. ion may be due t o

t h e i n i t i a l a s s u ~ n p t i o n m d e namely t h a t f o r a i)ivi,n conc re te , t h e r a t e

o f c reep a t al ly t i r r ~ e i s p r o p o r t i o ~ t a l t o 1 1 ~ ~ ~IIIUIIII~, o f po~0111-. ia l c reep

s t i l l t o appear l e s s s c h a r a c t e r i s l i c o f visco-clI,:~;tic I t o w . l hns

Where t - to = duration of loading

C = - limiting or ultimate creep

A = constant.

Integration of (3.10) gave rise to (3.9) above. Experimental

investigations into thin fibre-reinforced concrete slab in flexure by

Fattuhi [37] and Gardiner [MI, gave load-displacement curves with

corresponding pre-crack and post crack behaviour; without any corresponding

theoretical back-up.

In view of the fore-going therefore, it is necessary to formulate

an equation predicting the creep (time-dependent) behaviour of a polymer

fibre-reinforced concrete in form of a thin slab laminate.

From conclusion drawn on equation (3.8), a Rheological model

is conceived such as to depict the desired visco-elastic movement.

~ a i i c equilibrium equations based on the model are formed,

combined and solved to obtain a general equation characterising

creep of composite so formed.

Experimental investigation in both flexure and direct tension

of the laminate is performed and deformation-time graphs plotted.

Numerical values are obtained for the constants in equation

derived in (b) from experimental results.

(e) A curve o f bes t f i t i s obtained f o r t h e experimental r e s u l t s

by us ing Gauss Least Squares method i n

( i ) f i n d i n g a curve o f bes t f i t f o r a. and a

( i i ) f i n d i n g a curve o f bes t f i t f o r a. and em

(iii) f i n d i n g a curve o f b e s t f i t f o r oo and c0 .

( f ) Wi th the data i n (e) an emperical equat ion i s obtained

f o r a curve p r e d i c t i n g the Creep ( s t ra in - t ime) r e l a t i o n s h i p

f o r t h e laminate w i thout a c t u a l l y performing experiments.

Fig. 3% Kelvin Model

Fig.40~ Maxwell model

Figure 40(a) A typical Cornposit Cell of Matrix and fibres placed 4. . between 2 cartesian axes x and y .

I I A l ' i I Foul1 8 .. -. 118

RIIEOLOGICAL MOUCL ANALYSIS, --- THEORY - AN0 EXPERIMENIS ........... ........... .-

1.1 l l h e o l o g i c a l Model : - Fro111 d e r i v a t i o n s i n e q u a t i o n (3.!1) o f Chapter Three,

the v i s c o e l a s t i c n a t u r e o f t h e c o ~ i ~ p o s i t e was p r e d i c t e d as e~lrbodied i n t h e

? x p o n e n t i a l e q u d t i o n and cu rve ' a s a l r e a d y de?.cvibc$l i n 1 i t.erat.urt?. It i s

:he re fo re necessary t o s i n ~ r r l a l c a s i tua t . i o n wheri: ttic? conc:rcttc ( b r i t i l e

l a t e r i a l ) i s r e i w f o r c e d w i t h a polylner f i b r e (ducl. i l e i l ~ a t e r i a l ) ' a n d loaded

r i t h a c o n s t a n t l o a d i n t e n s i o n w i t h o u t i n v o l v i n g any r e c o v e r y .

Cons ider t h e r e f o r e t h e Rheo log ica l i ~ ~ o d e ' l shown -in F i g . 41 made up o f

i y s t e ~ m ( 1 ) and ( 2 ) -in p a r a l ' l c l . S y s t c ! ~ ~ ~ ( I ) i s a t n v d i f i e d Maxwe l l ' s n~ode l ,

:he ~ a o d i f i c a t . i o t i b e i n g t l ~ a t t h e r e i s a n o r - t r n valuc? i r l t h e dash-pot t o

l e l ~ i c t i r r e c o v e r d b i 1 i t y . T h i s n to i l i f i e t l Maxwc:II I!:; ~ i ~ u d a l r e p r e s e n t s So l i d

e i n f o r c e n ~ e n t s lmade up of aggregates dnd p o l y p r o p y l t w T i l m . l i l t ' who'le

ystem i s a i ~ ~ o d i f i e d for111 o r I 1 ~ ~ , y i i t ~ i ~ i g - ' S h o n ~ ~ ) ~ ~ ~ r ~ t.lodt?l ill111 may he c a l l e d t h e

h e o l o g i c d model . I n tilt! ensuin!) dna l y s i s , t:lic !.;~.rTi i x e s ( I ) and ( 2 )

e f e r t o system ( 1 ) antl ( 2 ) r r s p e t t i v e l y .

. ] ( a ) .. l t h e o l o y i c a l A n a l y s i s : ............

bihen t h e load u i s app' l ied, t.he f o ' l l o w i ~ ~ y eqr io t ions a r e o b t d i r ~ e d

rom t h e dynasiics o f t h e systetn:-

Fia. 41 : The Rheoloaical Model

( 7 equa l ions w l t l - I il nnkr~own~,)

When a i s constant = a say, (constant load) 0

then 4 . 7 becomes ' = 0 , then A1

where

0

and = 2 a

Solution o f 4. 8 i s generally of the form

Step Loading:

a ( t ) =

where H i s Heaviside step func

Integrate from t = 0- t o t = 0' t o get

C E ] = jump i n s t ra in

From 4. 9 , when t = 0 , then

0 = 0 I FI;[ l-l+r where r = 2

(3) = - Hr 2

Subst i tut ing i n 4 .9 gives

a ~ ( t ) = o - 1 at PT; F R ; e . . . . . . . . . . . . (4.10 )

= Em (I- &-at, where - '0 = - M* Em -

(s t ra in a t i n f i n i t y )

4.2 Solution f o r Constant a . ( ~ i m p l i f i e w

From 4.11,

Denoting y = -In ( l + r ) ( l - then

y = a t ................ ( 4 . 1 1 4

a i s determined by the method o f l e a s t squares; thus from formula

where ti and yi are experimentally determined values o f time t

and displacement y as given above.

Also a t t = 0 from 4.11

4.3 2nd Method O f Solution f o r a.

Let a

i = A; and

( ) = B from 4.10,; then l+r

( 4 .I 1 ) becomes

= A - Be -at

Applying d i f fe rent values o f E and t, then

-ati = A - Be

-at2 E~ = A - Be

N = 0, gives 2 .E (A - Be -at1

1x1 - 0

1

N $- = o gives 2 c (A - Be -at1 i = l

1 - ~1 -atl = 0

a N - = 0 gives 28 E ( A - Be -ati aa - € 1

) t i e -ati i = l = 0

( 4 . 5 , (3.26 and (4.17) a re 3 non- l inear equations t o be solved f o r

A , B and a f o r d i f f e r e n t values o f t and E.

4.4 T h i r d Method o f So lu t ion f o r a by Newton Raphson

u i s given by formula

a = a. - F(uo)

v where

a = i n i t i a l approximation 0

a = subsequent approximation and n

a i s found such t h a t the e r r o r E i s given

4.5 EXPERIMENTS

4.5.1 Test Apparatus and Mater ia ls

4.5. I (a) D i r e c t Tension Creep Equipn

= Minimum.

s c r i - The simple machine as shown i n f ig.42 i s b a s i c a l l y a mod i f i ca t i on

o f 'Tecquipment Creep Machine SM6' o r i g i n a l l y produced by Research &

Development Engineers, Nottingham, England f o r lead specimens. The

Modiflr d to rqulpmrnt L.T.D, England

main mod i f i ca t i on i s i n the jaw holder, E. A modi f ied jaw ho lder i s

produced w i t h a much wider a l o t t o take f i b r e concrete specimens.

The machine i s b a s i c a l l y a s i n g l e l e v e r t e n s i l e t e s t i n g machine

so made t o provide a steady uni form load throughout any t e s t . It

comprises a loading lever , A, p ivo ted on a b a l l journa l bear ing and

connected t o the lower jaw holder, E by two s tee l s t raps t o ensure t h a t

t he specimen load i s always app l ied v e r t i c a l l y . The upper end o f t he

specimen i s he ld i n jaws which are supported from the top ho r i zon ta l

cross member, a tension n u t being provided t o a l l o w t h e he igh t t o be

adjusted.

Loading i s achieved by a weight suspended from t h e load ing

l e v e r a t var ious leverages. The extension o f t he specimen i s d i r e c t l y

measured on a very sens i t i ve d i a l s t r a i n gauge.

The machine i s mounted on a t r i a n g u l a r c a s t i r o n base p l a t e

f i t t e d w i t h a co-planer s p i r i t l e v e l and th ree ad jus tab le l e v e l l i n g

screws.

4.5.l(b) Dimensions o f Apparatus

Length - 45.7 cms (18 i n s )

Width = 25.4 cms (10 i n s )

Height = 43.2 cms (17 i n s )

N e t t Weight = 16.8 kg (35 l b )

4.5.2 Experimental Procedure: The apparatus i s mounted on a very r i g i d

bench f r e e f rom v i b r a t i o n which i s l i k e l y t o cause e r r a t i c readings on

the d i a l gauge. The base p l a t e i s l e v e l l e d w i t h t h e l e v e l l i n g screws.

Care i s taken t o keep the temperature o f t h e room constant by a i r -

cond i t ioner .

The load ing l e v e r i s l i f t e d on t o i t s r e s t p o s i t i o n support. The

cross-p ins i n t h e jaw holders are withdrawn by unscrewing t h e knu r led

l o c k i n g nuts. One end o f t h e specimen i s i nse r ted i n t h e lower jaw

ho lder and t h e cross p i n i n t he knur led l o c k n u t i s replaced. The

upper end o f t he specimen i s f i t t e d i n t he t o p jaw holder , w h i l s t t h e

cross p i n and knur led n u t a r e replaced. The top tension n u t i s

ad jus ted t o take up a l l clearances, b u t n o t enough t o l i f t the l oad ing

l e v e r o f f i t s stop. The knur led n u t a t each end o f t he specimen a r e

t ightened.

The d i a l gauge i s s e t t o zero by r o t a t i n g the bezel. A p o s i t i o n i s

se lected on the loading l e v e r f o r t he weight. The breaking o r

f r a c t u r e 1 oads are determined.

I n app ly ing the l oad t o the specimen, t he l oad ing l e v e r end i s

l i f t e d very s l i g h t l y t o a l l o w the support t o be r o t a t e d c l e a r o f t he

lever . The l e v e r i s very c a r e f u l l y lowered u n t i l t he f u l l l oad i s

t rans fe r red t o t h e specimen and a t t he prec ise moment t h a t t h e specimen

receives the load. The p o s i t i o n o f t he load on t h e l e v e r i s v a r i e d i n

a s a t i s f a c t o r y t ime between the a p p l i c a t i o n o f t he l oad and f r a c t u r e of '

the specimen i s obtained.

Figure 43. Test Specimen for, Direct Tensile test

Other loads a r e app l i ed t o the specimen and de f l ec t i on readings

as recorded by t h e d i a l gauge a r e recorded f o r over a pe r iod o f s i x

months. The average ambient temperature i s recorded. Typical t e s t

r e s u l t s are obtained f o r var ious loadings and extension-t ime

curves a r e p l o t t e d l a s shown i n f ig .46 ,

4.5.3 Preparat ion of Test Specimens f o r D i r e c t Tension Test:

Mat specimens were prepared i n the l abo ra to ry by the hand lay-up

process. This invo lved preparat ion o f mould o f dimension shown i n f i g . p 3

and p l a c i n g of polypropylene f i b r e i n s i d e same mould on t o p o f t h i n

l a y e r o f concrete. Another l a y e r o f concrete i s placed on t o p o f t h e

f i b r e and then t rowe l l ed smooth. The concrete i n use was mnu fac tu red

t o a known mix design s t rength o f 25 N/lm2. The cement used i s o rd ina ry

Port land, t h e coarse aggregate i s I s i a g u g r a n i t i c type, 6-10mm maximum

size; the sand i s Opi River sharp sand; and water i s o rd ina ry s o f t p ipe

borne water. Appropr iate volume f r a c t i o n s o f t h e f i b r e i n t h e range o f

f i v e t o seven percent were used. The c h a r a c t e r i s t i c s o f t h e polypropylene

f i b r e used i s as shown i n Table 8. Special care was taken t o prepare

the end g r i p s of the specimen by thicknessing, w i t h allowance made f o r

poss ib le f a i l u r e by shear through the holes i n which the cross p i n s w i l l

pass. A i r - cu r i ng o f specimens was used f o r a pe r iod o f 28 days.

4.5.4 F lexura l Tests:

4.5.4(a) F lexura l Test Creep Equipment: - Descr ip t ion

The r i g as shown i n f i g .44 i s a mod i f i ca t i on o f t h a t used i n

the U n i v e r s i t y o f Shef f ie ld , England. The equipment i s f a b r i c a t e d main ly

from components o f s tee l angle, which forms the stands and braces.

FIGURE 44; CREEP ,RIG FOR FLEXURE

Figure 45: Typical Pol ypropyleno Reinforced Concr~te

laminato tested in flexure.

fig. 46- Variation of Extension with time for Various

The p la t fo rm f o r the specimen t o be loaded i s made o f angle i r o n s

w i t h broad flanges. A rod 6mm diameter runs h o r i z o n t a l l y and i s used

i n ca r ry ing the d i a l guage a t such a l e v e l t h a t t h e sp ind le o f t h e

guage r e s t s or touches the specimen t o be loaded. The p l a t f o r m

where the equipment i s mounted i s pe r fec t l y l e v e l l e d b y s p i r i t l e v e l

and padding on the f loor . Three t i e r s o f p la t fo rms a r e provided, each

w i t h i t s d i a l guage arrangement, so t h a t a system of 3 t e s t s can be

conducted simultaneously. This device has the specia l advantage o f

n o t making use o f spr ings i n loading. The dead weights a r e app l i ed

d i r e c t l y . The m i r r o r f i x e d as show i s used i n measuring gauge lengths

from a r e f l e c t i v e system below.

4.5.4(b) Preparat ion o f Test Specimens f o r t h e F lexura l Test

Appropriate moulds t o accommodate specimens o f s i ze 30cm x 60 cm x

2.6cm were prepared, Polypropylene f i b r e mesh was c u t

t o the above appropr iate dimension and volume f r a c t i o n o f 5-7% and then

l a i d i n the mould, making allowance f o r the cover. Concrete o f mix design

s t rength o f 25 N/mm2 was placed i n the mould and v ib ra ted g e n t l y by tapping.

A i r -cur ing was used f o r t he specimens f o r a per iod o f 28 days. The

c h a r a c t e r i s t i c s of t he concrete components and f i b r e a r e as a l ready

described i n the d i r e c t tension t e s t specimen. Samples were made i n sets

o f th ree and tested i n F lexural C r ip Rig Equipmentdearl ier described

and shown i n f i g . 44.

4.5.5 Procedure f o r F lexura l Tests: Three specimens were mounted on the

creep r i g a t t h e same time i n d i f f e r e n t t i e r s . The r i g was s e t

p e r f e c t l y l e v e l w i t h t h e s p i r i t l eve l . Simple supports a t t he ends were

ensured by the special round m i l d s tee l bearings as a l ready described.

Constant dead loads were app l i ed t o the composite specimen by means o f

the load ing arrangement. De f lec t i on readtngs were recorded from the

s t r a i n gauges a t vary ing i n t e r v a l s o f time. The i n i t i a l s t r a i n read ing

o f t he i n i t i a l instantaneous load was c a r e f u l l y noted i n every case.

Gauge lengths were measured w i t h t h e demec gauge by t h e a i d of t he

m i r r o r shown i n the diagram.

The readings were taken f o r a per iod o f s i x months and shown

i n Tables 21 - 25 w i t h graphical p l o t s as i n Fig.47.

. .

136 CREEP - m m r lo-'

137

Table 21: Creep R e s u l t s for Flexural Tes ts - .L:E?!~>!:~J&. -- ~-

Constant Load, (lo - 14.3 K g

a = 2.9344065621 x I I J - ~

138

Table 22: . . Creep Results ~~ for Flexural - le:;ts ....... - k~~!.!t"taL

Constant Load, (1 - 13.4 kg (i

a -= 1.53663900484 x i n - " .~

T i lie

(h r s ) . ~~ ... . . .

0

% 0

100

200

400

600

i300

1 00U

1 %00

1400

1800

2000

2400

2600

3200

3640

4370

139

Table 2 3 : -. Creep Iksull;s for Flexural Tests - , Exper. . imental -...

Cans tdnt. Load ( I - 12.9ky I

140

Table 24 : Creep Results fo r ~ Flexural 'Tests - E&&e,rln!en&l.

.r -i me (h r s )

.. .

0

20

100

200

'iuu

600

fi00

1!lurl

1%00

I400

1000

IRUII

2000

2200

2400

2601)

37011

44011

. . . . . . ... . . . . . .- -

141

Table 25: -~ Creep - Results. fo r Flexural Tests - Experirnerrtal -

Const.an-1. Load r i :: 11 .II kq 1,

iu 1 1 0 l 7 0 1 4 x

CHAPTER FIVE

PREDICTION OF EMPIRICAL CREEP EQUATION FROM EXPERIMENT AND - RHEOLOGICAL T H X

5.1 Step 1 . Calculat ion of Constants u

Using Method i n 4.2-Simplif ied Method, a i s calculated f o r t yp ica l

load a, thus: -

For . a . = 14.3 kg 0

Strain

t

= 331.4 - -..... - .. - ..... ...... , '

S i m i l a r l y o t h e r us a r e c a l c u l a t e d f o r bl;he~. 'loiids dntl w s u l t s

A g r a p h i c a l p l o t o f ire a g d i t l ~ t {I showcd ;IrJ\N'(i!Y i l t l d t ,~ l y ,> l i t t e a r

r e l a t , i o n . Henct? a c u r v e o f b e s t F - i t wits chl:air.ted t~c'Ii1t:'itlq t h e two

coord. inates i n the l ' o r ~ i l o f ;in c q u s t i u r l usir\!j (;(IIJ,;s Ilictorcttt o f L e a s t

squares thus : -

Nor l~ la l equa t ions a r e o f t h e For111

5.2 P r e d i c t i o n o f Empirical Equat ionfor Tensile Creep from Experiment and Rheological Analysis

Having calculated d i f f e r e n t as f o r d i f f e r e n t a,, fit

a suitable curve t o the f i v e points denoted by

Normal equations a re o f the form

Here n = 5

EYj = 14.3 + 13.4+12.9+12.35+11.8 = 64.75

1 EX = 13.32+5.47+5.06+3.72+20.86 = 30.43 5

c x . y. = (14.3x3.65)t(13.4x2.34)t(1229~2.25)t(l2.35xi.93)t, J 1

Hence the normal equations are

From (5,7 ) and ( ' 5 . 8 )

a = 8.88

b = 1.7

and equation i s o f form

y = a t b x

When a = O , o = 8.88 0

Having got from experiments, d i f f e r e n t E~ f o r d i f f e r e n t oo, then

f i t a curve o f best f it t o the f i v e poitits,

€9 = 7 4 6 x 1 0 - ~ , o = 14.3

0 1

E = 6 9 9 ~ 1 0 - ~ , o -'2

= 13.4 0 2

E = 6 7 2 ~ 1 0 - ~ , o = 12.9 m3 0 3

E = 6 5 0 ~ 1 0 - ~ , a ='4

= 12.35 0 4

C - ms

625x10-~ , o = 11.8 0 5

Normal equations a r e o f the form

a n t bcx = c y j j

2 a c x . + b c x = c x . y

J j J j

Here n = 5

I ience t he normdl equa t ions dr'e

5a c 3392 x 10- ' 11 =- 64.75

i e . a 6 7 x 1 0 1 - 11.95

b 1 .YO9

rl - 13.00 - 6.81 x 1.909

- 13.00 - 13.00

: [I

t ler~ce e q u a t i o n ' is ol' f o r ~ u

i e . Li I ,.OU!l f O

~ - . .. ~~. ~- . .. .. .. . ... .. .. .. . . ~. . . . . . . . . .. .

I -ro~ri exper.irtlor~ts, i n t r o d u c e a curve o l be!;l. [ i t : t o 1:he

f o l l o w i n g :-

Normal equations

Hence the normal equations are

eqn. i s y = a c bx . . o := 6 .374 4. 2.683 i : , ~ 0 . -. .~ .- ....- - - -. .. .,...... ( 5 . 1 1 )

4.535 = .288!, I- (7

0

S u b s t i t u t i n g i n t he g e n e r a l r e s p o n s e e q u a t i o n ( 4 . 1 1 )

When t = 0 , and oo = 14.3

= 295.4 say 296

When t = a, o = 14.3 0

When oo = 13.4, t = 0, then

When o o = 8.88, t = 0,

CHAPTER S I X

RESULTS, DISCUSSION AND LIMITATIONS

6.1 Rheological Model o f Composite behaviour: Contrary t o t h e usual - t r e n d o f determining t h e type o f model t o be app l ied us ing experimental

data, t he p r e d i c t i o n here was done us ing a pure a n a l y t i c a l approach which

gave r i s e t o equat ion (3.8) shown as

Though t h i s equat ion was n o t solved, i t i s ev ident t h a t t h e terms

a character o f v i s c o - e l a s t i c i t y and

f ($1 = f(M) f(Et) denote

s t r a i n t ime re la t i onsh ip .

A model i s t he re fo re formed t o s imulate a v i sco -e las t i c behaviour

by us ing a Mod i f ied Maxwell ' s model and 1 i nea r sp r i ng i n p a r a l l e l . The

mod i f i ca t i on i n t he Maxwel l 's model i s i n t he use o f non-return value

w i t h i n t h e dashpot t o dep ic t t h e i r r e c o v e r a b i l i t y o f t h e composite.

Another mod i f i ca t i on i s t h a t the l i n e a r spr ings used i n p a r a l l e l

have d i f f e r e n t constants.

As shown i n f i g . 41 l inear s p r i n g i n systel l l 1, r e p r e s e n t s

t h e e l a s t i c r e a c t i o r o f t h e nun-ghrinkaqe p a r t o f t h e systel l l lnade up

o f re in fo rce lnen ts (aggregates and po lypropy1 ene F i b r e ) , w h i l s t

Syste l l~ 2 rep resen ted hy l l lodiCied Maxwc.11 ' 5 ttiodel r r p r e s e n t s t h e

s o l i d corllponr.nt o f catnienl: 1~asf.e. I h e celnc:r!t cornpotlent behaves

~ i ~ ~ o t ? l a s t i c a l ' l y w h i l s t re.inforc.enlent and dqrjrc!qa.l.c! colnl~onent behave

e l a s t i c a ' l l y . T h i s t h e r e f o r e a:;sullrs t h a t the f i b ~ c dws r io t reach

p i a s t - i c tilllit d t t i n ~ e equal t o i n f i n i t y . 'The nrotiel i s oil t h e whole

a a lod i f . ica t ion o f Poynting-'Thoinpson's w i th a s i l ~ i u l a t i o r i o f

i r r e c o v e r a b i l i t y denoted by t h e i n l r o d u c t i o r l o f n o n - r e t u r n v a l v e i n

system 2 . 'The a n a l y s i s o f t h i s model nrakes use of i i i i l~p l t? equat.ions

which c o r r e s p o n d i n g l y g i v ~ s r i s e to s imple f i r s l : o r d e r d i l ' fe ren t i a l

equa t ion g i v e r a 5 i n (4.8) as

The s o l u t i o n o f e q u a t i o n ( 4 . U ) above g i ves

1 - a t ) t(t) = ' . - ( I - - 1.5- " which ulay be regarded

I c i I 1 ~ i . d I I I I G ~ hc. c b s c i - i l m i a s

O ! W J , < L I ~ F , C I ~ ' L ? I , , : ~ L ~ I ' s 7 0 ; i ~ ~ o s i i i o n pt. i r ic- iplr ,in ~ 1 1 ~ i c h z.:.rain , i s a l inear

~ i : i i i t i a oT si.wss his.i.or.y I i . l) i c?.c:rit.od .in

.i11.;.(:9ral .Tcliuns as

6.2 - . Deic:ri~~ina .~.. .. - .. . t ion . . o f . . Coiist.ants: . By making use of expcr.imenta1

resu l t s , t h e constiint u i n t h e equation (4.8) i s determined fo r v a r y i n g

constant loi~ds. 'Two rac 'd~nd~ are uscd namely -

( a ) Tile Sirnpl i f i ed A p p r o a c h

( b ) IkwLon -- K a p h s o n ' s Apprmch.

The r e s u l t s a r e as shown i n t a b l e 26 .

a r e o b t a i n e d r e s p e c t i v e l y i l s tr I= 1 .7u -1. f3.88 . . . . . . .

(1

( 6

I r e s u l t s

. I )

w i t h g r a p h i c a l l ~ l o t s as s h o w i n f. igs. 48,!19,!.,Il~

From the clap i r i c a l c?qual..ioi~ o : 'I .<:I119 c , ' , , I:IK IIIII~LI 1115 0

o f e l a s t i c i t y o f the compos i te l a ~ i i i n a t e cnn be r l e te r i~ t i ne t l FI-OIII t h e s lope

o f t h e l i n e g i v e n by . t h i s e q u x t i o r ~ . T h i s ttc!wcv~:r i s f o r l i t ~ r i t i n g va lues

8

E

(r (kg) 4

2

Fig. 49: Relation between 7 and Eob

Fig. 50: Relation between '% , : a n d €0

o f load o and s t r a i n , E_. The ana lys i s L ~ e r e f o r e prov ides a simple 0

method o f f i n d i n g o v e r a l l moduli o f composite ma te r i a l s o f t h e t ype

inves t iga ted . Th is when compared w i th H i l l ' s method us ing tensor

ana lys i s i s much s imp ler and o f more p r a c t i c a l s i gn i f i cance .

6.3 P red i c t i on of Empi r ica l Equation f o r Tens i le - Creep o f Composite:

From equations (6.1),(6.2) and (6.3), an emp i r i ca l creep equat ion

i s obta ined and g iven by

The above equat ion expresses creep e x p l i c i t l y as a f u n c t i o n o f u and 0

t ime. Therefore a t any g iven t ime t and constant l oad oo, t he creep o f

the re in fo rced concre te laminate can be found w i thout a c t u a l l y preforming

an experiment.

The l i m i t i n g va lue o f l o a d i n t h e above equat ion a r e o o = 8.88kg

which i s the minimum. For loads below the value o f 8.88 kg, t h e power o f

the exponential s ign (e) i s p o s i t i v e and hence t h e curve ob ta ined w i l l be

t h a t o f e'. Furthermore, f o r t h i s minimum value o f l o a d oo = 8.88, a i s zerc

6.4 Graphical P lo t s : The creep curves ob ta ined both from experimental

r e s u l t s and p r e d i c t e d ana lys i s a re exponent ia l i n shape. The experimental

graphs f o r f l e x u r a l ,es ts a re shorn i n fig.!!7. The curves as pred ic ted f rom

ana lys i s a,re shown i n f i gs .51 - 55 and fit the experimental r e s u l t s f a i r l y

we l l . Computer p r i n t - o u t s o f same are shown i n t h e Appendix A1-A5. Resul ts

f o r d i r e c t t e n s i l e t e s t s a re equa l l y exponent ia l as shown i n f ig .57. The

few i r r e g u l a r i t i e s i n these curves may be due t o the v i b r a t i o n s from

adjacent surroundings and f l u c t u a t i o n s i n temperature. They may a l s o be

due t o shrinkage e f f e c t s since the concre te used was o f a comparative

smal l sect ion.

800 - I62

- - - - - - - - - - - - - -

\ 8 14.3 kg

Experimental Predicted Creep

300

200

100-

0

-

-

I I I I I I .--

1000 2000 moo 4000 MXX) 66ao Time hrs

Fig. 51: Predicted 8 Experimental Curves of ensile Creep of polypropylene fibre Reinforced Concrete Laminate for = 14.3 kg

0 1 I I I I I 1 1000 2000 3000 4000 UoOQ 6000

Time - hrs.

Fig. 52: Predicted B Experimental Curvm of Tensile Creep of polypropylene fibre . Reinforced Concrete Laminate for 13. 4 kg

mot

1 dij 8 1Z.Y Kg ---__ _ _ _ - - - - _ -

Predicted

I I I

3000 4000 5000 Time - hrr.

Fig. 53: Predicted 8 Experimental Curves of. Tensile Creep of Polypropylene fibre Reinforced Concrete Laminate

for 12.9 kq.

Fig. 64: Predicted 8 Experimental Curves of . Tensile Creep of polypropylene fibre

700

Reinforced Concrete Laminate

-

for K = 12 35 kg.

- - ------ -----T- Oi;. 1e.afj kg

Predleted

c E~pcrlmentol

200

loo-

0

-

- I I 6 I I

4000 a loo0 2000 3000

Time - hrs.

Time - hrs

Fig. 55: Predicted 8 Experimental Curves of Tensile Creep of polypropylene fibre Reinforced Concrete Laminate for @ 8 11.8kg.

167

Results for creep obtained experimentally are shown i n tables ,

21 - 25. Predicted r e su l t s compared w i t h experimental a re shown i n Tables

26 - 30. Computer plots of the curves f o r same are shown in the Appendix

A1 - A5. These curves a r e a l l exponential i n shape and since the equations

a re predicted from experimental data, the curves f i t f a i r l y very well.

Creep resu l t s from the empirical equation obtained i n equation (5.12) are

shown i n tables 27 - 31.

The corresponding curves a s shown in figs.51-55 show tha t f o r lower

values of' time upto approximately 800 hours, the predicted s t ra in i s lower

than the experimental s t r a in . Beyond t h i s time, the predicted s t r a in i s

higher than the experimental s t ra in . This may be due to the i n i t i a l

e l a s t i c behaviour of the composite before i t goes into p las t ic stage. The

values of s t ra in a f t e r 4400 hours tend t o a l imiting value fo r any constant

load appl ied which i s indicative of the exponential equation obtained.

Of the f i v e constant loads used in the flexural t e s t s , i t would

appear from f igs . 53, 54 and 55 tha t lower loads give curves whose predicteii

r e su l t s are f a i r l y very close to experimental resul ts . This may be a t t r ibu ted

t o the d i f f i c u l t i e s in determining precisely the upper limit of load by the

same empirical method used in the lower l imit . Generally, fo r the cases

tes ted , t h i s upper l imi t of load was determined by t r i a l and e r ro r , by which

method a suitable load tha t fa i led t o give immediate f racture and a t the same

time provide gradual s t r a in readings was chosen. In theory, equation (5.12)

would appear t o give i n f i n i t e s t r a in fop i n f i n i t e load, which i s not

practicable.

1 bo

Table 2 7 : Creep Reslr l ts I'ut' F ' lexu~ 'a ' l .. Tes ts ~. - P r e d i c t e d . and ~ E x p e r i m e n t a l

C o n s t t ~ n t Load, 1J = 14.3 K g 0

t = 2.1J3434065621 x 1 0 - I '

169 Table 2 8 : Creep Res~llts fc~r r l exur ' a l T e s t s - I)rf?clictetl and Experimental

~ ~" ---- ~- .~ - ..... -

1, I 9 ... Crcop - .. IKc!sul 1.5 for ..... iFlcxura l T e s t s _ - IYed _ _ . i c t e d and Exper i~nenta ' l

Cons tan t Load n = 12.9 kg (1

T a b l c 3 0 : CI.I:~,~I ilc!;;~ I t s f o r I - I ~ x u i - ; I I ' res ts - P I ~ I ic: t,rrtl a t ~ d Cxper i~nen t d I I ~ .... ~ - * ~ -~ .. , , .

Coilstant Load, 11 = 12.35 kg r3

Constan l Load i r = 11.8 Kg 0

1.8001370084 x ld ... ...~ , .

1:xl~c.r i ~ i l e r ~ t a l Creep

(l,l,ll) x I[)-% ... . . . . .~ .. . .. . .. . .

i O ! l

P I ?

250

30U

400

48U

5 3

560

5 ' I 5

5!J(!

0 0 C

fl[I!!

b l l

6211

625

625

6 2 5

0 1 !>

6%:~

In a l l the t e s t s , the f i b re volume fract ion e f fec t has negligible r

e f f ec t since these were kept f a i r l y constant during the t e s t s a t

about 5-7%. On the whole, the flexural resu l t s appear more r e l i ab l e

than the d i r ec t t ens i l e resu l t s . This may be due t o the re la t ive ly

la rger section of the composites t sted in the flexural investigation . as compared to the t h i n sections I the d i r ec t t ens i l e t e s t s , w i t h

the attendant g r ip problem.

Plots of coeff ic ient of creep or r e l a t i ve creep against time a re

shown i n f i g . 56 and that of creep compliance against time i s shown i n

f i g . 57, a l l f o r constant load. In each case, there i s an i n i t i a l

f a i r l y l inear region f o r upto 400 hours followed by a curve of shape

shown. Isochronous curves of load against creep a re shown i n f i g . 58.

These show tha t for lower times, Lhe graph i s l inear indicative of the

composite obeying Hookes' Law. For higher values of time, the shapes

obtained a r e no more l inear . Bu t the shape of curve obtained when

load i s plotted against l imiting values of creep is l inear and f a i r l y

well f i t ted by the curve a = 1.909 E_ as e a r l i e r described. 0

The plots of creep compliance against time show tha t f o r

higher values of time over 800 hours, creep increases with decrease

i n loads.

Creep Coefficient per ka/mm2

Creep Compliance < ( t )/T m m h

lable 31: ~ o a a - > v a i n Kelacion

f o r t = 600 hrs

Tab le 32: Load-S t ra in R e l a t i o n

for t = 2400 hrs

Table 33: . Load-Strain Relat ion

for t = 1400 hrs

S t r a i n

Table 34: Load-Strain Re la t ion

f o r t = 1000 h r s

(T

4)

11.8

12.35

12.9

13.4

14.3

S t r a i n

560

590

610

630

725

6.4(a) Durabi l i ty : I n terms of steel reinforced concrete, the

du rab i l i t y o f the composite may be influenced by

(a) Exposure conditions

(b) Qua l i t y o f Concrete

(c ) Cover t o reinforcement

(d) Width o f any cracks.

I n the case o f polypropylene f i b r e reinforced concrete, factors (c )

and (d) above are absent. This i s due t o the high resistance o f

the polymer t o environmental corrosion and deter iorat ion.

The l i fe-span o f such composite can be determined by the

method and predict ions made i n the present invest igat ion and hence

the du rab i l i t y may be f a i r l y determined. The response equations and

predicted formula enhance th is .

6.5 Limitat ions: The l im i ta t ions i n the present study are

(a) The basic assumptions made i n establ ishing the general equation

characterising the v isco-elast ic behaviour o f the composite wi th

par t i cu la r reference t o Newtonian f low ru le . This approximately

over-estimates the e las t i c and viscous behaviours o f the materials

used.

(b) The consistency o f the mix proportions o f the concrete used

i n the invest igat ion has not taken i n t o account the shrinkage e f f ec t .

(c) The model used assumes i r recoverab i l i t y , though t h i s does not

obtain i n practice.

(d ) Experimental a n a l y t i c a l data c o n f i r m the type o f Rheological

model t o be used and hence i f the experimental o r a n a l y t i c a l data

i s wrong, t he rheo log ica l ana lys is becomes equa l l y f a u l t y .

(el There i s a range o f loads w i t h i n which l i m i t s t h e empi r ica l

equations obtained are appl icable.

( f ) Temperature e f fec ts have l i m i t i n g e f f e c t s t o the present

i n v e s t i g a t i o n and where there i s no steady power supply, t h e

experiments my be i n e r r o r .

(g) The type o f equipment used i n t h e present t e n s i l e creep

i n v e s t i g a t i o n i f n o t p roper ly f a b r i c a t e d i s bound t o g i v e

inaccurate r e s u l t s .

(h) Since t h e empi r ica l equations obtained depend on experimental

resu l t s , these equations a re bound t o be wrong i f t h e

experimental r e s u l t s are wrong.

6.6 P r a c t i c a l Appl icat ions: The r e s u l t s o f laminate tes ted as a model

may be app l i ed t o a prototype.

The r e s u l t s o f t h e laminate s i ze 600 mm x 300 mm x 25 mm t es ted

as described may be app l i ed t o a protype dimension s i x t imes same s i z e

approximating t o 3600mm x 1800mm x 150mm. This assumption i s made f rom

t h e i n i t i a l assumption i n choosing the dimension o f laminate t o conform

approximately t o t h a t g iven i n CP 110 w i t h regards t o span-depth r a t i o

which can he determined from d e f l e c t i o n considerat ions. I n t h e model

chosen, the span-depth r a t i o i s 24. In t h e pro to type suggested above,

t h e span-depth r a t i o i s t he same. It may however be suggested t h a t such

composite (polymer r e i n f o r c e d concrete) slabs be used i n pavement

const ruc t ion w i t h minimum heavy veh icu lar t r a f f i c . Th is may be

analogous w i t h Net lon Co. Ltd.'s use o f same f i b r e i n s t a b i l i s i n g ea r th

i n pavement work. The h igh res is tance o f t h e f i b r e t o t e r m i t e and

fungal a t tack i nc lud ing chemical a t tack makes t h e use o f t h e composite

feasible. It may however be pointed ou t t h a t inaccuracies may a r i s e

due t o sca ler effect i n model - prototype r e l a t i o n . However, t he

method o f work and f i nd ings i n t h e above inves t iga t i on are i n d i c a t i v e o f

a d a p t a b i l i t y o f such composites and g i ve room f o r f u r t h e r work on a

b igger scale.

CIIAPTER SEVEN --. -- .

CONCLUSION I \ND RCCOMMENIIATIONS -- ['OR FUll'ltlEII _ WORK

1.1 -- ~ o n c l u s i o r l : ( i ) -- The c reep behav iou r oi' po ' lypropy lene r e i n f o r c e d

c o n c r e t e lanr inate can be p r e d i c t e d by a R l i ~ o l o g i c i l l Mode'l ~)lade up o f

V inear s p r i n g and r l i od i f i ed M a x n c l l ' s rnode'l. The e iod i f ' i ca l ; io r~ t o t h e

Maxwe'l 1 ' s nlotfel I i i r i t h e introdrrc:i:-iori o f nun-rc1.111-11 v a l vc! w i t h i n t h e

dashpot t o d e p i c t i r r e c o v e r a b ~ i l i Ly c ~ f c o n c r e o h e l ~ a v iouu.

( v i i )

The response -to t h e above 111odel i s g i ven by a d i f f e r e n t i a l

e q u a t i o n o f t h e form &- + d~. = C w i t h a s o l u t i o n o f t y p e a t

The va lues of ' t h e c o n s t a n t s i n the e q n i l t i o l ~ Tor response a r e

deternrincxl f rom e x p e r i ~ w r l l : i ~ l d a t a .

An en lp i r ica ' l equat,iori Lo p r e d i c L l . 1 ~ ( I W ~ oi' polypl-opy lerle

f i b r e r e i n f o r c e d c o n c r e t c la~~~i i ia . I . ( ! i s S V I I ~ I ~ l o I w

Tile eq i i a t i on i n i i v j d e f i n e s c reep ~ ( 1 : ) I i l l y i n t e r w s o f a 0

ant1 i: so h a t a t any t i l i le arid f o r a cot l?, tant giver^ 'load, t h e

c r e e p o f t h e c o l ~ l p o s i t e can be dt!ter~~tiric?tl w i i.lrcrt~t, p ~ r f o r ~ ~ l i r i ( j any

exper i i i lcn t .

Equat ion i n ( i v ) g i v e s c r w p cu rves I?X~JOII(! I I~ i d 1 III 1 1 i i l.ure.

The l i n l i t i n g o r 111-in.iniu111 l o a d c,~, o r 1:I11! c i i r ves -in ( v i ) t o a p p l y

i s deteral ined f r o m equat ior ) i n ( i v ) a s 8.8:) kg. The (?qua t ion i n

( i v ) may n o t be a p p l i c a b l e f o r d e t e r r l i i t ~ i i t i ~ r ~ o f p r i ~ c t - i c a l lnaxirilum

load, which value may bes t be d e t e M h e d e i t h e r exper imenta l l y

o r by dimensional at ialysis. I n t he present i nves t i ga t i on , t h i s

upper l i t n i t i s dbterf i ined exper imenta l ly as 14.3kg f o r a

laminate O f dimensions 600mm x 300mm x 25m.

( V i i i ) Wi th ih the range o f loads g iven i h ( v i i ) a. bears the f o l l o w i n Q

empi r ica l r e l a t i b n t o a , E_ and eo respec t i ve l y thU9:

(a ) o = 1.77~ t 8.68 0

(Ik) Equatloti i h ( V i i i ( b ) ) enables the modulus o f elasticity

d f the cotiiposite t o be dbtgrmined as t h e grad ien t 0; the

equat idn o f l l h e g iven by

( X I With equdtions i h ( ~ i i i ) , the values o f constants a , E_ arid

Em can be determined w i thout performing experiments f o r any

given oo.

(XI) Flexura l t e n s i l e t e s t s gave more r e l i a b l e r e s u l t s than d i r e c t

t e n s i l e t es t s .

( x i i ) The range o f constant loads w i t h i n which the equat ion i s

app l i cab le i s approximately 62.09% o f t h e u l t i m a t e load.

( x i l i ) The mathematical model o f determinat ion o f constant, a,

( S i m p l i f i e d Approach) seems more appropr ia te s ince i n g i v i n g

r e s u l t s o f bes t f i t , a l o t o f t ime i s saved i n ca l cu la t i on . .

( x i v )

(xv)

( x v i )

( x v i i )

186

The e f f e c t o f temperature f l u c t u a t i o n on the experimental

s ide o f t h i s i nves t i ga t i on i s very remarkable w i t h creep

increasing w i t h increase i n temperature.

The Rheological Model used i n the present i n v e s t i g a t i o n m y be

app l ied t o o ther polymer re in fo rced concrete laminates.

The p red ic t i on o f equat ion f o r long-term deformation o f

polypropylene f i b r e re in fo rced concrete laminate enables the

durabi I i t y o f t h e composite t o be determined.

Concrete laminate formed by polypropylene and concrete behaves

as a v isco-e las t ic mater ia l when loaded i n tension.

( x v i i i ) The increased e l a s t i c p roper ty induced i n the concrete i s

caused by t h e presence o f polypropylene.

( x i x ) Creep compliance and creep c o e f f i c i e n t s are es tab l ished f o r

such t h i n slabs.

(xx) A curve exponential i n shape i s app l icab le t o polypropylene

f i b r e re in fo rced concrete laminate. -MmTK%rYaRlO

mP' 7.2 Recomnendations f o r Fur ther Work: Fur ther areas

inves t iga ted are:-

(i) Solut ion o f ' the 4 t h Order Biharmonic equation (3.8)

t o determine the stress, s t r a i n and t iw func t ions

dep ic t i ng the v i sco -e las t i c behaviour.

(ii) Inves t iga t i on i n t o Recoverab i l i t y o f t he composite

system by s imula t ing an adequate Rheological model.

( i i i ) I nves t i ga t i on o f Creep behaviour o f polymer f i b r e

8 composite o f d i f f e r e n t s t r u c t u r a l dimension.

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ZONSVELD,J.J. (1970) 'The Marriage o f Concrete and P l a s t i c s ' , P las t i ca Volume 23, No.10, pp 474-484.

ZONSVELDJ. J. ( 1975) 'Proper t ies and Test ing o f Concrete Containing F ib res o ther than S tee l t , RILEM Symposium 1975, 'F ib re Reinforced Cement and Concrete', pp 217-226.

ZWEBEN,~. , ROSEN,B.W. ,,(1970) ' A s t a t i s t i c a l theory o f Ma te r i a l Strength w i t h a p p l i c a t i o n t o Composite m a t e r i a l s t . Journal o f Mechanics o f Physics and Solids,Vol . l8,

- Pergamon Press Ltd.,Gt.Britain, pp 189-209.

COMPUTER ANALYSIS AND PRINTOUTS

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PLATES, PHOTOGRAPHS.

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FIG 1 Increased load on the modified Tecqipment Creep Machine, n j n g the necking of specimen, during the direct Tension Test I

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FIG 2 Initial load on the modified 'Tecauiument Creeu Machine SM6' for the Direct . . Tension Test

FIG 3 Front view of the Creep Rig Equipment with loading arrangement for the Flexural Tests

PIG 4 increased loading arrangement on the specimens on the Creep Rig Equipment foi the Flexural Tests

University of Nigeria Creep of Polypropylene...in Civil Engineering Department of the University of...

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