University of Nigeria Creep of Polypropylene...in Civil Engineering Department of the University of...
Transcript of 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 for t = 2400 hrs
Load-Strain Relation for t = 1400 hrs
Load-Strain Relation fo r t = 1000 hrs.
F i g u r e - 1
2
3
4
5
6
7
8
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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
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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
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54
57
6 2
64
68
6 9
70
7 1
84
85
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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
Ordinary Portland Cement
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
experi- mental
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 pred- ic 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