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FERRO-FIBROCRETE: PROPERTIES AND APPLICATIONS
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1320385
ALSAYED, SALEH HAMED
FERRO-FIBROCRETE: PROPERTIES AND APPLICATIONS
THE UNIVERSITY OF ARIZONA M.S. 1983
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FERRO-FIBROCRETE: PROPERTIES AND APPLICATIONS
by
Saleti Hamed Alsayed
A Thesis Submitted to the Faculty of the
DEPARTMENT OF CIVIL ENGINEERING AND ENGIREERING MECHANICS
In Partial Fulfillment of the Requirements For the Degree of
MASTER OF SCIENCE WITH A MAJOR IN CIVIL ENGINEERING
In the Graduate College
THE UNIVERSITY OF ARIZONA
19 8 -2-M t >3
tW*
STATEMENT BY AUTHOR
This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.
SIGHED: sJ4 t>/cy/eJ
APPROVAL BY THESIS DIRECTOR
This thesis has been approved on the date shown below:
Mt G. S. RAMASWAMY
Professor of Civil Engineering and Engineering Mechanics
Ac 3 . f2-Date
ACKNOWLEDGMENTS
I wish to extend my appreciation to Professor G. S.
Ramaswamy for suggesting the topic of research and super
vising my work.
I also acknowledge gratitude to Professor Paul H.
King, Head of the Department of Civil Engineering, for the
facilities extended for undertaking this investigation.
I am indebted to Professor Edward B. Haughen for
several useful discussions.
The fibers used in the experimental program were
provided by the Bekaert Steel Wire Corporation. I extend
thanks to this firm for their assistance.
iii
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS vi
LIST OF TABLES vii
ABSTRACT viii
CHAPTER
1. METHODS OF IMPROVING TENSILE STRENGTH OF CONCRETE 1
1.1 Concrete as a Construction Material: Strengths and Shortcomings 1
1.2 The Concept of Ferro-Fibrocrete ... 1 1.3 Potential Applications of
Ferro-Fibrocrete 5
2. DEVELOPMENT OF A MATHEMATICAL MODEL FOR FERRO-FIBROCRETE IN TENSION AND FLEXURE ... 7
2.1 Development of a Mathematical Model for Ferro-Fibrocrete Under Uniaxial Tension 7
2.2 Monte Carlo Simulation to Predict First Crack Strength of Ferro-Fibrocrete Under Uniaxial Tension . . 10
2.3 First Crack Strength of Ferro-Fibrocrete in Flexure 18
2.4 Evaluation of Weibull Shape Parameter m . 20
3. EXPERIMENTAL INVESTIGATIONS 22
3.1 Outline of Experimental Program ... 22 3.2 Properties of Materials 22 3.3 Description of Test Specimens .... 23 3.4 Strain at First Crack 23 3.5 Discussion of Test Results 27 3.6 Conclusions and Recommendations ... 38
iv
TABLE OF CONTENTS -- Continued
v
Page
APPENDIX 44
REFERENCES 51
LIST OF ILLUSTRATIONS
Figure Page
1. Ferro-fibrocrete specimen 4
2. Ferro-fibrocrete suspended roof 6
3. Ferro-fibrocrete specimen submitted to a uniaxial tension 8
4. Uniform distribution for Ec and E. 11 r w
5. Uniform distribution for E 13 m
6. The computer output diagram for uniaxial specimen stresses (total = 2,000 specimens) 16
7. The uniaxial test specimen 24
8. Centrally loaded beam 25
9. Beam loaded at two points 26
10. Average relation between the stress and the strain for the uniaxial test 33
11. An example of failure pattern of uniaxial specimens . 34
12. Average relation between stress and deflection for the beam shown in Figure 8 37
13. A typical failure pattern for the beam shown in Figure 8 39
14. Average relation between the deflection and the stress for the beam sketched in Figure 8 41
15. A typical failure pattern for the beam shown in Figure 9 42
vi
LIST OF TABLES
Table Page
1. The chi square test for the 2000 specimens under uniaxial tension 17
2. Split cylinder tests 28
3. Uniaxial test results 29
4. Average deflection from testing six specimens (see Figure 8) 35
5. Failure loads for six specimens loaded as in Figure 8 36
6. Relation between the deflection and the load for Figure 9 40
vii
/
ABSTRACT
Many attempts have been made in the past to over
come the weakness in tension of concrete. Some of the at
tempts made include reinforcing it with steel, incorporating
random fibers and using it in the form of ferrocement or
prestressing it. Reinforced concrete tends to be a wasteful
form of construction because the concrete below the neutral
axis is ineffective. Prestressing does not improve the
tensile strength of the concrete, and incorporation of fibers
does not significantly improve the tensile strength. This
paper presents a new material resulting from a combination
of the above techniques. The resulting material will be
described as ferro-fibrocrete. A model for predicting the
behavior of this material in tension and flexure is presented.
Some possible applications are outlined.
viii
CHAPTER 1
METHODS OF IMPROVING TENSILE STRENGTH OF CONCRETE
1.1 Concrete as a Construction Material: Strengths and Shortcomings
Concrete is a construction material with many
applications. It is monolithic and can be cast into any
shape. Also, cement, the main ingredient contained in con
crete, is produced by a less energy-intensive technology,
involving only 1.0 Gj/T as compared to 47.5 Gj /T for steel
[1]. Also, concrete has a more advantageous cost-to-strength
ratio. In spite of these advantages, concrete tends to be
less competitive than steel for long spans because of its
very low strength in tension and high weight-to-strength
ratio. The competitive edge of concrete structures can be
improved if the tensile strength of concrete can be raised.
The attempts made in the past to correct its weakness in
tension are to strengthen it with steel reinforcement or
short random fibers or use it in the form of ferrocement
or precompress it by prestressing. But reinforced con
crete tends to be a wasteful form of construction because
the part of the section below the neutral axis (amounting
to as much as 2/3 of the section in beams) is not only
ineffective but also adds to the weight. Prestressing
1
2
does salvage the wasted part of the concrete section, but
it does not improve the poor tensile strength of concrete.
Use of short, random fibers, first introduced in the
late fifties and early sixties, as a result of the pioneering
contributions of Romauldi and Batson [2] and Romauldi and
Mandel [3], improves other properties such as cracking feat
ures and toughness to a more important extent than the
tensile strength. Thus Shah and Rangan [4] have shown that
a 1.2570 volume content of fibers doubled the tensile strength
but raised the toughness by as much as twenty times. Clearly,
random fibers cannot replace other forms of reinforcement or
prestressing. Having the properties of lightness, impermea
bility and improved extensibility, ferrocement is an improved
form of reinforced concrete. It has extended the use of
concrete structures to long spans in the form of tension mem
branes. But its properties can be further improved by the
addition of random fibers. From the brief preceding review
of the various efforts made since the turn of the century
to overcome the existing weaknesses of concrete, it is quite
clear that no single method is sufficient in itself to bring
about a marked improvement. A combination to emphasize the
strong points of each is clearly the answer. In what follows,
such an optimum combination of techniques will be referred
to as ferro-fibrocrete, or prestressed ferro-fibrocrete,
depending upon whether or not it is prestressed.
ir
3
1.2 The Concept of Ferro-Fibrocrete
The concept of ferro-fibrocrete is best illustrated
by reference to a beam (Figure 1). The higher tensile
strength and superior crack control properties of ferro-
fibrocrete (consisting of ferrocement strengthened with mesh,
regular reinforcing bars, and random fibers) are most advan
tageously used in the bottom-most fibers of the beam. Hence
it is logical to provide a thin ferro-fibrocrete ribbon to
serve as the tension flange. Plain concrete makes up the
compression flange. As is well known, the web of the beam
carries the shear in the form of principal stresses, with
their orientation varying from point to point. It would
therefore seem logical to use randomly oriented fibers to
carry principal stresses and use stirrups in addition, if
found necessary. That such partial replacement of stirrups
by fibers is probably borne out by experiments [5]. For
longer spans, the beam may be prestressed. Taking advantage
of the ferro-fibrocrete in the tension flange, with its
higher tension strength and superior crack-control quality,
the prestressing force to be used can be considerably
reduced. Thus, if designed as class 2 beams of the C.E.B.-
FIP classification [6], higher tensions can be permitted,
resulting in a reduction of the prestressing force required
compared to a normally prestressed beam. If design conforms
to class 3, reduction in the prestressing force to be applied
4
PLAIN CONCRETE
FERROCEMENT TENSION FLANGE WITH MESH,
REGULAR REINFORCEMENT
AND RANDOM FIBERS
Figure 1. Ferro-fibrocrete specimen.
5
is again possible because of the increased extensibility and
resulting crack control provided by the tension skin of
ferro-fibrocrete. If the beam is of reinforced concrete, the
use of a ferro-fibrocrete tension skin would permit the use
of higher-strength reinforcing steels because the limitations
of cracking and deflexion, which place an upper limit on the
tensile stress that can be permitted in the reinforcing steel
in a conventional reinforced concrete beam, can be relaxed.
Limited experimental evidence with fiber-reinforced beams
would point to such a possibility [7].
1.3 Potential Applications of Ferro-Fibrocrete
The following are some of the possible applications
that deserve detailed study:
1. Prestressed ferro-fibrocrete beams conforming to
class 2 and class 3 groups.
2. Water-retaining structures wherein ferro-fibrocrete
will chiefly be in hoop-tens ion.
3. Hybrid structures such as long-span thin hanging
ribbons of ferro-fibrocrete in which the structure
acts as a beam-cable and is subjected to both ten
sion and bending (Figure 2).
These uses demand a knowledge of the laws governing
the behavior of ferro-fibrocrete in tension and bending.
CHAPTER 2
DEVELOPMENT OF A MATHEMATICAL MODEL FOR FERRO-FIBROCRETE IN TENSION AND FLEXURE
2.1 Development of a Mathematical Model for Ferro-Fibrocrete Under-
Uniaxial Tension
Because of the variability resulting from the pres
ence of more than one type of reinforcement, the random
orientation of the fibers and the variation in the strength
of the mortar, it may be more suitable to look for a proba
bilistic rather than a deterministic model to predict the
behavior of ferro-fibrocrete composite under uniaxial tension.
In what follows, ferro-fibrocrete is considered as an improved
ferrocement, combining both mesh reinforcement and random
fibers. Behavior under uniaxial tension is the fundamental
case. In the next section, a method will be developed to
i relate the first crack strength in bending to the first crack
' strength found for uniaxial tension.
' Consider a specimen of ferro-fibrocrete of uniform
1 cross-sectional area submitted to uniaxial tension (Figure
i 3). Let the tensile stresses in the composite, mortar, mesh,
and random fibers be, respectively, oc, om, ow> and o^.
I Before the appearance of the first crack, it will be supposed
; that all the materials involved obey Hooke's law. Moreover, I it will be assumed that the fibers possess sufficient bond
! 7
8
Figure 3. Ferro-fibrocrete specimen submitted to a uniaxial tension.
9
so that debonding will not precede failure of the fiber in
tension. Using the method of mixtures, we may write:
VA = A°t»(1 - vf - V + vfA"f + V°w •
or
°c - "m'1 " Vf - V +V£°f + Vw (1)
where V,- and V are the volume fractions of the fiber and r w
mesh strength, respectively. This equation is independent
of the state of stress and would hold even in the post-
cracking stage. Also,
°m _ a f % _ ^_c ( 2 ) E " E- E E ^ m f w c
where E , ECt E„„, and E are the initial tangent moduli of m r' w' c
flexibility of mortar, fiber, mesh, and composite, respec
tively. The stress c?c = H in the composite at the instant
of cracking of the mortar in tension may be arrived at by
making use of Equation (1) and substituting for af and
from Equation (2). At that instant, let am = Ym.
E^ E. H - V1 - Vf - V + vftrY»+ s Ym
or
H - Yn[l + Vf(^ - i) + - 1)1 (3) m m
Y , the tensile strength of the mortar at fracture, may be Hr °
found either from split cylinder or direct tension tests on
10
unreinforced mortar. It will be assumed to have a normal
distribution, the mean and standard course being found from
test results on specimens. The moduli of elasticity, E^,
E , and E , will be assumed to have a uniform distribution.
2.2 Monte Carlo Simulation to Predict First Crack Strength of Ferro-Fibrocrete
Under Uniaxial Tension
The mean and standard deviation of the first crack
strength H of ferro-fibrocrete can be forecast by the power
ful Monte Carlo simulation technique [8], applied to Equation
(3). This involves the generation of a large number of syn-Ef thetic experiments varying Ym and the ratio Z = and finding m
the corresponding values of H. To select random values of
Ym and Z for each synthetic experiment, use needs to be made
of normally distributed random numbers for Ym and uniformly
distributed random numbers for Er and E . It can be assumed r m
without undue error that E,. and E have the same uniform r w
distribution. It is known that Ew and E^ vary within the f l f \
narrow range of 28 x 10 and 30 x 10 psi. Hence, the uni-
| form distribution shown in Figure 4 will be used in the
Monte Carlo simulation. For a given probability y, the value
of Ef = Ew = x may be found as:
Ef = x = (2y + 28) x 106 psi (4)
Similarly, for the mortar it is known from available data
11
y
1.0
0.0 28 x 10° 30 x 10'
Figure 4. Uniform distribution for E^ and Ew.
12 £
that Em varies between 2.5 and 3.0 x 10 psi. The uniform
variation for Em may therefore be represented as shown in
Figure 5.. It is easily proved that:
Em = x = (0.50y + 2.5) x 106 psi (5)
The mean y and standard variation s of Y were determined m
from a limited number of experiments on split cylinders.
These led to:
y = 445.64 psi and s = 13.68 psi
The results presented in this paper for the first crack
strength H of the ferro-fibrocrete composite are based on
2,000 Monte Carlo simulations carried out on a digital
computer. Before carrying out Monte Carlo simulations,
Equation (3) is rewritten as:
H - Ym[(l - Vf - V„) + (Vf + V„)z] (6)
Let
Vf + Vw - V
In the Monte Carlo trials, + V = V was assumed to be
0.0372. Hence Equation (6) may recast as:
H = Ym(0.9628 + 0.0372Z) (7)
The following step-by-step explanation will clarify the pro
cedure used.
2.5x10
Uniform distribution for E
14
Step 1:
A computer program is written to generate two sets
of uniform random numbers, and U^. These will later be
used to produce random values of E^ and E^, respectively,
for each synthetic experiment.
Step 2:
A computer program is written to generate one set
of random numbers, U^. The following two formulas are used
to generate two normally distributed random numbers from
each two consecutive numbers in the U-^ set:
X(m) = /-2LnU(m) cos27rU(m+l) (8)
X(m+1) = /-2LnU(m) sin2irU(m+l) (8)
The normally distributed random numbers so found will be
used to arrive at random values of Y needed for each Monte m
Carlo trial.
Step 3:
The method involved is best shown by describing how
the first synthetic experimental value of H is generated.
The first of the random numbers Ug = 0.78637 is substituted
in place of y Equation (5), and the represented value of x
is found. This is the random value of E^ = 2893185.71. m
Similarly, inserting the first of the random numbers =
0.29762 for y Equation (4) and solving for x, the random
15
value of E,e for the first Monte Carlo trial is found as E
Ef = 28595240.53. Knowing Ef and Em, Z = = 9.883652. m
Next, we need to choose the first random value of Y . It ' m
is computed as:
Ym = y + 13.68X m
where
X = 0.993539, the first of the normally distributed
random numbers
y = 445.64
The first random value so found is Y_ = 459.231611. m
We are now ready to compute the value of H resulting from
the first Monte-Carlo trial by substituting the values of Z
and Ym already found in Equation(7). The resulting value
of H - 610.994732.
The sequences shown are repeated 2000 times by writ
ing a computer program for the purpose. The results of the
computer output are represented in Figure 6 in the form of
a histogram. The mean value of H and its standard deviation
resulting from 2000 Monte Carlo trials were, respectively,
603.367776 psi and 20.9951 psi. The Central Limit Theorem
would lead us to expect that H will be normally distributed.
The chi-square test, when applied, indicated that normality
may be assumed with a confidence level of 95%. The details
of the chi-square test are given in Table 1. The method
Number of Occurrences
400 -•
300
200 • •
100
Stress (psi)
500 550 600 650
Figure 6. The computer output histogram for uniaxial specimen stresses (total = 2,000 specimens).
Table 1. The chi square test for the 2000 specimens under uniaxial tension.
(v -nP—)2 x o o No. (z^ f (z)1 (z)2 f (z)2 f(z)2-f(z)1 nP-o v— o nPo
1 00 0 -2.994 0.00135 0.00135 2.7 1 2.89
2 -2.994 0.00135 -2.52 0.005 0.01775 7.3 5 0.73
3 -2.52 0.005 -2.042 0.02275 0.016 32 30 0.12
4 -2.04 0.02275 -1.566 0.0548 0.034 67 69 0.06
5 -1.56 0.054 -1.09 0.13556 0.086 172 175 0.05
6 -1.09 0.13566 -0.613 0.27425 0.1385 277 275 0.01
7 -0.613 0.27425 -0.137 0.4365 0.1623 403 330 0.11
8 -0.137 0.4365 0.34 0.6404 0.2039 324 391 0.67
9 0.34 0.6404 0.816 0.7937 0.1533 307 308 0.00
10 0.816 0.7937 1.293 0.903195 0.1095 214 205 0.37
11 1.293 0.903195 1.77 0.961 0.0609 116 123 0.42
12 1.77 0.961 2.25 0.988 0.027 54 60 0.56
13 2.25 0.988 2.72 0.997 0.009 18 20 0.22
14 2.72 0.997 3.198 0.999 0.002 4 6 0.20
15 3.198 0.999 CO 1 0.001 2 2 0.00
6.41
No. of degree of freedom = 15-1 = 14 Pr{x>7.79) = 0.9 acceptable at 90% level Pr(x>6.57} = 0.95 acceptable at 95% level
i-1
/
18
outlined in the previous sections will let the first crack
strength to be estimated to a high degree of reliability.
The application of this data is in the design of class 2
prestressed concrete members in which tension is allowed but
no cracks. The presence of normal reinforcing bars, if
present, can be added to the volume fraction of mesh and
rnadom fibers to arrive at V, the total volume fraction of
all types of strength.
2.3 First Crack Strength of Ferro-Fibrocrete in Flexure
A relationship between the first crack strength in
tension and under bending may be found via the Weibull
Statistical Theory of the strength of materials [9] if the
onset of cracking can be regarded as failure or the limit
of the usable strength in design.
Let the first crack strength of a material in uni
axial tension be aand let the volume of the specimen be
V^. Using Weibull's theory, the probability P-^ of failure
of the specimen may be stated as:
Px = 1 - exp{-[ (a - ou)/ao]m (9)
where au and m are known as the location and shape para
meters of the Weibull distribution. The location parameter
au which represents the value of a below which the probability
of failure is zero may be assumed to be zero. Equation (9)
19
may therefore be written as:
P, = 1 - expM-f-)1" V-,} i a0
(10)
Next, let us consider a centrally loaded beam (Figure 8) of
length L, breadth B, and depth d, with a maximum tensile
fiber stress of a The probability P9 of failure of the IuaX •
specimen may be written as:
= 1 ~ exp{- f(a)dV) (11) •V
where f(a) is the function defining the stress at any point
(x,y). The stress at any point (x,y) may be expressed as:
c = 4/Ld a xy ' max 3
f(j)d V = b L rd /2 , a z 4 max \m j , xy) dx'dy
V
= 2b
0 0
L /2 rd /2 / o f 4 max vm, , (id ~oTxy) dxdy
(12)
(13)
Integration is carried out only from 0 to d/2 because there
is probability of failure only in tension but not in compression.
Simplifying and denoting the volume of the specimen by V2,
1 o f (a)dV = ( maxNm r vc (14)
2(m+l)'
Hence the probability of failure may be found from Equation
(15) as:
20
maxsm, (15)
If c0 is the value of a , the condition for equal proband max ^ r
bility of failure under uniaxial tension and flexural tension
is:
If the Weibull parameter m is found, the ratio of the first
crack strength in flexure to that in tension can be evalu
ated, knowing the volumes of the two specimens.
experiments on uniaxial tension specimens described in a
later section. It can also be found from the Monte-Carlo
simulation results. The procedure is as follows. The values
H, the first crack strength obtained from Monte Carlo trials,
are arranged according to ascending order of magnitude.
Along the y axis, plot log log (N + 1)/(N + 1 - n) where N
is the total number of observations and n is the position of
an observation in the table in which the observations are
ranked. Along the x axis Log(H - a ) is marked. For this
purpose au may be assumed to be zero. The slope of the
or
"°1
2 (m+1) (16)
(17)
2.4 Evaluation of Weibull Shape Parameter m
The value of Weibull parameter may be found from
/
21
resulting line gives the value of m. A regression analysis
was carried out on the Monte- Carlo results to find the best
straight line fitting the observations. The value of m so
found is 35.493581. In the experimental program described
in a later section, the ratio = 216/72 = 3.0. Hence
the fextural first crack strength is estimated as:
i c /^l\l/m °2 " 1'5al(v;)
from Equation (17). This result will be compared in a later
section, with the results obtained from tests on uniaxial
specimens of ferro-fibrocrete.
CHAPTER 3
EXPERIMENTAL INVESTIGATIONS
3.1 Outline of Experimental Program
The experimental program was made up of tests on
mortar cylinders, uniaxial unreinforced mortar specimens,
ferro-fibrocrete uniaxial specimens, ferro-fibrocrete beam
specimens, and reinforced concrete beam specimens with a
thin tension skin of ferro-fibrocrete.
3.2 Properties of Materials
Type 1 cement was used in building all the experi
mental specimens. Sand used had a"specific gravity of 2.64
and a density of 119 lb/ft . The mortar had a sand-to-
cement ratio of 2:1 and a water/cement ratio of 0.50. The
fibers used in the experiments were supplied by Bekaert
Steel Wire Corporation. They had hooked ends. The length
and diameter of the fibers were, respectively, 50 mm (1.9685
in.) and 0.5 mm (0.0197 in.) and hence the aspect ratio was
100. The fibers are made from steel wires with an ultimate
tensile strength of 170,000 psi. The mesh used in the test
specimens was 20 gage 4x4. The diameter of the wires was
0.035", and the yield strength was 60,000 psi. The #3
reinforcing bars used had a yield strength of 63,000 psi and
an ultimate strength of 96,400 psi.
22
23
3.3 Description of Test Specimens
The 12 split cylinder specimens were 6" diameter and
12" length. The details of the uniaxial test specimens may
be seen in Figure 7. Of a total of 24 specimens tested, six
were of plain mortar; six had two layers of mesh; another
set of six had 1% of random fibers effective in the direction
of the tension; and the last six specimens had 1% of random
fibers effective in the direction of the pull, 2 layers of
mesh reinforcement, and 2 #3 bars, making up a total volume
fraction of 3.72%. This is the percentage used in the Monte
Carlo trials.
The beam specimens cast for arriving at the first
crack strength (Figure 8) had approximately the same total
volume fraction of steel, consisting of 4 layes of mesh,
1% fibers, and one #3 bar, making up a total of 3.64%. The
same volume fraction of steel was used, both in the uniaxial
tension and flexural specimens so that the first crack
strengths in bending and tension arrived at experimentally
can be compared with the predicted analysis made in 2.4
3.4 Strain at First Crack
Assuming linear variation of the stress-strain curve
up to the onset of cracking, the strain at first crack
may be written as:
£ = H
E (1 - V) + VEj; (19> m f
A
5"
1 Sec. B-B
V
3"
1 Figure 7. The uniaxial test specimen.
Sec. A-A
to
Figure 8. Centrally loaded beam.
*
26
io" >J ">4" — <- 12"—>j <- 12"—>j
M = 12P
( 2 )
Sec. A-A
Figure 9. Beam loaded at two points. — (1) Ferro-fibrocrete flange with mesh, fibers and regular bars; (2) Plain concrete.
27
The expression assumes that the Modulus of Elasticity of the
composite may be found by using the method of mixtures
method of generating synthetic observations for H by Monte
Carlo simulation has already been cescribed in detail. The
matching values of Em(l - V) + VE^ can be computed to arrive
at E. The strain was found, as expected, to be normally
distributed. Application of chi-square test shows that
normal distribution may be assumed with a confidence level
of 95%. The mean and standard rule of e based on 2,000
Monte Carlo trials are found to be 0.00016202 and 0.000009845,
respectively.
3.5 Discussion of Test Results
The results of the split cylinder tests are presen
ted in Table 2. The mean value of the split cylinder
strength was found to be 445.64 psi and the standard deriva
tion worked out to 13.68 psi.
The results of the uniaxial tension tests may be
seen in Tables 3a-d.
The percentage increase in strength of the specimens
with fibers only over those which are unreinforced may be
worked out from Tables 3a and 3c. The average increase is
found to be (848 - 716)/(716) x 100 = 18.447o. This figure
agrees reasonably well with the prediction equation proposed
by Johnston and Coleman [10] which takes the form:
28
Table 2. Split cylinder tests.
Specimen Number P (kips) Stress (psi) " itLd * 1000
1 50.8 449 .2
2 49 433 .26
3 49 433 .26
4 53 468 .62
'5 50 442 .1
6 50 442 .1
7 52 459 .78
8 50 442 .1
9 48 424 .4
10 51 450 .94
11 49 433 .26
12 53
00 |
VO
|
.62
L = 12", D = 6"
Mean = = 445.64 psi
Standard = J =
_ 2 Variance = ^ = 187.12
s ./<X-*> 2= 13.68
Table 3. Uniaxial test results. -- (a) unrein-forced mortar tension specimens; (b) specimens with mesh; (c) specimens with fibers; (d) specimens with bars, mesh, and fibers.
Specimen Number Load
Stress at Failure
1 6300 700
2 6450 717
3 7000 778
4 6200 689
5 6700 644
6 6000 667
Average 716
Specimen Stress Number Load at Failure
1 7000 778
2 7100 789
3 6900 767
4 6590 732
5 6800 756
6 6500 722
Average 757
Table 3. -- Continued
Specimen Number Load
Stress at Failure
1 8800 978
2. 8000 889
3 9000 1000
4 11500 1278
5 8580 953
6 8900 989
Average 848
Specimen Stress Number Load at Failure
1 19,700 2189
2 21,500 2389
3 20,200 2244
4 22,000 2444
5 20,900 2322
6 23,000 2556
Average 2354
31
P = 0.005 W(L/D)3/2 (18)
where
Pa = percentage increase in strength over unstrength-
ening specimens
L/D = aspect ratio
W = percent concentration of fibers by weight
Noting that W = 3.25 and L/D = 100,
P/a = 0.005 x 3.25 x (100)3/2 = 16.25%
against the actual value of 18.44%.
The percent concentration by weight of steel in spe
cimens with mesh is 0.78. Had this reinforcement been in
the form of fibers, we may expect a strength improvement of:
P = 0.005 x 0.78 x (100)3/2 = 3.9% a
But the real increase in tensile strength compared to the
unreinforced specimens is worked out as:
7577l6716 x 100 = 5.73%
These results clearly show the superiority of aligned fibers
such as mesh over random fibers for equal volume/weight
concentration of reinforcement. This is an account of the
absence of debonding right up to failure, when aligned
fibers are used.
Extensions were measured on the uniaxial test speci
32
mens with mesh, fiber and regular reinforcement bars with a
mechanical extensiometer on a 2" gage length, and the
results are recorded in Table 3d. But the extension measure
ments are not considered dependable. Without an automatic
recorder available, it was not possible to arrive at a reli
able load-extension curve experimentally. As a result, the
average stress-strain curve presented in Figure 10 cannot be
considered to be very accurate. It was hoped that the first
crack stress can be found by identifying the point in the
stress-strain curve at which there is a kink and a change
in slope. But this proved difficult, and the average first
cracking strength had to be roughly estimated as 1522 psi.
This is very much higher than the first crack strength ob
tained by Monte Carlo simulation. An example of failure
pattern of uniaxial specimens is shown in Figure 11.
Tests on ferro-fibrocrete beam specimens involved
deflection measurements at the center of the span by means
of a dial gage. The loads, deflection, and maximum stresses
of these specimens are tabulated in Tables 4 and 5. The
average stress-deflection curve shown in Figure 12 was used
to identify the load representing the first crack. The
stress at first crack stress was found as 7200 psi. The
ratio of:
flexure tension at first crack uniaxial tension at first crack
33
Stress
2000
1000
500
Strain
0.001 0 . 0 0 2 0.003 0.004 0.005
Figure 10. Average relation between the stress and the • strain for the uniaxial test.
Figure 11. An example of failure pattern of uniaxial specimens.
34
35
Table 4. Average deflection from testing six specimens (see Figure 8).
Load Stress Deflection
200 900 0.3
400 1800 0.45
600 2700 0.51
800 3600 0.68
1000 4500 0.79
1200 5400 0.87
1400 6300 0.103
1600 7200 0.117
1800 8100 0.146
2000 9000 0.162
2200 9900 0.172
2400 10800 0.192
2600 11700 0.22
2800 12600 0.239
3000 13500 0.25
3200 14400 0.26
3400 15300 0.271
3600 16200 0.296
3600 16200 0.33
3800 17100 0.34
3800 17100 0.35
3800 17100 0.36
3850 17325 Failure
I = bh3 8x3
TT = ~TT = 2 in4
M = f = 9 P
Stress = = y = 4 . 5 P
Table 5. Failure loads for six specimens loaded as in Figure 8.
Specimen Number
Failure Load
Failure Moment lb. in. Stress
1 3900 35100 17550
2 3700 33300 16650
3 4000 36000 18000
4 4100 36900 18450
5 3800 34200 17100
6 3600 32400 16200
Average failure stress = 17,325 psi
37
Stress (psi) 20 ,000
16,000 •
12,000
8 , 0 0 0
Deflection (inches)
0.3 0.4 0 . 2 0 . 1
Figure 12. Average relation between stress and deflection for the beam shown in Figure 8.
38
works out at = 4.73 against the analytical prediction
of 1.5. A typical failure of this group of specimens is
shown in Figure 13.
The test results on the reinforced concrete specimens
with a thin skin of ferrocement are shown tabulated in Table
6. Central deflections were recorded by a dial gage. An
average deflection curve for this set of beam specimens is
shown in Figure 14. There is a kink in the load-deflection
curve at a load of 4000 lbs. The average stress in the
bottom fibers at this load worked out by treating the beam
as a layered member [11] works out to be 32,160 psi, indi
cating that the parameters of a thin skin of ferro-fibrocrete
can considerably enhance the first crack strength of rein
forced concrete beams. A typical failure for the beam
loaded at two points is shown in Figure 15.
3.6 Conclusions and Recommendations
1. The method of mixtures on which the Monte Carlo
simulation is based underestimates the first crack
strength of ferro-fibrocrete in uniaxial tension.
2. A continuous autographic recorder is necessary to
determine the beginning of cracking and identify the
load at first crack.
3. The Weibull theory leads to a reliable means of fore
casting the ratio between tensile strengths in bend
ing and uniaxial tension.
Figure 13. A typical failure pattern for the beam shown in Figure 8.
39
40
Table 6. Relation between the deflection and the load for Figure 9.
Deflection Deflection Deflection Average p Stress (1) (2) (3) Deflection
200 1608 0.030 0.030 0.030 0.030 400 3216 0.045 0.045 0.045 0.045 600 4824 0.055 0.060 0.060 0.058 800 6434 0.070 0.068 0.070 0.069 1000 8040 0.085 0.079 0.080 0.081
1200 9648 0.099 0.103 0.100 0.100 1400 11256 0.112 0.117 0.120 0.116 1600 12864 0.127 0.130 0.130 0.130 1800 14472 0.145 0.146 0.147 0.146 2000 16080 0.162 0.162 0.164 0.163
2200 17688 0.170 0.172 0.170 0.171 2400 19296 0.185 0.192 0.191 0.189 2600 20904 0.204 0.220 0.200 0.210 2800 22512 0.213 0.239 0.240 0.230 3000 24120 0.243 0.250 0.247 0.250
3200 25728 0.262 0.260 0.261 0.261 3400 27336 0.276 0.271 0.275 0.270 3600 28944 0.278 0.296 0.291 0.290 3800 80552 0.311 0.330 0.320 0.320 4000 32160 0.321 0.34,0.35, 0.345 0.344
0.36,0.37 4200 33768 0.344 0.374 0.360,0.370 0.340 4400 35376 0.362 0.382 0.375 0.370 4600 36984 0.376 0.410 0.384 0.400 4800 38592 0.392 0.465 0.430 0.450 5000 40200 0.412 0.500 0.480 0.490
5200 41808 0.459 0.555 0.520 0.540 5400 43416 0.470 0.635 0.600 0.620 5600 45024 0.775 0.636 0.710 5800 46432 0.810 0.780 0.800 6000 48240 0.830 0.810 0.820
6500 Failure 0.840 0.840
7000 Failure Failure Failure
Stress (psi)
48,000
40,000
32,160
24,000
16,000
6,400-.
Deflection (inches)
0.3 0.4 0.5 0 . 6 0.7
Figure 14. Average relation between the deflection and the stress for the beam sketched in Figure 8.
Figure 15. A typical failure pattern for the beam shown in Figure 9.
42
43
4. For equal volume/weight concentration of reinforce
ment, aligned fibers are superior to random fibers
in uniaxial tension.
5. The provision of a thin ferro-fibrocrete skin in the
bottom-most layers of reinforced concrete beam delays
cracking by enhancing the tensile strength.
APPENDIX
A COMPUTER PROGRAM FOR MONTE CARLO SIMULATION AND SHAPE PARAMETERS
44
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
DIMENSION HH(6000),X(6000),U(6000) NA=0.0 NB=0.0 NC=0.0 ND=0.0 NE=0.0 NF=0.0 NG=0.0 NH=0.0 NI=0.0 NJ=0.0 NK=0.0 NL=0.0 NM=0.0 NN=0.0 N0=0.0 NP=0.0 NQ=0.0 NR=0.0 HUM=0.0 FIG=0.0 HIG=0.0 SUM=0.0 ALA=0.0 DUM=0.0 G0=0.0 GP=0.0 GQ=0.0 GR=0.0 DD 43 11=1,2000 U(II)=RANF( )
43 CONTINUE DO 44. M=1,2000, 2 X (M) = (SQRT(-2.0*AU>G(U(M))))*COS(6.2 8 318 5 3*U(M+1)) X(M+l)=(SQRT(-2.0*ALOG(U(M))))*SIN(6.2831853*U(M+1)) ^
i_n
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
44 CONTINUE WRITE(6,61) WRITE(6,62) WRITE(6,63) WRITE(6,64) WRITE(6,65) WRITE(6,66)
61 FORMAT (' 62 FORMAT(' 63 FORMAT (' 64 FORMAT(' 65 FORMAT (' 66 FORMAT('
DO 10 1=1,2000 U3=RANF( ) U4=RANF( ) EM=(0.5*U3+2.5)*1000000.0 EF= (2.0*U4+28-.0) *1000000.0 W=X(I) YM=445.64+13.68*W 7=EF/EM H=YM*( 0.9628+0.0372*1) HH(I)=YM*(0.9628+0.0372*1) ECOMP=0.9628*EM+0.0372*EF STR=H/ECOMP WRITE(6,5) I,U3,U4,X(I),H,ECOMP,STR
5 FORMAT(5X,15,8X,F8.7,8X,F8.7,8X,FIO.6,8X,F15.5,8X,F15.5,8X *,F15.9) SUM=SUM+STR DUM=DUM+STR**2.0 HUM=HUM+H ALA=ALA+H**2.0 HIG=HIG+(h-603.36778)**2.0 FIG=FIG+(STR-0.00016202)**2.0 IF(H.LT.520.5) NA=NA+1 ^
CT>
71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
IF( H. GE. 520. 5) .AND. H .LT. 530. 5) NB=NB+1 IF( H. GE. 530. 5) .AND. H .LT. 540. 5) NC=NC+1 IF( H. GE. 540. 5) • AND. H .LT. 550. 5) ND=ND+1 IF( H. GE. 550. 5) .AND. H .LT. 560. 5) NE=NE+1 IF( H. GE. 560. 5) .AND. H .LT. 570. 5) NF=NF+1 IF( H. GE. 570. 5) .AND. H .LT. 580. 5) NG=NG+1 IF( H. GE. 580. 5) .AND. H .LT. 590. 5) NH=NH+1 IF( H. GE. 590. 5) .AND. H .LT. 600. 5) NI=NI+1 IF( H. GE. 600. 5) .AND. H .LT. 610. 5) NJ=NJ+1 IF( H. GE. 610. 5) .AND. H .LT. 620. 5) NK=NK+1 IF( H. GE. 620. 5) .AND. H .LT. 630. 5) NL=NL+1 IF( H. GE. 630. 5) .AND. H .LT. 640. 5) NM=NM+1 IF( H. GE. 640. 5) .AND. H .LT. 650. 5) NN=NN+1 IF( H. GE. 650. 5) .AND. H .LT. 660. 5) NO=NO+l IF( H. GE. 660. 5) .AND. H .LT. 67-. 5) NP=NP+1 IF( H. GE. 670. 5) .AND. H.LT. 680. 5) NQ=NQ+1 IF(H .GE. 680.5) NR=NR+1
10 CONTINUE WRITF(6,17) NA,NB,NC,ND,NE
17 FORMAT(10X,515) WRITE(6,21) NF,NG,NH,NI,NJ
21 FORMAT(10X,515) WRITE(6,22) NK,NL,NM,NN,NO
22 FORMAT(10X,515) WRITE(6,23) NP,NQ,NR
23 FORMAT(10X,315) MOTAL=NA+NB+NC+ND+NE+NF+NG+NH+NI+NJ+NK+NL+NM+NN+NO+NP+NQ+NR WRITE(6,33) MOTAL
33 FORMAT(2OX' MOTAL OR TOTAL =',15) EANl=HUM/2000.0 EAN2=SUM/2000.0 VARl=HIG/2000.0 VAR2=FIG/2000.0 WRITE(6,3) FANl,VARl,EAN2,VAR2
3 FORMAT(4F18.8)
106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140
STA1=SQRT(VAR1) STA2=SQRT(VAR2) WRITE(6,47)STA1,STA2
47 FORMAT(5X,'STA1=*,F15.6,10X,'STA2='.F15.9) WRITE(6,71) WRITE(6,72) WRITE(6,73) WRITE(6,74) WRITE(6,75) WRITE(6,76) WRITE(6,77)
71 FORMAT(' ') 72 FORMAT(1 ') 73 FORMAT(' ') 74 FORMAT(' ') 75 FORMAT(' ') 76 FORMAT(' ') 77 FORMAT(' ')
DO 99 L=1,1999 LL=2000-L DD 99 K=1,LL TF(HH(K) .LE. HH(K+1)) GO TO 99 HARD=HH(K) HH(K)=HH(K+1) HH(K+1)=HARD
99 CONTINUE DO 98 J=l,2000 GM=ALOG(ALOG(2001.0/(2001.0-J))) GN=ALOG(HH(J)) GO=GO+GM*GN GP=GP+GN GO=GO+GM GR=GR+GN**2 WRITE(6,15)J,HH(J),GM,GN
15 FORMATdOX, 15 ,10X, F15 . 6j, 10X, F15 . 6,10X, F15 . 6)
141 98 CONTINUE 142 SLOPE=(GO-(GP*GQ/2000.0))/(GR-(GP**2.0/2000.0)) 143 SAD=(1.0/SLOPE) 144 FACT=((2.0*((SLOPE+1.0)**2.0))**SAD)*(0.3333)**SAD 145 WRITE(6,16) SLOPE,FACT 146 16 FORMAT(20X,'THE SLOPE OF THE LINE=',F15.7,10X,'THE FACT =',F15.7) 147 STOP 148 END
4> MD
The first 30 outputs of the computer program.
NO U3 U4 X H E comp Strain
1 .7863684 .2970211 .993539 610. 99473 3940968. 52065 .000158729 2 .9378357 .5496564 .993539 609. 58822 3940968. 52065 .000154680 3 .7873670 .0051996 -.319285 583. 75461 3828025. 31875 .000152495 4 .8877574 .1403723 -.204356 584. 62836 3886410. 12566 .000150429 5 .7554344 .0744811 .662488 603. 24479 3817807. 53222 .000158008 6 .8617533 .9765190 1 .256262 621. 56439 3936101. 06594 .000157914 7 .4179474 .8135461 .049453 611. 29443 3710327. 69944 .000164755 8 .3696968 .8580569 -.549910 602. 20639 3690411. 48581 .000163181 9 .6493421 .9498875 1 .508051 632. 52824 3831864. 90736 .000165071 10 .5573299 .5204526 -.638879 590. 51057 3755620. 27439 .000157234 11 .6860465 .2863880 .580146 606. 27857 3800162. 62348 .000159540 12 .1292593 .7731734 .988352 638. 86250 3568349. 53111 .000179036 13 .5715333 .4900918 -1 .749124 569. 22246 3760198. 95145 .000151381 14 .5761491 .3266572 .845688 614. 99552 3750261. 45791 .000163987 15 .7263267 .3329533 .603338 606. 05938 3823025. 37337 .000158529 16 .6932406 .5431752 -1 .165713 577. 03658 3822738. 25188 .000150948 17 .5676343 .5369609 -1 .100841 581. 84913 3761809. 02300 .000154673 18 .2472214 .5028462 2 .224324 654. 15167 3605024. 15046 .000181456 19 .6683521 .3449025 -.134652 594. 40890 3796005. 46596 .000156588 20 .3628335 .9238407 -.070550 612. 26605 3692001. 78351 .000165836 21 .5051441 .9819527 -.191732 605. 93716 3764833. 63584 .000160947 22 .6071212 .7913188 .086859 605. 59850 3799742. 25671 .000159379 23 .9665268 .5827574 .357760 597. 62514 3957243. 14288 .000151021 24 .3830246 .6461591 1 .311215 634. 01433 3681062. 27175 .000172237 25 .2283098 .1287885 -1 .533343 •579. 62862 3568090. 20618 .000162448 26 .3277774 .4128344 -.611536 597. 02682 3637106. 92518 .000164149 27 .2967464 .6204883 -.159963 609. 09389 3637618. 05912 .000167443 28 .8554337 .8941557 -1 .105249 577. 45443 3926930. 96296 .000147050 29 .5982595 .1791307 2 .593508 644. 54426 3749929. 43504 .000171882 30 .3844725 .5548248 .682806 621. 05916 3674964. 04527 .000168997
REFERENCES
1. Kelly, A. "Prospects and Problems of Fibre-Reinforced Concrete." RILEM Symposium, 1975, Vol. 2 (Editor, Adam Neville), p. 464.
2. Romualdi, J. P. and G. B. Batson. "Mechanics of Crack-Arrest in Concrete." Journal of the Engineering Mechanics Division, Proceedings ASCE, June 1963. pp. 147-168.
3. Romualdi, J. P. and J. A. Mandel. "Tensile Strength of Concrete Affected by Uniformly Distributed and Closely Spaced Short Lengths of Wire Reinforcement." ACI Journal, June 1964. pp. 657-671.
4. Surendra, P. Shah and B. Bijaya Rangan. "Fiber Reinforced Concrete Properties." ACI Journal, February 1971. pp. 126-135.
5. Williamson, G. R. and L. I. Knab. "Full Scale Fibre Concrete Beam Tests." RILEM Symposium, 1975, Vol. I. pp. 209-214.
6. International Recommendations for the Design and Construction of Concrete Structures, Principles and Recommendations. Sixth F.I.P. Congress, Prague, June 1970. English Edition, Cement and Concrete Association, London.
7. Swamy, R. N. and K. A. Al-Noori. "Flexural Behavior of Fibgr* Concrete with Conventional Reinforcement."
8. Haughen, Edward B. "Probabilistic Approaches to Design." John Itfiley and Sons, Inc., 1968. Ch. 6.
9. Hudson, John A. and Charles Fairhurst. "Tensile Strength, Weibull's Theory and a General Statistical Approach to Rock-Failure." Structure, Solid Mechanics and Engineering Design," the Proceedings of the Southampton 1969 Civil Engineering Materials Conference. Paper 73, pp. 901-914.
51
52
10. Johnson, Colin D. and Ronald A. Coleman. "Strength and Deformation of Steel Fiber-Reinforced Mortar in Uniaxial Tension." Fiber Reinforced Concrete, ACI Publication SP44. pp. 177-193.
11. Ramaswamy, G. S. "Ferro-Fibrocrete: A New Structural Concept." International Symposium on Ferrofibrocrete, July 22-24, 1981. ISMES--Instituto Sperimentale Modelli e Strutture, Bergamo, Italia.