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ANALYSIS m nearer
By

Abdeslam Reklaoui —

A THESIS
Submitted to
Michigan State University
In partial fulfillment of the requirements
of the degree of
MASTER OF SCIENCE
Department of Civil Engineering

1988

ABSTRACT

STRUCTURAL APPLICATIONS OE STEEL EIEER REINEORCED CONCRETE
ANALYSIS AND DESIGN

By
Baklaoui hbdoslan

Structural analysis and design techniques were developed
for reinforced concrete elements incorporating steel fibers
as well as conventional reinforcement. The work.‘was
performed in three phases which were concerned with flexural,
shear and torsional fibrous reinforced concrete elements. In
each phase structural design guidelines were developed which
accounted for the effects of steel fibers on strength and
ductility of reinforced concrete elements. In the case of
flexural elements emphasis was made on the improvement in
ductility and the consequent relaxation of limits on the
tension and compression steel ratio in the presence of steel
fibers. As far as shear and torsional elements were
concerned, consideration was given to the improvements in
strength characteristics of elements resulting from steel
fiber reinforcement.

The developed. design. guidelines were verified. using
relatively large numbers of flexural, shear and torsional
test data reported in literature. The final guidelines were
used to assess the effects of steel fiber reinforcement on
the performance characteristics of reinforced concrete

structural elements.

TO N! BELOVED PARENTS

ii

ACNNO'LEDGNENTS

The writer wishes to express his sincere appreciation to
his academic advisor, Dr. Parviz Soroushian, Assistant
Professor of Civil and Environmental Engineering, for his
encouragement, guidance, and aid throughout the writer’s
master of science studies.’ Thanks also to the other members
of the thesis committee: Dr. R. Harichandran, Assistant
Professor of Civil and Environmental Engineering, and Dr. A.
Altiero Professor’ of‘ Metallurgy, Mechanics And. Materials
Science.

Thanks are also extended to the Government of Morocco
"Ministere de I/equipement, de la Formation Professionnelle
et de la Formation des Cadres” for the financial assistance.

Last, but not least, special. appreciation,
admiration, and love to the writer's wife and his daughter

for their continuous encouragement, and love.

iii

TABLE OF CONTENTS

Page

LIST OF TABLES .......................................... Vi

LIST OF FIGURES ......................................... Vii

NOTATIONS ............................................... xi
CHAPTER

1 INTRODUCTION ........................................

GENERAL 00......OOOOOOOOOOOOOOOOO...0.0.0.0000...
STEEL FIBER REINFORCED CONCRETE .................
STEEL FIBERS IN FLEXURAL REINFORCED CONCRETE
ELEMENTS 0.......00...0.000000000000000.0.000....
STEEL FIBERS IN REINFORCED CONCRETE ELEMENTS
SUNECTEDTOSHEAR FORCES OOOOOOOOOOOOOOOOOOOOOOO
STEEL FIBERS IN TORSIONAL REINFORCED CONCRETE
Ems .0...OO..0.0......0.0000000000000000...O 10

2 EFFECTS OF STEEL FIBERS ON FLEXURAL BEHAVIOR OF
REINFORCED CONCRETE BEAMS: REFINED ANALYSIS AND
DESIGN 0.0.000COOOOOOOOOOOOOOOOOOOOOOOOOOOO0.0.0.0... 13

H H Hwaw
O O
m <h umoH
o q wra h‘

2.1 INTRODUCTION oooooooooooooooooooooooooooooooooooo 13
2.2 REFINED FLEXURAL ANALYSIS ....................... 15
203 CONSTITUTIVE MODELS ooooooooooooooooooooooooooooo 17
2.3.1 COMPRESSION MODEL ......................... 17
20302 TENSION MODEL ooooooooooooooooooooooooooooo 20
COMPARISON WITH TEST RESULES oooooooooooooooooooo 22
PARAMETRIC STUDIES oooooooooooooooooooooooooooooo 24
SIMPLIFIED FLEXURAL.ANAL¥SIS AND DESIGN
GUIDELINES oooooooooooooooooooooooooooooooooooooo 38
2.6.1 ”ALTERNATIVE METHOD" ...................... 38
2.602 "ACI-BASED METHOD” oooooooooooooooooooooooo 41
2.7 COMPARISON OF THEORETICAL PREDICTIONS WITH TEST

RESULES ooooooooooooooooooooooooooooooooooooooooo 44
2.8 MAXIMUM STEEL RATIO IN FIBROUS REINFORCED

CONCRETE FLEXURAL MEMBER ........................ 47
2.9 EXAMPLE: FLEXURAL ANALYSIS OF A FIBROUS

REINFORCED CONCRETE SECTION USING THE

”ACI-BASED METHOD" oooooooooooooooooooooooooooooo 51
2.10 SUMMARY AND CONCLUSIONS ........................ 53

3 SHEAR ANALXSIS AND DESIGN OF REINFORCED CONCRETE
ELEMENTS INCORPORATING STEEL FIBER .................. 55

3.11mODUflION 0.0000000000000000000000000......0.. 55
3.2 STEEL FIBER EFFECTS ON SHEAR STRENGTH OF

”ROM
0 oo
molé

iv

REINFORCED CONCRETE ELEMENTS .................... 56
3.3 SHEAR STRENGTH EQUATIONS ........................ 58

3.3.1 THE PROPOSED METHOD ....................... 59
3.4 COMPARISON WITH TEST RESULTS .................... 62
3.5 PARAMETRIC INVESTIGATION OF STEEL FIBER EFFECTS

ON THE SHEAR STRENGTH OF REINFORCED CONCRETE

Emms OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 66

3.5.1 EFFECTS OF THE LONGITUDINAL AND TRANSVERSE

REINFORCMNT OOOOOOOOOOOOOOOOOOOOOOOOOOOOO 67

30502 EFFEflS OF a/d “TIC OOOOOOOOOOOOOOOOOOOOOO 70

3.5.3 EFFECTS OF FIBER REINFORCEMENT PROPERTIES . 71
3.7 SUM! MD CONCmSIONS OOOOOOOOOOOOOOOOOOOOOOOOO 74

4 DESIGN OF FIBROUS REINFORCED CONCRETE ELEMENTS IN
TORSION OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 76

401 INTRODUQION OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 76
flCKGROUND OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 76
4 O 2 O 1 PROPOSED APPROACH O O O O O O O O O O O O O O O O O O O O O O O O O 78
VERIFICATION OF TORSIONAL STRENGTH EQUATIONS .... 81
PARAMETRIC STUDIES O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 90
4.4.1 EFFECTS OF THE FIBER REINFORCEMENT

PROPERTIES O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 90
4.4.2 EFFECTS OF THE CONVENTIONAL REINFORCEMENT . 93
4 O 5 SUMY AN D concws IONS O O O O O O O O O O O O O O O O O O O O O O O O O 95

5 SW! AND coucms IONS O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 9 7
5.1 GENERAL OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 97
5O 2 rum ELEMENTS O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 98
5.3 SE“ Emms O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 99
5O 4 TORSION“ ELEMENTS O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 100
REFERENCES OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 102

APPENDICES

bub uh
#U N

APPENDIXAOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 106

TABLE

2.1
2.2

LIST OF TABLES

Page

Variables in Parametric Study......................... 25
Comparison of Theoretical Predictions With Test

ResultSOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Experimental vs. theoretical Shear Strengths of
Fibrous Concrete and Fibrous Reinforced Concrete

ElementSOOOOOOOOOOOOOOO...OOOOOOOOOOOOOOOOOOOOOOOOOOOO

Comparison
Reinforced

Comparison
Reinforced

Comparison
Reinforced

Comparison
Reinforced

Comparison
Reinforced

Comparison
Reinforced

Comparison
Reinforced

Average of

of Torsional Strength for Fibrous

Concrete Sections (Test vs Theory) Ref 31..

of Torsional Strength for Fibrous
Concrete Sections (Test vs Theory)

of Torsional Strength for Fibrous
Concrete Sections (Test vs Theory)

of Torsional Strength for Fibrous
Concrete Sections (Test vs Theory)

of Torsional Strength for Fibrous
Concrete Sections (Test vs Theory)

of Torsional Strength for Fibrous
Concrete Sections (Test vs Theory)

of Torsional Strength for Fibrous
Concrete Sections (Test vs Theory)

Teoretical / Experimental Torsional

Ref

Ref

Ref

Ref

Ref

Ref

20..

33..

32..

34..

31..

20. O

Strength Ratio...OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

vi

46

65

83

84

85

86

87

88

89

LIST OF FIGURES
FIGURES

1.1 Effects of Steel Fibers on Microcrack Propagation and
Pre-Peak Tensile Behavior of Concrete. a) Microcrack
Propagation, b) Pre-Peak Tensile Behavior ...........

1.2 Pull-Out Action of Steel Fiber and Post-Peak Tensile
Behavior of Steel Fiber Concrete. a) Pull-out
Action of fibers b) Post-Peak Tensile Behavior ......

1.3 Improvement in Concrete Material Properties
Resulting from Steel Fiber Reinforcement [45]. a)
Com ress ve and Flexural Ductility b) Impact

Res stance c) Fatigue Life c) Freeze-Thaw Durability..

1.4 Steel Fiber Effects on the Flexural Performance of
Reinforced Concrete Elements[4]. a) Steffness

b) Strength and ductility c) Crack Characteristics ...

1.5 ical Improvements in the Torsional Performance of
Re nforced Concrete Elements Resulting From Steel

Fiber Reinforcement [31]. .... .......................

1.6 Joint Action of Steel Fibers and Transverse

Reinforcement in Torsional Elements ..................
2.1 Refined Flexural Analysis Procedure ..................

2.2 Typical Moment-curvature Relationshi s................

a) Vf-l.5 % ; As-7.9 in2 b) vr-1.s ; Its-3.81 in2

2.3 General Form of The Compressive Constitutive Model for
Steel Fiber Reinforced Concrete.......................

2.4 General Form of The Tensile Constitutive Model for

Steel Fiber Reinforced Concrete.......................

2.5 Comparison of Analytical and Experimental Moment -

Curvature Relationships ..............................

2.6 Comparison of Analytical and Experimental Moment —

Curvature Relationships ..............................

2.7 Comparison of Analytical and Experimental Moment -

cuwature RelationShips OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO
2.8 The basic Rectangular Section in Parametric Study.....

2.9 Calculated Moment-Curvature Relationship for vf=0.5 %.
(variable-fy)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

vii

Page

6

9

11

12
16

19

21

22

23

23

28

2.10

2.11

2.12

2.24
2.25
2.26

Calculated Mement-Curvature Relationship for vf-1.5 %
(variable . fy)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Moment-Curvature Relationship for vf-0.5 %
(variable-f’c)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Mbment-Curvature Relationship for vf-1.5 %
(variable-t'c)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Moment-Curvature Relationship for vf=0.5 %
(variable . A's)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Mbment-Curvature Relationship for vf-1.5 %
(variable - A's)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Mement-Curvature Relationship for vf=0.5 %
(variable - As)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Moment-Curvature Relationship for vf=1.5 %
(variable-As)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Moment-Curvature Relationship for vf=0.5 %
(Variable - fiber diameter)..........................

Calculated Moment-Curvature Relationship for vf=1.5 %
(variable-tiber diameter)OOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Moment-Curvature Relationship for vf-0.5 %
(variwle-fiber length).........-......OOOOOOOOOOOO

Calculated Moment-Curvature Relationship for vf=1.5 %
(variable-fiber length)OOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Moment-Curvature Relationship for vf-l.5 %
(Variable-Fiber Type)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Ultimate Moment as Function of Vf for
Different Fiber mngthOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Calculated Ultimate Moment as Function of Vf for
Different Fiber Diameter.............................

”Alternative" Method of Flexural Analysis ...........
"AC1 Based" Method of Flexural Analysis .............
Comparison of Theoretical Predictions With Test

Results OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

Relation Between Curvature Ductility (Qu / Qy) and
Tension Steel p / Pb Ratio .........................

Example ”Dbl” .OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

viii

28

3O

3O

31

31

32

32

34

34

35

35

36

36

37
39
44

45

50
51

3.1 Simplified Method of Shear Failure in Fibrous
Reinforced Concrete .................................. 60

3.2 Experimental vs Analytical Shear Strength Values
(Reference 2)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 63

3.3 Experimental vs Analytical Shear Strength Values
(Reference 27)OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 63

3.4 Experimental vs Analytical Shear Strength Values
(Reference 43)OOOOOOOOOOOOOO0.0000000000000000000000000 64

3.5 The Basic ”Standard" Reinforced Concrete Element...... 67

3.6 Effects of Fiber Volume Fraction on Shear Strength
for Beams With Different Tension Steel Areas.;........ 68

3.7 Effects of Fiber Volume Fraction on Shear Strength
for Beams With Different Stirrup Sapacings ........... 68

3.8 Effects of Fiber VOlume Fraction on Shear Strength
for Beams With Different Transverse Steel Yield
strengthOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO 69

3.9 Effects of Fiber Volume Fraction on Shear Strength
for Beams With Different Span-to-Depth Ratio (a/d).... 70

3.10 Effects of Fiber Volume Fraction on Shear Strength
for Beams With Different Fiber Lengths............... 72

3.11 Effects of Fiber Volume Fraction on Shear Strength
for Beams With Different Fiber Diameters............. 72

3.12 Effects of Fiber Volume Fraction on Shear Strength
for Beams With Different Fiber Types................. 73

4.1 Mechanism of Resistance Against Torsion in Fibrous
Reinforced Concrete Elements a) Skew Bending of
Rectangular section b)Cross-Section .................. 79

4.2 Torsional Strength Comparisons Ref 31 ................ 82

4.3 Torsional Strength Comparisons Ref 20 .......... ...... 83

4.4 Torsional Strength Comparisons Ref 33 ................ 84

4.5 Torsional Strength Comparisons Ref 32 ................ 85

4.6 Torsional Strength Comparisons Ref 34 ................ 86

4.7 Torsional Strength Comparisons Ref 31 ................ 87

4.8 Torsional Strength Comparisons Ref 20 ................ 88

ix

4.9 The Basic ("Standard”)Reinforced Concrete Element.....

4.10

Effects of Fiber Content (Vf) on Torsional Strength
at Different Fiber Lengths.... .......................

Effects of Fiber Content (Vf) on Torsional Strength
at Different Fiber Diameters.... ....................

Effects of Fiber Content (Vf) on Torsional Strength
at Different Transverse Steel Leg Areas..............

Effects of Fiber Content (Vf) on Torsional Strength
at Different Transverse Steel Yield Strengt..........

Effects of Fiber Content (Vf) on Torsional Strength
at Different Transverse Steel Spacings...............

91

92

92

93

94

94

NOTAIION

4"

00'?
m
'0

Fees"

compressive stress block depth

cross section area of longitudinal reinforcement to
resist torsion

area of tension steel

cross section area of transverse reinforcement
resisting torsion

total area of shear reinforcement supplied by
stirrup

area of the compression steel

width of the rectangular section

neutral axis depth-compression zone

stirrups spacing

beam web width

compressive force of fibrous concrete

compressive force of steel bars

diameter of fiber

effective depth, distance from compression face to
centroid of tension steel

distance from compression face of the member to
centroid of compression steel

modulus of elasticity of concrete

modulus of elasticity of steel fiber

curvature

modulus of rupture (7.5(f’c)°°5 for plain concrete)

tension steel stress

xi

yield strength of longitudinal steel

yield strength of transverse steel '

split cylinder tensile strength

the ultimate flexural strength of steel fiber
tensile strength of steel fiber reinforced concrete
post-cracking tensile resistance of steel fiber
reinforced concrete

compression strength of Concrete, measured at 28
days after casting

compressive steel stress

compressive strength of the steel fiber Reinforced
Concrete

the height of the rectangular section

length of fiber _

nominal moment at section

reinforcement ratio for the balanced strain
condition

reinforcement ratio, for tension steel

tension force of fibrous concrete

tension force of steel bars

total torsional strength

shear strength provided by concrete

volume content of fiber

shear force at section

torsional strength provided by concrete
torsional strength provided by transverse steel
width of the cross section

xii

MR 1
INTRODUCTIOI

1. 1 0mm

Steel Fiber Reinforced Concrete applications have grown
rapidly in the 1970' s and 80's. Steel Fiber Reinforced
Concrete is now a standard construction material for some
major secondary structures, including overlays on industrial
floors, bridge decks and parking structures, mine tunnel
lining, and dams. There are also major potentials for the
application of steel fiber reinforced concrete to primary
structural elements subjected to flexural, shear and
torsional forces. Steel fibers can improve the ductility,
strength and serviceability of structural elements. There is
however, a strong demand for the development of structural
analysis. and design procedures for facilitating the
commercial applications of steel fiber reinforced concrete to
primary load-bearing structural elements.

Development of structural analysis and design guidelines
has been the main thrust of this research. A brief
background on the effects of steel fibers on the concrete
material properties, and the flexural, shear and torsional
performance characteristics of reinforced concrete elements
is presented in this chapter. The available structural
analysis and design techniques for fibrous reinforced
concrete elements are introduced and critically evaluated

using the reported experimental data for flexural, shear and

1

torsional elements in chapters 2, 3 and 4, respectively.
The structural analysis and design methodologies developed in
this research for reinforced concrete elements subjected to
flexural, shear and torsional forces are also described and

verified in these chapters.

1.2 STEEL 31382 :3th 00!“!!!

Concrete is weak in tension and fails in a brittle
manner under tensile and impact loads. These deficiencies
generally result from the ease of initiation and propagation
of microcracks, and also from the lack of post-cracking
tensile resistance in conventional concrete materials [44].
microcracks are generally initiated in concrete at the
aggregate-cement paste interfaces as a result of the drying
shrinkage, bleeding and settlement of cement paste in
concrete. Under external load and environmental effects,
microcracks tend to propagate and interconnect, causing the
brittle failure of concrete. The propagation of microcracks
in concrete can be effectively hindered through the use
of closely-spaced short fibers, as shown in Figure 1.1a.
The arrest of microcracks by steel fibers leads to
improvements in the pre-peak tension behavior and tensile

strength of concrete as shown in Figure l.lb [7]*.

 

*Figures in brackets indicate reference number

 

  

 

 

 

 

Tensile Screen
\ \ ~
‘ \
\
\\
~‘ ‘ s. Flm
i
i .
\
Q
\
I
\
Aggregate “
I
Lin; 3
Steet Fiber 'g
f .‘ PM."
OI ' ‘\
MicrocraCK I
. 5
4m» 3
MI“. 5mm

a) Microcrack Propagation b) Pre-Peak Tensile Behavior

Figure 1.1 Effects of Steel Fibers on microcrack Propagation
and Pre-Peak Tensile Behavior of Concrete.

Upon the formation of a non stable microcrack system,

which usually marks the achievement of the ultimate tensile

strength in concrete, tensile deformations tend to

concentrate in a limited number of macrocracks. Steel fibers

bridge these cracks and restrain their widening through pull-

out resistance as shown in Figure 1.2a . Steel fiber are

thus highly effective in enhancing the post-peak ductility

and energy absorption capacity of concrete under tensile
stresses as shown in Figure 1.2b.

 

 

PM“!

 

 

 

 

Terrell. Strain

a) Pull-Out Action of fibers b) Post-Peak Tensile Behavior
Figure 1.2 Pull-Out Action of Steel Fibers and Post-Peak
Tensile Behavior of Steel Fiber Concrete.

The crack-stabilization action of steel fibers in
concrete also results in major improvements in the
compressive and flexural performance characteristics,
impact resistance, fatigue life and freeze-thaw durability
of steel fiber reinforced concrete as shown in Figure 1.3a,

l.3b, l.3c and Figure 1.3d respectively.

 

Fleur. Lees

 

Heron.

Hun

 

 

 

Flu. Dene

(a) Compressive and Flexural Ductility

 

.Oab-‘O-I.’ “€5.03-

’4’7' 1'14“
2*//////
,, / /' '4
,,//// /

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(b) Impact Resistance

 

/

F Ibroue

Plan

.‘3--.‘fl OI- ..—0¢O

 

 

 

Max S"...

(c) Fatigue Life

 

.5
O

Fuoroue

Dram

o: —¢ooK o-i-a~<o oe-a-o-OD

 

 

 

No or Freeze-Thaw Cycm

(d) Freeze-Thaw Durability

Figure 1.3 Improvements in Concrete Material Properties
Resulting from Steel Fiber Reinforcement [45].

1.3 STEEL FIBERS IN ILEEURRL REINFORCED CONCRETE ELEMENTS

The crack-aresting action of steel fibers in concrete

results in improved performance of fibrous reinforced
concrete flexural elements under service: conditions.
Elements incorporating steel fibers have higher flexural
stiffness ( reduced deflections ) and smaller crack widths
when subjected to service loads [9].
Figure 1.4a presents test results which are indicative of the
increase in flexural stiffness resulting from the application
of steel fibers to conventional reinforced concrete elements.
Steel fibers also improve the strength and especially post-
peak ductility and toughness of reinforced concrete flexural
elements as shown in Figure 1.4b. The flexural failure mode
of fibrous reinforced concrete elements involves the
formation of more closely-spaced cracks with reduced crack
widths when compared with conventionally reinforced concrete
elements as shown in Figure 1.4c.

These advantages of the use of steel fibers in
reinforced concrete flexural elements produce opportunities
for reducing the required compressive steel ratios and
increasing the maximum allowable tensile steel area ( set to
ensure the ductility of failure ), and also for using high-
strength materials ( which generally tend to encourage
brittle failure modes ), and reducing the safety factors in

flexural elements.

 

 

 

 

 

 

 

 

 

 

M
O
M
E
N
T
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til
at
o L l I L
5 1O 15 20 25
W HIM (3'0,
a) Stiffness
5
q: I ... ......
4 - -
E
3 ‘- ,. ‘ Flare dent m
L Fiber length 37.5mm
: FIB. VG... one-m
C .0
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2' fl 2 a-
f; z. ‘ ‘ \h 0.5 s
'- o s
1 " .0 1.
' z. v I. e nee .-
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o 1 l l
0.5 I 1.5 2
Oettectlee m

b) Strength and Ductility

 

 

 

 

 

 

 

 

\, \ § ‘
//;// I {MNW , a 7A; 1‘ f 6‘) ; i: t:

Beam a: no fiber reinforcement Beam b: with l % fiber
reinforcement

c) Crack Characteristics
Figure 1.4 Steel Fiber Effects on The Flexural Performance of
Reinforced Concrete Elements [4].
1.4 STEEL RIDERS IN REINFORCED CONCRETE ELEMENTS SUEJECTED TO
SHEAR FORCES
Shear failure modes in reinforced concrete elements are
generally brittle. Relatively high safety factors are used
in shear design of reinforced concrete elements, which lead
to the need for large ratios of transverse shear
reinforcement. Congestion of shear reinforcement, and the
cost and time involved in the placement of transverse steel
are some ‘undesirable aspects of :reinforced. concrete
structures.
The improvements in tensile resistance, ductility and

shear friction ( across cracks ) in concrete resulting from

10

steel fiber reinforcement lend themselves particularly well
to enhancing the performance characteristics of reinforced
concrete elements under shear. Improvements up to 70 % or
more in the shear resistance of reinforced concrete beams
have been achieved through steel fiber reinforcement [27].
References [21, 22] have reported test results which
indicate that fibers can change the failure mode of typical
reinforced concrete beams from a brittle shear mode to a
ductile flexural one. With the introduction of steel fibers
the amount of costly and labor intensive transverse steel can
be reduced, which also relaxes the congestion of reinforcing
bars in structural elements. It has been suggested that the
best results can be achieved through the combined use of
steel fibers with a relatively small amount of transverse

reinforcement [25].

1.5 STEEL FIBERS IN TORSIONRL REINFORCED CONCRETE ELENENTS

Significant improvement in the strength and ductility
of reinforced concrete elements subjected to torsional forces
can be achieved through steel fiber reinforcement as shown in
Figure 1.4 [31]. In the presence of steel fibers, the
widening of cracks in torsional elements tends to be
restrained a condition which leads to the formation of
well-distributed ( i.e. , closely spaced ) cracks with
controlled widths [20, 31, 32].

11

 

140

 

120

   
  
   

_—d 'o\
12m

 

 

0|
I
\O ’

 

53

W - 28 No Transverse Steel

3-' eo-x OCO‘O-‘l

4O

20 Vt - o I No Treneveree Steel

 

 

 

60 100 160 200 250 300 350
Unlt Angle at Met (18-3 Deg/In)

Figure 1.5 Typical Improvements in the Torsional Performance
of Reinforced Concrete Elements Resulting' From
Steel Fiber Reinforcement [31].

The improvements in torsional performance of reinforced
concrete elements resulting from fiber reinforcement tend to
be more pronounced at higher volume fractions of fibers with
larger aspect ratios [31, 34]. Optimum results seem to be

achieved through the use of a combination of steel fibers and

transverse reinforcement as shown in Figure 1.6.

12

 

 

 

 

 

 

140
72 In 6 ln
.3
120 .' 0‘ 12 In
100

  

Vf-1S S-3.5ln

     

6C

6C

3'“ “‘0“ OCD‘O-‘I

4C Vl - 1.5 ‘5 No Transverse Steel

20

 

 

 

50 100 150 200 250 300 350
Unlt Angle ol Twlet (16-3 Deg/in)

Figure 1.6 Joint Action of Steel Fibers and Transverse
Reinforcement in Torsional Elements.

EFFECTS OF STEEL FIBERS ON FLEZURRL EEEIVIOR OF REINFORCED
CONCRETE BEANS 8 REFINED RNRLYSIS AND DESIGN

2.1 INTRODUCTION

The advantages of steel fiber reinforced concrete can be
beneficial in structural elements subjected to flexural loads
for improving the ductility of failure and thus releasing the
strict limits on conventional reinforcement ratios ( e.g
maximum limit on tensile steel ratio ) required to ensure a
ductile failure mode. Steel fibers can facilitate the
effective use of high strength materials in structural
elements.

A review of the existing literature on the flexural
behavior of reinforced concrete beams incorporating steel
fibers reveals the following effects of fiber reinforcement
on flexural behavior:

Fibers are effective in increasing the flexural
stiffness and reducing deformations of cracked reinforced
concrete sections. The increase in flexural rigidity results
from the role of the fibers in inhibiting the growth and
widening of cracks [1, 8, 13].

The load at which flexural cracks appear in reinforced
concrete beams are twice the corresponding loads for the beam
without steel fibers [1, 13].

The flexural cracks in steel fiber reinforced concrete

members are more closely spaced, and have smaller widths

13

14

when compared with those in conventional concrete elements
subjected to flexural loads [2, 14].

The post-cracking behavior and ductility of the flexural
reinforced concrete elements are significantly enhanced
through the addition of steel fibers. This is a direct
result of the desirable toughness and ductility of steel
fiber reinforced concrete in compression and in tension [5 ,
6, 7, 8]. Steel fibers, through preventing a brittle
crushing of concrete in compression, also provide the
compression steel with lateral restraint against buckling at
large strains, and thus further enhance the ductility
flexural behavior in reinforced concrete elements.

The improvements in flexural strength resulting from
steel fiber reinforcement are not large enough to give steel
fibers the potential to fully substitute continuous
reinforcing bars in reinforced concrete elements [15] .
Optimum conditions in flexural elements may be achieved
through the use of steel fibers together with conventional
steel bars.

The purpose of this phase of research is to study the
influence of steel fibers on the strength and ductility of
reinforced concrete members subjected to flexural loads, and
to develop design techniques for reinforced concrete flexural
members incorporating steel fibers.

The flexural behavior of fibrous reinforced concrete
elements is investigated using a refined modeling technique.

Based on the results of this investigation, practical

15

design guidelines are produced which take advantage of the
improvements in flexural ductility of reinforced concrete

elements resulting from the presence of steel fibers.

2.2 REFINED FLEEURRL RNRLFSIS

A refined analysis technique was developed for
predicting the complete moment-curvature relationship of
reinforced concrete sections in the presence of steel fibers.
The proposed analysis procedure accounts for the post-
cracking tensile resistance and the improved ductility of
steel fiber reinforced concrete under tensile and compressive
stresses [7].

The developed flexural analysis procedure has been based
on the assumption that plane sections normal to the beam
longitudinal axis remain plane after bending. This
assumption indicates that the flexural strain distribution is
linear across the section as shown in Figure 2.1. The
tensile and compressive constitutive models of steel fiber
reinforced concrete are assumed to be known ( the
constitutive models used in this study will be presented
later ).

In the developed flexural analysis procedure the
complete moment-curvature relationship is constructed by
finding the value of moment corresponding to different
curvature input values. For each curvature, through a trial
and adjustment technique, the neutral axis position at which

the equilibrium of compressive and tensile forces at the

16

cross section are satisfied is decided. With the neutral
axis location and value of curvature known, the strains and
thus the stresses at the cross section can be calculated as
shown in Figure 2.1 and the moment of forces acting on the
section corresponding to the input value of curvature can be
computed. ' A computer program was developed to perform the
above flexural analysis procedure; this program is described
in Appendix ”A" where a listing of the program is also given.

Typical moment-curvature diagrams generated using the

developed program are given in Figures 2.2a and 2.2b as.

 

. ............ x- ““7? ........

 

 

 

 

 

 

 

Creee Seetlen muln Dletrlhutlen Streee Dletrllmlon

Figure 2.1 Refined Flexural Analysis Procedure.

17

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

7 3500
6*- M‘-
M
o 0'.
t-—-0 —- t
I!
I5. ,--‘--,x',.. N 2600 "*:.: m
l '0'“. ° \ ‘
K " :'\ z .0. ‘:’,‘ .0.
' ... s \ ’~ M n " ~
g 4)- Tee-.00. I 2000~ :“tu- F".
g— i“. “
I
n D
3? e 1600'
Oreo-e0 meeeu vet-ens. uneven- h-eeee ma- nned-e! Me-
I .00.... mn- mee mm ' «meal-none «unease-nee-
I
O
0 2’ n ‘m-
: lean-Cum Dune-ea...-
: “New lea-emu
S g.- leeeeou uneeeeew m . W ”I“ I”.
o 1 1 l L l o 1 J n 1 1
1 2 3 4 6 O l 2 3 4 5
Caveturelvlnll‘E-Ol Oswellnnllh-el
a) v, - 1.5 t; A, - 7.9 1:12 b) v, - 1.5 3; As - 3.8 in2

Figure 2.2 Typical Moment-curvature Relationships.

2.3 CONSTITUTIVE MODELS

The refined flexural analysis procedure introduced in
the previous section requires the compressive and tensile
constitutive models of steel fiber reinforced concrete. The

models used in this investigation are described below.’

2.3.1 COMPRESSION MODEL
The compressive stress-strain relationship presented in
References [18, 19] for steel fiber reinforced concrete was

selected for use in this study. This constitutive model is a

18

function of the following parameters.

The matrix compressive strength

Fiber reinforcement index Vflf / d:

The model consists of 3 segments, a curvilinear
ascending portion followed by a linear descending branch and
a flat tail as shown in Figure 2.3. The equations for this

model are given below:

f - - f’ct (€/€p)2 + Zf'cf (e/ép) for €_<_ p
r-z(€-€p)+t'cfzt'r fore; p

Where :
f'c a concrete compressive stress

6 a concrete compressive strain
6p - strain at peak stress
f’cf - compressive strength of the steel fiber

Reinforced Concrete

19

STRESS

 

 

t-felllédrelflfl

 

tut-don! H;

 

 

 

STRAIN '

Figure 2.3 General Form of the Compressive Constitutive Model
for Steel Fiber Reinforced Concrete.

2 , f'r a Coefficient to be derived, together with f’cf
and 6p, empirically in terms of the matrix
compressive strength and the fiber
reinforcement index.

The empirical expressions for coefficients of the above

compressive constitutive model are given below

f’cf a f'c + 994 Vflf
d:
f'r 3 0.12 f'cf + 2000 Vflf
5:
z -343 f’c ( 1 - 0.64 (vféfW-S ) 5 o
f

20

p - ( 0.00079 + 1‘1; ) Vflf + 0.0021
t'c df
Note : f’c , f’cf and f’r are in psi.
Where :
d: - diameter of fiber
f’c - concrete compressive strength at 28 days

Vf a volume content of fiber
11 - length of fiber

2.3.2 TENSION MODEL

The tensile constitutive model of steel fiber reinforced
concrete used in this study is one proposed in References
[16, 17]. This model is shown in Figure 2.4 and involves

four variables:

ftf - tensile strength
Ef - modulus of elasticity
fpf - post-cracking tensile resistance provided by the

pull-out resistance of fibers.

21

STRESS

 

"I

 

let

 

 

O
l l 1 l L L_Ju l 1 J l l g

STRAIN

 

Figure 2.4 General Form of the Tensile Constitutive Model for
Steel Fiber Reinforced Concrete.

The values of peak tensile stress (ftf) and post-peak
tensile resistance in the tensile constitutive model of

Figure 2.4 can be obtained using the following expressions:

f =1! (1-2 )+o.5*o.41*7’* y 1
tr ... 105 1.5.;

fpf - o.5*q.41*7'* 25 1f
1 f

The modulus of elasticity in tension was taken equal to
that in compression, and the effects of steel fibers on

elastic modulus were assumed to be negligible:

22

ac - 57000 (t'c)°-5 psi

2.4 COMPARISON 'ITE TEST RESULTS

The analytical moment-curvature relationships of steel
fiber Reinforced Concrete ( obtained using the developed
computer program and the constitutive models described
earlier ) are compared in Figures 2.5 - 2.7 with. the
experimental moment-curvature relationships reported in
References [3, 1] in order to verify the accuracy of the
analytical techniques. The comparison between experimental
and analytical result is observed to be reasonable, as shown

in Figures 2.5, 2.6 and 2.7.

 

 

 

 

 

l_'
\
25. ' '
o ' C"!
I , '
O

N
O
T

i
(I
v

   
 

e-eem Q-eln
“-038an Ago-0mm:
ly-mnl Ill-1.27!
«comm No.08 In

It! - 37 tel Stelont l=l0er
let-1mm lot-48m
Mme: - 25.3 Mn Fl - as 5-00

:0 —- eo-X dZMIOK

6
V

 

 

 

2 4 0 8 l0 :2
comm: lmnl (16-4)

Figure 2.5 Comparison of Analytical and Experimental Moment-
Curvature Relationships (Reference 3).

23

 

 

 

 

 

 

 

 

30
25"
u 20"
o
M
E
N
r
7 ‘°'
2 ,‘ a-aam‘ ca-ua
i ,' “roam: «9-.qu
“ 'OI ty-voue w-ees
shalom tl-l.2in
"I'SOISI mm”
5 let-sew let-57m
“Mozaaltln horse-00
o l l L J

 

2 4 6 8 10 12
cumme (mm le-tl

Figure 2.6 Comparison of Analytical and Experimental Mement-
Curvature Relationships (Reference 3).

 

200

 

 

 

 

 

 

160'-
ll
0
U
a
I
7
., 100»
l
I
I
.1 e-eeoln cap-nan
Measure Ase-2U!!!
Fy-S7ltel w-u
50l-
ol-mesm ll-tSSlfl
It! 0 .34 I00 Owned Plbev
let-Mm tor-0m

m '159.3 kJn Fl ' 0.7 E-04

 

 

 

o l l l 1 A

2 4 S S IO 12
CUMURE (Mn) (Ed)

Figure 2.7 Comparison of Analytical and Experimental Moment-
Curvature Relationships (Reference 1).

24

2.5 RRRRMETRIC STUDIES

The flexural performance characteristics of reinforced
concrete sections incorporating steel fibers are mainly
dependent on the steel fiber and conventional reinforcement
conditions, and also on the concrete strength. The
developed analytical techniques were used in this
part of the study to assess the effects of these factors on
the flexural moment-curvature relationships of typical
fibrous reinforced concrete sections.

The variable used in this parametric study are: a) fiber
volume fraction (Vt); b) fiber length (If); c) fiber
diameter (df): d) fiber type (Straight, Hooked and Crimped)
e) tensile steel area (As): f) compression steel area (A's):
g) rebars yield strength (fy): and h) concrete compressive
strength (f'c). The ranges of these variables considered in
this parametric study are given in Table 2.2.

The typical rectangular reinforced concrete beam section
used as the basic section in the parametric study on flexural
behavior is shown in Figure 2.8. The parametric study was
performed by analytically deriving the moment-curvature
relationship while one of the variables was changed with
the others kept constant. The effects of each variable on
moment-curvature relationship could thus be investigated

separately.

 

25

lel. 2.1 VRRIRELES IN FRRRMETRIC STUDIES

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sec. VariablelFiber properties (3)
number 8 value --------------------
(1) (2) IVf 8|lf inldf inltffi
10 tf* - 1.5 1 .013 str
11 tf* - 1.5 1 .013 erg
22 fy 40 .35 1 .013 str
23 fy 80 .35 1 .013 str
24 fy 120 .35 1 .013 str
25 fy 140 .35 1 .013 str
26 fy 40 1.5 1 .013 str
27 fy 80 1.5 1 .013 str
20 fy 120 1.5 1 .013 str
29 fy 140 1.5 1 .013 str
16a f'c 3 .4 l .013 str
17a f'c .4 1 .013 str
18a f’c 6 .4 1 .013 str
16b f'C 3 1.5 1 .013 str
17b f'c 4 1.5 1 .013 str
18b f'c 6 1.5 1 .013 str
19a A's 0 .5 1 .013 str
20a A's 1.81 .5 l .013 str
21a A's 2.74 .5 1 .013 str
19b A's 0 1.5 1 .013 str
20b A's 1.81 1.5 1 .013 str
21b A's 2.74 1.5 1 .013 str
1 As .79 .35 1 .013 str
2 As 1.57 .35 1 .013 str
3 As 3.95 .35 1 .013 str
4 As 6.32 .35 l .013 str
5 As .79 1.5 1 .013 str
6 As 1.57 1.5 l .013 str
7 As 3.95 1.5 1 .013 str
8 As 6.32 1.5 1 .013 str
13a df .02 0.5 1 .02 str
14a df .013 0.5 1 .013 str
15a df .01 0.5 1 .01 str
13b df .02 1.5 1 .02 str
14b df .013 1.5 1 .013 str
15b df .01 1.5 l .01 str

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

[Longitudinal steel(4)| f’cf ftf Fig.
---------------------- ksi ksi number
|As in2lA's in2|fy ksil (5) (6) (7)
2. 37 .24 60 5.1 .32

2. 37 .24 60 5.1 .32 2.9
2. 37 .24 60 5.1 .32

2.37 .24 40 4.2 .27

2.37 .24 80 4.2 .27 2.10
2.37 .24 120 4.2 .27

2.37 .24 140

2.37 .24 40 5.1 .32

2.37 .24 80 5.1 .32 2.11
2.37 .24 120 5.1 .32

2.37 .24 140 5.1 .32

2.37 .24 60 3.3 .24

2.37 .24 60 4.3 .27 2.12
2.37 .24 60 6.3 .33

2.37 .24 60 3.3 .24

2.37 .24 60 5.1 .32 2.13
2.37 .24 60 7.1 .38

2.37 0 60 4.3 .27

2.37 1.81 60 4.3 .27 2.14
2.37 2.74 60 4.3 .27

2.37 0 60 5.1 .32

2.37 1.81 60 5.1 .32 2.15
2.37 2.74 60 5.1 .32

.79 .24 60 4.2 .27

1.57 .24 60 4.2 .27 2.16
3.95 .24 60 4.2 .27

6.32 .24 60 4.2 .27

.79 .24 60 5.1 .32

1.57 .24 60 5.1 .32 2.17
3.95 .24 60 5.1 .32

6.32 .24 60 5.1 .32

2.37 .24 60 4.2 .27

2.37 .24 60 4.4 .28 2.18
2.37 .24 60 4.5 .28

2.37 .24 60 4.7 .3

2.37 .24 60 5.1 .32 2.19
2.37 .24 60 5.4 .35

 

 

 

 

 

 

 

 

Table 2.1 (cont'd)

 

 

 

 

 

 

26

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Sec. VariablelFiber properties (3)|Longitudina1 steel(4)| f'cf ftf Fig.
number 5 value ------------------------------------------ ksi ksi number
(1) (2) Iv: %|lf inldt inltttlns inZIA's in2|fy ksil (5) (6) (7)

32a 1! l .5 l .013 str 2. 37 .24 60 4.3 .27
333 If 1.5 .5 1.5 .013 str 2.37 .24 60 4.6 .29 2.20
34a 11 2 .5 2 .013 str 2.37 .24 60 4.7 .3
32b 1! l 1.5 1 .013 str 2.37 .24 60 5.1 .32
33b If 1.5 1.5 1.5 .013 str 2.37 .24 60 5.7 .36 2.21
34h if 2 1.5 2 .013 str 2.37 .24 60 6.2 .4
Note
tf* - Type of steel fiber.
crp - mped fiber
hok - Hooked fiber
str - Straight fiber.
1 in - 25.4 mm
1 ksi - 6.895 MPa
1 kip. in - 0.113 KN.m

 

27

Rectangular cross section (20 in height * 10 in width )

 

 

 

A8 - 2.37 in2 1: - 1 in d' - 2 in
Asp - 0.24 1n2 0, - 0.013 in d” - 2 in
fy - 60 ksi Vf - 1.5 % dp = 18 in
f'c - 4 ksi Type of Fiber a Straigth
B
0 ‘ I, - l 10'?
ltA.'--’
00 ;.:‘ : H
l \ I,
'\ M- I
i F 1 I Id. n.

 

 

 

 

Figure 2.8 The Basic Rectangular Section in Parametric Study.

Figures 2.9 and 2.10 show the effects of the
longitudinal steel yield strength on flexural performance for
conditions with Vf a 0.5 t and Vi a 1.5 8 respectively. The
ultimate moment capacity is observed to increase with the
increase in the yield strength of longitudinal steel. At the
lower fiber volume fraction of 0.5 % in Figure 2.9, the
increase in the reinforcing bar’ yield strength ‘tends to
damage the ductility of flexural behavior (by limiting post-
peak deformations prior to failure). At high fiber volume
fraction (1.5 t in Figure 2.10), however, the section
remains ductile in spite of major increases in the
reinforcing bar yield strength ( up to the very high value of
120 ksi ).

28

 

 

 

 

 

 

6
ea
(Or-«eh.
4 h #‘
try-teens
3
3 III, a
I} 3’ Ira-IO
e x"
f /
I ,f
I
e 2 " ,
O
O .f
3 -',...-—-——~--« ——88
g ' III-«III
' In
“an ea the euvee eerreeeeee h the
eeetlee ae-eet at eel-e l e! m
o l l I; l l

l 2 0 4 6 0
m "III. It'll

Figure 2.9 Calculated Moment-Curvature Relationship for

 

 

 

 

Vf- 0.5 8 ( variable - fy ).
8
6' 20
Mad
I
a
$4“ (WIS
'.‘
e
.___27
:3" /,// 01-40..
7 .
0
3
:21
E ’3’ ---------------- Ween-a
1.- ; Weathemveeeeneeeeeetetne
.' “hmwdeelueletm
o L l l L l l.

l 2 3 4 6 0 7
wmm (WI new)

Figure 2.10 Calculated Moment-Curvature Relationship for
= fo 1.5 t ( variable = fy ).

29

Figures 2.11 and. 2.12 show"the effects of concrete
compressive strength on flexural behavior of reinforced
concrete sections with 0.5 t and 1.5 % fiber volume fractions
respectively; The increase in fiber 'volume fraction is
observed to provide the section with higher flexural
ductility even in conditions with relatively high concrete
compressive strengths.

A comparison between the effects of compression steel
area on flexural behavior of sections with 0.5 8 and 1.5 8
fiber volume fractions in Figures 2.13 and 2.14,
respectively, indicates that at higher fiber volume fractions
there is a lesser need of compression reinforcement for
improving the flexural strength and ductility of reinforced
concrete sections.

Figures 2.15 and 2.16 show the effects of tension
reinforcement area on the flexural behavior of reinforced
concrete sections at 0.5 % and 1.5 8 fiber volume fractions,
respectively; The increase in fiber 'volume fraction is
observed to provide the section with desirable ductility
characteristics, even with high tension steel areas which
tend to encourage brittle failure modes prior to the yielding
of tension steel in reinforced concrete sections with zero or

low steel fiber contents.

3O

 

2000'

CM la.
"I
{ft-4“

‘0
if" I“

I. - I “R‘O‘

 

Weatheeuveemteue
eeeleeeeebereteeleeelelml

 

 

 

l I A I A
1 2 S 4 6 O 7
was: IVIIII tut-cl

Figure 2.11 Calculated MOment-Curvature Relationship for
Vf- 0.5 t ( variable - f’c ).

 

e.-- “08

 

Weefieeuveeeerreeeeeehtne
Mumwdmlelun

 

 

O __4 l 1 L 1

l 2 3 4 5 0 7
cum-wee want (IE-OI

Figure 2.12 Calculated Moment-Curvature Relationship for
Vfa 1.5 8 ( variable a f’c ).

31

 

 

 

 

2000
O
1600'-
H
I
r
‘ 1000b
I
O
i
O
600'
Ila-bereetDeeUIeeeerleeeaeI‘eue
”mended-ate!”

 

Figure 2.13 Calculated Moment-Curvature Relationship for
Vf- 0.5 % ( variable - Asp ).

 

 

 

 

2ND
2.
tam-enh-
‘mb hm
.__:ee
10 treat-at
o
a
S
r
7 1000-
a
O
i
I
600'
Westereeetneelveeeeneeeeeeletae
«dun-badeeh-teom
O ‘ ‘ ‘ 1 1

 

 

 

I 2 a 4 a o 7
mm (M II“!

Figure 2.14 Calculated Moment-Curvature Relationship for
Vf‘ 1.5 t ( variable - ASp ).

32

 

ee-I «OE

........4 8-:

 

 

 

 

Figure 2.15 Calculated Moment-Curvature Relationship for
Vf- 0.5 t ( variable a As ).

 

"lflEOE

 

 

1‘ / .

......O'Q .-.

   

 

. ’ ..m ....... - ...... IO... 4
r' “I..." h!)

Weathermeefl'eeeeeehtte
eee‘tleeae-ber‘ eteetuel 1‘91“

1 2 3 4 6 0
mm mm (soot

 

 

 

Figure 2.16 Calculated Moment-Curvature Relationship for
Vf- 1.5 % ( variable a As ).

33

In general, one may conclude from the above discussion
that the addition of steel fibers is effective in improving
the ductility of reinforced concrete sections. Hence, the
limits on maximum tension steel area can be relaxed ( leading
to the increase in the upper limits of the flexural strength
which can be achieved using sections of specified dimensions)
the need for compression steel can be reduced, and high -
strength concrete and steel can be used without loss in
ductility. Besides improving flexural ductility, steel
fibers also provide the section with increased flexural
strength.

Figures 2.17 and 2.18 clearly show that fibers with
lower diameters tend to be more effective in improving the
flexural strength and ductility of reinforced concrete
sections.

Figures 2.19 and 2.20 show that the increase in fiber
length results in enhanced flexural strength and ductility of
reinforced concrete sections.

Figure 2.21 shows the influence of different fiber types
( Straight, Crimped and Hooked ) on the flexural load-
deflection relationship of the base reinforced concrete
section. Hooked fibers are observed to result in a slightly
higher flexural strength than crimped and straight fibers.

Figures 2.22 and 2.23 show typical effects of fiber
volume fraction for different fiber lengths and diameters

respectively, on the flexural strength of reinforced concrete

34

 

2&0

 

 

«me:

O. -I

WOMWMbM
eeeleeu-eereleelueletm

 

 

 

L

o 1 A L 1 l
‘l 2 3 4 6 0 7
more (we teal

Figure 2.17 Calculated MOment-Curvature Relationship for
Vf- 0.5 t ( variable - fiber diameter ).

 

2000

 

16M’

room

no. «no;

 

Weathermeerreeeeeebtae
“nun-nautam

 

 

 

o 1 1 1 1 L 1
1 2 3 4 5 C 7
WW ""0. 4‘°‘|

Figure 2.18 Calculated Moment-Curvature Relationship for
Vf- 1.5 8 ( variable - fiber diameter ).

35

 

2000
840
mu
8‘
~__< em I
..., t—m‘L

 

 

’{1000:
e
I
i
O
000-
We-eereeetaeeuveeeeneeeeeehlhe
muewdeeluatetult
o t T

 

Y Y Y 7
l 2 3 4 5 0 7
m (tn-l "I...

Figure 2.19 Calculated Moment-Curvature Relationship for
Vf‘ 0.5 2 ( variable - fiber length ).

 

2000

 

 

 

1500'I

O. -I dlfl‘o‘
l

500‘

 

 

 

 

Figure 2.20 Calculated Moment-Curvature Relationship for
Vf- 1.5 2 ( variable - fiber length ).

36

 

2600-
" u
(mm-l
zooo- ‘°
11 Mme"
1600*
IMOP
600- Weekeelheeeneeeerreeeeeatetae
mumwueeh-eletml.

 

 

 

2 3 4 a a

Figure 2.21 Calculated Moment-Curvature Relationship for
Vf= 1.5 % ( variable = fiber type ).

 

2000

7
Z

ZZZZZZZZZZZZZZZZZZZZZ

 

 

 

 

 

 

 

 

Figure 2.22 Calculated Ultimate Moment as Function of Vf for
Different Fiber length.

37

 

Z
%

ZZZZZZZZZZZZZZZZZZfiV

 

 

 

 

 

/

 

9

0.02 0.013 0.
(“METER OF Flfill (In)

 

l-w-eas -Vl-lI [aw-101i

Figure 2.23 Calculated Ultimate Moment as Function of Vs With
Different Fiber Diameter.

38

2.6 SIMFLIFIED FLEXURRL ANALYSIS AND DESIGN GUIDELINES

Two methods for flexural analysis and design of
reinforced concrete sections incorporating steel fibers are
introduced in this section. The first one, referred to as
the "alternative method", is based on the observations of
flexural stress distributions made at different stages of the
refined flexural analysis. The second called the "ACI-based
method" , follows flexural analysis procedures which are
modified versions of those proposed by the American Concrete
Institute (ACI 318-83) for conventional concrete. The
predictions of these two methods are compared with the
results of experimental studies and refined flexural
analysis. Conclusions are made regarding the practicality of
these methods in application to flexural design of fibrous

reinforced concrete sections.

2.6.1 “ALTERNATIVE" METHOD

The method introduced in this section has been based on
the examination of flexural stress and strain distributions
of peak flexural load obtained in refined analysis of fibrous
reinforced concrete sections.

For the fibrous reinforced concrete section of Figures
2.24(a) and 2.24(b), Figure 2.24(c) shows the proposed linear
flexural strain distribution at ultimate moment, which has
been based on the fundamental assumption of this study which
implies that plane section normal to the beam longitudinal

axis remains plane after bending. A maximum compressive

39

strain value of 0.002 is proposed as the criterion for the
achievement of maximum bending moment. Figure 2.24(d)
presents a theoretical stress obtained using the strain
distribution of Figure 2.24(c) and the constitutive models of
the material in tension and in compression. The simplified
flexural stress distribution at peak bending moment, used in
the "Alternative method“ of flexural analysis, is presented
in Figure 2.24 (e) . This stress distribution together with
the proposed value of 0.002 for extreme compressive fiber
strain at peak moment form the basis for the "Alternative"
method of flexural analysis and design of fibrous reinforced

concrete sections.

 

 

 

 

 

 

 

 

 

 

 

1 [_AI__
Ia-

 

 

a-Element at length of member b-Fleolangular aeotlon

0.002 f'Of l‘cl

 

 

 

 

 

Hf

c-Straln d-Actualelreaa e-ldeallzea atreaa l-Reaultant

Figure 2.24 "ALTERNATIVE" Method of Flexural Analysis.

40

The flexural strength of fibrous Reinforced Concrete
sections can be obtained in the ”alternative" method through
the performance of the following steps:

1 - Guess a value for c ( the depth of the neutral axis in
Figure 2.24(c))
2 - Find the strain and stress values: for compression and

tension steel:

68 - dig at 0.002 (2-1a)
E's- gig * 0.002 (2-23)
ts - as 68 - Es (gig) * 0.002 < fy (2-3a)
f's- Es 6'8- as (géQL) * 0.002 < fy (2-4a)

Compute the compression force C applied on the section:
C - Cc + Cs 8 f'cf (gab) + A’s f’s (2-5a)
compute the tension force T:

T - Tc + Ts -£r (h-c)b + A318 (2-6a)

Where :
Cc - compressive force of fibrous concrete

Cs - compressive force of steel bars
Tc - tension force of fibrous concrete
Ts - tension force of steel bars

t' - r + 994 9 y 1 ( si) (2-7a
Cf c 105 d§ p )

41
fr . 0.5 t 0.41 *7'* 2 lg (Psi) (2-8a)
106 d,

Then compare T and C as components of the internal couple,
they must be equal.

3 - If the difference between the compressive force (T) and
the tension force (C) is close to zero (within acceptable
limits) then the selected neutral axis depth is acceptable
and the analysis can be completed by computing the ultimate
moment corresponding to the flexural stresses, as described
in the next step. Otherwise, the neutral axis depth should
be adjusted and the procedure repeated from step 2 (increase
c If Cc + Ca < TO + T3, and vice versa ).

4 - Once the location of the neutral axis has been
established, the bending moment capacity of section ( Mn )
can be obtained by taking the moment of all the forces
applied on the section around the extreme compression fiber

of the cross section see (Figure 2.24):

xn = cc (ggg) + c. (d-d') - fr (h-c) b (£25) (2'98)

2.6.2 " ACI - BASED ” METHOD

This method follows the conventional reinforced concrete
flexural analysis procedure [20, 21], with some refinements
which reflect the effects of steel fiber reinforcement. The
maximum strain at fiber when the peak bending moment is
assumed to be 0.003 as shown in Figure 2.25(c) for the cross

section presented in Figure 2.25.

42

(a) and (b). The strain distribution is assumed to be
linear. The simplified compressive stress distribution

( which assumes a uniform compression stress ) is shown in
Figure 2.25(e). The steps followed in deciding the neutral
axis depth and calculating the bending moment at peak
flexural load, which are comparable with those in the

previous method, are given below:

1 - Guess a value for the neutral axis depth, c.

2 - Compute the depth of the stress block: a - B * c,
and compute es and 5's using the strain distribution of
Figure 2.25(c), and then find fs and f’s:

Es - gig * 0.003 (2-1b)
6’3' 9:4 9 0.003 (2-2b)
c

f, - E3*Es - 38 gig * 0.003 g ty (2-3b)

.f's- 2396!,- E8 §§QL* 0.003 5 {Y (2-4b)
Compute the compressive force C:

c - cc + c8 - 0.35 t'cf 0.85cb + A’s t's (2-5b)
compute the tension force T:

T - TO + T8 'fr (h-c) b + As fs (2-6b)

Where : ,
Cc - Compressive force of fibrous concrete

Cs - Compressive force of steel bars
Tc - Tension force of fibrous concrete

Ts - Tension force of steel bars

43

t’CI :- to + 994 * 1%531: (2-7b)

tr - 0. 5 * 0. 41* “3:3 (2-8b)

3 - If C and T balance each other (within the acceptable
error) then go to the next step and find the flexural
strength: otherwise adjust c and repeat from step 2.

4 - Once the location of the neutral axis has been
established, the bending moment capacity of section can
be obtained by taking the moment of all the forces

applied on the section about the tension steel axis. (See
Figure 2-25(c)):

an -- 9., (9:1) + c. (d-d'l - :, (h—cl b ( 12:3,) (2-9b)

44

 

 

I"(\ IO‘T

 

 

0' “\’l H

 

 

 

 

 

 

 

 

‘ - I" J»

a-Element ol length ol memOer o-Bectangular sectlon

 

 

0.003 0.80l'cl

Cs
‘—
.850 r__
Cc
........... H -----..---.--.

 

 

 

 

 

 

 

 

 

J H! :

c-Straln 0-Actual stress e-laeallzed stress l-Resultant

Figure 2.25 "ACI-Based Method” of Flexural Analysis

2.7 COMFARISON OF THEORETICAL FREDICTIONS MITH TEST RESULTS

The flexural test data presented in References [1, 3 and
20] were used to verify the "alternative" and "ACI -based"
methods for predicting the flexural strength of reinforced
concrete elements incorporating steel fibers. The cross-
sectional properties of tested elements are given in Table
2.2 and Figure 2.26.

Table 2-2 also compares the theoretical flexural
strengths as predicted by the refined computer analysis, and
"alternative" and "ACT -based" methods with the flexural
strength test results. While the refined flexural analysis

45

shows the best comparison with test results (average error of
only 2 4), both the ”alternative" and 'ACI-based" procedures
also predict test results with a reasonable accuracy
( average errors less than 10 t ).

Considering the fact that the 'ACI-based' method
proposed for flexural analysis and design of reinforced
concrete sections is similar to the conventional reinforced
concrete flexural analysis procedure, this method is

suggested for use in design of fibrous reinforced concrete

flexural members.

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Results.

46

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47

2 . a MAXIMUM STEEL euro nt unease 113111102030 concurs
new use

An important contribution of fiber reinforcement to the
flexural behavior of reinforced concrete elements is related
to the improvement in ductility at failure. There are
important potentials for taking advantage of this aspect of
fiber reinforcement effects on flexural performance.

An, important restriction. in flexural design. of
conventional reinforced concrete beams is the maximum limit
specified for the tension steel area to ensure yielding of
steel prior to crushing of concrete for achieving a ductile
failure. Fiber reinforced concrete has a desirable ductility
in compression and it thus has relatively large strain
capacities beyond the peak compressive stress. Hence, more
tension steel can be used in reinforced concrete sections
incorporating steel fibers without reducing the ductility
below desirable levels.

The maximum. limit on the tension steel ratio of
conventional reinforced concrete sections without compressive
steel is 0.75 times the balanced steel ratio. The balanced
steel ratio (pb) is the one for which yielding of steel takes

place simultaneously with the crashing of concrete in

compression:
‘ 9.115514% ( £19.09.
pb f 87000 + f
Y Y
Where:

9-1333

48

B - ratio a - ratio of the depth of rectangular stress
0 ,

block to the neutral axis depth.
5 - 0.85 for I": _<_ 4000 psi
8 I 0.85 - 0.05 ( f'c - 4000 ) / 1000 Z 0.65
for f’c > 4000 psi
Pb - reinforcement ratio for the balanced strain
condition
- tension steel ratio
- area of tension reinforcement

width of web

‘1 A! Jr 13
I

- effective depth: distance from compression face to
centroid of tension steel

fy - yield strength of steel

f’c- compressive strength of concrete, measured at 28

days after casting

The objective of this part of the study was to decide

the maximum limit on tension steel ratio in fibrous sections,

at which the flexural ductility corresponds to that of

conventional reinforced concrete sections with p - 0.75 Pb’

For this purpose a curvature ductility was defined as

the maximum curvature at failure ( Where the refined flexural

analysis by computer shows a major drop in flexural

resistance in the post-peak region ) divided by the curvature

corresponding to flexural yielding Figure 2.27 presents the

flexural ductility of sections similar to the one shown in

Figure 2.8, but with different tension steel ratios and fiber

volume fractions. This figure which has been generated using

49

the results of refined flexural analysis clearly shows the
improvement in flexural ductility resulting from steel fiber
reinforcement. The data presented in Figure 2.27 indicate
that a ductility of about 1.2 is achieved at p / Pb - 0.75 in
beams with a minimal fiber reinforcement volume fraction of
0.35 8. This level of ductility can be achieved in the
presence of 1 8 (Vflf / df - 70) fiber volume fraction at p
/pb - 0.9, and with the presence of 1.5 8 (Vflf/ df - 105)
fiber volume fraction at p / Pb - 1.05.

Hence, in the presence of steel fibers the limits on
maximum tension steel for ensuring flexural ductility can be
relaxed, and smaller cross sections can be used in flexural
elements to generate the required load capacity. The
corresponding weight reductions and increase in useful height
of the buildings and other structures can be an important
economic advantages encouraging structural applications of

steel fiber reinforced concrete.

50

 

2.5+

1.5 L -

 

 

0.5 .... ._.. -1-.. ”1-.....“ ”1.-.-.. .-....m -11..-.)

~<Q\xm30 ~<~—-—-—ooco ®~c~m<ec0

 

 

 

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
P/Pb Ratio

 

—‘— Vf=1.5 % + Vf=1% + Vf=.35 %

Figure 2.27 Relation Between Curvature Ductility (Qu / Q!)
And Tension Steel p / Pb Ratio.

51

2.9 EXAMPLE: FLEEURAL ANALYSIS OF A FIBROUS REINFORCED
CONCRETE SECTION USING THE ”AC1 - BASED" METHOD

For the section given in Figure 1.32 find the flexural

 

 

 

 

 

strength.
Given:
B - 10 in H - 20 in
d a 18 in dp - 2 in
As a 1.57 in; ASp - 0.24 inz
df - 0.013 in l - 1 in
ftf - .32 ksi Sgraight Fiber
8
v If
s ‘f' T ' 1h.
I .‘V' l
I . ‘
do , Q ‘ H
l \ I,
J '\ A8- I
. ‘ . ' Bf L

 

 

 

 

Figure 2.28 Example Problem

Solution:

Check tension steel ratio

p / pb - 0.24 < 1.05 or

Find flexural strength:

Assume 1.’8 and f’s S fy .

Assume c ( a trial value for the neutral axis depth ):
guess a value of c = 4 in .

Using equations presented below (2-1b: 2-2b: 2-3b: 2-4b:

52

2-5b and 2-6b) compute respectively 68; 6'5; f8: f’s: C

and T, respectively.

68 - g;; 9 0.003 (2-1b)
c
6',- g;g 9 0.003 (2-2b)
c
fs - £396s - Es 9 n;; 9 0.003 < 1y (2-3b)
c
f's- Es*€’s- Es 9 g-g'9 0.003 < :Y (2-4b)
c
Compressive force:
0 - cc + cs - 0.85 t'cf 0.359c9b + A’s 9 f’s (2-5b)
Tension force:

Compare C and T as explained earlier.
C - 217 kips and T - 15.6 kips
Since C exceeds T, c needs to be reduced
Assume a smaller value of c a 2.2 in
Repeat steps 1 to 5.
C a 83 kips and T - 107 kips
Since T exceeds C, c should be increased, Assume a larger
value of c - 2.4, and repeat steps 1 to 5.
C a 104 kips and T = 107 kips
The two values are close enough (error less than 5 8).
The analysis can thus be continued. Compute the ultimate

bending moment by summing the moments of all forces acting on

53

the cross section about the tension steel centroid:

M a Ca (ggg) + C. (d-d') - fr (h-c) b ( gzg ) (2-9b)
The calculated value of flexural strength is:

M a 1653 kips . in

2.10 SUMMAR! AND CONCLUSIONS

.A refined flexural analysis procedure was developed for
reinforced concrete sections incorporating steel fibers. The
analysis procedure was verified using test results reported
in the literature, and it was used for an analytical
parametric study on the effects of different design variables
on flexural strength and ductility of fibrous reinforced
concrete sections. Based on the results of this parametric
study simplified flexural analysis and design guidelines were
developed for reinforced concrete elements incorporating
steel fibers.

From the analytical studies conducted on the flexural
behavior of fibrous reinforced concrete sections it was
concluded that:

1 - Steel fiber reinforcement results in improvements in
the flexural strength and especially the ductility of
reinforced concrete sections.

2 - High strength steel and concrete can be efficiently
used in flexural elements incorporating steel fibers with no
significant adverse effects on the ductile behavior.

3 - Steel fiber' reinforcement reduces the need for

54

[compression steel to improve the ductile behavior in flexural
elements. High tension steel ratios can be also used in
flexural reinforced concrete elements incorporating' steel
fibers, resulting in higher flexural strengths, with the
ductility being maintained at desirable levels.

The developed flexural analysis / design procedure is
simple and conforms to the design guidelines of the American
Concrete Institute (ACI 318 - 83). It accounts for the
improvements in flexural strength resulting from steel fiber
reinforcement (using a flexural stress distribution at
failure which considers the improvements in tensile and
compressive properties of the fibrous concrete) .

The developed flexural design guideline also accounts
for the positive effects of fiber reinforcement .on the
ductility of flexural behavior by increasing the maximum
limit on tension steel area in the presence of fibers.
Results of the proposed flexural analysis / design procedure

compare reasonably well with test results.

SHEAR ANALYSIS AND DESIGN OF REINFORCED CONCRETE ELEMENTS
INCORPORATING STEEL FIBER

3.1 INTRODUCTION

Conventional reinforced concrete elements generally fail
in a brittle manner under the action of shear forces. In
order to safely resist shear, reinforced concrete elements
should be provided with sufficient amounts of transverse
shear reinforcement, a requirement that occasionally leads
to undesirable congestion of steel and time consuming
construction efforts. Steel fiber reinforcement is an
alternative method of improving the strength and ductility of
reinforced concrete elements subjected to shear forces.
There are potentials for important gains in construction
speed and material costs through partial substitution of
conventional shear reinforcing bars with steel fibers.

This chapter summarizes the results of a study on the
effects of steel fiber on the shear behavior of reinforced
concrete elements. A new shear design equation is developed
and it is used in an analytical parametric study on the shear
strength of fibrous reinforced concrete elements as

influenced by different design variables.

55

56

3.2 STEEL FIBER EFFECTS ON SHEAR STRENGTH OF REINFORCED
CONCRETE ELEMENTS

A review of the published data is indicative of the
effectiveness of steel fibers as shear reinforcement for
improving the strength and ductility of reinforced concrete
elements subjected to shear forces.

Steel fibers improve the shear behavior of elements
mainly by increasing the cracking and tensile strength of
concrete [21 and 22] . The fact that steel‘ fibers provide
tensile resistance in all directions is specially important
in restraining shear cracks which develop with different
orientations. Following the formation of cracks, steel
fibers bridge the cracks and provide concrete with post-
cracking tensile resistance.

Reference [23] suggests that conventional shear
reinforcement (stirrups) can be substituted with steel
fibers. The use of steel fibers as shear reinforcement in
some reinforced concrete beams has been observed to double
the shear strength and substantially improve the ductility
under shear. Reference [27] suggests that shear strength of
reinforced concrete elements can be increased almost linearly
with the increase in fiber volume fraction.

Tests show that stirrups together with fiber
reinforcement at volume fractions as low as 0.44 percent can
be used effectively to provide sufficient shear capacity for
beams to fail in a ductile flexural mode [24, 23]. Reference

[25] reports that in reinforced concrete beams without

57

stirrups a 45 percent increase in shear capacity can be
achieved through reinforcement with 1.6 volume percentage of
straight fibers (but the fiber reinforced concrete beam still
failed in shear). The same Reference also reports that
reinforced concrete beams without stirrups but with 1. 1
volume percentage of steel fibers with deformed ends had an
increase in shear capacity of 45 to 67 percent when compared
with beams without fibers: and the beams failed in flexure.

The effectiveness of steel fibers in improving the shear
performance of concrete elements seems to be dependant on the
shear span-to —depth ratio (a / d) of the element. It has
been shown in Reference [28] that the shear capacity of
reinforced concrete beams without stirrups can be increased
by 130 percent through reinforcement with 1.0 volume
percentage of fibers in elements with an a / d ratio of 1.5.
At an a / d ratio of 3.0, there is about 108 percent
increase in shear strength of otherwise comparable
reinforcement concrete beams resulting from reinforcement
with 1.0 volume percentage of steel fibers [27].

The studies reviewed so far strongly indicate a fraction
of the web reinforcement can be replaced by steel fibers,
thus eliminating the need for large amounts of stirrups.

In spite of all the available evidence reported
regarding the effectiveness of steel fibers as shear
reinforcement, there are insufficient theories and equations
available for predicting the shear capacity of reinforced

concrete elements incorporating steel fibers.

58

3.3 SHEAR STRENGTH EQUATIONS

External shear forces are resisted in reinforced
concrete elements by several internal mechanisms : shear
strength of the uncracked concrete in compression, aggregate
interlock, dowel action of flexural steel and transverse
reinforcement. Steel fibers enhance the performance of these
internal mechanisms. It is however, difficult to quantify
the contribution of fibers to each mechanism based on the
available test results [27].

A relatively simple semi-empirical shear design equation
has been developed for fibrous reinforced concrete elements
by simply modifying the design equations commonly used for
conventionally reinforced concrete beams [31 and 32]. This
simple approach considers that the factors influencing shear
strength and formation of inclined cracks are so numerous and
complex that a definitive conclusion regarding the correct
mechanism of inclined cracking and shear resistance in the
presence of fibers is difficult to establish [29]. The
modification of ACI shear design equation in this approach
basically involves substituting (f’c,)°°5 in the ACI equation
for the concrete contribution to shear strength with .133fr

( fr' Modulus of rupture of steel fibers reinforced concrete)

Vc - 2(f'c)°°5bw d for conventional reinforced concrete (3-1)

Vc - .266fr bw d for fibrous reinforced concrete (3-2)

59

Or in the alternative more complex but also more accurate

equation:

vc - ( 1.9 f'c + 2500 pw yud ) bwd (3-3)

( for conventional reinforced concrete )

vc - ( 0.253 it + 2500 pw yhd ) bwd (3-4)
( for fibrous reinforced concrete )
Where :

Vc - shear strength provided by concrete

d - effective depth of the beam

bw - web width of a rectangular beam

pw ' As / bw d

V“ - factored shear force at section

Mu - factored moment at section

fr - modulus of rupture (7.5 (f'a)°°5 psi for plain

concrete )

f’c - compressive strength of concrete

3.3.1 THE PROPOSED METHOD

The following method, which is a refined version of the
one introduced in Reference [22], was developed in this study
based on a simplified physical model of shear failure shown
in Figure 3.1, noting that the effects of transverse

reinforcement will be considered later.

60
.1

f
[5.

Figure 3.1 Simplified Model of Shear Failure in Fibrous
Reinforced Concrete.

 

 

 

 

T

k.!ct

A shear strength equation can be derived based on the
physical model of Figure 3.1, using the following
assumptions:

- The shear diagonal crack extends through the beam depth.
- The main diagonal crack governing the shear failure of

the element initiates at the load point A.

- Fibers provide the diagonal crack with post-cracking
tensile resistance.
- The direction of tensile stresses in fibrous concrete is

assumed to be perpendicular to the crack plane.

Using the above assumptions, an expression for ultimate
shear strength can be derived as describe below:

Take the moment of applied forces in Figure 3.1 about point A

61

Mu-Vu as tys Asd+k1 fctbd2+vdd (3-5)
Where :
fy, - steel yield strength
k1 - constant depending on the fraction of fibrous
concrete tensile strength acting across the

fracture plane.

vd - p b2 (V1 fcu + V2) (3-6)

which represents the contribution of dowel bars to shear
strength in fiber reinforced concrete [30]
Where :
Va and V2 - constant coefficients: and
fcu - the ultimate flexural strength of steel fiber
reinforced concrete
The ultimate shear force can thus be derived using

equation (2-1):
Vu'tysAs g + klfct 2 bd + 90/1:cu +112»? 2 (3-7)

Regression analysis was carried out using the data
reported in Reference [22] to find the unknown coefficients
(V1 and V2 ) in the equation. using these empirical values
of the coefficients, the following expression was obtained
for predicting the shear strength of steel fiber reinforced

concrete, accounting for the effects of transverse steel._

vu - v,3 + vs (3-3)

62

Where:
V a 0.77 + 0.9 - 9.6 f bd 3-9
G ( ( plpgggt) r ( )
Vs '.Av fysd (3-10)
s
Where:

Av - total leg area of stirrups:
s - stirrups spacing:
fr - modulus of rupture of steel fiber reinforced

concrete: and

f -67.62 1
Ct 105 d:

3.4 COMPARISON IITH TEST RESULTS

Table 3.1 and Figures 3.2 through 3.4 present
comparisons between test results and the predictions of the
modified ACI (equation 3-2) and the proposed (equation 3-8)
expressions. The test results were selected from a number of
references. Obviously, the proposed equation compares
favorably with test results. While the average ratio of ACI-
based predictions to test results is 1.85, the average ratio

of the proposed equation predictions to test results is 1.09.

 

63

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TEST NUMBER

AOOOOE

 

 

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I

 

 

 

 
 

SHEAR Klps

Figure 3.2 Experimental vs Analytical Shear Strength Values
(Reference 2).

METHOD pnopossol

TEST NUMBER

ACI 000E

 

 

-
C2] 7551'

Figure 3.3 Experimental vs Analytical Shear Strength Values
(Reference 27).

64

 

 

   

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Figure 3.4 Experimental vs Analytical Shear Strength Values

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(.02(.02(3.40 (32 (7.5(21.7(13.5(1.30
(.02(.02(3.40 (32 |3.5(31.9(14.3(2.20
(.02(.02(3.40 (32 |6.8|32.6(14.4|2.10
(.02(.02(3.40 (32 (3.3(1s.5(13.1(1.17
(.02|.02(6.4o (32 (7.7(13.0(12.3(1.03

(7.44 (0.5 (1.13(.02(.04(4.30 (51 (4.1(30.5(24.5|1.23
(.02|.04(4.30 (51 (4.7(33.1(3o.2(1.19

(7.44 (0.5 (2
(7.44 (1
(7.44 (1 (z
(7.44 (1

(7.44 (1 (2

(1.13(.02|.04(4.30 (51
(.02|.04(4.30 (51
(1.13(.02(.04(4.30 (s1
(.02|.04(4.30 (51
(7.44 (1.5 (1.13(.02(.04(4.30 (51

(4.2(40.5(27.2(1.50
(4.7(50.7(29.9(1.70
(4.7(29.4(2s.7(1.14
(4.7(33.3|23.3|1.29
(4.3(4s.0(27.3(1.32

(nernoo
(rv (Ksil(1) (<2) |t1)/(2)|

(3)

3.1 EXPERIMENTAL VS. THEORETICAL SHEAR STRENGTHS OF
FIBROUS CONCRETE AND FIBROUS REINFORCED CONCRETE

(33100

((1)/(3)|
(11.30(o.2 (1.13(.02|.51(14.50(33 (3.5(27.3(13.3(1.30
(11.30(0.3 (1.13(.02(.s1(14.50(33 (3.3(27.3(13.2(1.30
(11.30(o.s (1.13(.02(.51(14.50|33 (7.3(27.1(13.7|1.30
(11.30(0.2 |1.13(.02(.s1(14.50(33 (7.3(37.4(27.3(1.37
(11.30(0.3 (1.13(.02(.51(14.50|3s (3.3(39.3|25.7(1.54

AVERAGE

I
I
|
I
1.6 |
|
I
I
|
l

.50(1.2 (.02(
.so(1.z (.02(
.50(1.2 (.02(
.50(1.2 (.02(
.50(1.2 (.02(
.75(1.2 (.02(
.75(1.2 (.02(
.75(1.2 (.02(
.75(1.2 (.02(
.75(1.2 (.02(
.75(1.2 (.02(

(7.3(19.5|4.9 (3.90
(7.3(12.3(4.3 (2.50
(7.3(11.5(4.3 (2.40
(7.3(9.45(4.3 (1.30
(7.2(23.4(s.79(4.00
(7.2(20.0(s.93(3.30
(7.2(19.3(s.93(3.20
(7.2(21.3(s.79(3.30
7.2(15.2(s.43(2.70
(7.0(3.s9|4.71(1.30
(7.0(11.3(4.75(2.40
(7.0(3.27(4.71(3.0o
(7.0(14.3(4.72(3.20
(7.0(19.2(s.79(3.30
(7.0(11.3(s.79(2.o1

I |"0 l
( | | (8 in (H in
(F ( 1 ( (8.25 (12
(l | 2 | (8.25 (12
(8 ( 3 (2 (8.25 (12
(R | 4 ( (8.25 (12
(0 ( 5 | (8.25 (12
(S ( 1 ( (7 (10
l I 2 I l7 l10
(R ( 3 ( (7 (10
IE I 4 I l7 l10
(l ( S (27 (7 (10
I" I 6 | l7 I10
(F ( 7 ( (7 (10
(0 ( 8 ( (7 (10
IR l 9 | l7 l10
(C (10 ( (7 (10
ID I 1 I I‘ I8
I I 2 I l‘ l8
lc l 3 l I4 I8
(O ( 4 (43 (4 (8
I" I 5 l l4 I8
It I 6 l l4 l8
IR I 7 l I4 I8
IF I 1 I I4 l8
I! l 2 l I4 l8
I3 l 3 l l6 (8
lR I 4 I l‘ I8
l0 l 5 I l4 l8
lU l 6 l l9 I8
ls l 7 I l4 l8
( | 8 (22 (4 (8
Ic l 9 I l4 I8
(0 (10 | (4 (8
IN I11 l I4 I8
(C (12 ( (4 (8
(R (13 | (4 (8
IE l14 I l‘ I8
(T (15 | (4 (8
Note:
1 in a 25.4 mm

1 kip 8 4448.2 N
1 ksi a 6.895 HPa

 

66

3.5 RARANETRIC INVESTIGATION OF STEEL FIBER EFFECTS ON THE
SHEAR STRENGTH OF REINFORCED CONCRETE ELEMENTS

The proposed shear strength equation was used to perform
a parametric study on the effects of fiber reinforcement
on shear strength of reinforced concrete elements with
different material and geometric properties. The basic
"standard" section used in the parametric study has the same
cross-sectional properties as introduced in Figure 1.8, with
No 3 stirrups placed at a spacing of 4 inches (102 mm).

Figure 3.5 presents the geometric and material
properties of the "standard” section. Further discussions on
the trends observed in the fiber reinforcement effects on
shear strength of reinforced concrete elements with different
characteristics are presented below.

Figures 3.5 though, 3.11. present. the results of’ the
parametric study. Fiber reinforcement is observed in these
figures to increase the shear strength more than it increases

the flexural strength.

GD

 

67

A, - 2.37 1n2 1, - 1 in
Asp - .24 in2 d, - .013 in
fy - 60 ksi Vf - 1.5 8
1'6 - 4 ksi, Av - 0.04 in2

Type of Fiber - Straight

 

1‘ \ - 0|: Id. (

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.5 The Basic ("Standard") Reinforced Concrete
Element. .

3.5.1 EFFECTS OF THE LONGITUDINAL AND TRANSVERSE
REINFORCEHENT

Variations in the longitudinal and transverse steel
ratios and yield strengths, as shown in Figures 3.6, 3.7 and
3.8, respectively, have small effects on the improvements in
the absolute values of shear strength resulting from steel

fiber reinforcement.

68

 

200
M0790Ifl2

§

00"! "fi-ODODO ‘OC’U
. '\ I

M - 3.18 (1'12

3

Al - 1571112

A. - 791112

8
j
I
I

o
.a
."
.-
.0

0.5 1 1.5 2 2.5 3 3.5
Float comet" montage .

 

 

 

Figure 3.6 Effects of Fiber vo1ume Fraction on Shear Strength
for Beams with Different Transverse Steel Areas.

 

 

 

 

 

7O
S-Zlfl
50" /\ 3‘4"!
./\
8 s-om
2
: Q", 5.3m
C
I
D
I
c I
I r
I
7 3(1-
I
I
D
I 20..
10”
O 1 L 1 A I 1 .I
0.5 1 1.5 2 25 3 3.5

“DOV COflIOflI DO'COOIIOO

Figure 3.7 Effects of Fiber Vo1ume Fraction on Shear Strength
for Beams with Different Stirrup Spacings.

69

 

 

 

 

100
60
a X 131-10014“.
O
I
r 1110801131.
c 60
C
D -
': ,~ Iy-SOKII.
I
y ‘0 ..." ..o’\
7 .- (y-Aom
D
3 fl.
20"
o 1 l l J l l
. 0.6 1 1.5 2 2.5 3 3.5

FIDO content percentage .

Figure 3.8 Effects of Fiber Volume Fraction on Shear Strength
for Beams with Different Transverse Steel Yield

Strengths.

70

3.5.2 EFFECT OF : RATIO

Among the parameters studied the shear span-to-depth
(a / d) ratio ( which describes the ratio of bending moment
to shear force at a section) seems to be the one with
relatively significant effects on the shear strength of plain
'and fibrous elements. Figure 3.9 shows the effects of fiber
reinforcement on shear strength of elements with different
a / d ratios. The trends in fiber reinforcement effects on
shear strength (as far as the absolute values of improvements
in strength are concerned) seem to be largely independent on

the shear span-to-depth (a / d) ratio.

 

60

Ila-1.5.
810.2.5

50'

old-35.

8/005.

\‘\. \ \ J

 

20“'

' .O—X "G-ODUIO “.0308

10*

 

 

 

O .L l 1 l l 1
05 1 15 2 25 3 35
F . oer con 13m percentage

Figure 3.9 Effects of Fiber Volume Fraction on Shear Strength
foti Beams with Different Span-to-Depth ( a / d )
Ra os.

71

3.5.3 EFFECTS OF FIBER REINFORCEHENT PROPERTIES

Figures 3.10, 3.11 and 3.12 show the effects of volume
fraction of fibers with different lengths, diameters and
mechanical deformations on the shear strength of the
"standard" reinforced concrete element. The increase in
fiber volume fraction is observed to improve the shear
strength rather significantly.
Longer fibers with smaller cross section dimensions, and
those with mechanical deformations, are observed to give
better theoretical results (if the practical problems with
workability and fiber dispersability can be overcome).
Typical improvements of the order of 50 8 in shear strength
of reinforced concrete elements are observed in Figure 3.10 -
3.12 to result from the addition of 2 % volume fraction of
steel fibers. .

72

 

80

11-2111.

11-15111.

  
 

//~_——_.n-1m.

40

.‘-I’ “IO—0.0.0 “.03“

. 20(-

 

 

 

o L l I L L

08 1 18 2 20 3 38
Hananumnuunuun

Figure 3.10 Effects of Fiber Volume Fraction on Shear
Strength for Beams with Different Fiber Lengths.

 

70

«0.01111.
31.9131...

I'f/ba.‘M'n'

 

' ..-? ‘00-'05...“ “.03“

20’

10’

 

 

 

o L 1 l l 1 1
05 1 15 2 25 3 35
Fleet 1:00.111 weemege .

Figure 3.11 Effects of Fiber Volume Fraction on Shear
Strength for Beams with Different Fiber Diameters

73

 

 

30('

 

uo—r xn-ODOIO ‘00:“

20(

 

1 J 1 l L J
0.5 1 1.5 2 2.5 3 3.5
9 10.! comem genomes.

Figure 3.12 Effects of Fiber Volume Fraction on Shear
Strength for Beams with Different Fiber Types.

74

3 . 6 SUIDIAR! AND CONCLUSION

Steel fiber reinforcement was found to be effective
in improving' the strength and ductility of“ reinforced
concrete elements subjected to shear forces. Best results
seem to be achieved when a combination of steel fibers and
transverse reinforcement is used to provide shear resistance.

A shear strength equation was developed in this study
for fibrous reinforced concrete elements based on the
physics of shear failure in the presence of steel fibers.
The proposed equation was shown to predict test results far
better than the previous equations (which are simply modified
versions of the conventional reinforced concrete equations).

The proposed shear strength equation was used in an
analytical parametric study on_ the fiber effects of
reinforcement on shear strength of reinforced concrete
elements. The results indicated that improvements of the
order of 50 It in shear strength can be expected in typical
reinforced concrete elements as a result of reinforcement
with 2 3 volume fraction of steel fibers. Longer fibers with
smaller cross-sectional dimensions and especially those with
mechanical deformations were theoretically more effective in
increasing the shear strength of reinforced concrete elements
(ifthe practical problems with fresh mix workability and
fiber dispersability can be overcome). The conventional
reinforcement properties on The shear span-to-depth ratio of
reinforced concrete elements did not significantly influence

the absolute value of improvement in shear resistance

75

resulting from fiber reinforcement.

CHAPTER 4
DESIGN OF FIEROUS REINFORCED CONCRETE ELEHENTS IN TORSION

4.1 INTRODUCTION

The effectiveness of steel fibers in enhancing the
torsional strength of fibrous reinforced concrete elements
is discussed in this chapter, and a practical method for
predicting the torsional strength of fibrous reinforced
concrete sections is introduced. The predictions of this
method are compared with a variety of experimental results,
and also with the predictions of other analytical methods.
The developed analytical method is also used for a numerical
parametric study on the torsional strength of fibrous
reinforced concrete elements. The principle variables
in this parametric study are the fiber reinforcement
properties and the longitudinal and transverse reinforcement

ratios.

4 .2 EACEGROUND

From the experimental data reported on the torsional
behavior of fibrous reinforced concrete elements it may be
concluded that elements with both steel fibers and
conventional torsional reinforcement perform better in
torsion than elements containing only the conventional
reinforcement [32, 33, 34] . The presence of fibers has
been observed to increase the ductility, rotational

capacity, stiffnesses, and damage tolerance of concrete

76

77

elements under the action of torsional forces [31]. The type
and aspect ratio of steel fibers are found to be important
factors deciding their effectiveness in torsional elements
[31] . The presence of both steel fibers and conventional
transverse reinforcement has also been reported to be much
more effective than steel fibers or transverse reinforcement
alone [20, 31, 32].

It has been proposed in Reference [35] that. the
torsional strength of fibrous reinforced concrete elements
can be adequately predicted using modified versions of the
American Concrete Institute code ( ACI 318-83 ) equations,
which have been originally developed for conventional
reinforced concrete elements.

The contribution of concrete to the ultimate torque in

elements without fiber reinforcement is given by [38] :

T - 6 (x2+10) y (11°)0-5 (4-1)
Where:

x and y are the smaller and larger dimensions of the

rectangular section respectively.

References [39, 40] suggest that the term (f’c)°°5
in Equation (4-1) can be expressed in terms of the modulus of
rupture ( fr ) or the splitting tensile strength ( fct ) of

concrete:
T - 1.56 ( x2+10 ) y (fr2)°°5 (4-2)
or T - 1.6 ( x2+10 ) y (fct2)°°5 (4-3)

These equations, according to References [39, 40] are

78

directly applicable to steel fiber reinforced concrete as far
as the values of fr or fct for the fibrous concrete are
available.

Reference [38] has suggested a modified version of the
above equations for predicting the ultimate torsional

strength of fibrous concrete:
T - xzy/3 (0.35 1,) (4-4)

An alternative equation for The ultimate torque of

fibrous concrete is given in Reference [34, 42]:
T - ny/a (0.71 fr) (4-5)

A comprehensive examination of the accuracy of these
alternative ultimate torsional strength equations is
presented below following a brief discussion of the approach
adopted in this study for torsional analysis and design of

fibrous reinforced concrete elements.

4.2.1 PROPOSED APPROACH

Resistance against the action of torsional forces in
fibrous reinforced concrete elements ( Tu ) is provided by
both the fiber reinforced concrete ( Tc ) and the

conventional reinforcement ( Ts ), as shown in Figure 4.1

‘79

x -ol111en3101101 - -. 0'

Comoreulonm

 

 

 

 

I
z ' 1,"’, . umlntenelonm

A1110 01 mm ,’
1’ [IA IO! 31... bending
I

I
o

 

 

 

 

 

 

 

a) Skew Bending of Rectangular Section b) Cross-Section

Figure 4.1 Mechanism of Resistance Against Torsion In Fibrous
Reinforced Concrete Elements.

Based on a comprehensive verification of the torsional
strength equations presented above using test results on
torsional behavior of fibrous reinforced concrete elements
reported in References [20, 31, 32, 33, 34], Equation (4-5)
was selected as a simple and sufficiently accurate
representation of fibrous reinforced concrete contributions
to the torsional strength of reinforced concrete elements.

The contribution of conventional reinforcement to
torsional strength of fibrous reinforced concrete elements is

suggested to be similar to this contribution in non-fibrous

80

concrete elements. The torsional strength of fibrous
reinforced concrete elements ( Tu ) can therefore be

calculated as follows:

Tst ' axlylAtty

Tu - total torsional strength
Tct - torsional strength provided by concrete
Tst = torsional strength provided by hoop steel
- (0.66 + 0.33 yl/xl ) 5 1.50
- smaller cross sectional dimension
y - larger cross sectional dimension
x1 - smaller dimension of the confined core of the
section subject to torsion
Y1 - smaller dimension of the confined core of the
section subject to torsion
- yield strength of steel
At :- cross sectional area of transverse reinforcement
resisting torsion
- spacing of transverse reinforcement
fr - modulus of ruptur of fiber reinforced concrete
- 490 vflf/df + 0.97fm11-vf)
Where:

Vf - fiber volume fraction

81

If - fiber length
df - fiber diameter

The minimum transverse reinforcement recommended is:

231; 2 52b":
‘11

The minimum longitudinal reinforcement recommended is:

A13[4002t;§-2At](x1+y1)/s
Y
Where:

b" - width of web
A1 - area of longitudinal reinforcement to resist

torsion

4.3 VERIFICATION OF TORSIONAL STRENGTH EQUATIONS

Torsional strength test results reported in References
[20, 31, 32, 33 and 34] were used to verify the torsional
strength expressions previously presented in Equations (4-1),
(4-2), (4-3), (4-4), (4-5). The proposed equation (4-6) is
also checked against test results on concrete specimens
reinforced with both fibers and conventional rebars. The
results of this comparison are shown in Tables 4.1 through
4.7, and in Figures 4.2 trough 4.8. A summary of the
comparison is presented in Table 4.8. As can be seen
Equation (4-6) (and also Equation (4-5), (which represents
the concrete contribution in Equation 4-6)) show good
comparisons with torsional strength test result fbr fibrous

concrete elements with and without conventional reinforcement

82

T8131. 4. 1 CONFARISON OF TORSIONAL STRENGTH FOR FIEROUS
CONCRETE SECTION. (TEST VS THEORY) .

( ( (Vf2(df (If (f'c(fr (ft (fctl Eq.1 ( Eq.2 ( Eq.3 ( Eq.4 ( Eq.5*( Test |

( ( | (In (in (ksilksi|ksi(ksi(kip.Inlkip.in(kip.Inlkip.in(k1p.In(kip.in ( (
(1 (0 (0 (0 (4.2(.38(.29(.48(53.40 (45.60 (56.60 (46.90 (39.20 (56.70
(2 (1.5(.02(1.18(4.7(.98(.48|.81(55.50 (84.70 (80.10 (119.00l99.40 (83.20
(3 (1.5(.02|1.18(4.7(.98(.48|.81(55.30 (84.70 (80.10 (119.00(99.40 (89.10
(4 (1 (.02(2 (4.3(.55|.42(.71(53.40 (57.80 (72.90 (67.00 (55.90 (65.70
(0 (0 (0 |4.2(.4 (.31|.51(53.40 (46.70 (59.10 (48.80 (40.70 (53.80
(6 (1 (.02(2 (4.3(.55(.42(.71(53.90 (57.80 (72.90 (67.00 (55.90 (65.70
(7 (2 (.02(1.18(4.9(1.1(.58(.97(56.30 (91.90 (90.40 (134.00(111.90(39.60
(8 (2 (.02(1.18(4.9(1.1(.58(.97(56.30 (91.90 (90.40 (134.00(111.90(89.60
(9 (1 (.02(1.13(4.3(.62(.42(.71(53.70 (62.80 (72.90 (76.10 (63.50 (33.70

—-—_a_—-—
.e
13
5
M

 

( AVERAGE THEORY / TEST (0.77 (0.91 (0.98 (1.18 (0.98 ( (
Note:
Section width 3 12 in 1 in a 25.4 as I kip.in a 0.113 KN.m
Section height: 6 in 1 ksi a 6.895 HPe * This study
120

100

80

"K2030"

60

3—- OO-X

40

 

 

 

 

 

 

 

 

 

 

 

 

1 2 3 4 5 6 7 5 9
TEST SECTION NUMBER

 

C3 603 - E05 - 1331
51:5 (T1113 study)

Figure 4.2 Torsional Strength Comparisons.

 

S3

TIDI. 8.2 CONPARISON OF TORSIONAL STRENGTH FOR FIEROUS
CONCRETE SECTION. (TEST VS THEORY) .

(Ref(Test(Fibe1' detail(Conc Prop ksi ( TQSIGIAL STRENGTH Kipedn ( (
(116 (116 ------------------------------------------------------------------------ FIG I
( ( (VfX(df (If (f'clfr (ft (fct( Eq.1 ( Eq.2 ( Eq.3 ( Eq.4 ( Eq.5'| Test (
( ( ( (in (in (ksi(ksi(ksi(ksi(kip.in|Itip.inlkip.inlkipdnlkipdnlkip.in | |
( (T1 (0 (016(1 (5.5(.44(.29( (58.50 (50.10 (41.06 (54.31 (45.36 (86.80 (
(20 (12 (0.5(016|1 (5.2(.51(.38( (57.30 (55.11 (48.70 (62.70 (52.40 (120.50 ( 4.4
( (T3 (1.5(016l1 (5.1(.64(.50( (58.40 (63.90 (58.40 (78.20 (65.32 (125.30 (

( AVERAGE TflEORY I TEST (0.52 (0.51 (0.45 (0.59 (0.49 ( (
Note:

Section width I 12.2 in

Section heights 5.98 in

1 in 8 25.4 11111

1 ksi - 6.895 1193

1 kip.in a 0.113 ”1.111

1 This study

 

140

120'-

chnO-t

80"

3- OO—X

40"

 

 

 

 

 

 

1 2 3
TEST SECTION NUMBER

D 602 - E05 - T33!

 

£05 (T1113 study)

Figure 4.3 Torsional Strength Comparisons.

 

84

leII 4.3 CONPARISON OF TORSIONAL STRENGTH FOR FIEROUS
CONCRETE SECTIONS (TEST VS THEORY) .

(Ref(Test(Fiber detaillConc Prop ksi ( TORSIOIAL STRENGTH Kips.in | (
(No (No ------------------------------------------------------------------------ FIG I
I ( (foldf (If (f'c(fr (ft (fctl Eq.1 ( Eq.2 ( Eq.3 I Eq.4 | Eq.5*( Test ( No (
( ( | (in (in (ksi(ksi(ksilksi(kip.inlkip.in(kip.In(kip.in(kip.in(kip.in (
( (1 (.75|016(1.20(5.7(.57(.50( (16.71 (16.65 (16.41 (15.39 (12.86 ( 13.45 (
(33 (2 (.75(016(1.20(5.7(.57(.50( (16.71 (16.65 (16.41 (15.39 (12.86 ( 13.23 ( 4.5
( (3 (.75(016(1.20(5.7(.57(.50( (16.71 (16.65 (16.41 (15.39 (12.86 ( 13.59 (

( AVERAGE THEORY/TEST (1.24 (1.23 (1.22 (1.14 (0.95 ( I

Note:
Section width 8 6 In
Section height: 4 in
1 in a 25.4 m
1 ksi a 6.895 1193
1 kip.in a 0.113 Ku.m
' This study

 

20

mCO‘DO—t

3- .O-X

 

 

 

 

 

 

 

TEST SECTION NUMBER

 

:3 E03 - Eqfi - Test

6115 (TnIe study)

Figure 4.4 Torsional Strength Comparisons.

85

TIDIO 4.4 CONPARISON OF TORSIONAL STRENGTH FOR FIEROUS
CONCRETE SECTIONS (TEST VS THEORY).

 

|Ref|Test(Fiber detail|Conc Prop ksi | TMSIONAL STRENGTH Kipe.in ( |
(No (No FIG |
( | (VfZIdf (If |f'c|fr (ft (fct( Eq.1 ( Eq.2 ( Eq.3 ( Eq.4*( Eq.5 ( Test ( No |
| ( | (in (In (ksi(l<si(|tsi(Itsi(kip.in|ItIp.In(Itip.In(Itip.in(|tip.in(|tip.in ( |

 

 

(91 (0 (0 (0 (5.9(.74( 30(.50(59.30 (70.50 (53.10 (90.60 (75.67 (77.70
(92 (.7 ( 02(1.13(5.3(.73(.40(.67(59.50 (69.30 (70.60 (39.40 (74.67 (33.70
(93 (0 (.02(2 |6.3(.72(.42(.71(61.20 (69.20 (73.40 (33.10 (73.53 (77.70

 

 

I | |
I | I
| I I
(32 (P4 (1.5|.02|1.18(5.2(.93(.40|.67(57.40 (81.90 (70.60 (113.0 (94.38 (76.20 ( 4.6 (
| (PS (2 (.02(1.18(4.2(.75|.37(.61(53.40 (70.90 (66.30 (91.60 (76.50 (65.70 ( (
( (P6 (2 (.02(2 (6.9(1.2(.41|.63(63.10 (97.20 (71.10 (147.00(122.80(89.60 ( (
| (P7 |1.5(.02(1.18(5.8|.74(.37(.61(59.50 (70.30 (66.30 (90.20 (75.34 (92.60 | |
( (P8 (2 (.02(2 (6.6(1.2(.42|.70(62.10 (96.40 (72.70 |145.00(121.12(89.60 ( (
I

( AVERAGE THEORY / TEST (0.72 (0.95 (0.84 (1.30 (1.09 ( | (
Note:

Section width x 12 in 1 in I 25.4 I. 1 Itip.in a 0.113 KN.111

Section height: 6 in 1 ksi 8 6.895 In ' This study

 

mCODO-i

:— (so—X

 

 

TEST SECTION NUMBER

 

[:I Eqa - E05 - Test

E05 (ThIe study)

Figure 4.5 Torsional Strength Comparisons.

86

T8331. 4 . 5 CONPARISON OF TORSIONAL STRENGTH FOR FIEROUS
CONCRETE SECTIONS (TEST VS THEORY).

(ReflTestlFiber deteillConc Prop ksi ( TGRSIGNAL STRENGTH K1ps.in ( (
(No (No ------------------------------------------------------------------------ FIG (
( ( (Vf%(df (If (f'c(fr (ft (fct( Eq.1 ( Eq.2 ( Eq.3 ( Eq.4 ( Eq.5*( Test ( No (
( ( ( (in (in (ksi(ksilksi(ksi(kip.inlkip.inlkip.in(kip.in(kip.in(kip.in (
( (A1 (.75(016(1.20|4.9(.64|.31( (10.62 (12.07 (7.95 (11.65 (9.57 (3.54 (
( (12 (.75(016|1.20(4.9(.64(.31( (10.62 (12.07 (7.95 (11.65 (9.57 (3.30 (
( (A3 (.75(016(1.20(4.9(.64(.31( (10.62 (12.07 (7.95 (11.65 (9.57 (8.97 (
( (31 (.75(016(1.20(4.9(.64(.31( (15.96 (13.11 (11.90 (17.46 (14.53 (14.32 (
(34 (32 (.75(016(1.20(4.9(.64|.31( (15.96 (13.11 (11.90 (17.46 (14.53 (14.63 ( 4.7
| (83 (.75(016(1.20(4.9(.64(.31( (15.96 (18.11 (11.90 (17.46 (14.58 (14.42 (
( (c1 (.75(016(1.20(4.9(.64(.31( (21.23 (24.10 (15.90 (23.23 (19.44 (13.73 (
( (CZ (.75(016(1.20(4.9(.64(.31( (21.28 (24.10 (15.90 (23.28 (19.44 (18.22 (
( (C3 (.75(016(1.20(4.9(.64(.31( (21.28 (24.10 (15.90 (23.28 (19.44 (18.55 (

 

| AVERAGE THEORY / TEST (1.14 (1.29 (0.85 (1.25 (1.04 ( |
Note:
Section width x 4 in .
Section height: 4 In for (A1, A2 and A3) 1 in I 25.4 ll
8 6 in for (81, 82 end 83) 1 ksi I 6.895 HPe
I 8 in for (C1, C2 and C3) 1 kip.in 8 0.113 KN.-
* This study
25
20"
T
3
13»
2
'f
2 10*
1
11
OH

 

 

 

 

 

 

 

 

 

 

 

 

 

1 2 3 4 3 8 7 8 9
rest SECTION NUMBER

(C3813 -E05 -Teet'

5% (TN. 81001)

 

Figure 4.6 Torsional Strength Comparisons.

87

lel. 4.6 CONPARISON OF TORSIONAL STRENGTH FOR FIEROUS
CONCRETE SECTIONS (TEST VS THEORY).

|Ref|Test|Fiber deteil(Conc Prop ksi (Hoop deteillTORS STRENGTH ( (
(No (No -------------------------------------------------------- FIG I
( ( (foldf (If (t'c(fr (ft (fct(Astlfy ( S ( Eq.6‘( Test (No (
( ( ( (in (In (ksi(ksilksilksilinZIRsilin (kip.in(kip.1n ( (

( (1 (0 (0 (0 (4.2(.38(.29(.48(.11|46 (7 (91.40 (33.20 ( (
( (2 (1.5(.02(1.18(4.7(.98(.48(.81(.11|46 (7 |151.60(149.00 ( (
(31 (3 (1 (.02(2 (4.3(.55(.42|.71(.11(46 (7 (129.00(179.00 (4.3 (
( (4 (2 (.02(1.18(4.9(1.1(.58|.97(.11(46 (7 (164.10|125.00 ( (
( (5 (1 (.02(1.18(4.3(.62(.42(.71(.11(46(3.5(168.00|146.00 ( (
................................................................. I
( AVERAGE THEORY I TEST (1.14 ( ( (
NOIC:

Section width 8 12 in
Section heights 6 in
1 in 8 25.4 mm

1 ksi a 6.895 N93
1 kip.In a 0.113 KN.m
* This study

 

200

150(

 

 

 

 

 

 

 

 

 

 

 

 

1 2 3 4
TEST SECTION NUMBER

1:] 563 - Eq5 - Test

 

E05 (True Study)

Figure 4.7 Torsional Strength Comparisons.

88

Table 4.7 CONPARISON OF TORSIONAL STRENGTH FOR FIEROUS
CONCRETE SECTIONS (TEST VS THEORY).

 

(ReflTest(Fiber detail|Conc Prop ksi (Hoop deteillTORS STRENGTH |
(No (No FIG I
| | (VfX(df (If (f'clfr (ft |fct(Ast(fy ( S ( Eq.6'( Test (No |
( | ( (in (in (ksi(ksi|ksi|ksi(in2(ksi|in (k1p.In(kip.in |

 

 

( (11 (0 (016(1 (5.5(.44(.29( (.04(66 (4 (32.34 (36.30

 

 

| I

(20 (T2 (0.5(016|1 |5.2(.51|.38( (.04(66 (4 (89.38 (120.50 (3-7 (

| (13 (1.5(016(1 (5.1(.64|.50| (.04(66 (4 (65.32 (125.30 ( (

I

| AVERAGE THEORY / TEST (0.92 | | (
Note:

Section width = 12.2 In
Section height= 5.98 in
1 in = 25.4 mm

 

1 ksi = 6.895 MP3

1 kip.in = 0.113 KN.m

' This study

140

§
0
U
E
K
1
D
8
i
n

 

 

 

 

2
TEST SECTION NUMBER

 

I: E03 - Eq5 - Test

605 (This study)

Figure 4.8 Torsional Strength Comparisons.

89

TSDIO 4.8 AVERAGE OF THEORETICAL I EXPERIMENTAL TORSIONAL
STRENGTH RATIO.

(E0 (10 (31c (10 (arc (1c (arc (1c (arc (1c (arc (10 (arc (

(Eq 1 (0.72( ( ( (1.24( (1.14( (0.77( (0.97( (

(Eq 2 (0.95( ( ( (1.23( (1.29( (0.91( (1.09( (

(Eq 3 (0.34( ( ( (1.22( (0.35( (0.93( (0.97( (

(Eq 4 (1.30( ( ( (1.14( (1.25( (1.13( (1.22( (

(Eq 5**(1.09( ( ( (0.95( (1.04( (0.93( (1.02( (

1' FC 8 Fibrous Concrete

9 FRC 1- Reinforced Fibrous Concrete

'1' Eq 5 used in this study for Fibrous Concrete

.. Eq 6 proposed in this study for Reinforced Fibrous Concrete

90

4.4 RARANETRIC STUDIES

The developed equation for the prediction of torsional
strength of fibrous reinforced concrete elements ( Equation
(4-6)) was used to investigate the effects of fiber
reinforcement properties and conventional reinforcement on

torsional strength of elements, as described below.

4.4.1 EFFECTS OF FIBER REINFORCENENT PROPERTIES

Figures 4.10 and 4.11 show the effects of the volume
fraction of straight-round steel fibers with different
lengths and diameters on the torsional strength of a
"Standard" reinforced fibrous element shown in Figure 4.9.
Maj or improvements in torsional strength, of the order of
100 t , are observed to result from reinforcement by 2 15
volume fraction of steel fibers. Longer fibers with‘ smaller
diameters (as far as dispersed satisfactorily in concrete
matrix) seem to be more effective in increasing the torsional

strength of fibrous reinforced concrete elements.

91

A, - 2.37 in? 1, - 1 in x - 10 in
A,p - .24 in2 d, - .013 in x1 - 3 in

t, - 60 ksi v, - 1.5 t Y - 20 in
:16 - 4 ksi At - 0.04 in? 21 - 13 in

Type of Fiber - Straight

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I l I 11 5'4
I \1
I ‘1,
s I ‘ ..
II I ’ ‘ c I
y 11 11 1 ‘ .‘ 1 ‘ ’.I . ‘
a I
'I \‘ I 4‘ I I I \I
|
- ' I ' ‘ \ ‘ ’ I \ \ I
I. ’ " I .
:1 r
H

Figure 4.10 The Basic ("Standard") Reinforced Concrete
Element.

92

 

1200

 

I II 210
O
1
I
- I- . .
a 18111
1‘!
I
I I
I
' 4 11-1111
V I
I
h
h
K
I
D
8
I
11 2C?“-

 

 

 

O.8 1 1.8 2 2.5 3 3.5
F1331 content percenteoe

Figure 4.10 Effects of Fiber Content (Vf) on Torsional
Strength at Different Fiber Lengths.

 

1000

OI I .01 1n

at-O‘Bin.

  
 

1-.O In
300 .-—° 2

400

3“ ’0’! a—oao'w-o. ~.30—.*O-§

200 "

 

 

J

 

o l 1 1 l 1
0.5 1 1.8 2 2.8 3 3 5
FIDO 6M1." ”COIN”.

Figure 4.11 Effects of Fiber Content (Vf) on Torsional
Strength at Different Fiber Diameters.

93

4.4.2 EFFECTS 01' m comma-tom 2318703681138!

Figure 4.12, 4.13, and 4.14 show the effects of
transverse reinforcement leg area, yield strength and
spacing on the relationship between torsional strength of the
"Standard" element and fiber volume fraction. The torsional
strength is observed to increase with the increase in leg
area and yield strength, and decrease in spacing of the
transverse steel. It should be noted that test results
indicate that steel fibers are most effective in improving
torsional performance when they are used together with a

small amount of transverse reinforcement.

 

1200

I?

-"L_u-o.nm

 

GOG-v".

200 *-

 

 

 

0.5 1 1.5 2 2.5 3 3. 5
F foo: content percentage

Figure 4.12 Effects of Fiber Content (Vf) on Torsional
Strength at Different Transverse Steel Leg Areas

94

 

 

amafi

3- ‘0‘? 3‘30“”. -.30-.'~O-4

 

 

 

o J l L l— A

0.5 1 1.6 2 2.5 3 3.5
Floor COMO!" Dementia.

Figure 4.13 Effects of Fiber Content (Vf ) on Torsional
Strength at Different Transverse Steel Yield

 

 

 

 

 

 

 

Strengths.
8L0

(’I’,a-a4u
/___ O-Ih

T

? 8-7h

I

I

O

"I

l

I

l

I

I

O

n

h

k

I

D

. 2mi-

i

n

i
o 1 1 1 1 ‘ ‘ n
0.5 1 15 2 25 3 J5

F'DO' content Dementia.

Figure 4.14 Effects of Fiber Content (Vf ) on Torsional
Strength at Different Transverse Steel Spacings.

95

4.5 BUIHIR! AND CONCLUSION

A torsional strength equation was developed for
predicting the torsional strength of fibrous reinforced
concrete elements. The proposed equation was verified using
torsional test results for fibrous concrete and fibrous
reinforced concrete elements with wide ranges of material
properties.

The proposed equation for predicting the torsional
strength of fibrous elements, after being verified, was used
in an analytical parametric study on the effects of fiber
reinforcement and conventional steel properties on the
torsional strength of reinforced concrete elements. The
results showed that the torsional strength of reinforced
concrete elements increase almost linearly with increasing
fiber volume fraction. In the presence of a typical 2 %
volume fraction of steel fibers with an aspect ratio of 70,
the torsional strength of reinforced concrete elements
increases by about 100 % over that of non-fibrous elements.
The results also showed that the longer fibers with smaller
diameters, as far as they are uniformly dispersed in the
concrete matrix, are more effective in increasing torsional
strength.

Conventional reinforcement properties seem to have
relatively small effects on the effectiveness of fibers in
increasing the torsional strength of reinforced concrete
elements. Test results indicate that best results can be

achieved when fibers are used in combination with relatively

96

small amounts of conventional reinforcement.

97

CHAPTER 5
BUNNLN! IND CONCLUSIONS

5.1 GENERAL

Steel fiber reinforced concrete has found growing
applications in secondary structures. The material is
currently under serious consideration for use in primary
load-bearing reinforced concrete elements. Such applications
of steel fiber reinforced concrete can be encouraged through
the development of structural design guidelines. This
investigation has been concerned with developing structural
analysis and design techniques for fibrous reinforced
concrete elements. Due consideration has been given to the
improvements in the flexural , shear and torsional strength,
ductility and cracking characteristics of reinforced concrete
elements resulting from steel fiber reinforcement. The
proposed equations and. guidelines provide designers with
practical tools for optimizing the structural design of
reinforced concrete elements through the use of steel fibers.

It is also worth mentioning that much is left to be
understood regarding structural performance characteristics
of fibrous reinforced concrete elements. The equations and
procedures presented in this study can be the basis for
further developments as more experimental data become
available in this area.

This investigation on the development of structural

98

procedures for fibrous reinforced concrete elements has been
performed in three phases, which were concerned with
flexural, shear and torsional elements. A summary of the
activities conducted in each phase of the study and the

corresponding outcomes and conclusions are given below.

5.2 FLIXURAL NLNNBNTB

A refined flexural analysis procedure was developed for
reinforced concrete sections incorporating steel fibers. The
analysis procedure was verified using test results reported
in the literature, and it was used for an analytical
parametric study on the effects of different design variables
on flexural strength and ductility of fibrous reinforced
concrete sections. Based on the results of this parametric
study, a simplified flexural analysis and design guidelines
were developed for reinforced concrete elements incorporating
steel fibers.

From the analytical studies conducted on the flexural
behavior of fibrous reinforced concrete sections it was
concluded that:

1 - Steel fiber reinforcement results in improvements in
the flexural strength and especially the ductility of
reinforced concrete sections.

2 - High strength steel and concrete can be efficiently
used in flexural elements incorporating steel fibers with no
significant adverse effects on the ductile behavior.

3 - Steel fiber reinforcement reduces the :need for

99

compression steel to improve the ductile behavior in flexural
elements, High tension steel can be used in reinforced
concrete flexural elements incorporating steel fibers,
resulting in higher flexural strength, withthe ductility
being maintained at a desirable level.

4 - The developed flexural analysis and design procedure
is simple and conforms to the design guidelines of the
American Concrete Institute (ACI 318 - 83). It account for
the improvements in flexural strength resulting from steel
fiber reinforcement using a flexural stress distribution at
failure which neglects the improvements in tensile and
compressive properties of the fibrous concrete. The
developed flexural design guideline also considers the
positive effects of fiber reinforcement on the ductility of
flexural behavior by increasing the maximum limit on tension
steel area. The result of the proposed simplified flexural
analysis and design procedures compare reasonably well with

test results.

5.3 88313 BLNNNNTB

Steel fiber reinforcement was found to be an effective
approach for improving the strength and ductility of failure
in reinforced concrete elements subjected to shear forces.
Best results seem to be achieved when a combination of steel
fibers and transverse reinforcement is used to provide shear
resistance.

A shear strength equation was developed in this study

100

for fibrous reinforced concrete elements based on the physics
of shear failure in the presence of steel fibers.
The proposed equation was shown to predict test results far
better than the previous equations (which are simply modified
versions of the conventional reinforced concrete equations).
The proposed shear strength equation was used in an
analytical parametric study on the fiber effects of
reinforcement on shear strength of reinforced concrete
elements . The results indicated that improvements of the
order of 50 8 in shear strength can be expected in typical
reinforced concrete elements as a result of reinforcement
with 2 4 volume fraction of steel fibers. Longer fibers with
smaller cross-sectional dimensions and especially those with
mechanical deformations were theoretically more effective in
increasing the shear strength of reinforced concrete elements
(as far as the practical problems with fresh mix workability
and fiber dispersability can be overcome). The conventional
reinforcement properties on The shear span-to-depth ratio of
reinforced concrete elements did not significantly influence
the absolute value of improvements in shear resistance

resulting from fiber reinforcement.

5.4 TORSIONAL names

A torsional strength equation was developed for
predicting the torsional strength of fibrous reinforced
concrete elements. The proposed equation was verified using

torsional test results for fibrous concrete and fibrous

101

reinforced concrete elements with wide ranges of material
properties.

The proposed equation for predicting the torsional
strength of fibrous elements, after being verified, was used
in an analytical parametric study on the effects of fiber
reinforcement and conventional steel properties on the
torsional strength of reinforced concrete elements. The
results showed that the torsional strength of reinforced
concrete elements increase almost linearly with increasing
fiber volume fraction. In the presence of a typical 2 %
volume fraction of steel fibers with an aspect ratio of 70,
the torsional strength of reinforced concrete elements
increases by about 100 % over that of non-fibrous elements.
The results also showed that the longer fibers with smaller
diameters, as far as they are uniformly dispersed in the
concrete matrix, are more effective in increasing torsional
strength.

Conventional reinforcement properties seem to have
relatively small effects on the effectiveness of fibers in
increasing the torsional strength of reinforced concrete
elements. Test results indicate that best results can be
achieved when fibers are used in combination with relatively

small amounts of conventional reinforcement.

102

1. Swamy, R.N. and Sa'ad A - Al-Ta'an "Deformation and
Ultimate Stren in Flexure of Reinforced Concrete Beams
made with Stee Fiber Concrete" ACI Journal / September -
October 1981. pp 335-405

2. Craig, R.J. "Light Weight Reinforced Fiber Concrete
Behavior and uses" New Jersey Institute of Technology .USA
RILEM 1986 VOL 1.

3. Kormeling, H.A., Reinhardt,H.W., and Shah, S.P. ”Static
and Fatigue Properties of Concrete Beams Reinforced with
Continues Bars and with Fibers" ACI Journal / January -
February 1980. pp 36-42

4. Swamy,R.N., Sa'ad A - Al-Ta'an and Sami A.R.Ali ”Steel
Fiber for Controlling Cracking and Deflection" Concrete
International / August 1979. pp 41-45

5. Byasi, 2. "Mechanical Pro erties and Structural
Application of Steel Fiber Re nforced concrete" Ph.D Thesis,
vo ume 2, Michigan State University 1987. ‘

6. Hannant, D.J., "Fiber Cements and Fiber Concretes':
John Wiley and Sons Ltd.

7. Soroushian, P. and Byasi, 2. ”Prediction of the Tensile
Strength of Fiber Reinforced Concrete Acritique of the
Composite Material Concept," ACI Convention on Fiber
Reinforced Concrete, Baltimore, November, 1986.

8. Soroushian, P. and Byasi, z. ”Mechanical Properties of
Fiber Reinforced Concrete” Seminar Procedings Composite
Material and Structures Center Michigan State University
February 1987.

9. Shah.S.P and Rangan.V. ”Effect of Reinforcements on
Ductility of Concrete" Journal of The Struct.Div ASCE
VOL,96,N..St6, June 1970 pp 1167-1184.

10. Craig,R.J., "Desi Procedures for Fibrous Concrete,
Shear, Moment and Tors on, New Jersy Institute of Technology
New York New, Jersy.

11. Park, R., and Panlay, T., ”Reinforced Concrete
Structures Department of Civil Engineering University of
Canterbury, Christchurch, New Zealand pp 207.

103

12. Batson,G.R, Clarkson Universit , Potsdam, New York,
USA C. Alguire, Sargent Lund , Ch cago Illinois, USA "Steel
Fiber as Shear Reinforcement n Reinforced Concrete T -
Beams".

13. Ot.Ibrahim "Control of Cracks In Reinforced Concrete
Using Steel fibers" University of Elminior. Egypt. RILEM 1986
VOL 1.

14. SA.AL-Ta’an and JR Al-feel "Role of Steel Fiber
Reinforcement in flexural Failure” University of Mosul,
Mosul, Iraq RILEM 1986 Vol 1.

15. Swamy, R.N. and Al-Noori, K.A. "Flexural Behavior of
Fiber Concrete With Conventional Steel Reinforcement", Proc.
RILEM Symp. on Fiber Reinforced Cement And Concrete
(Sept.1975). The Conctruction Press Ltd, Lancaster, 1975,
pp 137-195.

16. Shah,S.P. Stroven, D.Dalhuison, and p.Steke Lenburg, in
Testing and Test Method of Fiber Cement Composites, edited
by R.N Swamy (The construction Press, lancaster 1978), pp
399-408.

17. D.Fanella and A.Naaman, ACI Journal., 82, pp 475
(1985).

18. Fanella, D.A and Naaman, A.E., ”Stress-strain
Properties of Fiber Reinforced Mortar In Compression"
ACI Journal, July - August, 1985. pp 475-485.

19. Souroushian, P. and Cha - Don Lee.”Steel Fiber
Reinforced Concrete Under Com ression: Performance
characteristics and Constitut ve Modeling" Michigan State
University.

20. Abdul-wahad H.,M,S. and AL-Ausi and SH TH Tawfiq "Steel
Fiber Reinforced Concrete Member Under Combined, Shear and
Torsion " University of Technology, Baghdad, Iraq RILEM
1986 VOL 1.

21. Batson, 6., "Use of Steel Fiber for Shear Reinforcement
and Ductility" USA - Sweden joint seminar (NSF - STU), S.
Shah and A.SKarendahil, editors, Swedish cement and Concrete
Research Institute, Stockholm, 1985.

22. Takato Uomoto, Ranjan, Hitoshi Furukoshi, and Hideo
Fujino "Shear Strength of Reinforced Concrete Beams with
Steel Fiber reinforcement” RILEM 1986 VOL 1.

23. Batson , G; Jenkins, E.; and Spatney, R., "Steel Fibers
as Shear Reinforcement in Beams" ACI Journal, Proceeding v.69
N 10, Oct, 1972 pp 640 - 644.

104

24. ACI commitee 544 State - of - Art Report on Design
Consideration " Steel Fiber Reinforcement Concrete and
Morter' being written.

25. William son, G.R., ”Steel Fibers as Web Reinforcement in
Reinforced Concrete", Proceedings, U.S.Army Science
Conference, v.3, June 1978, West Point, New York.

26. Craig R.j, Unpublishid manuscript for text book of Fiber
reinforced Concrete, New Jersy Institute of Technology,
Newark, New Jersy, 1984.

27. Swamy R.N and Bahia H.M "The effectiveness of Steel
Fibers as Shear Reinforcement", Concrete International,
V.7,No.3, March, 1985, pp 35

28. Jindal R.L, "Shear and Moment Capacities of Steel Fiber
Reinforced Concrete Beams", Fiber Reinforced Concrete
International Symposium, Sp-81, American Concrete Institute,
Detroit, Michigan, 1984.

29. Chu - Kia wang Charees G.Salmon "Reinforced Concrete
design" Fourth Edition "Shear Strenght and Shear
Reinforcement” pp 121.

30. Swamy R.N and Bahia R.N "Influence of Fiber
Reinforcement on the Dowel Resistance to Shear" ACI. Journal,
Symposuim paper, Feb. 1979, pp 327 - 355.

31. Crai , R, John: Dunya, R.: Riaz, J.; and Shirazi, R.,
"Torsiona Behavior of Reinforced Fibrous Concrete Beams",
Fiber Reincorced Concrete, SP-81, American Concrete
Institute, Detroit, 1984, pp 17-49.

32. Craig,R.J., James A. Parr, Eddy Germain, Victor.
Mosquera and Stavros Kamilares ”Fiber Reinforced in torsion"
ACI Journal, November - December 1986 pp 934 - 942.

33. Mansur, M.A., and P. Paramasivan. ”Fiber Reinforced
Concrete Beams in Torsion, Bindin and Shear” ACI Jouran,
January -February 1985 pp 33 - 39.

34. Mansur, M.A., BSC Eng, MSC Eng, PhD, MASCE "Bending -
Torsion Interaction for Concrete Beams Reinforced with Steel
Fibers " Magazine of Concrete Research: Vol 34, No, 121 :
December 1982 pp 182 - 190.

35. Edgington, J., "Steel Fiber Reinforced Concret,”
Research Re ort Submitted to the Departement of Environment,
Jan.1974. A so PhD Thesis, University of Surry , 1974.

36. Edgington, J., "Steel Fiber Reinforced Concrete,"
Intermediate Report, Departement of the Environment, Grston,
Wat ford, March 1972.

105

37. Edgington, J., Hannant, D.J.: and Williams, R.I.T,
"Steel Fiber Reinforced Concrete," Fiber Reinforced
Materials, BRT Building Research Series, v.2 The Construction
Press, New York, 1978, pp 112 - 129.

38. Hsu, Thomas, T.C, "Torsion of Structural Concrete-Plain
Concret Rectangular Sections " Torsion of Structural Concret,
Sp.18 American Concrete Institute, Detroit, 1986, pp 203 -
238.

39. Paul, S.L. and Sinnamon, G.k., "Concret Tunnel Liners:
Structural Testing 0f ngented Liners ,"Final Report,
Departement of Transprtation, FRA and D-75-93 Augost, 1975.

40. Criswel, M.E, "Shear in Fiber Reinforced Concret,"
National Structural Engineering Conference, Augost 1976,
Masison, Wisconsin American Society of Civil Engineers, New
York, pp 1-20.

41. Turner, L. and V.C Davies,"Plain and Reinforced Concrete
in Torsion, with Particular REferance to Reinforced Beams"
Selected Engineering Papers No.165, London, Institution of
civil Engineers, 1934 pp 31.

42. Swamy, R.N and Mangat p.s "Influence of Test Geometry on
the Properties of Steel Fiber Reinforced Concrete," V1.4 pp
451 - 465, 1975.

22. Murty, D.S.R., T.Venkatach Aryulu "Fiber REinforced
Concret Beams Subjected to SHEAR fORCE,” Proceedings of the
International Symposium on Fiber Reinforced Concret, December
16 -19, 1987, Madras, India.

44. Shah, S.P., ”A plication of Fracture Mechanics to
Cementitions Compos tes," Martinus Publishers, 1985.

45 Soroushian, P , and Bayasi, z. ( editors ) ”Fiber
reinforced Concrete Design and Application" Composite
Magerials and Structures Center, Michigan State University
Fe 1987.

APPENDIX A

AN DUBLIN! 01' m CORNER PROGRAM I'OR RBI'INBD I'LBXURAL
ANALYSIS OF IIBROUB RRINFORCRD CONCRETE BBCTIONB

The developed computer program performs a step-by-step
flexural analysis of reinforced concrete sections
incorporating steel fibers, following the refined flexural
analysis procedures described in chapter 2. Different parts
of the program are outlined below.

Part 1 - Data input (only the matrix and fiber
properties and not the composite properties are needed):

B - width of the cross section

H - height of the cross section

d - effective depth, distance from compression face

to centroid of tension steel

dp - distance from compression face of the member to

centroid of compression steel

As 9 area of tension steel

A's - area of compression steel

f'c - concrete compressive strength
- yield strength of steel
Vf a volume content of fibers
1: a length of fibers
df - diameter of fibers
Part 2 - Calculate the composite material properties:

106

107

Tensile and compressive strength and constitutive models of
steel fiber reinforced concrete.

Part 3 - Input the new value of curvature (which is an
increment higher than the previous value).

Part 4 - Assuming a linear strain distribution, through
trial and adjustment, find the neutral axis position which
satisfies (within acceptable errors) the equilibrium of
tension and compression forces at the section.

Part 5 - Find the moment of compression and tension
forces: Repeat from part 3 step 3 until flexural failure is
marked by a major drop in force resistance and in stability
of the section flexural behavior.

The completed flexural moment-curvature relationship of
fibrous reinforced concrete sections can be produced
following the above analysis procedure.

A complete print-out of the computer program which
performs the above tasks for flexural analysis of fibrous

reinforced concrete sections is given below.

108

10 DD! PI(300). HOOD). EPSCUOOO). CCC(300). TCC(300)

20 DIH 28(2)

30 CLS

40 KEY OFF

50 PRINT TAB(20) 'FLEXURAL ANALYSIS OF FIBROUS R/C BEAHS'
50 pann- ruao) “W0
70 PRINT:PRINT

80 INPUT 'DO YOU WANT A HARD COPY OF GRAPH (Y/N)';YYNN$

90 PRINT

100 PRINT

110 'COLOR 31.5

120 LOCATE 23.63

130 BEEP :BEEP :BEEP

140 PRINT “PLEASE WAIT'

150 'COLOR 7.0.

160 FOR I-I TO 10000: NEXT I

170 RBI ---------------------------------------------------------
180 CLS :PRINT ”ENTER SECTION GEONETRY AND REINFORCBIENT PATTERN“
190 PRINT " ................................................ I

200 PRINT: PRINT: INPUT 'ENTER THE RECTANGUIAR BEAN WIDTH (in)";B

210 PRINT: INPUT “ENTER THE BEAN HEIGHT (in)';H

220 PRINT: INPUT "ENTER THE EFFECTIVE DEPTH OF BEAM (in)";D

230 PRINT: INPUT “ENTER THE DIST. FRO! CENTER OF COMP. STEEL TO COHP. SURFACE (i
n)';DP

240 PRINT: INPUT "ENTER THE TENSION STEEL AREA (in2)";AS

250 PRINT: INPUT 'ENTER THE CM. STEEL AREA (in2)";ASP

260 PRINT: INPUT 'ENTER REBAR YIELD STRENGTH (psi)';FY

270 R34 ----------------------------------------------------------

280 'COLOR 31.1

290 BEEP :BEEP

300 LOCATE 23.66

310 PRINT "CHECK DATA'

320 'COLOR 15,4

330 IDCATE 24.10

340 INPUT 'INPUT O/K (Y/N) “:13

350 LOCATE 20.66

360 'COLOR 7,0

370 IF IS-‘N‘ OR IS-‘n' OR IS—‘NO' OR IS-‘no' THEN CLS:GOTO 200

380 CLS:PRINT “ENTER STEEL FIBER PROPRETIES'

390 PRINT " ---------------------------- "

400 PRINT: INPUT “ENTER FIBER VOL. FRACTION (8)";VF

410 PRINT: INPUT "ENTER FIBER LENGTH (in)';LF

420 PRINT: INPUT 'ENTER FIBER DIAHETER (in)';DF

430 PRINT: INPUT ”ENTER FIBER TYPE: Ski-eight), H(ooked) or C(rimped)‘;FT$
440 IF PIS-'8' THEN TU-320 ELSE IF FTS-“H' THEN Til-450 ELSE IF FTS-‘C' THEN TU-3
00

450 'COLOR 31.1

460 BEEP :BEEP -

470 mCATE 15.66

480 PRINT “CHECK DATA'

490 'cowa 15,4

500 LOCATE 24.10

510 INPUT 'INPUT O/R (Y/N)';L$

520 IDCATE 12.66

530 'COLOR 7.0

540 IF I.$-"N' OR 1.3-“n“ OR LS-‘NO' OR IS-‘no' THEN CLS:GOTO 400

550 REH -------------------------------------------------------------------
S60 CLS :PRINT "ENTER THE MATERIALS AND HECHANIQUE PROPERTIES“

570 PRINT " --------------------------------------------- "

580 PRINT: INPUT “CHECKING THE POSSIBILITY OF EXCEEDING CRITICAL FIBLER LENGTH (

Y/N)
590
psi)
5*.4
600
610

1(19

":YNS

IF YN$-'Y' OR YNS-"y' THEN PRINT: INPUT "ENTER FIBER ULT. TENSILE STRENGTH (
';PUP: LC-PUP*D/TU: IP L>-LC THEN PPP-PUF*.S*VF/100/(3.14*DP‘2/4) ELSE PPP-.
1*TU*VF/100*LP/DP

PRINTzPRINT: INPUT “DO YOU KNOW THE ULT. TENSILE STRENGTH OF FRC (Y/N)”;NU$
IF NUS-"N” OR NUS-”n" THEN PRINT: INPUT ”ENTER THE TENSILE STRENGTH OP CONC.

MATRIX (psi). ABOUT h*Pc“.5';PTH ELSE 640

620
*.41
630
640
650
660
670
680
690
700
710
720
730
740

IP YNS-"Y' 0R YNS-"y' THEN PTP-PTH*(1-VP/100)+PPP ELSE PTP-PTH*(1-VP/100)+.5
*TU*VP/100*LP/DP

IP NUS-"N" 0R NUS-”n” THEN 650

PRINT: INPUT "ENTER ULT. TENSILE STRENGTH OP PRC (psi)';PTP

PRINT: INPUT 'DO YOU KNOW THE POST-PEAK TENSILE STRENGTH OP PRC (Y/N)”;NP$
IP NP$-'Y' 0R NP$-'Y' THEN 690

IP YN$-'Y' OR YN$-'y' THEN PPP-PPP ELSE PPP-.5*.41*TU*VP/100*LP/DP

IP NPS-‘N' OR NUS-"n: THEN 700 I

PRINT: INPUT “ENTER THE POST-PEAK TENSILE STRENGTH 0P PRC (psi)';PPP

PRINT: INPUT “DO YOU KNOW THE CONP. STRENGTH OP PRC (Y/N)';NC$

IP NC$-'N' 0R NOS-”n“ THEN 740

PRINT: INPUT "ENTER THE COHP. STRENGTH OP PRC (psi)';PCP

IP NOS-“Y” 0R NCS-‘y' THEN 860

PRINT: INPUT "ENTER THE CONP. STRENGTH 0P CONC. MATRIX (psi)";PC: PCP-PC+994

*VF/100*LF/DF

750
760
770
780
790
800
810
820
830
840

850
860
870
880
890
900
910
920
930
940
950
960
970
980

'COLOR 31.1

BEEP :BEEP

LOGATE 20.65

PRINT “CHECK DATA-
'COLOR 15,1.

LOCATE 24,10

INPUT "INPUT O/R
LOCATE 12.66
'COLOR 7,0

REH -------------------------------------------------------------------------

(Y/N)':T$

IP TS-‘N' OR T$-'n' OR TS-‘NO' OR TS-‘no' THEN CLS :GOTO 580
CP-(AS*PY*(H-D)tASP*PY*(H-DP)+PCP*B*H*H/2)/(AS*PY+ASP*PY+PCP*B*H)
PR-.12*PCP+2000*VP/100*LP/DP
EP-(.00079+1.13/(PCP-99h*VP/100*LP/DP))*VP/100*LP/DP+.0021
Z--343*(PCP-994*VP/100*LP/DP)*(1-.64*SQRT(VP/100*LP/DP)):IP Z>O THEN 2-0
NPI-I: PI(NPI)-O: H(NPI)-0

SCREEN 2: KEY OPP

KHAN-.OOIOOOOOOIO: YHAX-20000000#*.5

UINDOU (~XHAX/10,-YHAX/10)-(XHAX.YHAX): CLS

LINE (0.0)-(0,YHAX): LINE (0,0)-(XNAX.0)

FOR NN-I T0 10

LINE (0.YHAX*NN/IO)-(XNAX/65,YHAX*NN/10)

LINE (XNAX*NN/10,0)-(XNAX*NN/10,YHAX/50)

IP NN-2 0R NN-b OR NN-6 OR NN-B THEN LOCATE 24,6+72*NN/10: PRINT USING ".##8

s';XHAX*NN/10;

990

/100
1000
1010
1020
1030
1040
1050
1060
1070
1080
1090

IP NN-2 OR NN-h OR NN-6 OR NN-8 THEN LOCATE 22.5-22.5*NN/10,1: PRINT YHAX*NN

00;

NEXT NN

LOCATE 25.60: PRINT 'CURVATURE (1/1n)";
LOCATE 1,10: PRINT 'HOHENT (Kips.in)';
LOCATE 3.12:PRINT 'B (in) -';B
LOCATE 3,32:PRINT “H (in) -";H
LOCATE 4.12zPRINT "d (in) -';D
LOCATE 4,32:PRINT 'dp (in) -';DP
LOCATE 5.12:PRINT “As (in2) -';AS
LOCATE 5.32:PRINT ”Asp (in2) -';ASP
LOCATE 6,12 PRINT 'Py (psi) -';PY

1100
1110
1120
1130
1140
1150
1160
1170
1180
1190
1200
1210
1220
1230
1240
1250
1260
1270
1280
1290
1300
1310
1320
1330
1340
1350
1360
1370
1380
1390
1400
1410
1420
1430

110

LOCATE 6,32:PRINT "VI (4) -';VF

LOCATE 7,12:PRINT 'df (in) -';DF

LOCATE 8.33

IF FT$-'S' THEN PRINT "STRAIGHT FIBER”

IF FT$-'H' THEN PRINT 'HOOKED FIBER”

IF FT$-'C“ THEN PRINT 'CRIHPED FIBER"

IF YNS-‘N' OR YN$-'n” THEN 1200

LOCATE 9,32

IF L>-LC THEN PRINT 'Lf >- Le“

IF L<LC THEN PRINT 'Lf < Lc'

LOCATE 8.12:PRINT 'th (psi) -';FTF

LOCATE 9.12:PRINT 'Pcf (psi) -';PCP

LOCATE 7.32:PRINT 'Lf (psi) -‘;LF

LOCATE 10,12:PRINT 'pr (in) -';FPF

Rm --------------- . ------------------------------
FIFIHAX-.001

DFIFI-.00002

FOR FIFI-.00002 T0 FIFIHAX STEP DFIFI

NFI-NFI+1

FI(NFI)-FIFI

EPSPl--FIFI*CP

EPSP2-FIFI*(H-CP)

NA-CP+EPSP1/PIPI

EPSSP--(H-DP-NA)/(NA-CP)*EPSP1
EPSS-EPSP1*(NA-H+D)/(NA-CP)

FS-AS*EPSS*29000*1000

IF ABS(FS)>FY*AS THEN FS-FY*AS*FS/ABS(FS)
FSP-ASP*EPSSP*29000*1000
EPSCX-(EPSPI)*(H-NA)/(CP-NA)

GOSUB 2390

EPSTX--EPSPI*NA/(CP-NA)

REH ---------------------------------------------------------------
GOSUB 2480
' PRINT 'NA,EPSPI.TC.CC,FS.FSP,TOTAL-':NA;EPSP1;TC;CC;FS;FSP;TC+C

C+FS+FSP

1440
1450
1460
1470
1480
1490
1500
1510
1520
1530
1540
1550
1560
1570

T1-TC+CC+PS+PSP

NA-CP+EPSP2/PIPI

EPSSP--(H-DP-NA)/(NA-CP)*EPSP2

EPSS-EPSP2*(NA-H+D)/(NA-CP)

PS-AS*EPSS*29000*1000

IF ABS(PS)>PY*AS THEN PS-PY*AS*PS/ABS(PS)
PSP-ASP*EPSSP*29000*1000

1F ABS(FSP)>PY*ASP THEN PSP-PY*ASP*PSP/ABS(FSP)
EPSCX-EPSP2*(H-NA)/(CP-NA)

GOSUB 2390

EPSTX--EPSP2*NA/(CP-NA)

REH -----------------------------------------------------------
GOSUB 2480
' PRINT ”NA.EPSP2.TC.CC.PS,PSP.TOTAL-';NA;EPSP2;TC;CC;FS;FSP;TC+CC

+FS+FSP

1580
1590
1600
1610
1620
1630
1640
1650
1660
1670

T2-TC+CC+PS+PSP

EPSP3-(EPSP1+EPSP2)/2

NA-CP+EPSP3/PIPI

IP NA > H THEN NA-H: EPSP3-PIPI*(NA-CP)

IP NA < 0 THEN NA-O: EPSP3-PIPI*(NA-CP)

1P ABS(CP-NA)/CP < .0001 THEN EPSP3-.55*EPSP1+.45*EPSP2: GOTO 1610
EPSSP--(H-DP-NA)/(NA-CP)*EPSP3: EPSS-EPSP3*(NA-H+D)/(NA-CP)
PS-AS*EPSS*29000*1000: IP ABS(PS)>PY*AS THEN PS-PY*AS*PS/ABS(FS)
FSP-AS*EPSSP*29000*1000: 1P ABS(PSP)>PY*ASP THEN PSP-PY*ASP*FSP/ABS(PSP)
EPSCX-EPSP3*(H-NA)/(CP-NA)

111

1680 GOSUB 2390

1690 EPSTXp-EPSP3*NA/(CP-NA)

1700 REH ------------------------------------------------------------------

1710 60803 2480

1720 '. PRINT 'NA.EPSP3,TC.CC.FS.FSP.TOTAL~':NA;EPSP3:TC;CC:FS;FSP;TC+C
C+PS+FSP

1730 T3-TC+CC+FS+FSP

1740 IF T1*T3 < 0 THEN T2-T3: EPSP2-EPSP3: GOTO 1780

1750 IF T2*T3 < 0 THEN T1-T3: EPSPl-EPSP3: GOTO 1780

1760 IF T3-T1 OR T2-T1 THEN EPSP3-EPSPI+.4*(EPSP2-EPSP1): GOTO 1610

1770 PRINT 'SOHETHING Is URONG, T3 13 NOT BETWEEN T1 AND T2-: STOP

1780 IF ABS(T3) < .002*(B*H*FCF+FY*AS+FY*ASP) THEN 1790 ELSE 1590

1790 GG-H

1800 BB-EP/FIFI+NA

1810 DD-FCF*FIFI/EP .

1820 EPSR-(FR- -FCF+2*EP)/2

1830 EE-EPSR/FIFI+NA

1840 IF EPscx <- EP THEN AR1--(GG 4/4- -NA*GG 3*2/3+NA 2*GG 2/2)*FIFI/EP+2*(GG 3/3
-NA*GG 2/2)+NA 4/12*FIFI/EP+NA 3/3

1841 IF EPscx <- EP THEN AR2--FIFI/EP*GC*(GG 2/3- -NA*GG+NA 2)+2*GG*(GG/2 NA)+FIFI
/EP*NA 3/3+NA“ 2

1842 IF EPscx <- EP THEN ARHCC-ARI/AR2

1850 REH ----------------------------------------------------------------

1860 IF EPscx > EP AND EPscx <- EPSR THEN ARI-DD*( FIFI/EP*BB 2*(BB 2/4- 2*NA*BB/
3+NA 2/2)+2*BB 2*(BB/3- NA/2)+FIFI/EP*NA 4/12+NA 3/3)+GG 2*2*(GG/3*FIFL NA*FIFI/2
-EP/2)+GG 2*FCF/2 BB 2*2*(BB/3*FIFI— oNA/2*FIFI EP/2)- -BB 2*FCF/2

1870 IF EPscx > EP AND EPscx <- EPSR THEN AR2-DD*( FIFI/EP*BB*(BB 2/3 NA*BB+NA 2
)+2*BB*(BB/2 NA)+FIFI/EP*NA 3/3+NA 2)+Z*GG*(GG/2*FIFI~NA*FIFL EP)+FCF*GG- -Z*BB*(B
B/2*FIFI~NA*FIFI~EP)~FCF*BB

1880 IF EPscx > EP AND EPscx <- EPSR THEN ARHcc-ARI/ARz

1890 IF EPscx > EPSR THEN ARHI-DD*(-PIPI*88‘2/EP*(88‘2/4-2*NA*88/3+NA‘2/2)+2*88‘
2*(88/3-NA/2)+PIPI*NA‘4/EP/12+NA‘3/3)+EE‘2*2*(EE*EIPI/3-NA*PIPI/2-EP/2)+EE‘2*PGP
/2-BB‘2*Z*(BB/3*FIFIoNA*FIFI/2-EP/2)-BB“2*FCF/2+FR/2*(GG‘2-EE‘2)

1900 IF EPscx > EPSR THEN ARH2-DD*(~PIPI/EP*88*(DE‘2/3-NA*88+NA‘2)+2*88*(88/2-NA
)+FIFI/EP*NA‘3/3+NA‘2)+Z*EE*(EE/2*FIFI-NA*FIFI-EP)+FCF*EE-Z*BB*(BB/2*FIFI-NA*FIF
I-EP)-FCF*BB+FR*(GC-EE)

1910 IF EPscx > EPSR THEN ARHCC-ARHI/ARHZ

1920 IF EPSTx <- PTP/(S7000!*PGP‘.5) THEN ARHcT-NA/3

1930 IF EPSTX > FTP/(57000!*FCF‘.5) THEN ARHGT1-570001*PGP‘.5*EIPI*(NA‘3/8-NA/2*
(NA~PTP/(57000!*PGP‘.5)/P1PI)‘2+(NA-PTP/(570001*PGP‘.5)/PIPI)‘3/3)+PPP/2*(NA-PTE
/(57000:*EGP‘.5)/PIPI)‘2

1940 IF EPSTx > FTP/(57000!*FCF‘.5) THEN ARNGT2-57000!*PGP‘.5*PIPI*(NA‘2/2-NA*(N
A-PTP/(57000!*PGE‘.5)/EIPI)+(NA-PTE/(57000!*PGP“.5)/PIPI)‘2/2)+PPP*(NA-PTP/(5700
0!*FCF‘.5)/FIFI)

1950 IF EPSTx > FTP/(57000!*FCF‘.5) THEN ARNGT-ARHGTI/ARHcTz

1959 ' PRINT 'NA.TC,CC.PS,PSP.ARNCC.ARNCT-';NA;TC;CC;FS:FSP;ARHCC;ARHCT
1960 H(NPI)--TC* ARHCTo CC*ARHCC- PS*(H- -D)- PSP*(H- DP)

1970 REM ----------------------------------------------------------------------
1980' PRINT 'PI, H -" .FI(NPI); H(NFI)

1990 LINE (PI(NPI-1).H(NFI- -1))- (FI(NPI). H(NF1))

2000 IP INT(PIFI/DFIFI)><INT(FIF1HAX/DP1PI/4) AND INT(FIF1/DF1P1)><INT(2*FIPIHAX
/DFIPI/4) AND INT(PIPI/DFIFI)><INT(3*PIPIHAX/DFIFI/4) AND,INT(PIPI/DPIPI)><1NT(P
IFIHAX/DPIPI-2) THEN 2180

2010 VIEU ((.1*XHAX+FIPI)*640/I.1/XHAX-16.80)-((.1*XHAX+P1PI)*640/1.1/XNAX+16.10
0)..l

2020 UINDOU (-FCP.0)-(PTP}H): LINE (0,0)-(0.H)

2030 FOR X-NA TO N STEP (H-NA)/20

2040 EE-(X-NA)/(H-NA)*ABS(EPSCX)

2050 IF EE<-EP THEN SS--(-FCP*(EE/EP)“2+2*FCP*EE/EP) ELSE 1F EE <- EPSR THEN SS-
°(PCP+2*(EE-EP)) ELSE SS--PR

2060
2070
2080
2090
2100
2110
2120
2130
2140
2150
2160
2170
2180
2190
2200
2210

112

IP x-NA THEN LINE (SS.X)-(SS,X) ELSE LINE -(ss,X)
NEXT x

FOR x-NA To 0 STEP -NA/2o

EE-(NA-X)/NA*ABS(EPSTX)

IF EE <- FTF/(S7000!*FCP“.S) THEN ss-EE*570002*PCP‘.5 ELSE SS-FPF

IF x-NA THEN LINE (SS.X)-(SS,X) ELSE LINE ~(SS,X)

NEXT x

VIEW

WINDOW (-XHAX/10,-YMAX/10)-(XMAX,YMAX)

EPSCU(FIFI)-EPSCX

CCC(FIFI)-CC

TCC(FIFI)-TC

NEXT FIFI

FOR IQ-l To NFI

IF IQ-I THEN HHAX-HLIQ): GOTO 2220

IF H(IQ) > HHAx THEN HHAx-H(IQ):EPscuo-EPSGU(IQ) CCCO-CCC(IQ):TCCO-TCC(IQ):

PIO-PI(IQ)

2220
2230
2240
2250
2260
2270
2280
2290
2300
2310
2320
2330
2340
2350
2360
2370
2380
2390
2400
2410
2420
2430
2440

NEXT IQ

LOCATE 6,53

PRINT 'EPSCU -";EPSCX
LOCATE 7.53

PRINT 'FI (l/in) -";FIO
LOCATE 3.53

PRINT ”AT max Moment";

LOCATE 4,53

PRINT " ------------- ”;

LOCATE 5,53

PRINT "Hmax (K.IN)-";MHAX/1000
LOCATE 8,53

PRINT ”CC (Kips) -";CC/1000
LOCATE 9,53

PRINT ”TC (Kips) -";TC/1000
IF YYNNS-“Y” 0R YYNNS-"y' THEN GOSUB 2520
END

DD-PCF*PIFI/EP

GG-H

BB-EP/FIFI+NA
EPSR-(PR-FCF+Z*EP)/Z
EE-EPSR/FIFI+NA

1P EPSCX<-EP THEN CC--DD*(-FIPI/EP*GG*(GC“2/3-NA*CC+NA“2)+2*GC*(CG/2~NA)+FI

FI/EP*NAA3/3+NA22)*B

2450

IP EPSCX>EP AND EPSCX<-EP+(PCP-PR)/Z THEN CC--DD*(-PIPI/EP*BB*(BBA2/3—NA*BB

+NAA2)+2*BB*(88/2-NA)+PIPI/EP*NAA3/3+NA22)*B-(Z*((GGA2/2-NA*GG)*FIPI-EP*GG)+FCP*
GG-Z*((BB‘Z/Z-NA*BB)*PIPI-EP*BB)~PCP*BB)*B

2460

IP EPSCX > EP+(PCF-PR)/Z THEN CC--DD*(-PIPI/EP*BB*(BBA2/3-NA*BB+NAA2)+2*BB*

(BB/2ENA)+PIPI/EP*NAA3/3+NAA2)*B-(Z*((EEA2/2-NA*EE)*PIPI-EP*EE)+FCF*EE-Z*((BBAZ/
2-NA*BB)*PIPI—EP*BB)-PCP*BB)*B-PR*(GG—EE)*B

2470
2480
2481
2483
2490
2500
2501
2510
2520
2530
2540
2550
2560

RETURN
Ec-57000!*PGP‘.5

FTFEC-FTF/EC

' PRINT 'EPSTX, FTP/EC-';EPSTX;FTFEC

IF EPSTx <- FTFEC THEN TG-NA“2/2*8*EG*EIPI

IF EPSTx > FTFEC THEN Tc-PTP*(PTP/Ec/PIPI)/2*8+PPP*(NA-PTP/EC/PIEI)*8
' PRINT 'TC.FTF,FPF,EC-';TC;FTF;FR;EC

RETURN

WINDOW (0,0)-(639,199)

WIDTH "LPTl:',255
LINESPACE9S-CHRS(27)+CHR$(65)+CHR$(8)
LINESPACE6$-CHR$(27)+CHR$(65)+CHR$(12)
CRAPH400$-CHR$(27)+CHR$(75)+CHR$(144)+CHR$(1)

2570
2580
2590
2600
2610
2620
2630
2640
2650
2660
2670
2680
2690
2700
2710
2720
2730
2740

113

LPRINT LINESPACE9S;

FOR JA-I TO 9

LPRINT

NEXT JD

FOR COLA-O TO 79

LPRINT SPACE$(9);

LPRINT GRAPH4OOS;

FOR ROWS-O TO 199

GET (8*COL3+7,ROW§)-(8*COLA.ROW§).24
LPRINT GHR$(Z§(2))+CHR$(ZS(2));
NEXT ROWE

LPRINT

NEXT COL.

FOR JO-I TO 10

LPRINT .

NEXT J8

LPRINT LINESPACE6S;

RETURN

HICH IES

11111111111