IJCIET Experimental Verification of Minimum Flexural Reinforcement Formulas for HSC Beams

International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
(Print), ISSN 0976 – 6316(Online) Volume 4, Issue 5, September – October (2013), © IAEME

TECHNOLOGY (IJCIET)

ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 4, Issue 5, September – October, pp. 152-167
© IAEME: www.iaeme.com/ijciet.asp
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IJCIET
©IAEME

EXPERIMENTAL VERIFICATION OF THE MINIMUM FLEXURAL
REINFORCEMENT FORMULAS FOR HSC BEAMS
M. SAID1 & T. M. ELRAKIB2
1

Lecturer, Civil Eng. Dep., Shoubra Faculty of Eng., Benha University, Egypt
2
Lecturer, Housing and Building National Research Center, Cairo, Egypt

ABSTRACT
The main objective of the current research is to establish experimental data for minimum
flexural reinforcement, ρmin, of high strength concrete (HSC) rectangular beams. Nine full-scale
singly reinforced beams with flexural reinforcement ratios varying from 50% to 100% of the
minimum limit specified by the ACI 363R-35 were tested in flexure. Concrete compressive strengths
of 52, 73 and 96.5 MPa were used. The test results including crack patterns, deflections and strains in
the tensile flexural steel bars show that a 25 % reduction of the ACI 363R-35 limit for the ρmin
would result in a satisfactory flexural beam behavior. The experimental results of this study are
compared with the limits specified by available codes and researches.
Keywords: High Strength Concrete, Beams, Minimum Reinforcement, Ductility.
INTRODUCTION
High strength concrete (HSC) provides a better solution for reducing sizes and weights of
concrete structural elements. The major part of the application of high strength concrete concerns
particular structures such as offshore platforms and the lower story columns of high-rise buildings. HSC
correlates with improvements in its other engineering properties (tensile strength, creep coefficient, etc.),
however, it fractures suddenly and forms a smooth failure plane [1, 2].
However, in designing a reinforced concrete member, it is important to insure that the
member will not exhibit brittle failure and will be capable of sustaining large deformations near
maximum load. This capability gives ample prior warning before failure. Because concrete
becomes increasingly more brittle as its compressive strength is increased, guaranteeing adequate
ductility represents one of the primary design concerns when HSC is involved. Moreover, in
some situations and for one reason or another, the concrete section dimensions are bigger than
required by strength consideration to the extent that the flexural element does not need tensile
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steel reinforcement to carry its service loads. If the amount of flexural reinforcement is very
small, the cracking moment (i.e. the flexural strength computed from the modulus of rupture of
concrete) will exceed the flexural strength of the reinforced concrete section and a suddenly brittle
failure may occur with the formation of first crack, [3-5].Accordingly, a l l current national and
international codes of practice of reinforced concrete structures s t i l l require a minimum amount
of flexural reinforcement to be provided in these elements, including beams. According to
most of these codes, the minimum flexural reinforcement ratio, ρmin is the one that would
result in the ultimate moment of the cracked section being at least equal or larger than the
cracking moment of the gross section (i.e. Mu /M cr≥1.0). This condition is stated to be adequate
enough to fulfill the previous requirements of crack control and prevention of the b r i t t l e
failure[2-5].According to the previous condition, the minimum flexural reinforcement ratio ρmin
is inversely proportional with fyand it should be pointed out that Mu would be higher if the strain
hardening of the reinforcing steel is considered. If this is taken into account, it will be possible
to reduce the required ρmin.
Boscoet al[1] Proposed a dimensional analysis criterion based on a fracture mechanics model
to compute the minimum amount of reinforcement for HSC beams in flexure. He concluded that the
minimum flexural reinforcement provided by the ACI Code expression is inadequate for HSC,
Eq.1.
ρmin=1.4/fy(MPa)

(1)

Equation [1] does not include the concrete strength as a parameter. With the extensive use of
higher strength concrete, it was shown that Eq. [1] would yield inadequate reinforcement ratios
because it was based on test results obtained from beams made of NSC. The expression given below
is provided by ACI 318-08 [6] and ACI 363R-35[7] to be effective for HSC as well as the NSC.
ρmin = 0.25(f’c)0.5/fy>1.4/fy(MPa)

(2)

The expression of the Canadian standards CSA (A23.3-04)[8] for ρmin is as follows:
ρmin = 0.20(f’c)0.5/fy(MPa)

(3)

The limit of validity is given as 20MPa >f’c>80MPa. It is clearly noted that the ACI 363R-35
expression results in 25% more reinforcement than the CSA (A23.3-04).
In addition, the formula of the Japanese Standards JSCE [9] for ρmin is:
ρmin = 0.058 (h/d)2(f’c)0.66/fy(MPa)

(4)

Where h is the total height of beam and d is the effective depth. JSCE code states that the previous
formula should be used only for f’c≥35 MPa with a fixed value of ρmin = 0.002 for lower values of f’c.
On the other hand, El-Saie, A. [10] tested seven T-section beams to determine the
minimum amount of flexural reinforcement in HSC with fcu equals only 80 MPa for all tested beams.
One of the beams was plain concrete, three beams reinforced with high grade steel and three beams
reinforced with mild-grade steel. The tensile strength of the HSC was measured by using the
modulus of rupture fr .Test results show that the tensile strength of HSC as measured by the modulus
of rupture fr =0.98 (f’c) 0.5MPa which is higher than the ACI 363 R-35 formula by about 48%. Based

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on a proposed empirical expression, Mu/Mcr=1.45, a new formula for the ρmin in HSC beams was
suggested as follows:
ρmin = 0.25{(f’c)0.5/fy}{1-0.5(1+[((B/b)-1)*(ts/t)2]/+[((B/b) -1)*(ts/t)]}

(5)

Where B= flange width, b= web width, ts = slab thickness and t= total beam height. Accordingly,
in case of rectangular section, ρmin = 0.125 (f’c) 0.5/fy which represents 50% of the ACI 363R-35
limit. However, the researcher didn't mention any results about the ductility of the tested beams.
Recently, Rizk et al [4] tested sex thick HSC slabs having small ratios of flexural
reinforcement. The main test variables included concrete compressive strength, reinforcement ratio
and slab effective depth. Based on the experimental test results, it was concluded that ACI 318-08[6]
and CSA (A23.3-04) [8] codes overestimate ρmin required for thick HSC slabs greater than
250 mm. It was also reported that the value of ρ min tends to be inversely proportional to the slab
effective depth. A new model that uses the fracture mechanics concepts to account for the
size effect was suggested as follows:
ρmin = 0.236(125/d)0.33(f’c)0.5/fy(MPa)

(6)

Where d is the effective depth. It was mentioned that the application of the new formula can
result in a saving of steel reinforcement. However, all the previous limits for ρmin are based on
empirical equations that sometimes are conservative and, hence, may be uneconomic.
In addition, Rashid M. et al [5] stated that, based on the current code provisions it can be
analytically observed that an increase in concrete strength leads to higher ductility. Experimental
evidence reported by other researchers [11-13] supports this prediction except for Ashour[2]. In his
study, [2], test results showed enhanced ductility for higher strength concrete beams but only up to a
concrete strength of around 80MPa then ductility decreases as the concrete strength are increased. So,
further experimental evidence embracing concrete with compressive strength greater than 80MPa is
therefore necessary.
The diversity of the previous limits and observations clearly reveals the need for
further research in t h i s area, in order to evaluate the validity of the available formulas from
the economic point of view, as well as try to establish a more accurate limit for the minimum
reinforcement ratio in HSC beams. This paper presents experimental results of nine full scale
rectangular HSC beams reinforced in flexure w i t h small amounts of steel bars to examine
the available formulas for ρmin in HSC. The tested parameters are flexural reinforcement
ratio, ρ, and the concrete compressive strength, fcu. The flexural reinforcement ratio ranges
from 0.50 to 1.0 times ρmin specified by the ACI 363R-35 and the target concrete compressive
strength is 50, 75 and 100MPa.
EXPERIMENTAL WORK
Test Specimens
The experimental test program included nine RC beams 250x400x3500 mm. All beams were
constructed in the laboratory of the Housing and Building National Research Center and tested under
two point loads. Details of the beams are shown in Fig.1. The shear spans were the same for all
beams, i.e. (a/d) =3.60. Since very low ratios of flexural reinforcement were used, the flexural
capacity of all test specimens was lower than the diagonal cracking capacity. Therefore, no shear
cracks were expected. However, steel stirrups, R8@150mm, were provided in the shear span as an
extra precaution, R denotes normal grade bars. These stirrups and the top bars holding the stirrups
154

were terminated at the boundaries of the constant moment region. Test specimens were grouped into
three series according to their concrete compressive strength (50, 75 and 100MPa). Each series consisted of three beams. In each series, three different reinforcement ratios were used. The properties of
the test specimens are summarized in Table 2. In some specimens, two bars different in size and yield
stress were used in the longitudinal reinforcement. In calculating the required minimum flexural
reinforcement ratio, the weighted average of the yield stress was used for these specimens. Concrete
clear cover of approximately 15 mm was provided for all beams. After casting, test beams were cured
for fourteen days by continuous spraying of water. In addition, six standard cubes 150*150*150 mm
were cast from each mix as control specimens to evaluate the actual concrete compressive strength
fcu. These cubes and beams were cured and tested on the same day of testing the corresponding
specimens.
Table 2 Details of test specimens
Series

Specimen

Target
compressive
strength
fcu, MPa

B501

^

+

Longitudinal
reinforcement,
As

Yield
stress
fy ,MPa

#ρmin*10-3

ρ*10-3

#(ρ/ρmin)

1.69

0.51

2 T 12

515

3.10

2.39

0.77

2 T 12 + 1 T 10

501

3.17

3.23

1.02

1 T 12 + 1 T 10

495

3.93

2.04

0.52

1 T 12 + 2 T 10

501

3.96

2.89

0.73

B753

2 T 12 + 2 T 10

495

3.93

4.05

1.03

B1001
3

3.31

B751
2

480

B503

1

2 T 10

3 T 10

480

4.68

2.48

0.53

4 T 10

480

4.68

3.37

0.72

3 T 12 + 1 T 10

506

4.44

4.45

0.99

B502

B752

B1002
B1003

50

75

100

+

T denotes high grade deformed bars, and the following number indicates the diameter in mm.
fy was calculated using the weighted average of the yield strength for different sizes of bars.
#According to ACI 363-R35

^

Materials
Three different concrete mixes were used in casting the test specimens. The mixes were
designed to obtain compressive strengths of 50,75 and 100MPa. Ordinary Portland cement, siliceous
sand, coarse aggregates size 10 mm, Silica fume and super-plasticizer type F [7] were used with the
quantities shown in Table 3 for each mix. Concrete compressive strength, fcu, given in Table 3for
each mix represents the average of six uniaxially loaded standard cubes. Two different sizes of
deformed steel bars were used as flexural reinforcement, 10 and 12 mm in diameter, having yield
stress of 480, 515MPa, respectively. Mild steel with fy=380 MPa, denoted R, was used for stirrups
and top bars.

155

Table 3 Mix Proportions of the Three Concrete Grades

Super
plasticizer
Liter/m3

Actual
compressive
strength, fcu
(MPa)

0.30

10

52

580

0.24

18

74

580

0.20

40

96.5

Series

Cement
(kg/m3)

Silica
fume
Kg/m3

Dolomite
Kg/m3

Sand
Kg/m3

W/C

1

450

15

1180

640

2

575

50

1100

3

750

100

1200

Instrumentation and Test Procedure
Test specimens were instrumented to measure the applied load, mid-span deflection, and strains
of longitudinal reinforcement in the constant moment region. A general view of the instrumentation is
shown in Fig.1. A Linear variable displacement transducer (LVDT) for measuring vertical deflection
was mounted at the bottom side of the midspan for each specimen. Two electrical resistance strain
gauges mounted on the bottom reinforcing bars were used to measure the strains up to yielding. The
locations of the strain gauges are also shown in Fig1. The load was distributed equally by a
spreader beam to two points along the specimen to generate a constant moment region at midspan.
At each load stage, the electrical strain gauges, load cells and (LVDT) voltages were fed into the data
acquisition system. The voltage excitations were read, transformed and stored as micro strains, force,
and displacement by means of a computer program that runs under the Lab View software. All
specimens were tested three months after casting.

HYDRAULIC JACK
P
LOAD CELL

250 R8@150 mm

400

400

2R8

*

As, refer to
Table 2

x

sec.x

1300
100

1300
3300

* strain gauge

250

y

sec.y

100

Elvation

LVDT

Fig. 1: Test Setup and Details of the Tested Beams
156

TEST RESULTS AND ANALYSIS
The adequacy of the flexural reinforcement for the tested beams will be evaluated by
considering both of the reserve strength beyond flexural cracking and displacement ductility index.
Presently, there are no generally accepted criteria for such an evaluation. A reserve strength parameter
can be defined as the ratio of the yield load to the cracking load (Py/Pcr) or as the ratio of the ultimate
load to the cracking load (Pu/Pcr). Since tested beams, having low flexural reinforcement, are under
reinforced, their behavior was expected to be ductile.
Behavior of Test Specimens
In all specimens, flexural cracks were observed first in the constant moment region. As the
load was increased, tension reinforcement yielded, which resulted in a significant increase in the
crack width and the deflection. No shear cracks were observed till the end of the test. Crack patterns
of the tested beams at the end of the test indicate that the final crack width increased as the
reinforcement ratio decreased. Table 4 presents the experimentally obtained cracking, yielding and
ultimate loads for the tested beams. The experimental cracking load, Pcr corresponds to the load at
which the initial cracking was observed on the test specimen and was confirmed by the deviation of
the load-deflection curve. The experimental yielding load, Py, corresponds to the load at which
yielding flat plateau is observed in the load-deflection curve. The experimental ultimate load, Pu is
the peak load reached during testing. The test results showed that the concrete compressive strength
had more influence on the cracking load than the flexural reinforcement ratio, and that the flexural
reinforcement had an obvious influence on the yielding and ultimate loads. At this very low range
of flexural reinforcement ratios for the tested beams, lower or around the ρmin limit of the
ACI 363R-35, it was noted that the initiation, the development, the distribution and the
widths of flexural cracks were highly sensitive to the flexural reinforcement ratio. In
addition, the beams behavior, the ratio between their ultimate to cracking loads were also
sensitive to the flexural reinforcement ratios provided, see in Table 4.
Beams B 5 0 1 , B 7 5 1 a n d B 1 0 0 1 w i t h flexural reinforcement ratio ρ≈ 0.5ρmin,
according to ACI 363R-35 formula, had ultimate loads of45.6, 50.7 and 66.5kN, after the
initiation of the first crack at a load of 37.5, 39.2 and 43.9 kN, respectively. The failure mode
of these beams was characterized by a few flexural wide cracks, as shown in Fig. 2. This
semi-ductile type of failure of t h e s e b e a m s and the low margin between its cracking and
ultimate loads, refer to Table 4, are clearly due to flexural reinforcement ratio which is
much lower than the minimum limit required by the ACI 363R-35.
Beams B 5 0 2 , B 7 5 2 a n d B 1 0 0 2 with flexural reinforcement ratio ρ≈ 075 ρmin,
according to ACI 363R-35, had ultimate loads of 76.3, 81.2and89.3kN, respectively. The first
crack loads were 44.5, 46.1 and 47.8 kN, respectively. Unlike the previous group (B 5 0 1 ,
B 7 5 1 a n d B 1 0 0 1 ) , the failure of these beams (B 5 0 2 , B 7 5 2 a n d B 1 0 0 2 ) was
characterized by a larger number of cracks with smaller widths, as shown in Fig. 2. Although the
flexural reinforcement ratios µ of (B 5 0 2 , B 7 5 2 a n d B 1 0 0 2 ) were still lower than the
minimum limit ρmin required by the ACI 363R-35, it can be noted that a better uniform crack
distribution and a higher margin between the cracking and the ultimate loads were the main
outlines of the behavior, see Table 4 . This gives an initial indication that the ρmin of the ACI 363R35 is overestimated and could be reduced by about 25 % without harmful effect on the beam
behavior.
Other tested beams with flexural reinforcement ratios ≈ ρmin, according to ACI 363R-35,
which are B 5 0 3 , B 7 5 3 a n d B 1 0 0 3 , had ultimate loads of 90.2, 114.9 and 134.3kN,
157

respectively. The cracking loads of these beams were 47.3, 50.5 and 53.4kN. The failure of these
beams was characterized by an increasingly growing number of cracks with smaller widths, as
shown in Fig.2. Such expected increased uniformity in the crack distribution as well as the increased
margin between the cracking and the ultimate loads were due to the increase in the corresponding
flexural reinforcement ratio, ρ that resulted in a better moment distribution and hence, a better beam
behavior. However, it was noted that the ratio of the yield load to the cracking load increased with
increasing the reinforcement ratio, ρ, see Table 4 .
Table 4: Experimental Results of the Tested Beams

Beam

Pcr,
(kN)

Py,
(kN)

Pu,
(kN)

Py,/Pcr

Pu,/Pcr,

∆y
mm

∆f
mm

*λ∆=∆f/∆y

B501
B502
B503

37.6
44.5
47.3

39.4
56.9
83.1

45.6
76.3
90.2

1.05
1.29
1.76

1.21
1.72
1.91

6.7
7.3
8.2

56.3
76.2
91.8

8.4
10.3
11.2

B751
B752

39.3
46.1

45.3
63.5

50.7
81.2

1.15
1.38

1.29
1.76

6.9
7.5

74.5
92.3

10.8
12.3

B753

50.5

103.5

114.9

2.01

2.23

8.6

115.2

13.4

B1001

43.9

53.8

66.5

1.22

1.51

9.8

63.7

6.5

B1002

47.8

78.2

89.3

1.63

1.87

11.8

99.1

8.5

B1003

53.4

113.7

134.3

2.13

2.51

12.6

123.5

9.8

*Displacement Ductility Index

Fig 2-a: Crack Pattern of Series 1
158


Fig 2-b: Crack Pattern of Series 2

Fig 2-c: Crack Pattern of Series 3

Load-Deflection Behavior
The applied load was plotted against the vertical deflection measured at midspan for all tested
beams as shown in Fig3. It may be seen that four distinctly different segments, separated by four
significant events that took place during the loading history, can idealize a typical load-deflection
curve. Labeled as A, B, C and D, these events may be identified as first cracking, yielding of tensile
reinforcement, initiation of concrete crushing and failure of concrete compression zone, respectively.
The first two events were associated with a reduction in beam stiffness, while the remaining two
events led to a reduction in the applied load. In between, a straight line may approximate the curve,
159

Fig. 3-a. The effects of different parameters on the load-deflection behavior of test beams are
presented in Figs. 3-b to 3-d.

C

Applied Load

B
A

ABCD-

F ir s t c r a c k
D
S t e e l y ie ld in g
I n itia ti o n o f c o n c r e te c r u s h in g
C o n c r e te C r u s h in g

D e f l e c t io n

Fig 3-a: Idealized Load- Deflection Curve
100

B503

80

Load, kN

60

B502

40

B501
B501,ρ=0.50ρmin

20

B502,ρ=0.75ρmin
B503,ρ=ρmin

0

0

20

40

60

80

100

Displacement, mm

Fig 3-b: Load-Deflection Relationship for Series 1, fcu=50MPa
120
B753
B751

100

Load, kN

80
B752
B751

60

40
B751
B751

20

B751,ρ=0.50ρmin
B752,ρ=0.75ρmin
B753,ρ=ρmin

0
0

20

40

60

80

100

120

140

Displacement, mm

Fig 3-c: Load-Deflection Relationship for Series 2,fcu=75MPa

160

140
B1003

120

Load, kN

100

80

B1002

60
B1001
40

B1001,ρ=0.50ρmin
20

B1002,ρ=0.75ρmin
B1003,ρ=ρmin

0
0

20

40

60

80

100

120

140

Displacement, mm

Fig 3-d: Load-Deflection Relationship for Series 3,fcu=100MPa
Fig. 4-a shows the reserve flexural parameter (Py,/Pcr) at first yielding beyond the
cracking strength as a function of the f lexural reinf orcement ratio ρ. Generally, it
was noted that the ratio (Py,/Pcr) was increased with increasing f lexural reinforcement
ratio µ. Examining Fig. 4-a, it can be concluded that as the concrete compressive strength
increases a higher reinforcement ratio is required to achieve a specific reserve flexural
capacity. ACI 318-08 requires providing at least one-third more flexural
reinforcement than t h a t required by analysis, i . e . , (Py,/Pcr) ≥1.30. Fig.4-b gives t h e
experimental minimum reinforcement ratios ρmin needed to assure t h i s one-third reserve
strength for the different concrete strength and it can be drawn that providing (ρ/ρmin)=0.75 will
lead to attain a reserve flexural parameter(Py,/Pcr)=1.29 for all concrete grades. In addition, Fig.5
shows the reserve ultimate flexural parameter (Pu,/Pcr)for ultimate load beyond the cracking
strength as a f unction of (ρ/ρmin)and it was observed that this reserve strength
parameter,(Pu,/Pcr,), increases for higher strength concrete.
2.50

Py / Pcr

2.00
1.50
1.00

fcu = 52 Mpa
fcu = 73 Mpa
fcu =96.5 Mpa
ACI318-08 limit

0.50
0.00
1.5

2

2.5
3
3.5
Flexural reinforcement ratio, ρ*10-3

4

4.5

Fig. 4-a: Relationship between the Actual Flexural Reinforcement Ratio ρ and the
Experimental Reserve Flexural Capacity (Py/Pcr)
161

2.50

Py / Pcr

2.00

1.50

1.00

fcu = 52 Mpa
fcu = 73 Mpa
fcu =96.5 Mpa
ACI318-08 limit

0.50

0.00
0.5

0.6

0.7

0.8

0.9

1

1.1

(ρ/ρmin), ACI 363-35

Fig. 4-b: Relationship between the ratio (ρ/ρmin) due to ACI363-R35 and the Experimental
Reserve Flexural Capacity (Py/Pcr)
3.00
2.50

Pu/Pcr

2.00
1.50
1.00

fcu = 52 Mpa
fcu = 73 Mpa

0.50

fcu =96.5 Mpa
0.00
0.5

0.6

0.7

0.8

0.9

1

1.1


Fig. 5: Relationship between the Ratio(ρ/ρmin)According to ACI 363-R35 and the
Experimental Ultimate Reserve Flexural Capacity(Pu/Pcr)
Ductility
Ductility of a structural member may be defined as its ability to deform at or near the
failure load without a significant loss in strength. In the case of a flexural member, sectional
ductility based on curvature and/or member ductility based on deflection is usually considered. In the
present study, the definition of displacement ductility is investigated. Displacement ductility index,
λ∆, may be defined as λ∆= (∆f/∆y) in which ∆f and∆y are the midspan deflections of the beam at
failure and at yielding of the longitudinal tensile reinforcement, respectively [5, 7]. Deflection at
yield load was calculated from the load-deflection curve as the corresponding displacement of the
intersection of the secant stiffness at a load value of 80% of the ultimate lateral load and the tangent
at the ultimate load. Also, failure is assumed to have occurred at a load equal to 80% of the ultimate
load in the descending branch of the load-deflection curve. Table 4 presents the values of the
deflections at yielding of tensile reinforcement ∆y and at failure load ∆f. In general, it was noted that
both of ∆y and∆f increase as ρ increases. Considering the displacement ductility index λ∆in Table 4, it
is shown that, everything else remaining the same, λ∆increased slightly as fcu increased from 52 to
73MPa, but then decreased obviously as fcu increased further from 73 to 96.5MPa, see Fig. 6. The
same trend has been reported by Rashid et al [4].According to ACI 363-R35, it is evident that
increasing f’c leads to increase the minimum reinforcement ratio, ρmin. Thus, for a certain flexural
162

reinforcement ratio, ρ, the ratio (ρ/ρmin) decreases as f’c increases. The variation of the displacement
ductility index, λ∆ , as a function of (ρ/ρmin) for different concrete grades is shown in Fig. 7 where
λ∆increases as (ρ/ρmin) increases. A displacement ductility index, λ∆,in the range of 5 is considered
imperative for adequate ductility, especially in the areas of seismic design and the redistribution of
moments [2,5]. Therefore, assuming that aλ∆value of 5 represents an acceptable lower limit to ensure
the ductile behavior of flexural members, it appears that beams with (ρ/ρmin)≈ 0.70-0.80, according to
ACI 363-R35, would meet that requirement Fig. 7. This gives another indication that the ρmin of the
ACI 363-R35 can be reduced by 25 % with attaining an acceptable ductility index.
14
fcu = 52 Mpa
fcu = 73 Mpa
fcu =96.5 Mpa

Ductility index, λ∆

12

10

8

6
1.5

2

2.5
3
3.5
Flexural reinforcement ratio, ρ*10-3

4

4.5

Fig. 6: Relationship between Flexural Reinforcement Ratio ρ and the Ductility Index λ∆
14


12

10

8
fcu = 52 Mpa
fcu = 73 Mpa

6

fcu =96.5 Mpa
4

0.5

0.6

0.7

0.8

0.9

1

1.1


Fig. 7: Relationship between the Ratio (ρ/ρmin) due toACI 363-R35 and the Ductility Index λ∆
Strains in the Longitudinal Steel Reinforcement
Strain measurements on the longitudinal reinforcing steel bars of all the tested beams
recorded tension steel strain values >4000 µ s at the ultimate load level which indicate that the
longitudinal reinforcement developed yielding, where µ s means micro strain= strain x 10-6 , see
Fig.8 .

163

140

120

100

Load kN

80

60

40
B501
B503
B1003
B753

20

B502
B1002
B751

0
0

2000

4000

6000

8000

10000

Strain *10 -6

Fig. 8: Load- Steel Strain Relationship for all Tested Beams
COMPARISON OF AVAILABLE FORMULAS FOR THE MINIMUM FLEXURAL
REINFORCEMENT IN HSC BEAMS
Fig.9 shows a comparison between the different formulas for the minimum flexural
reinforcement ratio versus the concrete compressive strength and it can be concluded that ACI
363R-35[7] formula represents the upper limit, i.e. the most conservative one, while El-Saie[10]
formula represents the lower limit. Fig.10 represents a comparison between all of the available
limits for the minimum flexural reinforcement ratio in the light of the experimental results and from
the ductility point of view. In this respect, Rizk[4], ACI 363R-35[7],CSA-A23.3-04[8], JSCE
[9],and the El-Saie [10] formulas are considered. If 5.0 is considered as an adequate ductility index
[2,5], then for a singly reinforced section it can be shown that all the formulas of the previous limits
lead to achieve a reasonable ductility index for all concrete tested grades. However, forfcu> 100 MPa
and (ρ/ρmin) =1,El-Saie[10] formula seem to lead to inadequate ductility.
5.00

ρmin*10 -3

4.00

3.00

2.00
ACI363-R35 [7]
JSCE [9]
ELASI [10]

CSA [8]
Rizk [4]

1.00
40

60

80

100

Concrete Compressive Strength, f’c MPa

Fig. 9: Relationship between the Concrete Compressive Strength and ρmin for Available
Formulas
164



12

11

(a) fcu=52MPa
10
ACI363-R35 [7]
CSA [8]
JSCE[9]
Rizk[4]
ELASI[10]

9

8
0.50

1.00

1.50

2.00

2.50

ρ/ρmin

Ductility index, λ ∆

13

12

(b)fcu=73MPa
Rizk[4]
CSA [8]
JSCE[9]
ACI363-R35 [7]
ELASI[10]

11

10
0.50

1.00

1.50

2.00

2.50

ρ/ρmin
11

Ductility index, λ ∆

10

9

(c)fcu=96.5MPa
8
ACI363-R35 [7]
CSA [8]
JSCE[9]
Rizk[4]
ELASI[10]

7

6
0.50

1.00

1.50
ρ/ρmin

2.00

2.50

Fig. 10: Relationship between the Ratio (ρ/ρmin) According to Different formulas and the
Ductility Index λ∆ : (a) fcu=52 , (b) fcu=73 and (c) fcu=96.5 MPa
165

CONCLUSIONS
Within the range of the investigated parameters and properties of the materials used in this
work, the following conclusions could be drawn:
1. The ACI 3 63 R -35 f o r m u l a for minimum flexural reinforcement ratio in HSC can be
reduced by 25% without any harmful affect on the flexural behavior and i t w i l l
achieve an adequate ductility index which may lead to good savings in the amount of
the flexural reinforcement.
2. Also, reducing the l i m i t of ρmin to 75% of the limit specified by the ACI 36 3R -35 will
lead to attain a reserve flexural paramete r (Py,/Pcr)≥ 1.29 for all concrete grades.
3. For the same longitudinal reinforcement, it was observed that the reserve strength parameters,
(Pu,/Pcr,) and (Py,/Pcr,) decrease for higher strength concrete.
4. For the same concrete compressive strength with low values of flexural reinforcement
ratio, ρ, the displacement ductility index increases as ρ increases.
5. The displacement ductility index λ∆ increases as concrete compressive strength increases for
the same ratio (ρ/ρmin) up to 75 Mpa and then decreases as fcu increases.
6. For concrete compressive strength> 100 MPa and (ρ/ρmin) =1, El-Saie [10] f o r m u l a for
minimum flexural reinforcement ratio, which represents50% of the ACI 36 3R - 35
limit, seem to lead to inadequate ductility index, i.e.λ∆<5. However, the other available
formulas achieved a ductility index λ∆>6 for all concrete grades.
ACKNOWLEDGEMENT
The researcher expresses his great gratitude to the staff of both "Building Materials and Quality
Control Institute" and " Reinforced Concrete Institute", Housing & Building National Research
Center for their great effort through all experimental stages of this research.
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