Ijciet 10 02_041

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International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 02, February 2019, pp. 382-392, Article ID: IJCIET_10_02_041
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=10&IType=02
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication Scopus Indexed
SUGGESTING DEFLECTION EXPRESSIONS
FOR RC 2-WAY SLABS
Khattab Saleem Abdul-Razzaq, Abbas H. Mohammed and Taha K. Mohammedali
Department of Civil Engineering, University of Diyala, 32001, Diyala, Iraq
ABSTRACT
The purpose of the experimental work presented in this study is to study the effect
of concrete compressive strength and steel reinforcement ratio on capacity and
deflection of reinforced concrete two-way slabs. Three steel reinforcement ratios are
considered which are minimum, maximum and average of them in addition to two
concrete compressive strength values of 20 and 30 MPa. The results from
experimental work show that increasing the reinforcing steel ratio leads to increase the
ultimate capacity of the slab in addition to decrease the maximum deflection. For slabs
with = 20 MPa, increasing the reinforcing steel ratio from the minimum to the
maximum, i.e. 600 %, leads to increase ultimate capacity by about 156 % and decrease
maximum deflection by about 52 %. Wheras, For slabs with = 30 MPa, increasing
the reinforcing steel ratio from the minimum to the maximum, i.e. 900 %, leads to
increase ultimate capacity by about 155 % and decrease maximum central deflection
by about 27 %. In addition, matmatical expresions for load-deflection relationships are
presented in the current study.
Keyword head: Two-way slab, Deflection, RC, simply supported, ultimate capacity,
expression.
Cite this Article: Khattab Saleem Abdul-Razzaq, Abbas H. Mohammed and Taha K.
Mohammedali, Suggesting Deflection Expressions for Rc 2-Way Slabs, International
Journal of Civil Engineering and Technology, 10(02), 2019, pp. 382–392
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=02
1. INTRODUCTION
Engineering structures are constructed from materials that deform slightly when subjected to
stressing. As a result of thisideformation, the structures undergo certain movement called
deflection. When the elastic limit of the material is not exceeded and the resulting deflections
disappear in case of stress removing, this type of deformation or deflection is called elastic,
otherwise, it is inelastic.

On of the most important designicriteria is the control of deflections andicracking at service
loads, i. e. serviceabilityirequirements. For crack control, limiting theispacing of bars and the
minimumiarea of reinforcement can avoid excessiveecracking of slabs.
For deflectionicontrol, the selected slab thickness (usuallyibased on practicality) should be
checkediso that the calculatedddeflections are within allowable limits. The calculation
ofideflections (short-term) of reinforced concrete (RC) slabs issunpopular with
designerssbecause of the complexitiesiinvolved in performing such calculations [1].
In concrete structures, deflections are divided into two categories; immediate and long-
term. ACI-318 provides limits and methods to calculate both kinds of these deflections.
Immediate deflections are those that occur as soon as the load is applied to the structure. They
are calculated using usualimethods or formulae for elasticcdeflections as provided by structural
analysis. They may be used either with a constant value of Ec * Ig (stiffness) along the length
of the member for uncracked members or by more exact means of calculation for members
cracked at one or more sections or even if the member depth varies along the span [2].
In the last decades, a variety of studies [3-13] have focused on the study of the factors
affecting the capacity and deflections of RC two-way slabs. Hossain et al. [3] tested a series of
two-way edge-supported slabs. All the slabs were 214 cm square and 8 cm thick. Two slabs
(S-8 and S-11) from this series were analyzed. S-8 and S-11 were isotopically reinforced with
10 mm bars to provide 5.24 and 4.36 cm2 /m steel in each direction, respectively. The two
slabs differ in concrete strengths which were 15.9 for S-8 and 22.0 for S-11 MPa. Short-term
load-deflection results from finite element analysis and experiment were compared for the
central nodes of two slabs.
Husain et al. [4] tested nine orthotropically RC rectangular slabs having various boundary
restraints at the edges under uniformly distributed load. All the slab models were rectangular.
The slab models were reinforced with 2.6 mm diameter smooth steel wires. The ratio of steel
reinforcement provided in the short span was + = 0.36 % for positive moment region and -
= 0.72 % for negative moment regions. For the long direction, the ratio of steel reinforcement
was + =0.26 % for positive moment region and - b=0.52 % for negative moment regions.
The main aim of these tests is to show that when some or all edges of a slab are restrained
against rotation and horizontal translation, the ultimate load carrying capacity of the slab will
be enhanced greatly above that suggested by the simple Johansen’s yield line theory [5]. The
results of tests show that for restrained slabs, the enhancement in load above Johansen’s load
ranges between 50 % and 100 % depending on the number and positions of the slab restrained
edges.
The purpose of the current experimental work presented in this study is to study the effect
of concrete compressive strength and steel reinforcement ratio on capacity and deflection of
RC two-way slabs. Three steel reinforcement ratios are considered which are minimum,
maximum and the average of them, besides two concrete compressive strengths are used; once
= 20 MPa and the second = 30 MPa. Load-deflection relationships are suggested in order
to be used by the designing engineers quickly and easily.
2. EXPERIMENTAL PROGRAM
In the current study, six two-way slab specimens have been cast in the Structural Engineering
Laboratory, College of Engineering at the University of Diyala. The dimensions of the tested
slabs are 1000*1000*80 mm with concrete cover of 13mm. Every specimen is loaded
concentrically through its center. The slabs are designed to fail in flexure. They were tested as
simply supported, Figure 1.

Suggesting Deflection Expressions for Rc 2-Way Slabs
Figure 1 Loading and geometry of the tested slabs
The specimens are divided into two groups; A and B. Every group has a different concrete
compressive strength; 20 MPa and 30 MPa. Every specimen in each group has a different
reinforcement steel ratio; minimum, maximum and the average of them. Table 1 shows the
specimens in detail.
Table 1 Experimental work plan
Group Designation
Dimension
s
(m)
(MPa)
Bottom Reinforcement
Top
Reinforcemen
t
Steel
Reinforcement
Steel
Ratio
(ρ)
A
S20-min
1*1*0.08
20
6(φ5) 0.00147 6(φ5)
S20-ave 7(φ10) 0.00687 7(φ5)
S20-max 8(φ12) 0.0113 8(φ5)
B
S30-min
30
7(φ5) 0.00171 7(φ5)
S30-ave 9(φ10) 0.00883 9(φ5)
S30-max 11(φ12) 0.01555 11(φ5)
Three sizes of steel bars are used. Bars of φ5 mm are used as a bottom mesh reinforcement
with 13 mm concrete cover for the two specimens S20-min and S30-min. For the other two
specimens S20-avg and S30-avg, φ10 mm bars are used as a bottom mesh reinforcement. For
the last two specimens S20-max and S30-max, φ12mm bars are used. Bars of φ5 mm size are
used for top mesh reinforcement with 13 mm concrete cover for all slabs.
As mentioned earlier, two normal concrete mixes were made in the current study. The first
was 20 MPa compressive strength, while the second was 30 MPa. Table 2 shows the mix
design.

Table 2 Details of the concrete mixes used in slabs
Group
(MPa)
Cement
(kg/m3
)
Sand
(kg/m3
)
Gravel
(kg/m3
)
Water
content
(kg/m3
)
W/C
Ratio
A 20 375 750 792 206.25 0.55
B 30 400 750 792 204.8 0.512
The specimens are vibrated by rod vibrator. The cubes and prisms molds are filled with
concrete in two equal layers, while the cylinder specimens are cast in equal three layers. Figure
2 shows the reinforcement of slab S30-ave as a reinforcement cage sample.
Each slab is loaded directly at the top face with a concentrated load. A circular bearing
plate of 110 mm in diameter is used at loading point to avoid local direct load concentration,
Figure 1. The displacement control loading incrimination technique has been adopted in this
study.
a) Bottom reinforcement b) Top reinforcement
Figure 2 Reinforcement of slab S30-ave
3. TEST RESULTS AND DISCUSSION
3.1. Compressive Strength Test
150*300 mm cylinders and 150 mm cubes specimens have been used to determine the
compressive strength of concrete. The compressive tests have been done according to ASTM
C39 [14] and B.S 1881 [15] using MATEST compression machine, with capacity of 2000 kN
at loading rate of 5.3 kN/sec for cylinders and 6.8 kN/sec for cubes. The average of three
cylinders and cubes were recorded. These tests have been conducted at the age of 28 days. The
average compressive strengths are listed in Table 3.
3.2. Splitting Tensile Strength Test
The splitting tensile strength test has been carried out according to ASTM C496-2004 [16]
using MATEST machine at 2.1 kN/sec rate load. Cylindrical specimens of 150*300mm have
been used for this test. The average of three cylinders has been recorded. This test is conducted
at the age of 28 days. The values of the tensile strength are listed in Table 3.

Table 3 Compressive, splitting tensile and flexural strength tests
Group
Specimen
Designation
Compressive
Strength f'c, MPa
Splitting tensile
Strength ft,
MPa
Flexural
Strength fr,
MPa
*Ec ,
MPa
A
S20-min
20.5 2.8 4.2 21280S20-ave
S20-max
B
S30-min
30.6 3.6 5.1 26000S30-ave
S30-max
* = 4700 ′
3.3. Flexural Strength Test
Concrete flexural strength has been measured on two prisms of 100*100*500mm dimensions
in conformity with ASTM 293-2004 [17]. Using an electrical ELE machin, the prisms have
been tested under two point load with 0.2 kN/sec load rate and 600 kN capacity, Table 3.
3.4. Crack and Deflection Behavior of Slabs
Table 4 shows the test results in detail.
Table 4 Two-way slab test results
Group
Specimen
Designatio
n
Ult.
Load
,
Pu
(kN)
Increase
in
Pu (%)
Mid-span
Deflection
(mm)
Decrease
in Mid-
span
Deflection
(%)
Failure
Type
A
S20-min 45 - 26.99 - Flexural
S20-ave 81 80 14.58 46 Punching
S20-max 115 155.6 13.02 51.8 Punching
B
S30-min 55 - 20.27 - Flexural
S30-ave 115 109.1 15.32 24.4 Punching
S30-max 140 154.5 14.81 27 Punching
The load-deflection curves for all slab specimens of Group A were almost similar as shown
in Figure 3. Nevertheless, changing reinforcing steel ratio had changed the mode of failure
from flexural (minimum reinforcement) to punching (average and maximum reinforcement).
All slab specimens showed flexural cracking that began near the load stub center and spread
towards the corners of the slab specimens through approximately 45 degrees angles, Figure 4.
It is seen in Table 4 that load capacity Pu is higher and mid-span deflection is less for S20-max
than that for S20-min by about 155.6 % and 51.8 %, respectively. Besides, S20-min has the
minimum load capacity and the maximum deflection by about 80 % and 46 %, respectively
when compared with S20-ave.

Failure occurred by punching failure at loading point for the specimens S20-max and S20-
ave, while flexural failure took place for the specimen S20-min.
Figure 3 Load- deflection curves for Group A slabs
S20-min S20-ave S20-max
Figure 4 Crack propagation and failure mode for all slabs of Group A
The load-deflection curves for all slab specimens of Group B are shown in Figure 5.
Changing reinforcing steel ratio has changed the mode of failure from flexural (minimum
reinforcement) to punching (average and maximum reinforcement). All slab specimens showed
flexural cracking that began near the load stub center and spread towards the corners of the
slab specimens through approximately 45 degrees angles, Figure 6. It is seen in Table 4 that
load capacity is higher and mid-span deflection is less for S30-max than that for S30-min by
154.5 % and 27 %, respectively. Besides, S30-min has the minimum load capacity and the
maximum mid-span deflection by about 109.1 % and 24.4 %, respectively when compared with
S30-ave.
Failure occurred by punching failure at loading point for the specimens S30-max and S30-
ave, while flexural failure took place for the specimen S30-min.

Figure 5 Load- deflection for Group B slabs
S30-min S30-ave S30-max
Figure 6 Crack propagation and failure mode for Group B slabs
3.5. Effect of Compressive Strength and Steel Reinforcement Ratio on Capacity
and Deflection of Slabs
From Figure 7, it is shown that increasing compressive strength from 20 MPa to 30 MPa, i.e.
50 %, had affected the values of load and deflection as follows:
• For slab with minimum reinforcement ratio, increasing compressive strength leads
to increase load capacity and decrease deflection by about 22.2 % and 24.9 %,
respectively.
• For slab with average reinforcement ratio, increasing compressive strength leads to
increase load capacity and deflection by about 41.97 % and 5.1 %, respectively.
• For slab with maximum reinforcement, increasing compressive strength leads to
increase load capacity and deflection by about 21.74 % and 13.75 %, respectively.
0
20
40
60
80
100
120
140
160
0 5 10 15 20 25
Pu(kN)
Centaral Deflection (mm)
S30-max
S30-ave
S30-min

Figure 7 Load- deflection curves for slabs
3.6. Effect of steel reinforcement ratio on capacity and deflection of slab
specimens
Figures 8 and 9 show the load-steel reinforcement ratio relationships for Groups A and B slabs,
respectively. From these Figures, it can be seen that:
• For Group A slabs, increasing steel reinforcement ratio from minimum to average,
i.e. 300 %, leads to increase load capacity and decrease deflection by about 80 %
and 46 %, respectively.
• For Group A slabs, increasing steel reinforcement ratio from minimum to
maximum, i.e. 600 %, leads to increase load capacity and decrease deflection by
about 155.6 % and 51.8 %, respectively.
• For Group B slabs, increasing steel reinforcement ratio from minimum to average,
i.e. 450 %, leads to increase load capacity and decrease deflection by about 109.1
% and 24.4 %, respectively.
• For Group B slabs, increasing steel reinforcement ratio from minimum to
maximum, i.e. 900 %, leads to increase load capacity and decrease deflection by
about 154.5 % and 27 %, respectively.
Figure 8 load-steel reinforcement ratio relationships for Group A slabs
0
20
40
60
80
100
120
140
160
0 5 10 15 20 25 30
Pu(kN)
Central Deflection (mm)
S20-min
S20-ave
S20-max
S30-min
S30-ave
S30-max
0
20
40
60
80
100
120
140
0 0.005 0.01 0.015
Pu(kN)
Steel reinforcement ratio

Figure 9 load-steel reinforcement ratio relationships for Group B slabs
3.7. Suggested relationships for load- deflection:
The load-deflection curves that shown in Figure 7 are transformed to mathematical expressions
as shown in Table 5. These suggested expressions can be used quickly and easily by the
designing engineers.
Table 5 Suggested Load-Deflection relationships
Group
Specimen
Designatio
n
Load-Deflection relationships R2
A
S20-min P=45.70997+(0.1676266-45.70997)/(1+(∆/1.548623)1.254945
)1.02019
) 0.9966
S20-ave P = 0.0008∆5
- 0.0311∆4
+ 0.4287∆3
- 2.6228∆2
+ 14.219∆ + 0.4984 0.9985
S20-max P = -0.0123∆4
+ 0.3004∆3
- 2.275∆2
+ 14.503∆ + 1.0309 0.9976
B
S30-min P = -0.002∆4
+ 0.0992∆3
- 1.7266∆2
+ 13.805∆ + 2.6931 0.9919
S30-ave P = -0.0391∆3
+ 0.5604∆2
+ 7.9818∆ + 2.553 0.9998
S30-max P = -0.0027∆4
+ 0.0605∆3
- 0.1738∆2
+ 7.3352∆+ 0.6566 0.9998
Where:
(P ) is the load (kN)
(∆) is the deflection at mid-span (mm)
4. CONCLUSIONS
An experimental program was presented in this study. The question was to investigate the
ultimate strength capacity and maximum deflection of RC slabs with different reinforcing steel
ratios according to ACI 318-14. The variable that is taken into consideration in the
experimental program of this study is using minimum steel reinforcement ratio ,
maximum steel reinforcement ratio and the average steel reinforcement ratio ( ).
These different ratios are repeated two times with different concrete compressive strengths;
once with f = 20 MPa and the second with f = 30 MPa.
The following conclusions are drawn based on the experimental investigations:
0
20
40
60
80
100
120
140
160
0 0.005 0.01 0.015 0.02 0.025
Pu(kN)
Steel reinforcement ratio

1- When f = 20 MPa, increasing the reinforcing steel ratio leads to increase the ultimate
capacity of the slab in addition to decrease the maximum deflection. Therefore, increasing the
reinforcing steel ratio from the minimum to the average of the minimum and the maximum,
i.e. 300 %, leads to increase ultimate capacity by about 80 % and decrease maximum deflection
by about 46 %. From the other hand, increasing the reinforcing steel ratio from the minimum
to the maximum, i.e. 600 %, leads to increase ultimate capacity by about 155.6 % and decrease
maximum deflection by about 51.8 %.
2- When f = 30 MPa, increasing the reinforcing steel ratio from the minimum to the
average of the minimum and the maximum, i.e. 450 %, leads to increase ultimate capacity by
about 109.1% and decrease maximum deflection by about 24.4 %. From the other hand,
increasing the reinforcing steel ratio from the minimum to the maximum, i.e. 900 %, leads to
increase ultimate capacity by about 154.5 % and decrease maximum central deflection by about
27 %.
3- The load-deflection curves for the 20 and 30 MPa concrete slabs that reinforced from
minimum to maximum reinforcement ratios of ACI 318-14 are transformed to mathematical
expressions. These suggested expressions can be used quickly and easily by the designing
engineers.
REFERENCES
[1] Al-Nu'man, B. S., Analytical Model for Estimating Long-Term Deflections of Two-Way
Reinforced Concrete Slabs. Journal of Engineering and Sustainable Development, 11(1),
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[2] Gullapalli, A. V. L., ACI 318 Code Provisions for Deflection Control of Two-way Concrete
Slabs. 2009.
[3] Hossain, T. S., Uddin, T. and Siddiquee, S., Prediction of short and long-term deflection of
two-way edge-supported reinforced concrete slabs using artificial neutral network. Journal
of Civil Engineering, 35(2), 2007, pp. 119-128.
[4] Husain, H. M., AL-Hassani, H. M. and AL-Badri, S. S. A. Q., Experimental tests on
orthotropically rectangular slabs having various restrained edges and subjected to uniform
load. Engineering and Technology Journal, 27(5), 2009, pp.913-929.
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of Structural engineering, ASCE, 122(2), 1996, pp. 150-159.
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RC slabs. The Indian Concrete journal, 78(7), 2004, pp 19-25.
[9] Sarkar P., Govind M. and Menon D., Estimation of Short-term deflections in Two-way RC
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[10] Scanlon, A. and Murray, W. D., Practical calculation of Two-way slab deflection, Concrete
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slabs. Canadian Journal of Civil Engineering, 25(3), 1998, pp.451-466.

[12] Varma, M., and Pendharkar, U., Equivalent load method for Short-term Deflection of
Simply Supported Two-way RC Slabs. International Journal for Earth Sciences and
Engineering, 3(3), 2010, pp 685-690.
[13] Kadhum, A. K. and Khattab Saleem Abdul-Razzaq, Effect of Seismic Load on Reinforced
Concrete Multistory Building From Economical Point of View. International Journal of
Civil Engineering and Technology (IJCIET), 9(11), 2018, pp. 588–598.
[14] ASTM C39, Test Method for Compressive Strength of Cylindrical Concrete Specimens.
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[15] B.S.1881, part116, Method for Determination of Compressive Strength of Concrete Cubes.
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Specimens. Annual Book of ASTM Standards, Vol. 04-02, 2004, pp.259-262.
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