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ORIGINAL CONTRIBUTION
A Comparative Study of Strength of Two-Way Rectangular Slabswith and without Openings
M. Ravindra1 • V. Rakesh2 • K. Rambabu2
Received: 21 March 2016 / Accepted: 26 August 2016 / Published online: 27 September 2016
� The Institution of Engineers (India) 2016
Abstract The present work uses yield-line theory to find
the strength of uniformly loaded rectangular reinforced
concrete slabs with and without rectangular openings. Five
positions of openings are considered, i.e. the slab centre,
the slab corner, the centre of a short side, the centre of a
long side and the opening eccentric to the slab centre. All
possible admissible yield line patterns are considered for
all given configurations of the slab subjected to uniformly
distributed load keeping in view the basic principles of
yield line theory. The ratios of the corresponding lengths of
the sides of the opening and the slab are different and sizes
of opening up to 0.49 the length of the slab sides are
considered. Symmetric edge conditions like continuous
slab, simply supported, two long sides continuous and two
short sides continuous are considered for various sizes of
openings in order to plot the design charts for isotropic
reinforcement coefficients only. Affine transformation is
also performed for slab with openings.
Keywords Affine transformation � Aspect ratio �Isotropic slab � Two-way slabs � Yield-line theory
List of Symbols
Continuous edge
Simply supported edge
Free edge
Negative yield line
Ast Area of steel
b Width of the slab (1000 mm)
CS A slab supported on all sides
continuously (restrained)
d Effective depth
fck Characteristic compressive
strength of concrete
fy Characteristic strength of steel
I1 and I2 Negative moment coefficients in
their corresponding directions
I1mult Negative ultimate yield moment
per unit length provided by top
tension reinforcement bars placed
parallel to x-axis
I2mult Negative ultimate yield moment
per unit length provided by top
tension reinforcement bars placed
parallel to y-axis
Kx1mult Positive ultimate yield moment per
unit length provided by bottom
tension bars placed parallel to X-
axis
Ky1mult Positive ultimate yield moment per
unit length provided by bottom
tension bars placed parallel toY-axis
Lx, Ly Slab dimensions in X and
Y directions respectively
mult Ultimate yield moment per unit
length of the slab
& M. Ravindra
1 Department of Architecture, Andhra University College of
Engineering, Visakhapatnam, India
2 Department of Civil Engineering, Andhra University College
of Engineering, Visakhapatnam, India
123
J. Inst. Eng. India Ser. A (June 2017) 98(1-2):1–14
DOI 10.1007/s40030-016-0176-9
Mu Moment of resistance of a section
Mulim Limiting moment of resistance of a
section without compression
reinforcement
r Aspect ratio of slab defined by
Lx/Lyr1, r2, r3, r4 Non dimensional parameters of
yield line propagation
SS A slab simply supported on all sides
TLC A slab restrained on two long
edges and other two sides simply
supported
TSC A slab restrained on two short edges
and other two sides simply supported
UDL Uniformly distributed load
Wll Live load/imposed load per unit area
Wdl Dead load per unit area
Wult Ultimate uniformly distributed
load per unit area of slab
xumax Limiting value of depth of neutral
axis
a, b Coefficients of opening in the slab
l Coefficient of orthotropy =K0xþI1½ �
K0yþI2½ �
Introduction
Two-way reinforced concrete slabs often contain openings
of considerable size for service lines like ducts, pipes and
other purposes. The ultimate strength of such slabs may be
conveniently determined by using the yield-line theory
proposed by Johansen [1, 2]. Many design codes [3–5]
recommend Yield-Line theory as one of the possible
methods of slab design. Many researchers [6–13] have
produced work equations for uniformly loaded two-way
rectangular slabs with and without openings for different
edge conditions. Five possible positions of the opening are
treated (Fig. 1) the slab centre, the slab corner, the centre
of a short side, the centre of a long side and an opening
eccentric to the slab centre with various size of openings
for Continuous Slab (CS), Two Short edges Continuous
(TSC), Two long edges continuous (TSC), and simply
supported (SS) slab. Strength against aspect ratio of slabs is
considered in plotting the design charts for isotropic
coefficients only.
The concept of yield-line analysis was first presented
by A. Ingerslev in 1921–1923, K. W. Johansen devel-
oped modern yield-line theory. The method involves
postulating a yield-line pattern (failure mechanism)
which is compatible with the boundary conditions and
then using the principle of virtual work to compute the
ultimate load carrying capacity. However, due to the
upper-bound nature of the yield-line method, a range of
yield-line patterns will often need to be explored. Yield-
line theory is particularly suitable for obtaining an ulti-
mate limit-state solution for an irregularly shaped slab,
slabs with openings and slabs of various shapes [1].
Hillerborg’s strip method [1, 3, 6, 14] which is a lower
bound method can also be used to compute the ultimate
load carrying capacity.
Lx
Ly
Lx
Ly
(a) (b) (c)
(d) (e) (f)
Lx
LyαLx
βLy
Lx
LyαLx
βLy
αLx
βLy
Lx
LyαLx
βLy
Lx
LyαLx
βLy
I1
I2
I3
I4
Fig. 1 Slabs with and without Openings. a Solid slab, b central opening, c corner opening, d Opening at centre of short side, e opening at centre
of long side, f opening eccentric to the slab centre
2 J. Inst. Eng. India Ser. A (June 2017) 98(1-2):1–14
123
Virtual Work Equations: Formulation of VirtualWork Equations for Slab with Different Openings
There are several possible yield line patterns associated
with different openings of the slab (Appendix A). For any
opening of slab, all the possible admissible failure yield line
patterns are considered keeping in view the basic principles
of yield line theory. These admissible failure yield line
patterns are obtained basing on the yield line principles
[1, 6, 14] for the given configuration of the slab. These
failure patterns and corresponding equations may be
investigated using a computer program. In order to solve the
complicated virtual work equations, a computer program
was developed in FORTRAN for all the cases separately
which gives the least value ofWultLy2/mult for the given input
data (Kx1, Ky
1, I1, I2, I3 and I4). It is difficult to solve virtual
work equations to analyse and design orthogonal slabs for
every input data. The design charts presented in this paper
simplifies the analysis and design of orthogonal slabs.
Many researchers [6–13] have produced work equations
for uniformly loaded two-way rectangular slabs with and
without openings for different edge conditions. The slab is
subjected to ultimate uniform distributed load (Wult) and
supported on different boundary conditions. Note that the
slab is not carrying any load over the area of the opening.
The ultimate load equations are derived for the assumed
possible admissible failure yield patterns using the virtual
work method for continuous edge (CS) condition of slab.
To get equations for other edge conditions of the slabs,
modifications have to be carried out in the numerator of the
work equations [9–13].
Affine Transformation (Using Theorems VIand VII [1, 6, 14])
‘Affine transformation’ based on theorems VI and VII of
Johansen [1, 6, 14] is a technique that transforms an
orthogonal slab into that of an equivalent isotropic slab,
whose strength is same as that of original orthotropic slab.
The transformation of orthotropic slab to that of related
isotropic slab is presented with specific examples. In order
to perform the transformation, charts are required and these
charts are prepared using isotropic (affine) reinforcement
(coefficients), which means that in each direction the
reinforcement is same and unity i.e. Kx1 = Ky
1 = I1 =
I2 = I3 = I4 = 1.0. These charts show the minimum value
ofWultLy2/mult versus aspect ratio of slab for various sizes of
openings. A computer program is developed in FORTRAN
for all the cases separately which gives the least value of
WultLy2/mult for affine coefficients (Kx
1 = Ky1 = I1 =
I2 = I3 = I4 = 1.0).
According to theorems VI and VII of Johansen
[1, 6, 14], any orthogonal slab can be transformed into an
equivalent related isotropic slab provided that the ratio of
negative to positive moments in the slab is same. When the
ratio of negative to positive moment in the orthogonal slab
is same and unity, then such a slab can be transformed into
an equivalent isotropic slab directly as shown in Fig. 2b,
which means that the given orthogonal slab solution can be
obtained by analyzing the isotropic slab with modified
dimensions in the X-direction.
Example Transform an orthotropic continuous slab to an
equivalent isotropic slab in which the ratio of negative
moment to positive moment in both directions is same and
unity using affine theorem.
Since I1/Kx0 = I2/Ky
0 = 1.0 (refer Fig. 2), the transfor-
mation of the given orthotropic slab (Fig. 2a) to an
equivalent isotropic slab (Fig. 2b) by simply dividing the
longer span with square root of coefficient of orthotrophyffiffiffi
lp� �
as per Theorems VI and VII of Johansen [1]. This
principle is illustrated in Fig. 2 using the above method-
ology. Using affine transformation the value of WultLy2/mult
can be obtained from design charts.
Development of Analysis and Design curves
Design charts have been prepared for continuous slab (CS),
two short edges continuous (TSC), two long edges contin-
uous (TLC), and simply supported (SS) slab for all type of
the openings and are shown in Appendix A. Design charts
(Chart 1–64 of Appendix A) are plotted for WultLy2/mult
against aspect ratio, r for different size of openings. These
charts are prepared based on affine coefficients (Kx1 =
Ky1 = I1 = I2 = I3 = I4 = 1.0). The values of r are taken
between 1.0 and 2.0 and sizes of opening of up to 0.4 of the
length of the slab sides are considered. Since two-way slabs
are considered, the aspect ratio of slab, r is limited to 2.0.
With increase in value of r, the value ofWultLy2/mult of a slab
decreases. And size of opening is limited to 0.4 so that the
slab behaviour doesn’t change to cantilever action from
two-way action particularly near openings. If the sizes of
opening are changed the value of WultLy2/mult of a slab may
increase or decrease depending upon the type of opening.
While preparing the design charts, the least value of
WultLy2/mult given by all the failure patterns is considered
for the corresponding opening. For example in case of a CS
slab with opening a = 0.1 and b = 0.4 (Chart 4 of
Appendix A), it consists of six lines. Each line represents
the least value of WultLy2/mult of a slab with opening’s (five
nos.) and without opening (solid slab). Using these charts
one can directly design/analyze a slab with an opening at
J. Inst. Eng. India Ser. A (June 2017) 98(1-2):1–14 3
123
different locations. If the designer knows the strength of the
solid slab, one can directly read the strength of the slab
with opening from these charts.
Analysis and Design Problems
Analysis Problem
Determine the safe uniformly distributed load on a rect-
angular simply supported (SS) two way slab with corner
opening for the following data:
A slab 6 m 9 5 m with corner openings of size
1.2 m 9 0.5 m is reinforced with 10 mm diameter bars
@200 mm c/c perpendicular to long span and 10 mm
diameter bars @178 mm c/c perpendicular to short span.
Thickness of the slab is 120 mm. Characteristic strength of
concrete is 20 MPa and yield stress of steel is 415 MPa.
Solution: As per IS 456:2000 [5],
Mu ¼ 0:875fyAstd 1� fyAst
fckbd2
� �
ð1Þ
Assuming thickness of the slab = 120 mm and effective
cover of slab = 15 mm.
Effective depth of slab in short span direction =
100.00 mm.
Effective depth of slab in long span direction =
90.00 mm.
Area of the steel perpendicular to long span =
392.7 mm2.
Area of the steel perpendicular to short span =
441.23 mm2.
The ultimate moments in short and long span directions
can be found using the Expression (1).
Therefore,
Mu parallel to long span = K0ymult = 14.144 kN-m/m
Mu parallel to short span = K0xmult = 14.273 kN-m/m
Assume K0y = 1.0, then K0
x = 14.144/14.273 =
0.991 & 1.0.
For aspect ratio of slab, r ¼ 6:05:0 ¼ 1:2 and taking
mult = 14.144 kNm/m, the orthogonal coefficients will be
K0x = I1 = K0
y = I2 = 1.0.
From Chart 53 of Appendix A for a = 0.2, b = 0.1 and
r = 1.2, we get WultLy2/mult & 38.7
Wult ¼ð38:7� 14:144Þ
4:02¼ 34:211 kN/m2
Wdl = (dead load including finishing) = (0.12 9 25) ?
0.5 = 3.5 kN/m2
Wult ¼ 1:5 Wll þWdlð Þ ¼ 34:211 kN/m2
Wll ¼34:211
1:5� 3:5 ¼ 19:307 kN/m2
The intensity of live load on the slab is 19.307 kN/m2.
Design Problem
Design a Continuous Slab (CS) of 4.8 m 9 3.84 m with
short side openings of 0.96 m 9 1.152 m to carry a uni-
formly distributed live load of 4.5 kN/m2. Use M20 mix
and Fe 415 grade steel.
Given data:
Aspect ratio of slab(r) = Lx/Ly = 4.8/3.84 = 1.25,
aLx = 0.96 m, bLy = 1.152 m, a = 0.2 and b = 0.3.
Assuming K0x = I1 = 0.5, K0
y = I2 = 1.0. The value of
WultLy2/mult is obtained by performing affine transformation.
WultLy2/mult = 31.060 (refer Sl. No. 2 of Table 1)
Assuming overall thickness of slab = 110 mm
Lx=7
αLx=2.βLy=2
Ly=
βLy=2αLx=3.0
Ly=6
Lx=10.1
(a) (b)K1
x=I1=0.5, K1y =I2=1, α=0.3, β=0.2,
I1/ K1x = I2/ K1
y = 1.0, μ = 0.5,r=Lx/Ly=7.2/6=1.2,∑K=3.0Inorder to check on affine theorem a computer program is used to evaluate the value of WultLy
2/mult and the value is 32.543
As per affine theorem the transformed factors are:K1
x =I1=, K1y=I2=1, α=0.3, β=0.2,
Lx=Lx/√μ=7.2/√(0.5)=10.182m, r=10.182/6≈1.7,μ = 1.0,∑K=4.0, The value of WultLy
2/mult is obtained from Chart 4, for r=1.7.Also computer program gives the same value for r=1.697
Fig. 2 Affine transformation for continuous slab (CS). a Orthotropic slab, b equivalent isotropic slab
4 J. Inst. Eng. India Ser. A (June 2017) 98(1-2):1–14
123
Dead load of slab = 120 9 25 = 3.0 kN/m2
Dead loads including finishing’s = 4.5 kN/m2
Total load = 9.0 kN/m2
Ultimate total load = 1.5 9 9.0 = 13.5 kN/m2
mult ¼13:5� 3:842
31:060¼ 6:409 kNm/m
The orthogonal moments are
K0x mult = I1 mult = 0.5 9 6.409 = 3.205 kNm/m
K0y mult = I2 mult = 1.0 9 6.409 = 6.409 kNm/m
As per IS 456:2000 [5],
Mu lim ¼ 0:36xumax
d
� �
1� 0:42xumax
d
� �� �
bd2fck ð2Þ
6:409� 106 ¼ 0:36 0:48ð Þ 1� 0:42 0:48ð Þð Þð1000Þ d2� �
20
) d ¼ 48:195mm ) d ¼ 48:195mm
Effective depth required, d = 48.195 mm
Adopt cover 15 mm and effective depth as 100 mm and
overall depth as 120 mm
Positive/Negative short span moment = 6.409 kNm/m,
Ast (required) = 205.88 mm2
Provided 10 mm diameter bars @300 mm c/c =
261.80 mm2
Positive/negative long span moment = 3.205 kNm/m,
Ast (required) = 89.94 mm2
Provided 8 mm diameter bars @300 mm c/c =
167.55 mm2
Conclusions
• Design charts for two-way slabs with openings at dif-
ferent locations for four edge conditions are presented
for different size of opening and different aspect ratios.
• The charts developed can help any design engineer or
architect to pick up once the choice of location of
opening depending upon the plan or required strength
criteria. A comparative study of ultimate strength of
these cases is presented with the help of charts.
• The charts can be used either for analysis or design of
two-way slabs with different size of openings and
different openings.
• Few numerical examples are presented based on
theorem of VI and VII of affine theorem of Johansen
[1] for orthotropic slabs with unequal openings.
• Any designer can also select the slab with other
opening for the same strength of the given original
opening of the slab for different boundary conditions.
• The strength of corner opening slab is less compared to
other slabs irrespective of the size of the opening,
aspect ratio in case of continuous slab and two long
sides continuous slab (except few cases).
Table 1 Affine transformation examples for continuous slab
Example (CS) Orthogonal moment
co-efficient
Aspect
ratio, r
Strength
WultLy2/mult
Aspect
ratio, r*
Refer
chart* no.Sl. no. Openings
1 a = 0.4, b = 0.2 Kx1 = 0.25, Ky
1 = 1.0,
I1 = 0.25, I2 = 1.0,
l = 0.25,P
K = 2.5
1.0 29.320 2.0 14
2 a = 0.2, b = 0.3 Kx1 = 0.5, Ky
1 = 1.0,
I1 = 0.5, I2 = 1.0,
l = 0.5,P
K = 3.0
1.25 31.060 1.77 7
3 a = 0.1, b = 0.4 Kx1 = 0.67, Ky
1 = 1.0,
I1 = 0.67, I2 = 1.0,
l = 0.67,P
K = 3.34
1.2 33.626 1.50 4
4 a = 0.3, b = 0.4 Kx1 = 1.5, Ky
1 = 1.0,
I1 = 1.5, I2 = 1.0,
l = 1.5,P
K = 5.0
1.5 41.328 1.22 12
5 a = 0.2, b = 0.3 Kx1 = 2.0, Ky
1 = 1.0,
I1 = 2.0, I2 = 1.0,
l = 2.0,P
K = 6.0
1.7 46.637 1.20 7
6 a = 0.4, b = 0.4 Kx1 = 3.0, Ky
1 = 1.0,
I1 = 3.0, I2 = 1.0,
l = 3.0,P
K = 8.0
1.8 42.7 1.04 16
r* equivalent isotropic slab aspect ratio, I1/Kx1 = I2/Ky
1 = 1.0, I1 = I3, I2 = I4
* Charts are shown in Appendix A
J. Inst. Eng. India Ser. A (June 2017) 98(1-2):1–14 5
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• The variation in strength of the slab is increasing with
increase in size of opening when compared to the
maximum and minimum value at a particular aspect
ratio for all the four edge conditions.
Acknowledgment The authors acknowledge sincere thanks to
Department of Civil Engineering, Andhra University College of
Engineering, Visakhapatnam, India for their continuous encourage-
ment and valuable suggestions.
Appendix A
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