the effect of interior corner opening in orthotropic ... · other researchers are used stepped...
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International Journal of Civil Engineering and Technology (IJCIET) Volume 9, Issue 9, September 2018, pp. 1113–1123, Article ID: IJCIET_09_09_107
Available online at http://www.iaeme.com/ijciet/issues.asp?JType=IJCIET&VType=9&IType=9
ISSN Print: 0976-6308 and ISSN Online: 0976-6316
© IAEME Publication Scopus Indexed
THE EFFECT OF INTERIOR CORNER
OPENING IN ORTHOTROPIC RECTANGULAR
REINFORCED CONCRETE SLAB UNDERTHE
CONDITION OF OLD AND TAC USING YIELD
LINE THEORY
D.Subrahmanyam*, V.Divya Sri, Dr.G.Thirupathi Naidu
Department of Civil Engineering Aditya Institute of Technology and Management College,
Tekkali, Srikakulam, Andhrapradesh, India
ABSTRACT
Attempt is made to determine collapse load on orthotropic slab with shorter edge
interior corner opening by modified affinity theorem and principles of yield line theory
possible yield line patterns considered for one long edge discontinues(OLD) and two
adjacent edges continues (TAC)subjected to uniformly distributed load (udl) by
computer programme to minimize the equations of virtual work. A methodology has
been brought out with the segment equilibrium method with various conditions of the
slab are used to analyze the slab and varying of strength with respect to aspect ratio is
shown in Tables and charts. For two different conditions of slab for interior corners
openings are presented using affine theorem.
Keywords: One long edge discontinues (OLD), Aspect ratio, Shorter side Corner
opening, two adjacent edges continues (TAC), Modified affinity theorem, Orthotropic
slab.
Cite this Article: D.Subrahmanyam, V.Divya Sri and Dr.G.Thirupathi Naidu, The
Effect of Interior Corner Opening In Orthotropic Rectangular Reinforced Concrete Slab
Under the Condition of Old and TAC Using Yield Line Theory, International Journal
of Civil Engineering and Technology, 9(9), 2018, pp. 1113–1123.
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=9&IType=9
1. INTRODUCTION
Johansen has analyzed most of the isotropic slabs and presented a simple "affinity theorem" to
solve orthogonal slabs. Other researchers are used stepped yield line theory which is proposed
by Johnson, reinforced slab with TAC and, OLD. Various design charts are presented for Two-
way slabs with shorter side corner openings subjected to uniformly distributed load for different
edge conditions. Analysis of simply supported orthotropic slab has a central opening id
D.Subrahmanyam, V.Divya Sri and Dr.G.Thirupathi Naidu
http://www.iaeme.com/IJCIET/index.asp 1114 [email protected]
presented by Zaslavsky. He has presented some design charts. Later, Islam was presented with
isotropic slabs supported with continuous edges with openings at different positions and
different shapes are used to analyzing slabs. He has presented some design charts. An
orthogonal slab with interior corner opening for One Long edge discontinuous condition and
two adjacent edges continues has been chosen in this thesis work.
2. RESEARCH SIGNIFICANCE
Dr.T.V.S.VaraLakshmi took conditions of two long side continues condition Johanson took
condition of opening at centre with different conditions Goli Invented linearity in limit state
design of orthotropic slab Veerendra kumar was taken a condition of two adjacent discontinues
Islam K Rambabu taken many conditions on orthotropic slabs with different aspect ratios
Aslavsky presented transformation of orthotropic slab into isotropic slab using modified affine
theorem. In this study slab with shorter side interior corner opening, are taken. The aspect ratio
of the opening is quite different from that of the aspect ratio of the slab. The slab with interior
corner openings on the short side is subjected to uniformly distributed load for different edge
conditions is analyzed using virtual work equations considering all possible admissible yield
line failure patterns. And graphs are drawn based on the results untrained from the computer
programme for the taken conditions.
3. METHODOLOGY
The method of finding the ultimate load by using virtual work as proven as useful tool for
structural engineer despite gives an upper bound value. The virtual work equations are formed
by equating work done by the external load and internal energy absorbed by yield is given as,
( ) ( )∫∫ ∑ += 0yyult,0xxult,ult xθmyθmdxdyyx,δw
Eq. (1)
In order to reduce the long and lengthy equations to find the energy absorbed by each
segment. By using computer programme has utilized and non dimensional parameters are given
as in put and failure pattern and ultimate strength is obtained immediately this methodology is
used to get so many different aspect ratios are obtain with in no time are presented on graphs.
4. FORMATION OF VIRTUAL WORK EQUATIONS
4.1.1. Virtual Work Equations for One Long Edge Discontinuous Slab (OLD) and two
adjacent edges continue (TAC)
Twenty-eight possible failure patterns are considered for two edge conditions of the slab. Some
of the governing failure patterns for different edge conditions and for different data Consider
for the failure pattern-1 &let delta is virtual displacement at A and B figure 1(a) and figure 1
(b) unknown value of c1,c2,c3 are need to find ot to define yield lines completely in the
directions of lx and ly respectively.
The Effect of Interior Corner Opening In Orthotropic Rectangular Reinforced Concrete Slab Under
the Condition of Old and TAC Using Yield Line Theory
http://www.iaeme.com/IJCIET/index.asp 1115 [email protected]
4.1.1. One long Edge DiscontinuesFailurePattern1:
Fig. 1(a)
4.1.2. Two Adjacent Side Edges Continues Failure Pattern1
Fig. 1(b)
The external work done by segment A:
( )
+++
−
+−=
+−+−+−+−
25r1βr
6
3β
2
6r
2β
2
26
βr
21
3r
23
r23r
1r
1
16r
1ylxluw
3
6X
5X
12C
uw6
Xyβl
2
8X
5C
1C
uwyβl8X
3
2X
1X
12C
yβl2Xuw
2
4X
5C2
yβl4x
uw3
uw
1Cyl
2
1
Eq. (2)
The external work done on segment B:
( )
+−
+−+−−+
2
yβl
6C
3C
yβl3Xuw
3C
/3yβl6C
2Xyluβw2
1
3
uw
3C2C2
1
2
uw
3C2C1Cxl3
uw
3C1C2
1
( ) ( )
−−+
++
+−
+−=
1
3
1
6356
1
36
3
2133
yxur
βr
r
rrrβ2r
r
βr
3
βr
2
βr
r
1
r
1
3r
1
2r
1llw α
Eq. (3)
The external work done on segment C:
26r
ylxluw
3
uwyl
2C
2
1=
Eq. (4)
D.Subrahmanyam, V.Divya Sri and Dr.G.Thirupathi Naidu
http://www.iaeme.com/IJCIET/index.asp 1116 [email protected]
The external work done on segment D:
( ) ( )( )
( )( ) ( ) uw
3Cyl
3
yβl
6C
yβl6
X2
1
2
yβl
6C
3Cyl
yβl7Xuw
6
2C
3Cyl
uw3
Cyl2C
1CXl
2
uw
3
uw
1C
3Cyl
2
1
−
+
−+−
−−
+−−−+−
Eq. (5)
Total work done = work done on segments (A+B+C+D):
( )
( )( ) ( )
( )( )
−
++
−
−−
+++
−−−−
−=
213r13r
23r
3β
6rβ2
13r1r
6r23βr
13r2
5rα3r2β
5rα13r
6r3βr
23r
1
13r
1
2
1
3r
13r
ylXluw
( ) ( ) ( )( ) ( )
−
+−
−
+−−
−−
−
+
−+
13
r
3r
2
5rα3
r2β
13
r
5rα3
r
6r
3βr
6
β
13
r2
6r
13
r2
6rβ
13
r1r
23
r2β
( ) ( ) ( ) ( )
−
++++
−
+−−
−
−−
−
−+−=2
13r
6rβ
1r
6r23βr
13r
23r
3β2
5r1βr213r
11
16r
23r
3β
12
13r
1
12r
23r6r
2β
12
13r
1
12r
26r
23βr
2r
1
r
1
6
1
2
1r
2yluw
Eq. (6)
Energy absorbed by yield lines:
( ) ( ) [ ]
−++++
−−+−+−
3Cyl
yl4I
2C
yl3I
3C
Xl2I
1C
yl1I
um
3Cyl
1
4XXlum 1yk
2C
1ylum
1Xk
3C
1
2XXl1ykum
1C
1y2ββyl
1Xkum
Eq. (7)
Equating energy observed by yield lines to total work done by segments the
equations
( )( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( )( ) ( )
−
+−
−
+−−
−−
−
+
−+
−
++++
−
+−−
−
−−
−
−+−
++−
++−
−−
++−+−
=
13r
3r
2
5rα3r2β
13r
5rα3r
6r3βr6
β
13r2
6r
13r2
6rβ
13r1r
23r
2β
213r
6rβ
1r
6r23βr
13r
23r
3β2
5r1βr
213r
11
16r
23r
3β
12
13r
1
12r
23r6r
2β
12
13r
1
12r
26r
23βr
2r
1
1r
1
6
1
2
1
r
2r3I
r
1r1I
13r
4I3rr
2I3rr13r1r
3βr1
13r
3rr1yk
r
2r1
k
1r
3βr13rr
1yk2β1
r
1r1k
r
1
ultm
2ylult
w
x
x
Eq. (8)
4.2.1. Other Possible Failure Patterns for OLD
[ ] [ ]( ) ( )
[ ]2r3I1r1Ir
1
13r
4I
2I3rr13r1r
3βr113r
3rr1yk
r
2r1Xk
β3r1r
1r
3rr1yk2β1
r
1Xk
1r ++−
++−
−−
++−+−=
The Effect of Interior Corner Opening In Orthotropic Rectangular Reinforced Concrete Slab Under
the Condition of Old and TAC Using Yield Line Theory
http://www.iaeme.com/IJCIET/index.asp 1117 [email protected]
4.2.3 Other Possible Failure Patterns of TAC
D.Subrahmanyam, V.Divya Sri and Dr.G.Thirupathi Naidu
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5.1. Minimization of Virtual Work Equations:
The value WultLy2/Mult of above equation consists of non dimensional parameters untrained
from computer programme these are r1,r2,r3 are defined as position of node points of yield line
in order to find the minimum values of WultLy2/Mult for the yield line failure cracks are
considered. Using the above equation can develop charts based on orthogonality it is used in
analysis and design.
5.1.1. Some of Admissible Failure Pattern For Two Adjacent Edges Continue Condition:
Table 1
Orthotropic coefficients as input:
KX =1.2, KY=0.8, I1 =1.2, I2=0.8, I3&I4=0 ∑K = 4.0, µ = 1.5
R
Variables
α β r5 r6 Failure pattern attained
as output result
1.6 0.15 0.15 0.1 0.15 Pattern – 1
1.8 0.1 0.1 0.1 0.05 Pattern – 2*
1.2 0.1 0.2 0.1 0.1 Pattern – 3
2.0 0.1 0.1 0.1 0.3 Pattern – 4
2.0 0.1 0.1 0.1 0.2 Pattern – 6
1.7 0.05 0.2 0.2 0.1 Pattern – 7
1.2 0.1 0.2 0.1 0.1 Pattern – 9
1.9 0.1 0.1 0.1 0.2 Pattern – 11
1.3 0.1 0.06 0.1 0.1 Pattern – 12*
The Effect of Interior Corner Opening In Orthotropic Rectangular Reinforced Concrete Slab Under
the Condition of Old and TAC Using Yield Line Theory
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5.1.2 Some of the governing admissible failure patterns for one long edge Discontinuous
conditions
Table 2
Orthotropic coefficients as input:
KX =1.2, KY=0.8, I1 =1.2, I2=0.8, I3=1.2 I4=0 ∑K = 4.0, µ = 1.5
R
Variables
α β r5 r6 Failure pattern attained
as output result
1.6 0.15 0.15 0.1 0.15 Pattern – 1
2.0 0.1 0.1 0.2 0.2 Pattern – 2
2.3 0.2 0.1 0.1 0.2 Pattern – 6
2.3 0.1 0.1 0.2 0.1 Pattern – 7
2.8 0.05 0.2 0.1 0.1 Pattern – 13
2.8 0.05 0.2 0.1 0.1 Pattern – 13
0.9 0.2 0.1 0.1 0.2 Pattern– 18
1.6 0.1 0.1 0.1 0.1 Pattern– 19
2.6 0.05 0.2 0.2 0.1 pattern–3
5.1.3. ODL with Orthotropic Coefficient
KX =1.2, KY=0.8, I1=1.2, I2=0.8, I3=1.2, I4=0 ∑K = 4.0, µ = 1.5
wultly2/Mult =22.01
Fig. 1.1
5.1.4 TAC with Orthotropic coefficients
KX =1.2, KY=0.8, I1=1.2, I2=0.8, I3&I4 =0 ∑K = 4.0, µ=1.5 wultly2/Mult =17.48
D.Subrahmanyam, V.Divya Sri and Dr.G.Thirupathi Naidu
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Fig 1.2
5.2. RESULTS FOR TAC CONDITION STRENGTH VS ASPECT RATIO v
(I3&I4=0):
The Effect of Interior Corner Opening In Orthotropic Rectangular Reinforced Concrete Slab Under
the Condition of Old and TAC Using Yield Line Theory
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5.3. RESULTS FOR OLD CONDITION STRENGTH VS ASPECT RATIO:
(I4=0)
D.Subrahmanyam, V.Divya Sri and Dr.G.Thirupathi Naidu
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CONCLUSION
• The equations for orthotropic slabs with interior corner opening parallel to a shorter side whose
aspect ratio is different from the aspect ratio of slab.
• Charts are prepared using affine theorem for one long discontinuous slab for Wly2/Mu verses
aspect ratio r ranging from 1.0 to 2.0 different μ values and ∑k values.
• By using computer programme different failure patterns are obtained with different moment
coefficients are presented in graphs.
• From chart no. 1,3,4,5,6 the strength is same at 1.5 to 1.6. The strength is same for different
values of µ and value of α, β keeping r5 and r6 values as constant.
• From chart no. 2,7,8 the strength is same at 1.2 to 1.3. The strength is same for different values
of µ and value of α, β keeping r5 and r6 values as constant.
• From chart no. 9,10,11,12 the strength is same at 1.6 to 1.7. The strength is same for different
values of µ and value of α, β keeping r5 and r6 values as constant for the benefit of the practicing
engineers charts are prepared for one long edge discontinuous condition of the slab for affine
theorem coefficients for different α, β values and keeping r5 and r6 values as constant.
• It is observed from the charts that the strength of slab with the opening (for α = β) is generally
high
• From the charts, it is compared that the strength is less for all size of openings for µ = 4.0. The
strength of slab is same for all µ values for aspect ratio r = 1.2 and the strength is 28.34 from
chart 9. The strength of slab is same for all µ values for aspect ratio r = 1.5 and the strength is
20.14 from chart 11.
• Graphs are presented based on yield line theory for One Long Edge Discontinuous condition.
Fig 1(a)
• Graphs are presented based on yield line theory for Two Adjacent Side Edges continuous
condition Fig 1(b),
• Based on all 12 graphs Relatively OLD strength is more than TAC as per graphs.
The Effect of Interior Corner Opening In Orthotropic Rectangular Reinforced Concrete Slab Under
the Condition of Old and TAC Using Yield Line Theory
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NOMENCLATURES
A = α, B = β Coefficients of the opening in the slab mentioned in charts. C1 lx/r1=Yield line
propagation parallel to longer span lx. C2 lx/r2=Yield line propagation parallel to longer span
C3ly/r3=Yield line propagation parallel to shorter span ly.C4ly/r4=Yield line propagation parallel
to shorter spanly.C5lx r5 =Distance of opening from shorter edge. C6ly r6 =Distance of the opening
from longer edge.I1Mult, I3Mult -ve yield moment/unit length parallel to X-axis
I2Mult, I4Mult -ve yield moment/unit length parallel to Y-axis
KxM + ve yield moment/unit length parallel to X-axis KyMult + ve yield moment/unit length
parallel to Y-axis lx, ly Slab dimensions in X and Y directions.
Mult Yield moment/unit length of slab. r- Aspect ratio (lx/ly),
r1, r2, r3, r Non dimensional perametersr5,r6 Distances from supports. udlUniformly distributed
the load.wult Uniformly distributed the load/unit length αLx, βLy where α and β are coefficients
of the opening, ΣK Free edges. Sum of I1, I2, I3, I4. µCoefficient of orthotropy = ���
� � �� (��)�
���� � �� (��)�
Continuous edge.
Free edges.
Negative yield line.
Simply supported edge
REFERENCES
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Theorem to an orthotropic rectangular reinforced concrete slab with long side opening
symmetric about vertical Axis. Journal of mechanical and civil engineering volume 13
issue 13 page 30-38 2278-1684.
[2] Islam. Park, R.1971). Yield-Line Analysis of two-way slab with openings Journal of
Structural Engineer Vol 49, No 6 June 1971 pp 269/275.
[3] Johansen, K.W. (1962). Yield line Theory Cement and Concrete Association London 1962
pp 181.
[4] Rambabu, K. and Goli. (2007) HB Simplified of Approach Design Orthogonal Slabs using
Affine Theorem" Journal of Structural Engineering India Vol 33, No 5, December 2006,
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[5] Verander Kumar. (2009). Yield line analysis of Two way reinforced Concrete Slab having
two adjacent edges discontinuous with openings. Journal of Structural Engineering India
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[6] Vikranth, j.Reddy, P.V.R.K, and Rambabu, K. (2017) Application of affine theorem on
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