THE EFFECT OF INTERIOR CORNER OPENING IN ORTHOTROPIC ... · Other researchers are used stepped yield line theory which is proposed by Johnson, reinforced slab with TAC and, OLD. Various

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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


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


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)



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 ++−

++−

−−

++−+−=




4.2.3 Other Possible Failure Patterns of TAC



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*




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



Fig 1.2

5.2. RESULTS FOR TAC CONDITION STRENGTH VS ASPECT RATIO v

(I3&I4=0):




5.3. RESULTS FOR OLD CONDITION STRENGTH VS ASPECT RATIO:

(I4=0)



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.




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

[1] VaraLakshmi, T.V.S. Rambabu, k. and Srinivasarao, K.(2016). Application of Affine

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,

Jan 2007 pp 435-442. ISSN 0970-0137.

[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

Vol 36, NO 2, June 2009, Jul92007 pp 435-442. ISSN 0970-0137.

[6] Vikranth, j.Reddy, P.V.R.K, and Rambabu, K. (2017) Application of affine theorem on

orthotropic rectangular slab with interior corner opening. International Journal of

Engineering Research and technology volume 6 issu9, 54-103 ISSN 223-5381

Documents

THE EFFECT OF INTERIOR CORNER OPENING IN ORTHOTROPIC ... · Other researchers are used stepped yield line theory which is proposed by Johnson, reinforced slab with TAC and, OLD. Various