Yeild Line Theory

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Yield Line Theory

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DHARMSINH DESAI UNIVERSITYCIVIL ENGINEERING DEPARTMENT


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Introduction:Limitations of elastic analysis of slab.

1. Slab panels are square or rectangular.

2. One-way slab panels must be supported along two opposite sides only , the other two unsupported.

3. Two-way slab panel must be supported along two pairs of opposite sides , supports rem

4. Apply loads must be uniformly distributed.

5. Slab panels must not have large openings.


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History

Yield Line Theory as we know it today was pioneered in the 1940s by the Danish engin

K W Johansen.

As early as 1922, the Russian, A Ingerslev presented a paper to the Institution of Stru

London on the collapse modes of rectangular slabs.

Authors such as R H Wood, L L Jones, A Sawczuk and T Jaeger, R Park, K O Kemp, C T Mand many others, consolidated and extended Johansens original work so that now

theory is well established making Yield Line Theory a formidable international design to

In the 1960s 70s and 80s a significant amount of theoretical work on the application o

to slabs and slab-beam structures was carried out around the world and was widely rep


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DEFINITION:

A yield line is a crack in a reinforced concrete slab across which the reinforcing barand along which plastic rotation occurs.

Yield Line Theory is an ultimate load analysis. It establishes either the moments in a

a loaded slab) at the point of failure or the load at which an element will fail. It may

many types of slab, both with and without beams.


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BASIC PRINCIPLE OF YIELD LINE THE


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ASSUMPTIONS:1. The elastic deformations of a part of a slab between the yield lines negligible as c

plastic deformations along the yield line. It is assumed that the deformations tak

along the yield lines and the individual parts of the slab between the yield line re

2. As the intersection between the inclined planes are the straight lines it follows thlines are straight.

3. Yield line passes through the intersection of the axes of the rotation of the adjac

elements.


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4. Axes of the rotations generally lie along the lines of supports and pass over the colu

negative yield lines for the continuous or restrained slab must also form along the li

support before the latter can act as axes of rotation and the transform the slab into

mechanism.

5. Yield line ended at slab boundaries so that a complete mechanism is formed to alloof slab elements about the axes of rotation.


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Yield Line Patterns:


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Yield Line Patterns:


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Yield Line Moments: Total work done by the Mband Mt.

W=MbL(xCOS+ySIN)+MtL(-xSIN+yCOS)..*1+

Work done by MBand MTalong line AB is equal to

work done by their component

along projection of line AB.

MbL=MXLSIN SIN + MYLCOS COS

Mb=M

xSIN2+ M

yCOS2.*2+

MtL=MxLSIN COS -MyLCOS SIN

Mt=(Mx-My)SIN COS .*3+

Put Mband Mtfrom [2] and [3] in [1].

W=MxLSIN y+MyLCOS x

W=MxLy y+MyLxx


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Special condition at edges & corn


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Taking moment about the line AC and equating to zero,

My(Lx+dLx)COS(-d)-MyLxCOS(-d )+MxLySIN(-d )-MxLySIN(-d )- VdLxSIN(-d)=0

VSIN (-d )=MyCOS(-d )

V(SIN COSd -COS SINd )=My(COS COSd +SIN SINd )V=MyCOT

When =90,

V=0,means that the yield line intersects the free edge at right angle.


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Method of yield line analysis:1) Method of segmental equilibrium:

Steps:-

1. Predict possible trial yield line patterns and axes of rotation.

2. Draw free body diagram of each element of the slab.

3. Compute nodal forces keeping in view that:

Resultant of nodal forces is zero at the junction or node of yield lines.

A free edge is considered to be a yield line of zero strength.

Nodal forces appears when a yield line intersects a free edge at angle other than 90

4. Formulate equilibrium equations for each element considering the load on it, and the

and nodal forces on the bounding yield lines.


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2) Method of virtual work:

This method is based on principle of virtual work.

Steps:-

1. Predict possible yield line patterns and axes of rotation.

2. Apply infinitesimal increase in load to cause the structure deflection.

Work done by the load to cause small virtual deflection=internal work done by thmoments to cause the rotation in accommodating the virtual deflection.

W = M l

where,

W=Collapse load,

=Vertical deflection through which the collapse load W moves,

M=Moment capacity of section per unit length,

=Rotation of the slab segment satisfying the compatibility of deflection,l=Length of the yield line.


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Analysis of two way slab(pattern1) Method of segmental equilibrium:

for segment 1,moment of the all the forces and

moment of segment about edge AB is zero.

(2)(Lx/2)(Ly)W(Lx/6)-V(Lx/2)-MnxLy-MpxLy=0

WLxLy-12V=24(Mpx+Mnx)(Ly/Lx)*1+


moment of segment about edge BC is zero.

(2)(Ly/2)(Lx)W(Ly/6)+V(Ly/2)-MnyLx-MpyLx=0

WLxLy+12V=24(Mpy+Mnx)(Lx/Ly)*2+

adding [1] and [2],neglecting V,

W=12{(Mpx

+Mnx

)/Lx2+(Mpy

+Mny

)/Ly

2}.......[3]


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TEW=4(2)Lx(Ly/2)W(/3)=WLxLy(/3).*1+

TIW=2(Mpx+Mnx)Lyx+2(Mpy+Mny)Lxy

=4 (Mpx+Mnx)(Ly/Lx)+(Mpy+Mny)(Lx/Ly)-.*2+

equating [1] and [2],

WLxLy(/3)= 4 (Mpx+Mnx)(Ly/Lx)+(Mpy+Mny)(Lx/Ly)}

W=12{(Mpx+Mnx)/Lx2+(Mpy+Mny)/Ly

2}......[3]


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Case:-1

Square and simply supported slab,

Lx=Ly=L, Mnx=Mny=0, Mpx=Mpy=MpMp=WL

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Case:-2

Square and Clamped slab,

Mpx=Mpy=Mnx=Mny=MpMp=WL

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Analysis of two way slab(pattern:1) Method of segmental equilibrium:


moment of segment about edge AB is zero.

(2

)(Ly

)W(X/3)-(Mpx

+Mnx

)Ly

=0

W=6(Mpx+Mnx)(X2)*1+


moment of segment about edge BC is zero.

(2)X(Ly/2)W(Ly/6)2+W(LX-2X)(Ly/2) (Ly/4)-(Mpy+Mny)Lx=0

W=24(Mpy+Mny)Lx/{2XLy2 +3Ly

2(Lx-2X)-*2+

equating [1] and [2],6(Mpx+Mnx)/X2=24(Mpy+Mny)Lx/{2XLy

2+3Ly2(LX-2X)}

4(Mpy+Mny)Lx2+4(Mpx+Mnx)Ly

2X-3((Mpx+Mnx)LxLy2 = 0

Continue..


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TEW=2W1+2(W21+W22+W23)

=2W(2)XLy(/3)-2W(2)X(Ly/2)(/3)-2W(Lx-2X)(Ly/2)(/2)-

=(W/6)(3LxLy-2xLy).*1+

TIW=2(Mpx+Mnx)Lyx+2(Mpy+Mny)Lxy

=2 (Mpx+Mnx)(/X) Ly+2(Mpy+Mny)(2/Ly) Lx.*2+

equating [1] and [2],

(W/6)(3LxLy-2xLy)= 2 (Mpx+Mnx)(/X) Ly+2(Mpy+Mny)(2/Ly) Lx

W={12(Mpx+Mnx)Ly2+24(Mpy+Mny)XLx}/Ly

2(3XLx-2X2)........[3]

To get the minimum collapse load,

dW/dX = 0

4(Mpy+Mny)Lx2+4(Mpx+Mnx)Ly

2X-3((Mpx+Mnx)LxLy2 = 0


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Case:-

Rectangle and clamped slab (Interior panel),

Lx=1.1Ly , X=Lx/2, Mny=1.33Mpy , Mnx=1.33Mpx, Mpx=0.85MpyShort span coefficients y =0.032 (Positive)

y =0.043 (Negative)

Long span coefficients x =0.027 (Positive)

x =0.036 (Negative)


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References:Yield line analysis for slabs : Module 12, Lesson 30,31,32 (NPTEL)

IIT Kharagpur

Practical Yield line design : Perencanaan-Praktis-Garis-Leleh1

Concrete structure : By V N Vazirani & M.M RatwaniDesign of reinforced concrete structure : M L Ghambhir

Design of reinforced concrete structure : B C punamia

I S 456 : Plain & reinforced concrete code of practice

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Yeild Line Theory