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Plain & Reinforced

Concrete-1CE-313

Flexural Analysis and

Design of Beams

(Ultimate Strength Design ofBeams)

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a n e n orceConcrete-1

Ultimate Strength Design of Beams(Strength Design of Beams)

Strength design method is based on the philosophy of dividingF.O.S. in such a way that Bigger part is applied on loads andsmaller part is applied on material strength.

fc

0.85fc

Stress

Strain

Crushing

Strength

0.003

favg

favg = Area under curve/0.003

If fc 30 MPa

favg = 0.72 fc

1 = Average Strength/Crushing

Strength

1 = 0.72fc / 0.85 fc = 0.85

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a n e n orceConcrete-1

Ultimate Strength Design of Beams (contd)

Cc

T = Asfs

la = d a/2

N.A.

cu =0.003

StrainDiagram

ActualStress

Diagram

Internal ForceDiagram

In ultimate strength design method the section is alwaystaken as cracked.

c = Depth of N.A from the extreme compression face atultimate stage

a = Depth of equivalent rectangular stress diagram.

s

h

cd

b 0.85fc

fs

0.85fca

EquivalentStress

Diagram/Whitneys

Stress

Diagram

a/2

fs

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a n e n orceConcrete-1

Ultimate Strength Design of Beams (contd)

ActualStress

Diagram

0.85fc 0.85fc

Equivalent StressDiagram/

Whitneys StressDiagram

cCc Cca

The resultant of concretecompressive force Cc, acts at thecentriod ofparabolic stressdiagram.

Equivalent stress diagram ismade in such a way that it hasthe same area as that of actualstress diagram. Thus the Cc, willremain unchanged.

a/2

ab'0.85fcbf cav =

a'0.85fc'0.72f cc =

c'0.85f

'0.72fa

c

c =

ca 1=

l i i f d

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Plain & ReinforcedConcrete-1

Ultimate Strength Design of Beams (contd)

Factor 1

1= 0.85 for fc 30 MPa

Value of 1decreases by 0.05 for every 7 MPa

increase in strength with a minimum of 0.65

0.65'0.00714f1.064 c1 = 85.0

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a n e n orceConcrete-1

Determination of N.A. Location at UltimateCondition

CASE-I:Tension Steel is Yielding at Ultimate Condition

ys

orys ff

CASE-II:Tension Steel is Not Yielding at UltimateCondition

yf

y s

y

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a n e n orceConcrete-1

CASE-I:Tension Steel is Yielding at UltimateCondition

ysss fAfAT ==

ab'0.85fC cc =2

ada =l

abffA cys = '85.0

For longitudinal Equilibrium

T = Cc

bf

fAa

c

ys

=

'85.0 1

ac =and

Cc

T = AsfsInternal ForceDiagram

a/2

la

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a n e n orceConcrete-1

CASE-I:Tension Steel is Yielding at UltimateCondition(contd)Nominal Moment Capacity, M

ndepending on steel = T x

la

= 2Mna

dfA ys

Design Moment Capacity

=

2M bnb

adfA ys

Nominal Moment Capacity, Mn depending on concrete = Ccx la

=

2

adab0.85fc'Mn

=2

adab0.85fc'M bnb Design Moment

Capacity

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a n e n orceConcrete-1

Minimum Depth for Deflection Control

I

1

3(Depth)1

For UDL

4L

( ) 3LLDeflection Depends upon Span, end conditions, Loads and fy

of steel. For high strength steel deflection is more and moredepth is required.

Pl i & R i f d

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Plain & ReinforcedConcrete-1

Minimum Depth for Deflection Control (Contd)ACI 318, Table 9.5(a)

SteelGrade

SimplySupported

One EndContinuous

Both EndContinuous

Cantilever

300 L/20 L/23 L/26 L/10

420 L/16 L/18.5 L/21 L/8

520 L/14 L/16 L/18 L/7

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Concluded