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8/14/2019 Flexural Analysis and Design of Beamns
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Flexural Behavior of Beams Under Service LoadWhen loads are applied on the beam stresses are producedin concrete and steel reinforcement
If stress in steel bars is less than yield strength, steel is in
elastic range If stress in concrete is less than 0.6fc concrete is assumed to
be within elastic range
Following are important points related to Elastic Range: Loads are un-factored
Materials are in elastic range
Analysis and design are close to allowable stress analysis anddesign
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Assumption for the Study of Flexural Behavior
Plane sections of the beam remains plane after bending.
The material of the beam is homogeneous and obeys hooks law
Stress Strain Perfect bond exists between steel & concrete so whatever strain
is produced in concrete same is produced in steel
All the applied loads up to to failure are in equilibrium with theinternal forces developed in the material
At the strain of 0.003 concrete is crushed
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Assumption for the Study of FlexuralBehavior(contd)
When cracks appear on the tension face of beam itscapacity to resist tension is considered zero
Stress and strain diagrams for steel and concrete aresimplified
Strain
Stress
Steel
Strain
StressConcrete
0.6fc
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Flexural Behavior BeamsGeneral Procedure for the Derivation of Formula
Step # 1 Draw the cross section of beam with reinforcement
Step # 2 Draw the strain diagram for the cross section
Step # 3 Draw the stress diagram
Step # 4 Show location of internal resultant forces
Step #5 Write down the equation for given configuration
C
Tla
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Flexural Behavior Beams (contd)
2. When Cracks are Appeared on tension Side
C
Tla
N.A.
fc
c
fs
StrainDiagram
StressDiagram
ResultantForce Diagram
When the tension side is cracked the concrete becomesineffective but the strains goes on increasing. The steelcomes in to action to take the tension.
s
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Flexural Behavior Beams (contd)
3. When Compression StressesCross Elastic Range
C
Tla
N.A.
0.85f
c
c
fs
StrainDiagram
StressDiagram
ResultantForce Diagram
It is clear that the stress diagram is infect obtained by rotatingthe stress strain diagram of concrete.Strains keeps on changing linearly in all three cases.
s
fc0.85fc
Stress
Strain
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Flexural Behavior Beams (contd)
Final Equation for Calculating Moment Capacity
Mr= T x l
a= C x l
a
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Flexural/Bending Stress Formula
f = My/I (Valid in Elastic RangeOnly)
f = M/(I/y)
f = M/Sf = Flexural Stress
S = Elastic Section Modulus
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Shear Stress Formula
= VAY/(Ib)(Valid in Elastic Range Only)
= VQ/(Ib) = Shear StressQ = First moment of area
First Moment of Shaded Area, Q = (b x d ) h
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b
d
h
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Notationb As (Compression
Face)
As(TensionFace)
h d, Effective Depth d
bw
b
hf
fc = concrete stress at any load level at any distance formthe N.A
fc= 28 days cylinder strength
c = Strain in concrete any load level
cu = Ultimate concrete strain, 0.003
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Notation(contd)fy = Yield strength of concrete
fs = Steel stress at a particular load level
s = strain in steel at a particular level, s = fs/Es
y = Yield strain in steel
Es = Modulus of elasticity of steel
Ec = Modulus of elasticity of concrete
,Roh = Steel Ratio, = As/A
c= A
s/(bxd)
T = Resultant tensile force
C = Resultant compressive force
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Notation(contd)
N.A
h c=k
d
d
C
T
la = jd
jd = Lever arm j =la /d (valid for elastic range)
kd = Depth of N.A. from compression face, k = c/d
j and k are always less than 1.
b
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Concluded