This document discusses the cracked transformed section method for analyzing reinforced concrete beams. It explains that when cracking occurs in the tension side of the beam, the concrete is no longer effective and the steel takes the tensile stresses. The effective cracked transformed section is used, where the depth is reduced by a factor k. Equations are provided to calculate k in terms of the steel ratio ρ. The depth and neutral axis location for the cracked transformed section model are then used to calculate the resisting moment capacity of the beam.

1. Plain & Reinforced
Concrete-1
CE3601
Lecture # 6
21rd
Feb 2012
Flexural Analysis and
Design of Beams

2. Plain & Reinforced Concrete-1
Cracked Transformed Section
When tensile stress cross fr, crack appear on the tension face
of beam.
 Tensile strength of concrete is neglected.
 All the concrete on the tension side of N.A. is neglected.
Only the Shaded area of cross-section is considered effective.
N.A
Concrete
Steel

3. Plain & Reinforced Concrete-1
Cracked Transformed Section (contd…)
Cc
T = Asfs
la = jd
N.A.
fc
εc
fs
Strain Diagram
Stress Diagram
Internal Force
Diagram
When the tension side is cracked the concrete becomes
ineffective but the strains goes on increasing. The steel comes in
to action to take the tension.
εs
Cracked
Transformed Section
h
kd
d
nAs
b

4. Plain & Reinforced Concrete-1
Cracked Transformed Section (contd…)
Calculations for“k”, taking top face as reference
kd = [b x kd x kd/2 + nAsd]/ [b x kd + nAs]
b x (kd)2
+ nAskd = b x (kd)2
/2 + nAsd
b x (kd)2
/2 = nAsd - nAskd
b x (kd)2
/2 = (1-k) nAsd
bd2
/2 k2
= (1-k) nρbd2
k2
= 2(1-k) nρ
Cracked
Transformed Section
kd
nAs
b
d
ρ = As / bd

5. Plain & Reinforced Concrete-1
Cracked Transformed Section (contd…)
k2
+ 2nρ k - 2nρ = 0
k = [ -2nρ ± √ (4n2
ρ2
+ 8nρ) ] /2
k = [ -nρ ± √ (n2
ρ2
+ 2nρ) ]
k = √ (n2
ρ2
+ 2nρ) -nρ
Taking only positive value as distance can’t be negative
Cracked
Transformed Section
kd
nAs
b
d

6. Plain & Reinforced Concrete-1
Cracked Transformed
Section (contd…)
Value of “j”
jd = d – kd/3
J = 1 – k/3
Cc = volume of stress diagram
Cc = ½ kd x fc x b
Cc = fc/2 x kdb
T = Asfs = nfcAs
b
kd
fc
T = Asfs
Cc
jdN.A
d

7. Plain & Reinforced Concrete-1
Cracked Transformed Section (contd…)
Resisting Moment Capacity
For longitudinal equilibrium
T = Cc
Mr = T x jd = Cc x jd
Mr = Asfs x jd = fc/2 x kdb x jd ------------- 1
FOR MAXIMUM VALUE RESISTING MOMENT
Put in eq-1
fs = 0.5 fy for grade 300 steel
fs = 0.4 fy for grade 420 steel
fs = Maximum allowable tensile stress in steel
fc= 0.45 fc’
fc = Maximum allowable concrete stress

8. Concluded