PRC-I Slides by Dr. ZAHID AHMAD SIDDIQUI

1. Plain & Reinforced
Concrete-1
CE53601
Lecture # 10
1st March 2012
Flexural Analysis and
Design of Beams
(Ultimate Strength Design of Beams)

2. Plain & Reinforced Concrete-1
Balanced Steel Ratio, ρb
It is corresponding to that amount of steel which will cause
yielding of steel at the same time when concrete crushes.
At ultimate stage:
ys εε  ys ff and
0.003εcu 
From the internal Force diagram
bc ab'0.85fCc 
ab = depth of equivalent rectangular stress block when balanced steel
ratio is used.
εcu= 0.003
Strain Diagram
ε
y
cb
d- cb

3. Plain & Reinforced Concrete-1
Balanced Steel Ratio, ρb (contd…)
ybys fd)b(ρfAT 
For the longitudinal equilibrium
cCT 
bcyb ab'0.85ffd)b(ρ 
d
a
f
'f
0.85ρ b
y
c
b  (1)

4. Plain & Reinforced Concrete-1
Balanced Steel Ratio, ρb (contd…)
From the strain diagram εcu= 0.003
Strain Diagram
εy
cb
d- cb
B C
A
E D
ADE&ABCΔs
b
y
b cd
ε
c
0.003


y
b
ε0.003
0.003d
c


s
s
s
y
b
E
E
d
E
f
0.003
0.003
c 



5. Plain & Reinforced Concrete-1
Balanced Steel Ratio, ρb (contd…)
d
f600
600
c
y
b 


d
f0.003E
0.003E
c
ys
s
b 


b1b Cβa As we know
d
f600
600
βa
y
1b 

 (2)
Put (2) in (1)
y
1
y
c
b
f600
600
β
f
'f
0.85ρ



6. Plain & Reinforced Concrete-1
Types of Cross Sections w.r.t. Flexure at Ultimate
Load Level
1. Tension Controlled Section
A section in which the net tensile strain in the extreme tension steel is
greater than or equal to 0.005 when the corresponding concrete strain
at the compression face is 0.003.
εcu= 0.003
Strain Diagram
εs=>0.005
c
d- c
cd
0.005
c
0.003


d
0.008
0.003
c 
d
8
3
c  d
8
3
βa 1and

7. Plain & Reinforced Concrete-1
Types of Cross Sections w.r.t. Flexure at Ultimate
Load Level (contd…)
2. Transition Section
The section in which net tensile strain in the extreme tension steel is
greater than εy but less than 0.005 when corresponding concrete strain
is 0.003. εcu= 0.003
Strain Diagram
εy<εs<0.005
c
d- cd
8
3
βa 1 baa 

8. Plain & Reinforced Concrete-1
Types of Cross Sections w.r.t. Flexure at Ultimate
Load Level (contd…)
2. Transition Section (contd…)
εcu= 0.003
Strain Diagram
0.004
c
d- c
To ensure under-reinforced behavior, ACI code
establishes a minimum net tensile strain of 0.004
at the ultimate stage.
cd
0.004
c
0.003


d
7
3
c  d
7
3
βa 1

9. Plain & Reinforced Concrete-1
Types of Cross Sections w.r.t. Flexure at Ultimate Load
Level (contd…)
Both the “Tension Controlled Section” and “Transition Section” are
“Under-Reinforced Section”
In Under-Reinforced Sections steel starts yielding before the crushing
of concrete and:
ρ < ρ b
It is always desirable that the section is under-reinforced otherwise
the failure will initiate by the crushing of concrete. As concrete is a
brittle material so this type of failure will be sudden which is NOT
DESIREABLE.

10. Plain & Reinforced Concrete-1
Types of Cross Sections w.r.t. Flexure at Ultimate Load
Level (contd…)
3. Compression Controlled Section (over-reinforced section)
The section in which net steel strain in the extreme tension
steel is lesser than εy when corresponding concrete strain is
0.003.
 Capacity of steel remain unutilized.
 It gives brittle failure without warning.
baa  bCC 

11. Plain & Reinforced Concrete-1
Strength Reduction Factor (Resistance Factor), Φ
 Tension Controlled Section, Φ = 0.9
 Compression Controlled Section
Member with lateral ties, Φ = 0.65
Members with spiral reinforcement, Φ = 0.75
 Transition Section
For transition section Φ is permitted to be linearly
interpolated between 0.65 or 0.75 to 0.9.

12. Plain & Reinforced Concrete-1
Strength Reduction Factor (Resistance Factor), Φ
 Transition Section (contd…) Φ
0.65
0.9
εy 0.005
ε
Compression
Controlled
Tension
Controlled
Transition
 yt
y
εε
ε0.005
0.25
0.65Φ 


For members with ties
For members with Spirals
0.25
 yt
y
εε
ε0.005
0.15
0.75Φ 


εt
εt = strain in extreme tension steel
when concrete crushes.
Φ

13. Plain & Reinforced Concrete-1
Maximum Steel Ratio, ρmax
For
T = C
ab'0.85ffA cys 
ab'0.85ffbdρ cy 
d
a
f
'f
0.85ρ
y
c

For tension controlled section d
8
3
βa0.005ε 1s 
So






 1
y
c
max β
8
3
f
'f
0.85ρ

14. Plain & Reinforced Concrete-1
Maximum Steel Ratio, ρmax (contd…)
For transition section
d
7
3
βa0.004ε 1s 






 1
y
c
max β
7
3
f
'f
0.85ρ

15. Plain & Reinforced Concrete-1
Minimum Reinforcement of Flexural Members
(ACI – 10.5.1)
 The minimum steel is always provided in structural members
because when concrete is cracked then all load comes on steel, so
there should be a minimum amount of steel to resist that load to
avoid sudden failure.
 For slabs this formula gives a margin of 1.1 to 1.5.
 This formula is not used for slabs.
yy
c
min
f
1.4
4f
'f
ρ 

16. Concluded