Dr. M.E. Haque, PE
1
Dr. M.E. Haque, PE
2
A Simply Supported Reinforced Concrete Beam
-
N.A.
P = 0
A
A
A
-
A
Dr. M.E. Haque, PE
3
N.A.(Zero
Stress
Line)
P
Compression
Tension
R (Radius of
Curvature)
Three stages before collapse:
1. Un
-
cracked Concrete stage
2. Cracked Concrete (tension zone)
-
Elastic Stage
3. Cracked Concrete (tension zone)
-
Ultimate Strength Stage
Dr. M.E. Haque, PE
4
Typical Stress
-
Strain Curves for Concrete and
Reinforcing Steel
Dr. M.E. Haque, PE
5
N.A.(Zero
Stress
Line)
P
Compression
Tension
R (Radius of
Curvature)
1. Un
-
cracked Concrete stage
Dr. M.E. Haque, PE
6
1. Un
-
cracked Concrete stage
P
Compression
Tension
Dr. M.E. Haque, PE
7
h
b
d
Compression
zone
Tension Zone
Strain
Diagram
Stress
Diagram
Tensile Stress
Compressive Stress
f
c
'
f
t
= f
r
= 7.5
f
c
'
f
c
f
t
= f
c
Stress
-
Strain Diagram for Concrete
1. Un
-
cracked Concrete Stage
f
t
< f
r
M < M
cr
f
c
= f
t
<< f
c
'
Dr. M.E. Haque, PE
8
C=T ; f
c
= f
t
M = 0.5f
c
x (b x 0.5h) x (2/3 h)
= 1/6 f
c
x b x h
2
f
c
= f
t
= 6M/(bh
2
)
C=0.5f
c
x (b x0.5h)
T=0.5f
t
x (b x0.5h)
2/3 h
b
1/2 h
1/2 h
f
t
f
c
Stress diagram
M
Section 1
-
1
f
c
= f
t
= Mc/I
g
where c = 0.5h
I
g
= bh
3
/12
OR
At f
t
= f
r
, where modulus of rupture, f
r
= 7.5
fc’
Cracking Moment Capacity, M
cr
= f
r
x I
g
/(0.5h) = (f
r
x b x h
2
)/6
Dr. M.E. Haque, PE
9
2. Cracked Concrete (Tension Zone)
-
Elastic Stage
f
y
0.5f
y
c
< 0.003
s
= f
s
/E
s
0.45f
c
'
h
b
d
Compression
zone
Tension Zone
Concrete Cracked
Strain Diagram
Stress
Diagram
Tensile
Stress
Compressive Stress
f
c
'
f
t
= f
r
= 7.5
f
c
'
f
s
=0.5 f
y
Stress
-
Strain Diagram for
Concrete in Compression
Stress
-
Strain Diagram for
Reinforcing steel in Tension
E
s
0.003
f
t
> f
r
M > M
cr
f
c
= 0.45f
c
'
f
s
=0.5 f
y
Dr. M.E. Haque, PE
10
3. Cracked Concrete (Tension Zone)
-
Ultimate Strength Stage
f
t
> >f
r
M > >M
cr
f
s
= f
y
f
c
= entire stress
block until
compression
failure
h
b
d
Compression zone
Tension Zone
Concrete Cracked
Strain Diagram
Stress Diagram
Compressive Stress
f
c
'
T = A
s
f
y
f
y
Stress
-
Strain Diagram for
Concrete in Compression
Stress
-
Strain Diagram for
Reinforcing Steel in Tension
c
= 0.003
s
= f
y
/E
s
E
s
0.003
Dr. M.E. Haque, PE
11
COMPRESSION
TENSION
Figure 4 Manipulated Image visualization for flexural failure.
(Digital image from Northridge Collection, Earthquake Engineering
Research Center, University of California, Berkeley)
Dr. M.E. Haque, PE
12
f
c
M
C
T
b
1
1
c
Neglect concrete in tension
Concrete in compression
Reinforcing Steel in tension
Dr. M.E. Haque, PE
13
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