# Dr. M.E. Haque, PE 1

Urban and Civil

Nov 25, 2013 (4 years and 6 months ago)

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Dr. M.E. Haque, PE

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Dr. M.E. Haque, PE

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A Simply Supported Reinforced Concrete Beam
-

N.A.

P = 0

A

A

A
-

A

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N.A.(Zero
Stress
Line)

P

Compression

Tension

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

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Typical Stress
-
Strain Curves for Concrete and
Reinforcing Steel

Dr. M.E. Haque, PE

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N.A.(Zero
Stress
Line)

P

Compression

Tension

Curvature)

1. Un
-
cracked Concrete stage

Dr. M.E. Haque, PE

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1. Un
-
cracked Concrete stage

P

Compression

Tension

Dr. M.E. Haque, PE

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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

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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

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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

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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

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COMPRESSION

TENSION

Figure 4 Manipulated Image visualization for flexural failure.

(Digital image from Northridge Collection, Earthquake Engineering
Research Center, University of California, Berkeley)

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f
c

M

C

T

b

1

1

c

Neglect concrete in tension

Concrete in compression

Reinforcing Steel in tension

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