Dr. M.E. Haque, PE 1

siennatearfulUrban and Civil

Nov 25, 2013 (3 years and 8 months ago)

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Dr. M.E. Haque, PE

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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

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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

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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

R (Radius of
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

Dr. M.E. Haque, PE

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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



Dr. M.E. Haque, PE

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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)

Dr. M.E. Haque, PE

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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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