Dr. M.E. Haque, PE 1

Dr. M.E. Haque, PE 2

A Simply Supported Reinforced Concrete Beam -

N.A.

P = 0

A

AA - 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 stage2. Cracked Concrete (tension zone) - Elastic Stage3. 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

fc

ft

M

C

T

b

1

1

fc

ft

fc

ft

M

C

T

b

1

1

Dr. M.E. Haque, PE 7

h

b

d

Compression zone

Tension Zone

Strain Diagram

Stress Diagram

Tensile Stress

Compressive Stress

fc'

ft = fr = 7.5 fc'fc

ft = fcStress-Strain Diagram for Concrete

1. Un-cracked Concrete Stage ft < fr

M < Mcr fc = ft << fc'

Dr. M.E. Haque, PE 8

C=T ; fc = ft

M = 0.5fc x (b x 0.5h) x (2/3 h) = 1/6 fc x b x h2

fc = ft = 6M/(bh2)

C=0.5fc x (b x0.5h)

T=0.5ft x (b x0.5h)

2/3 h

b

1/2 h

1/2 h

ft

fc

Stress diagram

M

Section 1-1

fc = ft = Mc/Ig

where c = 0.5h Ig = bh3/12

OR

At ft = fr , where modulus of rupture, fr = 7.5 fc’Cracking Moment Capacity, Mcr = fr x Ig/(0.5h) = (fr x b x h2)/6

Dr. M.E. Haque, PE 9

2. Cracked Concrete (Tension Zone) - Elastic Stage

fy

0.5fy

c < 0.003

s = fs/Es

0.45fc'

h

b

d

Compression zone

Tension Zone Concrete Cracked

Strain Diagram Stress Diagram

Tensile Stress

Compressive Stress

fc'

ft = fr = 7.5 fc'

fs =0.5 fy

Stress- Strain Diagram for Concrete in Compression

Stress- Strain Diagram for Reinforcing steel in Tension

Es

0.003

ft > fr

M > Mcr fc = 0.45fc'fs =0.5 fy

Dr. M.E. Haque, PE 10

3. Cracked Concrete (Tension Zone) - Ultimate Strength Stage

ft > >fr

M > >Mcr fs = fy

fc = entire stress block until compression failure

h

b

d

Compression zone

Tension Zone Concrete Cracked

Strain Diagram Stress DiagramCompressive Stress

fc'

T = Asfy

fy

Stress-Strain Diagram for Concrete in CompressionStress-Strain Diagram for

Reinforcing Steel in Tension

c = 0.003

s = fy/Es

Es0.003

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

fc

M

C

T

b

1

1

c

Neglect concrete in tension

Concrete in compression

Reinforcing Steel in tension

Dr. M.E. Haque, PE 13