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madam rafia firdous is a lecturer and instructor in University of South Asia LAHORE,PAKISTAN and She gave lecture us via in this slide
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Plain and Reinforced Concrete- 1
Doubly Reinforced Beams By Engr. Rafia Firdous
Plain & Reinforced Concrete-1 Doubly Reinforced Beams
“Beams having both tension and compression reinforcement to allow the depth of beam to be lesser than minimum depth for singly reinforced beam”
• By using lesser depth the lever arm reduces and to develop the same force more area of steel is required, so solution is costly.
• Ductility will be increased by providing compression steel.
• Hanger bars can also be used as compression steel reducing the cost up to certain cost.
• For high rise buildings the extra cost of the shallow deep beams is offset by saving due to less story height.
Plain & Reinforced Concrete-1 Doubly Reinforced Beams (contd…)
• Compression steel may reduce creep and shrinkage of concrete and thus reducing long term deflection.
• Use of doubly reinforced section has been reduced due to the Ultimate Strength Design Method, which fully utilizes concrete compressive strength.
Doubly Reinforced Beam
Behavior Doubly Reinforced Beams
Tension steel always yields in D.R.B.
There are two possible cases:
1. Case-I Compression steel is yielding at ultimate condition.
2. Case-II Compression steel is NOT yielding at ultimate condition.
Behavior Doubly Reinforced Beams
Cc
T = Asfs
N.A.
εcu= 0.003
Strain Diagram Internal Force Diagram
εs
h
c
d
b 0.85fc
a
Whitney’s Stress Diagram
(d-d’)
fs
d
εs’ fs’ Cs
d – a/2
T = Asfs
Cs=As’fs’
Cc=0.85fc’ba
fs=Esεs
fs’=Esεs’
Behavior Doubly Reinforced Beams (contd…)
Case-I Both Tension & Compression steel are yielding at ultimate condition
fs = fy and fs’=fy
Location of N.A.
Consider equilibrium of forces in longitudinal direction
sc CCT
yscys f'Aba'f85.0fA
b'f85.0
f'AAa
c
yss
1β
ac and
Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…)
c
d'c
0.003
'εs
εcu= 0.003
Strain Diagram
εs
c εs’ d’
B
D E
C
A
Δ ABC & ADE
c
d'c0.003'εs
1
1s
β
β
c
d'c0.003'ε
a
d'βa0.003'ε 1
s
If εs’ ≥ εy compression steel is yielding.
If εs’ < εy compression steel is NOT yielding.
(1)
Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…)
Cc
T = Asfy
N.A.
Internal Force Diagram
(d-d’)
Cs
d – a/2
T = total tensile force in the steel
21 TTT
T1 is balanced by Cs
T2 is balanced by Cc
s1 CT
c2 CT
Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…)
Cc
T = Asfy
N.A.
Internal Force Diagram
(d-d’)
Cs
d – a/2
Moment Capacity by Compression Steel
'dd'f'A'ddCM yssn1
'ddT1
Moment Capacity by Concrete
2
adT
2
adCM 2cn2
2
adTTM 1n
2
2
ad'f'AfAM ysysn
2
Case-I Both Tension & Compression steel are yielding at ultimate condition (contd…) Total Moment Capacity
21 nnn MMM
2
ad'f'AfA'dd'f'AM ysysysn
Case-II Compression steel is not yielding at ultimate condition
fs = fy and fs’< fy
'εE'f ss
b'f85.0
'f'AfAa
c
ssys
1β
ac and
a
d'βa600'f 1
s
Location of N.A.
Concluded
