DOUBLY REINFORCED
RECTANGULAR BEAMS
If a beam cross section is limited because of architectural or
other considerations, it may happen that concrete cannot
develop the compression force required to resist the given
bending moment.
In this case, reinforcing steel bars are added in the
compression zone, resulting in a so called doubly
reinforced beam, that is one with compression as well as
tension reinforcement.
• Another reason for placing reinforcement in the
compression zone is that when beams span more than two
supports (continuous construction), both positive and
negative moments will exist as shown in the figure.
It has been found that the inclusion of some compression steel has
the following advantages:
It will reduce the long-term deflections of members.
It will set a minimum limit on bending loading
It act as stirrup-support bars continuous through out the beam
span.
Analysis
For analysis, the total resisting moment of the beam will be assumed to
consist of two parts or two internal couples:
I. The part due to the resistance of the compressive concrete and
tensile steel and
II. The part due to the compressive steel and additional tensile steel.
The total nominal capacity may be derived as the sum of the two internal
couples, neglecting the concrete that is displaced by the compression steel.
0.003 0.85fc’
c a 0.85fc’ab
Part I Part II
Maximum steel ratio,
• To ensure that the beam section will be ductile, the minimum steel strain as per NSCP is
0.004.
Maximum steel ratio,
[ ] 𝑓 = 𝑓 , 𝑠𝑡𝑒𝑒𝑙 𝑦𝑖𝑒𝑙𝑑
max 𝑓𝑜𝑟 𝑠𝑖𝑛𝑔𝑙𝑦
𝑀𝑎𝑥 𝑠𝑡𝑒𝑒𝑙 𝑟𝑎𝑡𝑖𝑜
𝑓𝑜𝑟 𝑑𝑜𝑢𝑏𝑙𝑦
𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑑 𝑏𝑒𝑎𝑚
Analysis for Doubly Reinforced Rectangular
Beams
"In the analysis of a doubly reinforced beam, assumptions are based
on whether the compression reinforcement yields or remains elastic.
The tension reinforcement is typically assumed to yield as a
conservative approach, ensuring ductile failure behavior."
Analysis for Doubly Reinforced Rectangular
Beams 0.003 − 𝜀 0.003
=
𝑑′ 𝑐
0.003𝑑′
𝑐=
0.003 − 𝜀
0.003𝑑′
c 𝑐=
𝑓
0.003 −
𝐸
0.003𝑑′
𝑐=
600 − 𝑓
20000
600𝑑′
𝑐=
600 − 𝑓
𝑖𝑓 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑏𝑎𝑟𝑠 𝑦𝑖𝑒𝑙𝑑 𝑓 = 𝑓𝑦
600𝑑′
𝑐=
600 − 𝑓𝑦
Analysis for Doubly Reinforced Rectangular
Beams
𝐴 =𝐴 −𝐴 ′
𝜌 = 𝜌 − 𝜌′
.
𝜌− 𝜌 ≥ compression bars yield
𝐶=𝑇
0.85𝑓 𝑎𝑏 = 𝐴 𝑓
Therefore
0.85𝑓 𝛽 𝑐𝑏 = 𝜌 − 𝜌 𝑏𝑑𝑓 𝜌− 𝜌 <
.
compression bars will not yield
0.85𝑓 𝛽 𝑐
𝜌− 𝜌 =
𝑑𝑓
0.85𝑓 𝛽 600𝑑′ Limit for compression
𝜌− 𝜌 =
𝑑𝑓 600 − 𝑓𝑦 bars to yield
Analysis for Doubly Reinforced Rectangular
Beams
Example 1
537.5
62.5
Analysis for Doubly Reinforced Rectangular
Beams
0.85𝑓 𝛽 3
𝜌 = ( )
𝑓 7
0.85(27.6)(0.85) 3
𝜌 = ( )
414.7 7
𝜌 = 0.0206
𝐷𝑒𝑠𝑖𝑔𝑛 𝑎𝑠 𝑑𝑜𝑢𝑏𝑙𝑦 𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑑 𝑏𝑒𝑎𝑚.
Analysis for Doubly Reinforced Rectangular
Beams 0.003
𝜖 ′
0.0206
𝜖
Analysis for Doubly Reinforced Rectangular
Beams
Example 2
Analysis for Doubly Reinforced Rectangular
Beams
0.85𝑓 𝛽 3
𝜌 = ( )
𝑓 7
0.85(34.56)(0.803) 3
𝜌 = ( )
414.6 7
𝜌 = 0.0244
𝐷𝑒𝑠𝑖𝑔𝑛 𝑎𝑠 𝑑𝑜𝑢𝑏𝑙𝑦 𝑟𝑒𝑖𝑛𝑓𝑜𝑟𝑐𝑒𝑑 𝑏𝑒𝑎𝑚.
177.57
𝑐= = 𝟐𝟐𝟏. 𝟏𝟑𝟑 𝒎𝒎
0.803
Analysis for Doubly Reinforced Rectangular
Beams
𝑑−𝑐
𝜀 = 0.003
𝑐
. .
𝜀 =
.
𝜀 = 0.00514 > 0.005
Analysis for Doubly Reinforced Rectangular
Beams
Example 3
Design Doubly Reinforced Rectangular
Beams
0.85𝑓 𝛽 3
𝜌 = ( )
𝑓 7
0.85(34.6)(0.8) 3
𝜌 = ( )
414.7 7
𝜌 = 0.0243
0.0249 0.0243
0.0243
Design Doubly Reinforced Rectangular
Beams
Analysis for Doubly Reinforced Rectangular
Beams
Example 4
0.85𝑓 𝛽 3
𝜌 = ( )
𝑓 7
0.85(27.6)(0.85) 3
𝜌 = ( )
414.6 7
𝜌 = 0.0206
0.0206
Analysis for Doubly Reinforced Rectangular
Beams
ρ′ 𝑓 ′ − 0.85f
𝜌 =𝜌 +
f
0.010496 282.45 − 0.85𝑥27.6
𝜌 = 0.0206 +
414.6
𝜌 = 0.0213
Analysis for Doubly Reinforced Rectangular
Beams
Design for Doubly Reinforced Rectangular
Beams
Design for Doubly Reinforced Rectangular
Beams
Design for Doubly Reinforced Rectangular
Beams
End……….
