Principle of Design of Beams
When external loads are applied, moments are generated, and the maximum moment developed varies in each beam depending on the support conditions and loading combinations. For the below-shown beam, let M be the maximum moment which is also termed as unfactored moment/service moment/working moment.
Factored External Moment = 1.5 x Working Moment
Mu = 1.5M
Design of Beams is done for the factored external moment (Mu)
Maximum internal resistance without failure or ultimate moment of resistance is called Moment of Resistance (MOR). The externally applied moment should be always less than or equal to the internal moment of resistance during the design of beams.
Where, MOR = CZ or TZ
C- Compressive Force
Z- Lever Arm
Principle of Design of Beams for a Singly Reinforced Beam
If Mu > MORlim, the principle of design of beams will not get satisfied. Therefore, for the design of beams as singly reinforced beams of under reinforced type, Mu < MORlim. Limiting the moment of resistance can be increased by increasing fck or by increasing the depth, but both will increase the cost of construction. Therefore, in such cases beams are designed as doubly reinforced beams.
Area of Steel
Ast=0.5 × fck/fy [1- √1- 4.6Mu/bd2fck]bd
Where, fck – Characteristic Compressive Strength of Concrete
fy – Yield Strength of Steel
Area of steel provided should be greater than the area of steel required.
Astmin < Astmin < Astmax
Maximum area of tension steel = Astmax = 0.04 x bD
Area of tension steel provided should be less than the maximum area of tension to avoid congestion during concreting.
Minimum area of tension steel = Ptmin % = Percentage minimum area of tensile steel
Ptmin= Ast min/bd ×100
Ast min= 0.85bd/fy
Area of tension steel provided should be more than the minimum area of steel to avoid sudden failure.
For the design of beams (singly reinforced and under reinforced beam).
Ast < Astlim
Ast < Astmax
Doubly Reinforced Beam
A beam is made doubly reinforced beam when a singly reinforced beam becomes over the reinforced beam, i.e, MOR > MORlim. An over-reinforced beam is sudden. In such a case, a beam is designed as doubly reinforced beam. Doubly reinforced beams are beneficial in case of stress reversal. The compression steel helps in bearing additional strain due to creep and shrinkage. Compression steel is found to reduce deflections. They are helpful in case of shock or impact loads. Overall more ductile compared to single reinforced beams.
Total Compressive Force C =C1 + C2
Where, fsc = Compressive Stress in compression steel
fcc = Stress in concrete at level of compression steel
fcc = 0.45 fck
Total Tensile Force T = 0.87 fy Ast
Doubly reinforced beams are not economical because compression steel is under stress.
Design of Doubly Reinforced Beam
Doubly reinforced beams are designed for moment equal to
Mu= MORlim+(Mu- MORlim)
Area of tension steel Ast = Ast1 + Ast2
Ast2= Mu-MORlim/0.87 fy (d-d')
Area of compressive steel
Asc= Mu-MORlim/(fsc-fcc) (d-d')
In case of monolithic casting of beams and slabs, part of slab behaves as a beam to take the compression, such beams are called flanged beams.
Monolithically Casted T Beam
Effective flange width of T beam
bf= bw+ [6Df+ Lo/6]
bw = Web width
bf = Effective flange width
Df = Flange depth/Depth of slab
Lo = Distance between points of contra flexure
Maximum effective flange width (bfmax) = bw + Lc
bf < bfmax
Where Lc = Clear distance between adjacent beams
Monolithically Casted L Beam
bf= bw+ [3Df+ Lo/12]
(bfmax) = bw + Lc