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Today we are going to learn about Doubly Reinforced Beam Design in Ultimate Stress Design (USD) method. Many complications in design always been seen in different tutorials and today we are going to show you how simply you can design a doubly reinforced beam for a structure.
Before going to Beam Design fist have a tour to Beam characteristics.
What is a BEAM?
A beam is generally a structural member that carries load along its transverse axis. It resists the load through bending and through the compressive stress carrying capacity of concrete and through the tension carrying capacity of reinforcement.
What is CONCRETE?
A Concrete is a stone like material which is artificially prepared by using a bonding material and inert materials. Bonding material are generally Cement(Ordinary Portland Cement (OPC) and Portland Composite Cement (PCC)) and inert materials are generally crushed stone,sand etc.
What is Reinforcement?
High yield reinforcement is a building material that is prepared from mild steel. Reinforcement in beam carries tension that comes in the total life of the structure.
What is SINGLY REINFORCED AND DOUBLY REINFORCED BEAM?
Singly reinforced beam is one in which the steel buried in the concrete lies only in the tension side of the section.
Singly reinforced beam section
Sometimes the upcoming moment from the load is such that only a singly reinforced beam can not support it. In this case, Reinforcement is also provided in the compression side of the beam. This type of beam is called doubly reinforced beam.
doubly reinforced beam section
Assumptions in Beam Design:
1. Plane sections remain plane after Bending.
2.Strain distribution is linear for both steel and concrete and varies directly with the distance from the Neutral Axis.
3. All of the compressive force are taken up by the concrete only and all of the tensile stress are taken up by the reinforcement only.(N.B. Concrete has a capacity of taking tensile stress which is almost 10-15 percent of its compressive strength carrying capacity)
Evaluation of Design Parameters:
Total compressive force, C = 0.85fc’ ba (Refer stress diagram)
Total Tensile force, T = As fy
C = T
or, 0.85fc’ ba = As fy
or, a = As fy / (0.85fc’ b)
= pd fy / (0.85 fc’) (putting p = As / bd)
Moment of Resistance, Mn = 0.85fc’ ba (d – a/2)
or , Mn = As fy (d – a/2)
= p bd fy [ d – (pdfyb / 1.7fc’) ]
= w fc’ [ 1 – 0.59 w] bd^2 (putting w = r fy / fc’)
Mn = Kn bd2 (putting Kn = w fc’ [ 1 – 0.59 w)
Mu = f Mn
=f Kn bd^2 , where f = Strength Reduction Factor
Balanced Reinforcement Ratio (Pb)
From strain diagram, similar triangles
cb / d = 0.003 / (0.003 + fy / Es) ; Es = 29x10^6 psi
cb / d = 87,000 / (87,000+fy)
Pb = Asb / bd
= 0.85fc’ ab / (fy. d)
= β1 ( 0.85 fc’ / fy) [ 87,000 / (87,000+fy)]
In case of statically determinate structure ductile failure is essential for proper moment redistribution. Hence, for beams the ACI code limits the max. amount of steel to 75% of that required for balanced section. For practical purposes, however the reinforcement ratio (r = As / bd) should not normally exceed 50% to avoid congestion of reinforcement & proper placing of concrete.
P =0.75 P b
Min. reinforcement is greater of the following:
As(min) = 3Öfc’ x bwd / fy or 200 bwd / fy
rmin = 3Öfc’ / fy or 200 / fy
For statically determinate member, when the flange is in tension, the bw is replaced with 2bw or bf whichever is smaller
The above min steel requirement need not be applied, if at every section, Ast provided is at least 1/3 greater than the analysis