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In singly reinforced simply supported beams reinforcing steel bars are placed near the bottom of the beam where they are most effective in resisting the tensile stresses.
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Reinforcement in simply supported beam
COMPRESSION b
STEEL REINFORCEMENT D d
TENSION
SUPPORT SECTION A - A
CLEAR SPAN
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STRESS DIAGARM FOR SINGLY R.C.BEAM ( IS : 456 – 2000 , P.69 )
0.0035 0.446 fck
C = 0.36 fck * b* xu
T
Z = Lever ArmN A
X = X u max
0.87 fy Ast0.87fy
ES+ 0.002
Ast
b
d
SECTION STRAIN STRESS
0.42Xu
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Xu max = Maxi. Depth of Neutral axis b = width of beam d = effective depth Z = d-0.42xu = lever arm D = overall depth Xu = depth of N.A Ast = area of steel Resultant force of compression C = average stress X area
= 0.36 fck b xu
Resultant force of tension T = 0.87 fy Ast
Force of compression should be equal to force of tension,
0.36 fck b xu = 0.87 fy Ast
xu =
Where Ast = area of tension steel
0.87 fy Ast0.36 fck b
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Moment of resistance with respect to concrete= compressive force x lever arm
= 0.36 fck b x z
Moment of resistance with respect to steel = tensile force x lever arm
= 0.87 fy Ast x z
Maximum Depth Of Neutral Axis (IS 456 – 2000 , P.70)
fy ( N/mm2 ) Xu max
250 0.53 d
415 0.48 d
500 0.46 d
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If Xu = Xu max then it is balanced section If Xu < Xu max then it is under reinforced section If Xu > Xu max then it is over reinforced section
Xu max Xu < Xu max Xu > Xu max
Balanced section Under reinforced Over reinforced
b
D
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Moment of resistance Mu with respect to steel ( for under reinforced section)
Mu = T X Z ( IS 456-2000, P.96, Cl. G1.1 (b) )
where , Z = d - 0.42 xu = 0.87 fy Ast x (d - 0.42 xu ) T = 0.87 fy Ast
= 0.87 fy Ast d ( 1- fy Ast / fck b d )
Mu with respect to concrete ( for balanced section) Mu = C x Z ( IS 456-2000, P.96, Cl. G1.1 (c) ) = 0.36 fck b xu ( d – 0.42 xu ) where , C = 0.36 fck b xu
= 0.36 fck b xu d (1 – 0.42 xu / d)
= 0.36 fck b xu/d d2 (1 – 0.42 xu / d)
= 0.36 xu/d (1 – 0.42 xu / d) fck b d2
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Limiting value of tension steel and moment of resistance
Since the maximum depth of neutral axis is limited, the maximum value of moment of resistance is also limited.
Mu lim with respect to concrete = 0.36 fck b x Z
= 0.36 fck b Xu (d – 0.42 Xu)
Mu lim with respect to steel = 0.87 fck Ast (d – 0.42 Xu)
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Limiting moment of resistance
Grade of concrete
Grade of steelFe 250 steel Fe 450 steel Fe 500 steel
For a limiting section 200 mm x 300 mm effective , determine the following , if it is reinforced with an effective cover of 50 mm
take M-20 conc. And Fe 250Sol :
b = 200 mm d = 300 mm
fck = 20 N / mm2
the given section is the given section is balanced section therefore , xu = xu max (IS 456 – 2000 , P 70)
b = 200
d= 300
Ast
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For Fe – 250 , xu max = 0.53 d = 0.53 x 300 = 159 mm xu = xu max = 159 mm 1. Maxi . Comp. stress in concrete = .446 x fck = .446 x 20 = 8.92 N / mm2
2 . Lever arm Z = d – 0.42 x
= 233.22 mm
3 . Total tension T = 0.87 fy Ast
Ast ;
xu = (IS 456 – 2000 , P.96)
Ast = 1052.6 mm2
0.87 fy Ast0.36 fck b
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T = 0.87 fy Ast
= 0.87 x 250 x 1052.6
= 228.96 kN
Total compression C = 0.36 fck b xu = 0.36 x 20 x 200 x 159 = 228.96 kN for limiting section C = T .
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Design of section
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Design a rectangular RC beam having width 250mm, simply supported with effective span of 4.5 m, it is loaded wit a udl of 20 kN/m including self weight . Use M20 concrete and Fe 415 steel. Check the beam for maximum and minimum steel and deflection Solution :
factored udl w = 1.5 x 20 b = 250 = 30 kN/m
Mu = wl2 / 2 = 75.94 kN/m Design as a balanced section , fy = 415 N/mmMu lim. = 0.138 fck bd2 75.94 x 106 = 0.138 x 20 x 250 x d2
AstD
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d= 331.75 mm provide d = 340 mm Therefore , effective size of beam is 250 mm x 340 mm to find Ast : pt = 50 fck / fy ( 1- (1- 4.6 Mu / fck bd2)1/2 ) pt = 0.896% Ast = pt bd /100 = 761.6 mm2 provide 4 – 16 dia. (Ast = 804 mm2) effective cover = 40 mm D = d + 40 = 380 mm
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Minimum steel in beam As / bd = 0.85 / fy As = 174.09 mm2 ( mini. Required) Ast > As therefore, ok Maximum steel in beam Ast = 0.4 bD = 0.4 x 250 x 380 = 3800 mm2 804 mm2 < 3800 mm2 therefore, ok
{ IS 456-2000 P.42 }
{ IS 456-2000 P.42 }
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Check for deflection actual l/d = 4500/340 = 13.23 Pt = 100 Ast / bd = 0.945 % Maxi l/d = 20 x M.F = 19.5 actual l/d < maxi. l/d therefore, ok o check for development length o Ld <= 1.3 M1 / V + L0
o d= 340 mm o 12 = 12 * 16 = 192 mmo taking larger of two values L0 = 340 mm o S F at support = wl/2 = (30 * 4.5) / 2 = 67.5 kN o Ast = 804 mm2