1. CE8501 Design Of Reinforced Cement Concrete Elements
Unit 1-Introduction
Analysis of singly reinforced beam
[As per IS456:2000]
Presentation by,
P.Selvakumar.,B.E.,M.E.
Assistant Professor,
Department Of Civil Engineering,
Knowledge Institute Of Technology, Salem.
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2. Singly Reinforced Rectangular beam โ Analysis
1. Types of section based on limiting moment
2. Moment of resistance as per IS 456:2000
3. Limiting moment of resistance as per SP-16
4. Area of steel (Ast) as per IS 456:2000
5. Area of steel (Ast) as per SP-16
6. Example problems #03,04
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3. Types of section based on limiting moment
โข If Mu < Mu,lim then it is under reinforced section
โข If Mu = Mu,lim then it is balanced section
โข If Mu > Mu,lim then it is over reinforced section
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4. Area of steel (Ast)
โข If Mu < Mu,lim then it is under reinforced section
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[Refer IS456 Pg.96]
Ast derived from
5. Area of steel (Ast)
โข If Mu = Mu,lim then it is balanced section
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[Refer IS456 Pg.96]
Ast derived from
6. Limiting Moment of resistance
โข If Mu > Mu,lim then it is over reinforced section
โข The section must be designed as doubly reinforced section.
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7. Limiting moment of resistance
[SP -16] (Alternate Method)
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For Fe 415 HYSD bars,
Mu,lim = 0.138 fck b d2 [SP -16, Pg:10, Table C]
(Ast unknown)
๐ฅ ๐ข
๐
=
0.87 ๐๐ฆ ๐ด๐ ๐ก
0.36 ๐๐๐ ๐ ๐
Not applicable
8. Area of tensile reinforcement (Ast)
[SP -16] (Alternate Method)
Ast =
๐ท๐
๐ ๐
๐๐๐
[SP -16, Pg:17]
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๐๐ข
๐๐2 value, Corresponding
pt for 415
Percentage of steel
Pt
[SP -16, Pg:48, Table โ 2]
9. Problem#03 Area of steel (Ast) type
โข Determine the area of reinforcement required for a singly reinforced concrete
section having a breadth of 300mm and effective depth of 675 mm to support a
factored moment of 185 kNm. Adopt M-20 grade concrete and Fe-415 grade
HYSD bars.
โข Given :
b = 300mm d = 675mm
M-20 โ fck = 20N/mm2 Fe415 โ fy = 415N/mm2
Ast = ? Mu= 185 kNm = 185 x 106 N.mm
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b= 300mm
d =675 mm
Ast
10. Step 1 : Limiting moment of resistance
[IS 456:2000]
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For Fe 415 HYSD bars,
Mu,lim = 0.36
๐๐,๐๐๐
๐
[1 โ 0.42
๐๐,๐๐๐
๐
] b d2 fck
Mu,lim = 0.36 * 0.48 [1 โ 0.42 โ 0.48 ] * 300 * 6752 * 20
= 377.15 x 106 N.mm
= 377.15 kN.m > 185 kN.m [Mu<Mu,lim Under reinforced section]
(Ast unknown)
๐ฅ ๐ข
๐
=
0.87 ๐๐ฆ ๐ด๐ ๐ก
0.36 ๐๐๐ ๐ ๐
Not applicable
11. Step 2 : Area of tensile reinforcement (Ast)
[IS 456:2000]
For under reinforced section, Ast derived from Mu
Mu = 0.87 fy Ast d [1 -
๐จ ๐๐
๐๐
๐ ๐ ๐๐๐
]
185 * 106 = 0.87 * 415 * Ast * 675 [1-
Astโ415
300โ675โ20
]
= (- 24.9 Ast
2 )+ (243.71 * 103 Ast ) โ (185 * 106)
Ast = 829.37 โ 830 mm2
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12. Step 1 : Limiting moment of resistance
[SP -16] (Alternate Method)
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For Fe 415 HYSD bars,
Mu,lim = 0.138 fck b d2 [SP -16, Pg:10, Table C]
Mu,lim = 0.138 * 20 * 300 * 6752
= 377.25 x 106 N.mm
= 377.25 kN.m > 185 kN.m [ Under reinforced section]
(Ast unknown)
๐ฅ ๐ข
๐
=
0.87 ๐๐ฆ ๐ด๐ ๐ก
0.36 ๐๐๐ ๐ ๐
Not applicable
13. Step 2 : Area of tensile reinforcement (Ast)
[SP -16] (Alternate Method)
For under reinforced section, Ast derived from Mu
Ast =
๐ท๐
๐ ๐
๐๐๐
[SP -16, Pg:17]
Ast =
0.409 โ300 โ675
100
Ast = 828.2 โ 830 mm2
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๐๐ข
๐๐2
=
185 โ 106
300 โ 6752
๐๐ข
๐๐2 = 1.35
Corresponding pt for 415
Percentage of steel
Pt = 0.409
[SP -16, Pg:48, Table โ 2]
14. Problem#04 Area of steel (Ast) and depth of beam
โข Design the minimum effective depth required and the area of reinforcement
for a rectangular beam having a width of 300mm to resist an ultimate moment
of 200 kNm, using M20 grade concrete and Fe-415 HYSD bars.
Given :
b = 300mm d = ?
M-20 โ fck = 20N/mm2 Fe415 โ fy = 415N/mm2
Ast = ? Mu= 200 kNm = 200 x 106 N.mm
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b= 300mm
d =?
Ast
15. Step 1 : Depth of the section
[IS 456:2000]
Mu,lim = 0.36
๐๐,๐๐๐
๐
[1 โ 0.42
๐๐,๐๐๐
๐
] b d2 fck
200*106 = 0.36 * 0.48 [1 โ 0.42 โ 0.48 ] * 300 * d2 * 20
d = 491.5 mm
d โ 500 mm
For Fe 415 HYSD bars,
Mu,lim = 0.138 fck b d2 [SP -16, Pg:10, Table C]
Mu,lim = 0.138 * 20 * 300 * d2
d=
๐ ๐ข
,
๐๐๐
0.138 ๐๐๐ ๐
=
200โ106
0.138 โ20 โ300
= 491.47 ๐๐
d โ 500 ๐๐
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Alternate method
[SP -16]
17. Step 2 : Area of tensile reinforcement (Ast)
[IS 456:2000]
For under reinforced section, Ast derived from Mu
Mu = 0.87 fy Ast d [1 -
๐จ ๐๐
๐๐
๐ ๐ ๐๐๐
]
200 * 106 = 0.87 * 415 * Ast * 492 [1-
Astโ415
300โ492โ20
]
= (- 24.97 Ast
2 )+ (177.64 * 103 Ast ) โ (200 * 106)
Ast = 1402 mm2 โ 1410 mm2
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18. Step 2 : Area of tensile reinforcement (Ast)
[SP -16] (Alternate Method)
For under reinforced section, Ast derived from Mu
Ast =
๐ท๐
๐ ๐
๐๐๐
[SP -16, Pg:17]
Ast =
0.995 โ300 โ492
100
Ast = 1409.5 โ 1410 mm2
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๐๐ข
๐๐2
=
200 โ 106
300 โ 4922
๐๐ข
๐๐2 = 2.75
Corresponding pt for 415
Pt = 0.995
[SP -16, Pg:48, Table โ 2]
19. Assignment#02
โข Determine the area of reinforcement required for a singly
reinforced concrete section having a breadth of 300 mm and an
effective depth of 500 mm to support a factored moment of
230 kN.m. Assume M25 grade concrete and Fe 145 steel.
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20. Assignment#03
โข Determine the minimum effective depth required and the
correspond ing area of tension reinforcement for a rectangular
beam having a width of 300 mm to resist an ultimate moment of
250 kN.m, Using M-30 Grade concrete and Fe-415 HYSD bars.
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