Singly reinforced beam ast - over reinforced, ast problems ,area of steel ,is456 ,sp16 ,beam ,mu ,mulim ,steel area ,anna university ,civil engineering

1. CE8501 Design Of Reinforced Cement Concrete Elements
Unit 1-Introduction
Analysis of Singly reinforced beam (Ast)
[As per IS456:2000] & SP -16
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 – 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 for over reinforced beam as per IS 456:2000
5. Example#05
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3. Types of section based on limiting moment
• If Mu < Mu,lim then it is under reinforced section
• If Mu = Mu,lim then it is balanced section
• If Mu > Mu,lim then it is over reinforced section
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4. Area of steel (Ast)
• For balanced and Under reinforced sections
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[Refer IS456 Pg.96]
Ast derived from

5. 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)
𝑥 𝑢
𝑑
=
0.87 𝑓𝑦 𝐴𝑠𝑡
0.36 𝑓𝑐𝑘 𝑏 𝑑
Not applicable

6. Limiting Moment of resistance
• If Mu > Mu,lim then it is over reinforced section
• The section must be designed as doubly reinforced section.
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7. Area of steel (Over reinforced section)
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Mu>Mu,lim Over reinforced section. Hence it is recommended to design as
doubly reinforced section
For a doubly reinforced section, reinforcement steel is necessary to provide
both in tension and compression zone.
• Ast = (Ast1 + Ast2 ) Area of steel at tension zone
• Asc – Area of steel at compression zone
Ast
Asc

8. Area of tension reinforcement (Over reinforced section)
Ast= Ast1 + Ast2 • Mu1 derived from
• Mu1 = Mu,lim
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Mu2 derived from
• Mu2 = 𝐌 𝐮 − 𝐌𝐮, 𝐥𝐢𝐦
• Mu – (Actual B.M at beam)
• Given in question
Ast formula (Ast1)
Mu1 = 0.87 fy Ast1 d [1 -
𝑨 𝒔𝒕𝟏
𝒇𝒚
𝒃 𝒅 𝒇𝒄𝒌
]
Ast =
𝑷𝒕 𝒃 𝒅
𝟏𝟎𝟎or
IS 456 SP 16
Ast formula (Ast2)
Ast2 =
𝑴 𝒖𝟐
𝟎.𝟖𝟕 𝒇𝒚 (𝒅−𝒅′)
Mu2 = 0.87 fy Ast2 (d-d’)
SP 16, pg:12
Mu1 Mu2

9. SP -16 Recommendation
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10. Area of Compression reinforcement (Asc)
Mu2 = 𝐴 𝑠𝑐 𝑓 𝑠𝑐 − 𝑓𝑐𝑐 𝑑 − 𝑑′
Asc =
𝑀 𝑢2
𝑓 𝑠𝑐
−𝑓𝑐𝑐 𝑑 −𝑑′
10
SP 16, pg:12
SP 16, pg:13
𝑓𝑐𝑐 = 0.446 𝑓𝑐𝑘
Asc =
𝑀 𝑢2
𝑓 𝑠𝑐
−0.446 𝑓𝑐𝑘 𝑑 −𝑑′
d’ is the distance from the extreme
compression fibre to the centroid of
compression reinforcement.

11. Problem#05 Area of steel (Ast) type
(over reinforced beam)
• A rectangular beam has b = 200 mm, d = 400 mm. If steel used is Fe 415
and grade of concrete is M20, find the steel required to carry a factored
moment of 120 kNm.
Given :
b = 200mm d = 400mm
M-20 – fck = 20N/mm2 Fe415 – fy = 415N/mm2
Ast = ? Mu= 120 kNm = 120 x 106 N.mm
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b= 200mm
d =400 mm
Ast

12. Step 1 : Limiting moment of resistance
[IS 456:2000]
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For Fe 415 HYSD bars,
Mu,lim = 0.36
𝒙𝒖,𝒎𝒂𝒙
𝒅
[1 – 0.42
𝒙𝒖,𝒎𝒂𝒙
𝒅
] b d2 fck
Mu,lim = 0.36 * 0.48 [1 – 0.42 ∗ 0.48 ] * 200 * 4002 * 20
= 88.30 x 106 N.mm
= 88.30 kN.m < 120 kN.m [Mu>Mu,lim Over reinforced section]
(Ast unknown)
𝑥 𝑢
𝑑
=
0.87 𝑓𝑦 𝐴𝑠𝑡
0.36 𝑓𝑐𝑘 𝑏 𝑑
Not applicable

13. Step:2 Calculation of Mu1 & Mu2
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Mu2 derived from
• Mu2 = 𝐌 𝐮 − 𝐌𝐮, 𝐥𝐢𝐦
Mu2 = 120 − 88.30
Mu2 = 𝟑𝟏. 𝟕 𝐤𝐍. 𝐦
• Mu1 derived from
• Mu1 = Mu,lim
Mu1 = 88.30 kN.m

14. Step 3 : Area of steel at tension zone (Ast)
For under reinforced section, Ast1 derived from Mu
Mu1 = 0.87 fy Ast1 d [1 -
𝑨 𝒔𝒕𝟏
𝒇𝒚
𝒃 𝒅 𝒇𝒄𝒌
]
88.30 * 106 = 0.87 * 415 * Ast1 * 400 [1-
Ast1
∗
415
200∗400∗20
]
= (- 37.46 Ast1
2 )+ (144.42 * 103 Ast1 ) – (88.30 * 106)
Ast1 = 762 ≈ 770 mm2
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Ast1

15. Step 3 : Area of steel at tension zone (Ast)
For under reinforced section, Ast2 derived from Mu
Ast2 =
𝑴 𝒖𝟐
𝟎.𝟖𝟕 𝒇𝒚 (𝒅−𝒅′)
=
31.7 ∗106
0.87 ∗415 (400−40)
Ast2 = 243.88 mm2 ≈ 250 mm2
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Ast2
𝑑′
𝑑
= 0.10
𝑑′
= 0.10 * 400
= 40 mm
𝑓𝑠𝑐 = 353 N/mm2

16. Step 3 : Area of steel at tension zone (Ast)
Area of tension reinforcement
Ast1 = 770 mm2
Ast2 = 250 mm2
Ast= Ast1 + Ast2
Ast= 1020 mm2
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Ast

17. Step 2 : Area of steel at Compression zone (Asc)
Asc =
𝑀 𝑢2
𝑓 𝑠𝑐 −0.44 𝑓𝑐𝑘 𝑑 −𝑑′
Asc =
31.7 ∗106
353 −0.446 ∗20 400−40
Asc = 255.91 mm2 ≈ 260 mm2
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𝑑′
𝑑
= 0.10
𝑑′ = 0.10 * 400
= 40 mm
𝑓𝑠𝑐 = 353 N/mm2
SP 16, pg:13
Asc

18. Area of Steel
At tension zone,
Ast = 1020 mm2
At compression zone,
Asc = 260 mm2
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Ast
AscAst
Asc

19. Assignment#04
• A reinforced concrete beam of rectangular section of size
250 x 550 mm overall is to be designed for a factored moment of
225 kNm. Compute the reinforcement required at the effective
cover of 50 mm. The concrete mix to be used is M20 and the grade
of steel is Fe415.
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20. Thank You
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