Doubly reinforced beam analysis

CE8501 Design Of Reinforced Cement Concrete Elements
Unit 1-Introduction
Analysis of Doubly reinforced beam
[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.
1

Singly Reinforced Rectangular beam – Analysis
1. Types of section based on limiting moment
2. Moment of resistance in tension zone per IS 456:2000
3. Moment of resistance in tension zone per IS 456:2000
4. Example#
2

Types of section based on neutral axis depth
• If xu < xu,max then it is under reinforced section
• If xu = xu,max then it is balanced section
• If xu > xu,max then it is over reinforced section
3

Neutral axis depth (xu)
4
𝑓 𝑐𝑐 = 0.446 𝑓 𝑐𝑘
xu =
0.87 𝑓𝑦 𝐴𝑠𝑡 − 𝑓 𝑠𝑐
−𝑓𝑐 𝑐
𝐴 𝑠𝑐
0.36 𝑓𝑐𝑘 𝑏

Ultimate Moment due to tension (Mut)
Mut= Mu1 + Mu2
• Mu1 derived from
• Mu1 = Mu,lim
5
Mu2 derived from
• Mu2 = Ast2 (0.87 fy) (d-d’)
SP 16, pg:12
Mu,lim = 0.36
𝒙𝒖,𝒎𝒂𝒙
𝒅
[1 – 0.42
𝒅
] b d2 fck
Ast2 = Ast – Ast1

Ultimate Moment of due to compression (Muc)
Muc= Mu1 + Mu2
• Mu1 = Mu,lim
6
Mu2 derived from
Mu2 = 𝑨𝒔𝒄 𝒇𝒔𝒄 − 𝒇𝒄𝒄 𝒅 − 𝒅′
SP 16, pg:12
Mu,lim = 0.36
𝒅
[1 – 0.42
𝒅
] b d2 fck
𝑓 𝑐𝑐 = 0.446 𝑓𝑐𝑘

Problem#08
• Determine the ultimate moment carrying capacity of a doubly
reinforced beam with b= 350mm, d’= 60mm, d= 550mm. Asc= 1690mm2,
Ast= 4310mm2, fck= 30 N/mm2, fy= 415 N/mm2.
7
b= 350mm
d =550 mm
Ast
Given :
b = 350mm
d = 550mm
fck = 30N/mm2
fy = 415N/mm2
Ast = 4310mm2
Asc = 1690mm2
To find
Ultimate moment by
compression
Muc= ?
Ultimate moment by tension
Mut= ?
d’ =60 mm
Asc

Step 1 : Depth of neutral axis
8
xu =
0.87 ∗415 ∗4310 − 353−13.38 ∗1690
0.36 ∗30 ∗350
xu = 260 mm
xu =
0.87 𝑓𝑦 𝐴𝑠𝑡 − 𝑓 𝑠𝑐
−𝑓𝑐 𝑐
𝐴 𝑠𝑐
0.36 𝑓𝑐𝑘 𝑏
Xu,max = 0.48 * d
Xu,max = 0.48 * 550
Xu,max = 264 mm
[xu<xu,max Hence it is under reinforced section]

Step 2: Limiting moment of resistance (Mu,lim)
Mu,lim = 0.36
𝒅
[1 – 0.42
𝒅
] b d2 fck
Mu,lim = 0.36 * 0.48 [1 – 0.42 ∗ 0.48 ] * 350 * 5502 * 30
= 438.20 x 106 N.mm
Mu,lim = 438.20 kN.m [Consider as Mu1]
9

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
]
438.20 * 106 = 0.87 * 415 * Ast1 * 550 [1-
Ast1
∗
415
350∗550∗30
]
= (- 14.27 Ast1
2 )+ (198.58 * 103 Ast1 ) – (438.2 * 106)
Ast1 = 2750 mm2
10
Ast1

Step 4: Ultimate Moment due to tension (Mut)
Mut= Mu1 + Mu2
Mu1 derived from
• Mu1 = Mu,lim
11
Mu2 derived from
• Mu2 = Ast2 (0.87 fy) (d-d’)
Mu1 = 438.20 kN.m
Ast2 = Ast – Ast1
= 4310 – 2750
Ast2 = 1560 mm2
Mu2 = 1560 (0.87 * 415) (550-60)
Mu2 = 276 x106 N.mm
Mu2 = 276 kN.m

Step 4: Ultimate Moment due to tension (Mut)
Mut= Mu1 + Mu2
12
Mu1 = 438.20 kN.m
Mut = 438.20 + 276
Mut = 714.2 kN.m
Mu2 = 276 kN.m Mut

Step 5: Ultimate Moment of due to compression
(Muc)
Muc= Mu1 + Mu2
• Mu1 = Mu,lim
13
Mu2 derived from
Mu2 = 𝑨𝒔𝒄 𝒇𝒔𝒄 − 𝒇𝒄𝒄 𝒅 − 𝒅′
Mu1 = 438.20 kN.m

Ultimate Moment of due to compression (Muc)
= 1690 353 − 13.38 550 − 60
= 281.23 x 106 N.mm
Mu2 = 281.23 kN.m
14
𝑓𝑐𝑐 = 0.446 𝑓𝑐𝑘
𝑓𝑐𝑐 = 0.446 ∗ 30
𝑓𝑐𝑐 = 13.38 N/mm2
𝑑′
𝑑
=
60
550
= 0.10
Hence from table F
𝒇 𝒔𝒄 = 353 N/mm2
Mu2 = 𝑨 𝒔𝒄 𝒇 𝒔𝒄 − 𝒇 𝒄𝒄 𝒅 − 𝒅′

Step 5: Ultimate Moment due to Compression
(Muc)
Muc= Mu1 + Mu2
15
Mu1 = 438.20 kN.m
Muc = 438.20 + 281.23
Muc = 719.43 kN.m
Mu2 = 281.23 kN.m
Muc

Result : Ultimate moment of resistance
At tension zone,
Mut = 714.2 kN.m
At compression zone,
Muc = 719.43 kN.m
16
Mut
Muc

Assisgnment#06
• Determine the ultimate moment carrying capacity of a doubly
reinforced beam with b= 280mm, d’= 50mm, d= 510mm. Ast= 2455 mm2,
Asc= 402 mm2, fck= 30 N/mm2, fy= 415 N/mm2.
17

Doubly reinforced beam analysis

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