CS3001 RC2 (Flexure Design of Beams)
CS3001 RC2 (Flexure Design of Beams)
CS3001 RC2 (Flexure Design of Beams)
Reinforced Concrete #2
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Dr. Rabin Tuladhar March 14, 2011
l
εc σc
εct: tension strain in concrete
dn at cracking
D f’cf : concrete flexure tensile
strength
M εst σst : tension stress in steel
εct σst
σct=f’ct.f
strain profile stress profile
ε st ε cu 0.003
= =
d − dn dn dn 3
dn=kud
d
M
εst σst
b γ ku d 2
εcu = 0.003 α2f’c
γ ku d C
γ ku d ku d
D d d − γ ku d 2
Ast
fsy T
εst > εsy
strain stress force
C =T
α 2 f c'bγku d = Ast f sy
Ast f sy M u = T × z = Ast f sy × (d − γk u d 2 )
ku =
α 2 f bγd
c
'
b γ ku d 2
γ ku d ku d γ ku d C
d d − γ ku d 2
Ast
σst< fsy T
D εst < εsy
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b) Compression failure – Over reinforced section
Here, C = α 2 f c'bγku d
1 − ku
And, T = Astσ st = Ast ε st E s since ε st < ε sy ⇒ σ st = E s × ε st = E s × × ε cu
ku
Equilibrium of horizontal forces
C =T
α 2 f c'bγku d = Ast ε st E s
1 − ku Solving this quadratic equation we can get
α 2 f c'bγku d = Ast E sε cu ku and we can find nominal flexural
ku strength
M u = T × z = Astσ st × (d − γku d 2 ) 7
b γ kub d 2
εcu = 0.003 α2f’c
kub d γ kub d C
D d d − γ kub d 2
Ast
fsy T
εst = εsy
strain stress
force
C =T
α 2 f c'bγkub d = Ast f sy
f c'
Ast = α 2 kubγbd
f sy
Ast f'
ρb = = α 2 c kubγ 9
bd f sy
ρb) for
Example #3 Evaluate reinforcement ratio at balance point (ρ
D500N steel assuming fc’ = 40 MPa) εcu = 0.003
kud
d
εst = εsy
strain
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Failure mode
a) ku = 0.15-0.3 b) ku = 0.45-0.55
ductile behaviour brittle behaviour
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Change in angle:
δθ =
(ε 0δx + ε st δx )
d
Hence,
δθ (ε 0 + ε st ) ε 0
κ= = =
δx d dn
For small deflections, the curvature equals the slope of the average strain diagram
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Ductility
- Ductility allows large deformations to occur under overload conditions
before eventual failure
- Ductile members give warning of the impending failure
- Under reinforced beams are more ductile
κu
Ductility: µ=
κy
where,
ε cu
κu =
dn
Moment-Curvature plot for an under reinforced beam
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Known sectional
properties
250mm γ ku d 2
εcu = 0.003 0.85f’c
ku d γ ku d C
450mm 400mm d − γ ku d 2
3D24
σst T
εst
strain stress
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