flexural analysis and design of beams 3-3 ~ 3-5 20121008
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Design of Reinforced ConcreteDesign of Reinforced Concrete
Chapter 3: Flexural Analysis and Design of Beams
Chapter 3: Flexural Analysis and Design of Beams
wcliao@ntu.edu.tw R803; 3366-4337;
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
Review
1. Introduction2. Bending of Homogeneous Beams3 R i f d C t B B h i3. Reinforced Concrete Beam Behavior4. Design of Tension-Reinforced Rectangular Beams5 Practical Considerations in Design of Beams5. Practical Considerations in Design of Beams6. Rectangular Beams with Tension and Compression
Reinforcement7. T-Beams
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
B di f H B
Review
Bending of Homogeneous Beams
Essential Assumptions
o Plane sections (before loading) remain plane (after loading) Compatibilityloading) Compatibility
o Development of stress and strain in the materials follows a given stress-strain diagram Constitutive
oEquilibrium- in any section all the time
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
Reinforced M
Review
Concrete Beam Behavior
M (III)
EIcr (II)
EI (I)
EIg (I)Sectional Analysis:I U k d ti Li B h iI. Uncracked section Linear BehaviorII. Cracked section Linear BehaviorIII. Cracked section Nonlinear Behavior
Compatibility, Constitutive, Equilibrium
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(I) Uncracked section Linear Behavior
Review
E steelconcrete
EnE
Modular ratioconcrete
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(I) Uncracked section Linear Behavior
Review
( ) ( 1)2 shbh n A dSol1: Method of Transformed Section
3 3 2
2( 1)
1 1 ( ) ( 1) ( )
s
s
ybh n A
I b b h A d3 2( ) ( 1) ( )3 3
g sI b y b h y n A d y
Mf y
cg
g
f yI
IM f
y
( )
cr rr
M fh y
fM E
y
cr ct
crg
M EEI h y h y Mcr:
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(II) Cracked section Linear Behavior
Review
f f f f fLimits of Linearity, fc< 0.5 fc and fs< fy
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(II) Cracked section Linear Behavior
Review
Sol 1: Method of Transformed Section
( ) ( )2 syb y nA d y
3 21 ( )3
cr sI b y nA d y31 ( )2 3
c yM b y f d2 3 M
crEI
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(II) Cracked section Linear Behavior
Review
Sol 2: Sectional Analysis
: c scompatibility
kd d kd1: ,2
c c c cconstitutive f E C f bkd,1 1:
s s s s s s
s sc s
f E T A f bdfEequilibrium C T f bkd bdf k
2
2 21 1 1 0
c sc c
q f fE
kk n k nk nk
2
2 2( ) 2
kk n n n
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(II) Cracked section Linear Behavior
Review
: 1 kd klocation of N A jd d jSol 2: Sectional Analysis
. . : 13 3
:
s s s
location of N A jd d j
MMoment M Tjd A f jd fA jd
22
1 1 2,2 2
s s ss
s c c c
j f j fA jd
MM Cjd f bkdjd f kjbd fkjbd 22 2s c c c
j f j f j fkjbd
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(III) Cracked section Nonlinear Behavior
Review
f : stress strain curvefc : , stress-strain curvec: ,stress-strain curve
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(III) Cracked section Nonlinear Behavior
Review
: u ccompatibilityd
: '
s
c
p yd c
constitutive C f bc
: '
s s
s s s
T A fA f f dequilibrium C T f bc A f c:
' '
( . .)
c s s c cequilibrium C T f bc A f c f b flocation of N A
: ( )
' ( )
s s
c
Moment M Tz A f d cCz f bc d cc
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(III) Cracked section Nonlinear Behavior
Review
Code formatRequirement of first yielding of steel fs=fy for limited steel
ys f df dq y g s y
amount
' '
ys
c c
fc cf f
2( ) ( ) (1 )' '
y y
n s y y y
f d fM A f d c bdf d bd f
f f c cf fNominal moment
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
(III) Cracked section Nonlinear Behaviorf d f
Review
2( ) ( ) (1 )' '
y y
n s y y yc c
f d fM A f d c bdf d bd f
f f
2 (1 0.59 )'y
n y
fM bd f
f
'y cf
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.3 Reinforced Concrete Beam BehaviorBalanced Reinforcement ratio
Balanced failure condition: simultaneously crushing of concrete and initiation of steel yielding y g
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.3 Reinforced Concrete Beam BehaviorBalanced Reinforcement ratio
yu ubc dc d c
; '
b b u y
c b b y
c d cC T f bc bdf
'
c b b y
c ub
f fff
b y u yf
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.3 Reinforced Concrete Beam BehaviorBalanced Reinforcement ratio(a) < b (under reinforced) , , ductile!!
2 (1 0.59 )'
yn y fM bd f f 'y cf
(b) > b (over reinforced) ,, brittle!! s s s yf E f
'
s s s y
s ss u
f f
A fd c cf b'
( )
c
s s
c f bM A f d c
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.3 Reinforced Concrete Beam BehaviorBalanced Reinforcement ratio
Why always < b ?
= b
y y b
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.3 Reinforced Concrete Beam BehaviorBalanced Reinforcement ratio
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.3 Reinforced Concrete Beam Behavior
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
Chapter Outline
1. Introduction2. Bending of Homogeneous Beams3 R i f d C t B B h i3. Reinforced Concrete Beam Behavior4. Design of Tension-Reinforced Rectangular Beams5 Practical Considerations in Design of Beams5. Practical Considerations in Design of Beams6. Rectangular Beams with Tension and Compression
Reinforcement7. T-Beams
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
Analysis:Analysis:A given section (reinforcement layout, material strength) Find capacity (Mn, Vn, Pn)
Design:A given load condition
Factored Load ()
A l i (M V P )Analysis (Mu, Vu, Pu)
Find section dimension reinforcement so thatFind section dimension, reinforcement so thatMn> Mu, Vn> Vu, Pn> Pu (: strength reduction factor)
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular BeamsSection Definition
d = distance from extreme compression'' sAbd
d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement
dt = distance from extreme compression fiber to centroid of extreme layer of longitudinal tension steellongitudinal tension steel
d = distance from extreme compression fiber to centroid of longitudinal gcompression reinforcement
As = area of nonprestressed longitudinal
sA
tension reinforcement
As= area of compression reinforcements
bd
Cover: 1.5 (4cm) to stirrup
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.1 Equivalent Rectangular Stress Distributionq g
Whitney Stress Block
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.1 Equivalent Rectangular Stress Distributionq g
Equivalent Stress Distribution same effect
:magnitude of resultantc' '
:
c c cC f cb f ab alocation of resultant
1
:
22 2
location of resultant
a a ac12 2
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.1 Equivalent Rectangular Stress Distributionq g
1; 2 ca12' 280 , 2 2 0.425 0.85
0 72
c kgffor f cmc c
1 1
0.72 0.852 0.85
,
c ca c
In general
1
,
0.65 0.85 n gene al
andc1f ' - 280
= 0.85; = 0.85 - 0.0570
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.1 Equivalent Rectangular Stress Distributionq g
1; 2 c
10.65 0.85 a
andc1f ' - 280
= 0.85; = 0.85 - 0.0570
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.2 Balanced Strain Condition
ubc d compare
10.85 ' 0.85 '
u y
b y c b c bT C f bd f ba f bc'
c ub
y u y
ff
1'0.85
c ub
y u y
ff
y u yf
1 1
c ca c
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.3 Under-reinforced Beams (Tension Failure)( )
b
()T( f =f ) T=C CT(, fs fy) T C C(c2>c1 c2>c1), (c2
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.4 Over-reinforced Beams (Compression Failure)
bcrushing (c1=0.003) ()
T T C CT T=C C()C (c2>c1)C, (c2>c1)(brittle)(no warning)
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.4 Over-reinforced Beams (Compression Failure)
:Analysis :
0 003
s s s yAnalysisf E f
d c0.003 0.003
0 003
ss
d cc d c c
d cf E E
11
0.003
0 003
s s s sf E Ecd aa c f E1
1
0.003
0.85 ' 0.003
s s
c s s s
a c f Ea
d aC T f ab A f E bd
2 21
0.85 '( ) 00.003
c
af a ad d find aE0.003
0.85 ' ( )2
s
c
EaM f ab d
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.5 ACI Code Provisions for Under-reinforced Beams
1. =b (NOT preferred) Mechanical properties of materials vary Amount of materials varies Strain hardening of steel Strain-hardening of steel Just yielding without ductility for warning
2. Code addresses The minimum tensile reinforcement strain allowed
at nominal strength The strength reduction factor accordingly The strength reduction factor accordingly
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.5 ACI Code Provisions for Under-reinforced Beams
1. (tension-controlled) =0 003; c=0.003;
t>0.005
2. (compression-controlled) c=0.003;
t
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.5 ACI Code Provisions for Under-reinforced Beams
0 003c 0 003c 0 003c0.0030.003 0.0020 600
tcd
0.0030.003 0.0040 429
tcd
0.0030.003 0.0050 375
tcd
0.600max
0.429min ( )
t code
0.375
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
ub
u y
c d 1
1
0.85 ' 0.85 '
'0.85
b y c b c b
c ub
T C f bd f ba f bcff
y u yf
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.5 ACI Code Provisions for Under-reinforced Beams
max0.004t
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.6 Minimum Reinforcement RatioObjective: To avoid brittle failureMethod:
n crM MStrength of RC beam with Strength of Plain Concrete beam minStrength of RC beam with Strength of Plain Concrete beam
2 'f f3
2
2 '
'12
r cf fbh
I f bhmin ( )2 n y aM bdf d 122 '
32
g ccr r cI f bh
M f f hymin ( )2
n yf
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.6 Minimum Reinforcement RatioFor typical sections with small
a h0.05 ; 1.12
a hassume dd
M M2'
( ) 0 95
n cr
c
M M
f bhaM bdf d bdf d M( ) 0.952 3' '0 333
n y y crM bdf d bdf d Mf fh 2 6.2' 210 kgff20.333 ( ) 0.425
0.95 c c
y y
f fhf d f
2
2
210
7.1' 280
cy
cy
fcm fkgffcm f
2
8.0' 350 y
cy
fkgffcm f
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.6 Minimum Reinforcement RatioACI code ( 401-100) takes Mn,min 2Mcr
0 8 ' 14fmin
0.8 14 cy y
ff f
min, 0.5% For beam
i
0.8 ' 14 cfA b d b d,min s w wy y
A b d b df f
R k M i R i f t R ti ( 0 004)Remarks: Maximum Reinforcement Ratio (t=0.004)
' ' 0.0030 85 0 85 c u cf fmax 1 10.85 0.85 0.003 0.004 y u t yf f
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.6 Minimum Reinforcement Ratio
For statically determinate T-beam with the flange in tension
bE: effective width of flange
0.8 ' 14 cf,min 0.8 14(2 ) (2 )
0 8 '
cs w w
y y
fA b d b d
f fsmaller
f,min
0.8 '( )
cs E
y
fA b d
f
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 ExamplespAnalysis:Given: section, reinforcement, material strength, , gFind: moment capacity
D iDesign:Given: required moment capacity, material strengthFind: section dimension reinforcementFind: section dimension, reinforcement
2 2(1 0.59 )'
yn y fM f bd Rbdf(1 0.59 )
''
c
yy
f
fR f
f ff( )
','( )
yc yc
ff ff
flexural resistance factor
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
max 1 0.004'0.85
( )c u
y u t
ff
A f
'0.85s y
c
A fa
f b
2 (1 0.59 )'
4066462
yn y
c
fM bd f
fk f
406646240.66
kgf cmtf m
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
2wLM
'0 85 c uf 8ultimate
M
0max . 190.85
0.0( )05for y u tf
0.004t
2 (1 0.59 )'y
n yc
fM bd f
f
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design AidgAppendix A of Design of Concrete Structures, NilsonTables A.1, A.2, A.4~6; Graph A.1 , , ; p
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design AidgAppendix A of Design of Concrete Structures, NilsonTables A.1, A.2, A.4~7; Graph A.1 , , ; p
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design Aidg
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
-
Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design Aidg
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
2 (1 0.59 )yf
M bd f
2 2
(1 0.59 )'
949 12 17.5 3487575
n yc
M bd ff
Rbd lb in
40.6 tf m
-
Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular BeamsDetailing
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design AidgDesign Algorithm 1
Optimum concrete sectionp
1. Mu=Mn= Rbd2
2. Choose , min<
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design AidgDesign Algorithm 2
Select concrete section find
1. Select b and d, find R=Mu/(bd2)
2. Find by Table A.5
3. Choose steel As= bd (Table A.1)
4 Check detailing (Table A 7)4. Check detailing (Table A.7)
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.5 Practical considerations in Design of Beams
OObjective: Translating theoretical requirement to practical design
D t ili f BDetailing of Beams:1. 5cm (2)2 2. 3. , steel #11 or smaller4 4.
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.5 Practical considerations in Design of Beamsa. 1.5 (4cm) clear cover to stirrupa. 1.5 (4cm) clear cover to stirrupb. Stirrup bar diameter
at least #3 tie for #10 or smallert l t #4 ti f #11 #14 #18at least #4 tie for #11, #14, #18
and bundledc. diameter of corner bar is
d t b l t d tassumed to be located to intersect the horizontal tangent to stirrup bend- For #11 or smaller, c=3/4
(2cm)- For #14 and #18, c=0.5db
d. Clear spacing db or 1 (2.5cm)or > 1.33 dmax () whichever is greater
e. Clear spacing 1 (2.5cm)
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.5 Practical considerations in Design of Beams
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Design of Reinforced Concrete 101-1Chapter 3. Flexural Analysis and Design of Beams
3.5 Practical considerations in Design of Beams
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