Flexural Analysis and Design of Beams 3-3 ~ 3-5 20121008

Design of Reinforced ConcreteDesign of Reinforced Concrete

Chapter 3: Flexural Analysis and Design of Beams

Chapter 3: Flexural Analysis and Design of Beams

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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


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


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


(I) Uncracked section Linear Behavior

Review

E steelconcrete

EnE

Modular ratioconcrete


(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:


(II) Cracked section Linear Behavior

Review

f f f f fLimits of Linearity, fc< 0.5 fc and fs< fy



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



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



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


(III) Cracked section Nonlinear Behavior

Review

f : stress strain curvefc : , stress-strain curvec: ,stress-strain curve



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



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


(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


3.3 Reinforced Concrete Beam BehaviorBalanced Reinforcement ratio

Balanced failure condition: simultaneously crushing of concrete and initiation of steel yielding y g



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


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



Why always < b ?

= b

y y b


3.3 Reinforced Concrete Beam Behavior


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


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)


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


3.4 Design of Tension-Reinforced Rectangular Beams3.4.1 Equivalent Rectangular Stress Distributionq g

Whitney Stress Block



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



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



1; 2 c

10.65 0.85 a

andc1f ' - 280

= 0.85; = 0.85 - 0.0570


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


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


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)


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


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



1. (tension-controlled) =0 003; c=0.003;

t>0.005

2. (compression-controlled) c=0.003;

t



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



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



max0.004t


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


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


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


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


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



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



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


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


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


3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design Aidg



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


3.4 Design of Tension-Reinforced Rectangular BeamsDetailing


3.4 Design of Tension-Reinforced Rectangular Beams3.4.7 Design AidgDesign Algorithm 1

Optimum concrete sectionp

1. Mu=Mn= Rbd2

2. Choose , min<


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)


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.


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)


3.5 Practical considerations in Design of Beams

Documents

Flexural Analysis and Design of Beams 3-3 ~ 3-5 20121008