Design of Reinforced Concrete Beams for Shear

8/6/2019 Design of Reinforced Concrete Beams for Shear

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Lecture 17 - Design of Reinforced

Concrete Beams for Shear

November 1, 2001

CVEN 444


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Lecture GoalsLecture Goals

Stirrup Design


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Uncracked Elastic BeamUncracked Elastic Beam

BehaviorBehavior

Look at the shear and

bending moment

diagrams. The acting

shear stress distribution

on the beam.


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BehaviorBehavior

The acting stresses distributed

across the cross-section.

The shear stress acting on

the rectangular beam.

Ib

VQ!X


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BehaviorBehavior

The equation of the shear stress for a rectangular beam is given as:

Note: The maximum 1st

moment occurs at the neutral

axis (NA).

Ib

VQ!X

avemax

2

max

3

5.1*2

3

84*2Q

InertiaoMoment12

XX !

!

!

!

!

bh

V

bhhbh

bhI


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BehaviorBehavior

The ideal shear stress distribution can be described as:

Ib

VQ

!X


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Uncracked Elastic Beam BehaviorUncracked Elastic Beam Behavior

A realistic description of the shear distribution is shown as:


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Uncracked Elastic Beam BehaviorUncracked Elastic Beam Behavior

The shear stress acting along the beam can be described with a

stress block:

Using ohrs circle, the stress block can be manipulated to

find the maximum shear and the crack formation.


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Inclined Crackingin ReinforcedInclined Crackingin Reinforced

Concrete BeamsConcrete Beams

Typical Crack Patterns for a deep beam.


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Flexural-shear crack - Starts

out as a flexural crack and

propagates due to shear

stress.

Flexural cracks in beams arevertical (perpendicular to the

tension face).


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For deep beam the cracks are

given as:

The shear cracks Inclined

(diagonal) intercept crack with

longitudinal bars plus verticalor inclined reinforcement.


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For deep beam the cracks are

given as:

The shear cracks fail due two

modes:

- shear-tension failure

- shear-compression failure


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ShearStrength ofRC BeamsShearStrength ofRC Beams

without Web Reinforcementwithout Web Reinforcement

vcz - shear in compression

zone

va

- Aggregate Interlock

forces

vd = Dowel action from

longitudinal bars

Note: vcz increases from(V/bd) to (V/by) as crack

forms.

Total Resistance = vcz + vay +vd (when no stirrups are used)


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Strength ofConcrete inShearStrength ofConcrete inShear

(No Shear Reinforcement)(No Shear Reinforcement)

(1) Tensile Strength of concrete affect inclined

cracking load

(2) Longitudinal Reinforcement Ratio,Vw

dbf

dbA

wccw

w

sw

2:0025.00075.0for

cracksrestrains

d$ee

!

V

V


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(3) Shear span to depth ratio, a/d ( /(Vd))

e ectlittlehasato2

d

a

requireddesigndetailmorespanssheardeep2

"

ed

a


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(4) Size of Beam

Increase Depth Reduced shear stress atinclined cracking

(5) Axial Forces

- Axial tension Decreases inclined cracking load

- Axial Compression Increases inclined crackingload (Delays flexural

cracking)


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Function andStrength ofWebFunction andStrength ofWeb

ReinforcementReinforcement

Web Reinforcement is provided to ensure that

the full flexural capacity can be developed.(desired a flexural failure mode - shear failure

is brittle)

- Acts as clamps to keep shear cracks fromwidening

Function:


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ReinforcementReinforcement Uncracked Beam Shear is resisted

uncracked concrete.

Flexural Cracking Shear is resisted by

vcz, vay, vd

bars.allongitudinfromActionDowl

forceInterlockAggregateofcomponentVerticalzonencompressioinShear

d

ay

cz

V

V

V


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ReinforcementReinforcement Flexural Cracking Shear is resisted by

vcz, vay, vd and vs

Vs increases as cracks

widen until yielding of

stirrups then stirrups

provide constant

resistance.


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Designingto Resist

Shear

Designingto Resist

Shear

Shear Strength (ACI 318 Sec 11.1)

demandcapacity u

u un VVJ

factorreductionstrengthshear0.85trengthhearominal

sectionatforceshearfactored

!

!!

J

n

u

VV

scn VVV !

entreinforcemshearby theprovidedshearominal

concretebyprovidedresistanceshearominal

!

!

s

c

V

V


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ShearStrength Provided by ConcreteShearStrength Provided by Concrete


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Lightweight Concrete:Shear Strength Provided by Shear Reinforcement

inimum Shear Reinforcement: (11.5.5)

cu VV 2

1whenquiredRe Ju

e

w

f

b1/2

t2.5

10"

oflargerhwithBeamsc

8.11)(definedonConstructiJoistConcreteb

Footings&SlabsaExcept


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Lightweight Concrete:Shear Strength Provided by Shear Reinforcement

inchesin,b50wmin sf

sb

y

w

v!

(provides additional 50 psi of shear strength)

concreteweightstandardforfor85.0substitutecanor

concretetlightweigh-allforfor75.0substitutecan2

for7.6/substitutecan1

cc

cc

ccct

ff

ff

fff e

Note: stirrups.forpsi60000eyf

(11.2)concrete

tLightweighforcV

strengthtensilesplitting!ctf


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Typical

Shear Reinforcement

Typical

Shear Reinforcement

Stirrup - perpendicular to axis

of members

(minimum labor - morematerial)

15)-11eqn( I;90

cossin

s

dfV

s

dfV

yv

s

yv

s

!!

!

E

EE


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Typical

Shear Reinforcement

Typical

Shear Reinforcement

Bent Bars (more labor -

minimum material) see

reqd in 11.5.6

5.6)-11( I41.1

;45

45cos45sin

s

dfV

s

dfV

yv

s

yv

s

!!

!

E


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Stirrup Anchorage RequirementsStirrup Anchorage Requirements

Vsbased on assumption stirrups yield

Stirrups must be well anchored.@Refer to Sec. 12.12 of ACI 318 for development

of web reinforcement. Requirements:

- each bend must enclose a long bar

- # 5 and smaller can use standard hooks 90o,135o, 180o

- #6, #7,#8(fy = 40 ksi)- #6, #7,#8(fy > 40 ksi) standard hook plus a min embedment

Also sec. 7.11 requirement for min. stirrups in beams with

compression reinforcement, beams subject to stress reversals, or

beams subject to torsion


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Design Procedurefor

Shear

Design Procedurefor

Shear

(1) Calculate Vu

(2) Calculate JVc Eqn 11-3 or 11-5 (no axial force)

(3) Check

pu

done.no,If

4)to(goentreinforcemebaddyes,If

2

1is cu VV J


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Design Procedurefor

Shear

Design Procedurefor

Shear

(4)

entreinforcemshear

minimumprovide,2

1If cuc VVV JJ ee

!! v

ysv

y

v Ab

fAs

f

sbA minfor

50or50 maxmin

Also:

(Done)

11.5.4"242

max eed

s


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Design Procedurefor

Shear

Design Procedurefor

Shear

(6) Solve for required stirrup spacing(strength) Assume

# 3, #4, or #5 stirrups

(7) Check minimum steel requirement (eqn 11-13)

15-11froms

ysv

Vdfs e

50

max

w

ysv

b

fs !


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Design Procedurefor

Shear

Design Procedurefor

Shear

(8) Check maximum spacing requirement (ACI 11.5.4)

(9) Use smallest spacing from steps 6,7,8

illegal8If:Note

"124

4If

"24

2

4If

c

maxc

maxc

dbfV

dsdbfV

dsdbfV

ws

ws

ws

du

eepdu

eepde

Note: A practical limit to minimum stirrup

spacing is 4 inches.


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Location ofMaximum Shearfor

Beam Design

Non-pre-stressed members:

Sections located less than a distance d from

face of support may be designed for same

shear, Vu, as the computed at a distance d.

Compression fan carries

load directly into support.


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Location ofMaximum Shearfor

Beam Design

Compression from support at

bottom of beam tends toclose crack at support


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ExampleExample:: Design ofStirrups to ResistShearDesign ofStirrups to ResistShear

fc = 4000 psi

fy = 60 ksi

wsdl =1.2 k/ft

wll= 1.8 k/ft

fys = 40 ksiwb = 0.5 k/ft

From flexural design:

will use either a #3 or #4 stirrup

Design of Reinforced Concrete Beams for Shear

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