CE 575-Lecture-4(Partially Prestressed Concrete Beams)

8/18/2019 CE 575-Lecture-4(Partially Prestressed Concrete Beams)

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LECTURE 5

Part ia l ly Prestressed Concrete Beams

2016CE 407-Prestressed Concrete Structures

1

PRESTRESSED CONCRETE STRUCTURES

(CE 407)

سم ا لرحن لرحيم

By

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Contents


2

Objectives of the present lecture

Full Prestressing

Partial Prestressing

Flexural stresses in full prestressed beams

Flexural stresses in partially prestressed beams Problems

Further reading

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Objective of the Present lecture


3

To define partial prestressing and learn the stepsfor the calculation of flexural stresses in partiallyprestressed beams.

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Full Prestressing

The kind of design, where the limiting tensile stressin the concrete at full service load is zero, is generallyknown as full prestressing.

Full prestressing is required in those cases in whichit is necessary to avoid all risk of cracking.

Such cases include tanks or reservoirs where leaksmust be avoided, submerged structures or those

subject to a highly corrosive environment wheremaximum protection of reinforcement must beinsured.

CE 407-Prestressed Concrete Structures

4

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Partial Prestressing

The kind of design in which flexural tensionand thus some cracking are permitted in theconcrete at normal service load is called

partial prestressing.


5

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A Classification

A prestressed concrete structure may be grouped in one of the threeclasses depending on the extent of cracking.

Class 1 Structures: Tensile stresses are not permitted in thesestructures, hence, no cracking under service loads. Such structures

may be referred to as fully prestressed structures.

Class 2 Structures: Limited tensile stresses are permitted but thereshould be no visible cracking under service loads. Tensile stressesshould be less than 3 MPa.

Class 3 Structures: Tensile stresses, and therefore, cracking underservice loads is permitted in these structures. Such members arereferred to as partially prestressed members.


6

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Partially Prestressed Members

Prestressed concrete members behave well in the post-cracking load range, provided they contain sufficient

bonded reinforcement to control the cracks.

A cracked prestressed concrete section under service

loads is significantly stiffer than a cracked reinforcementconcrete section of similar size and containing similarquantities of bonded reinforcement. Members that aredesigned to crack at the full service loads are calledpartially prestressed.

The width of surface cracks should not exceed 0.1 mm formembers in aggressive environments and 0.2 mm for allother members.


7

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Elastic Flexural Stresses in full prestressing(Uncracked Beams)

As long as the beam remains uncracked, and both

steel and concrete are stressed only within theirelastic ranges, then concrete stresses can be foundusing equations of mechanics.


8

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Stress distribution due to initial prestress alone


9

:effects bendingandaxialingsuperimpos byfound becanface bottomat theandmembertheof facetop at thestressconcretethe,forceng prestressiinitialtheonly tosubjectedismemebrtheIf

2

1

f f P i

2

2

1

/

1/

r

ec

A

P

I

ec P

A

P

c I

e P

A

P

Z

M

A

P f

r

ec

A

P

I

ec P

A

P

c I

e P

A

P

Z

M

A

P f

b

c

i

c

t i

c

i

bc

i

c

i

bc

ib

t

c

i

c

t i

c

i

bc

i

c

i

t c

it

cc

c

c

/A I r

I

A

e

gyrationof radius

sectioncrossconcretetheof inertiaof moment

sectioncrossconcretetheof area

centroidconcretethefrom

downwardmeasuredtyeccentricitendon

21

r

ec

A

P b

c

i

ye

Concretecentroid

ct

c b

Section

21

r

ec

A

P t

c

i

c

t i

I

ec P

c

i

A

P Stresses caused by initial prestress:

c

bi

I

ec P

c

i

A

P

Steel centroid

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Stress distribution due to initial prestress +self-weight


10

:ediatelyeffectsimm bendingandaxialingsuperimpos byfound be

canface bottomat theandmembertheof facetopat thestressconcretetheitself,

theof loaddeadtheand forceng prestressiinitialthetosubjectedismemebrtheIf

21 f f

P i

b

b

c

ib

t

t

c

it

Z M

r ec

A P f

Z

M

r

ec

A

P f

0

2

0

2

1

1

cc

c

c

/A I r

I

A

S

S

M

gyrationof radius

sectioncrossconcretetheof inertiaof moment

sectioncrossconcretetheof area

beamtheof surface bottomtherespect towithmodulusSection

beamtheof surfacetoptherespect towithmodulusSection

beamtheof weightself by thecausedmoment

2

1

0

ye

Concretecentroid

c1

c2

Section

21

r

ec

A

P t

c

i

t Z

M 0

2

11r

ec

A

P

c

i

b Z

M 0

t

t

c

i

Z

M

r

ec

A

P 02

1

b

b

c

i

Z

M

r

ec

A

P 02

1

Steel centroid

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Stress distribution due to final prestress +full service load


11

:effects bendingandaxialingsuperimpos byfound be

canface bottomat theandmembertheof facetopat thestressconcretethe

)(loadliveedsuperimpos)load(deadedsuperimpos)(weightself

theand forceng prestressieffectivethetosubjectedismemebrWhen the

21

0

f f

M M M

P

l d

i

l d t M M M M 0

ye

Concretecentroid

c1

c2

Section

t

l d

Z

M M

2S

M M l d

t

t

c

i

Z

M

r

ec

A

P 02

1

Steel centroid

b

t b

c

eb

t

t t

c

et

Z

M

r

ec

A

P f

Z

M

r

ec

A

P f

2

2

1

1

t

t t

c

e

Z

M

r

ec

A

P

21

b

t b

c

e

Z

M

r

ec

A

P

21

b

b

c

i

Z

M

r

ec

A

P 02

1

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Flexural Stresses in Partially Prestressed Beams


12

21

1

isentreinforcem barin thestrainecompressivtheand

;istendonin thestressthestage,At thisalone.

prestresseffectiveof napplicatiotoscorrespond(1)stageLoad

s s s

p

e pe p

e

E f

A

P f f

P

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Decompression Stage


13

0)(iscreep,andshrinkageof effects

theneglectingent,reinforcem barin thestressthestage,loadalhypotheticAt this

depth.entirethe

oughstrain thr concretezeroishereat which tconcrete,theof iondecompress

completetoingcorrespond(2)stageloadfictitiousaconsidertousefulisIt

22 s s s s E f

222

2

2 1

: propertiessectionconcreteuncrackedtheof basison thecalculated becanand

level,at thatconcretein thethatassametheistendonin thestraininchangeThe

p p p

cc

e p ε E f

r

e

E A

P ε

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CE 407-Prestressed Concrete Structures 14

R

Fe M e Fe M Fd e R

F F R

R

M F

f f A F

t t

t

p p p

centroid.concreteuncrackedtheaboveetyeccentriciwithapplied

forceresultanta bydrepresente becanloads,edsuperimposandweight-elf todue

momentexternalwith thetogetherforce,This.forceoppositeandequalan

applying bycancellednowisforcesingdecompressfictitiousthisof effectThe

)( :forceexternal

fictitiousawith pulled bemusttendontheconcrete,in thestatestresszero

the produce but to(2),stageatunstressedisentreinforcem barThe

21

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Equivalent concrete


15

c

s s

c

p

p

s s

p p

E

E n

E

E n

An

An

and

where,area by thereplacedisentreinforcem

bartheandconcretetensileof areaequivalentan byreplaced

istendonTheconcept.sectionedtransformtheusingfound becan

concrete,in thestresstheaswellasstresses,steellincrementaThe

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Depth of Neutral Axis

known

0)()(32

1-

0)()(32-

trials.successive bysolved becanthatforequationcubicainresults zero.This be

mustof actionof lineabout theforcesinternalallof momentthat thecondition

mequilibriuthefromfound becansurface,topthefromaxisneutralof depthThe

111

131313

y

ced An y

yd ced An

y

yd ce

y A

ced An y

yd f ced An y

yd

f ce y

A f

y

R

y

s s p p p

p

s sc p p p

p

cc

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Centroid of Cracked concrete


17

found. be

cansurface,topthefromsection,crackedtheof inertiaof momentandareaedtransformeffectivetheknown,isOnce

1

*

ct

ct

cid own centroabout its I A y

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Contd..

ct

s

ct

s s

ct

p

ct

p p

ct ct

c

I

cd e R

A

Rn f

I

cd e R

A

Rn f

/S e Re R I

ce R

A

R f

)(.

)(.

.stressBending&.momentBending: Note .

are(3),stageto(2)stagefrom passesloadingassought,stresseslincrementaThe

*

1

*

3

*

1

*

3

***

1

*

3


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Final Stresses


19

3

213

321

: beamtheof surfacetopat thestressConcrete

0: Note entreinforcemBar

:Tendon

cc

s s s s

p p p p

f f

f f f f

f f f f

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Problem(in US Customary Unit)


20The partially prestressed T beam shown incross section in Fig. below is subjected tosuperimposed dead and service live loadmoments of 38 and 191 ft-kips, in addition to amoment of 83 ft-kips resulting from its own weight. An effective prestress force of 123 kips

is applied using six Grade 250 ½ -inchdiameter strands.Two non-prestressed Grade 60 no. 8 bars arelocated close to the tension face of the beam,The elastic moduli for the concrete, tendonsteel, and bar steel are respectively 3.61 ×106

psi, 27 ×106

psi and 29 ×106

psi.The modulus of rupture of the concrete is 500psi.Find the stresses in the concrete, prestressedsteel, and bar reinforcement at the full serviceload.

Note:1ft-kips = 1.36 kN.mkips=4.45 kN1 ksi = 6.9 MPa1 inch =25.4 mm1 psi = 0.0069 MPa

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Solution


21

22

2

1

3

2

3

1

2

in103

in9.16

in1.13

in1290

in1664

in212

aresectioncrossuncrackedtheof Properties

uncracked.ismemebrtheassumingchecked, bewill beam

theof bottomat theconcretein thestresstensiletheFirst,

r

c

c

S

S

Ac

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Contd.

beam.d prestresse partiallyfordiscussedasoutcarried bewillanalysisThe

cracked. beenhassectionThe

psi)(500ruptureof modulus psi1186

psi1186

1290

12312000

103

9.169.111

212

123000

1

fiber bottomin thestressestensilederivedhadsection weuncrackedFor

2

2

2

2

22

f

f

S

M

r

ec

A

P f t

c

e

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Load stage 1


23

; psi143000863.0

123000

istendonin thestressthestage,At thisalone.

prestresseffectiveof napplicatiotoscorrespond(1)stageLoad

1 p

e pe p

e

A

P f f

P

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Decompression stage


24

depth.entirethe

oughstrain thr concretezeroishereat which tconcrete,theof iondecompress

completetoingcorrespond(2)stageloadfictitiousaconsidertousefulisIt

psi108000004.01027Therefore

0004.0103

9.111

1061.3212

1230001

: propertiessectionconcreteuncrackedtheof basison thecalculated becanand

level,at thatconcretein thethatassametheistendonin thestraininchangeThe

622

2

62

2

2

p p p

cc

e p

ε E f

r

e

E A

P ε

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in25.16133

9.1113312312

centroid.concreteuncrackedtheabovee

tyeccentriciwithappliedforceresultanta bydrepresente becankips,-ft312

of momenttotalwith thetogetherforce,This.forceoppositeandequalan

applying bycancellednowisforcesingdecompressfictitiousthisof effectThe

kips133)8.10143(863.0)( :forceexternalfictitiousawith pulled

bemusttendontheconcrete,in thestatestresszerothe produceTo

21

R

Fe M e

R

F

f f A F

t

p p p

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Transformed area


26

2

2

6

6

6

6

in61.2157.103.8 barstheof areadTransforme

in6.460.8637.48tendontheof areadTransforme

03.81061.3

1029 and7.481061.3

1027

where,area by thereplacedisentreinforcem bartheand

concretetensileof areaequivalentan byreplacedistendonThe

s s

p p

c

s s

c

p

p

s s

p p

An

An

E E n

E E n

An

An


27/33

Contd..


27

in1.140)15.327(61.1227

)15.325(46.625

15.33

)604(2

-

zero. bemustof actionof

lineabout theforcesinternalallof momentthat theconditionmequilibriu

thefromfound becansurface,topthefromyaxisneutralof depthThe

3

33

y y

y f

y

y f

y y

f

R

c

cc

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Contd..


28

in9347centroidownitsaboutsection,crackedtheof inertiaof momentand

in13561.1246.61.144512

4512areaedtransformEffective

are propertiesSectionsection.theof topthefromin75.7

centroidthelocatessurfacetopabout theareas partialtheof momentsTaking

2

2

1

ct

ct

s s p pct

*

I

A

An An y A

c

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Contd..


29

in90.1075.71.1325.16

issectionedtransformcrackedthe

of centroidtherespect towithforcetheof tyeccentriciThe

* e

R

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Contd..


30

psi161009347

25.1990.10133000

135

13300003.8

)(.

psi126009347

25.1790.10133000

135

13300048.7

)(.

psi-21909347

75.790.10133000

135

133000

.are(3),stageto(2)stagefrom passesloadingassought,stresseslincrementaThe

*

1

*

3

*

1

*

3

*

1

*

3

ct

s

ct

s s

ct

p

ct

p p

ct ct

c

I

cd e R

A

Rn f

I

cd e R

A

Rn f

I

ce R

A

R f

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Final Stresses


31

psi2190: beamtheof surfacetopat thestressConcrete

psi16100entreinforcemBar

psi1664001260010800143000:Tendon

3

3

321

cc

s s

p p p p

f f

f f

f f f f

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Further Reading


32

Read more about the partially prestressed concrete beams from:

• Design of Prestressed Concrete by A. H. Nilson, John Wiley and Sons, Second Edition, Singapore.

• Design of Prestressed Concrete by R. I. Gilbert and N. C. Mickleborough, First Edition, 2004, Routledge.

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Thank You


33

2016ALGHRAFY

CE 575-Lecture-4(Partially Prestressed Concrete Beams)

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