CE 407-Lecture-3(Flexural Analysis of Prestressed Concrete Beams)-Unprotected

8/18/2019 CE 407-Lecture-3(Flexural Analysis of Prestressed Concrete Beams)-Unprotected

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LECTURE 3

F lexural Analys is of Prestressed Beams-I I

CE 407-Prestressed Concerte Structures

1

PRESTRESSED CONCRETE STRUCTURES

(CE 407)

سم ا لرحن لرحيم

By

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Contents

2016CE 407-Prestressed Concerte Structures

2

Objectives of the present lecture

Cracking load and cracking moment

Flexural strength analysis

Failure of Prestressed beams

Flexural strength estimation by strain compatibility Unbonded tendons

Approximate equations for unbonded tendons

Code provisions for bonded tendons

Further reading

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Objectives of the Present lecture


3

To calculate cracking moment at a given section ofa prestressed concrete beam.

To estimate flexural strength of prestressed

concrete beams.

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Cracking Load


4

The relation between applied load and steel stress in a typical well- bonded pretensioned beam is shown in a qualitative way.Performance of a grouted post-tensioned beam is similar.

When the jacking force is first appliedand the strand is stretched betweenabutment, the steel stress is f pj . Upon

transfer of force to the concretemember, there is an immediatereduction of stress to the initial stresslevel f pi , due to elastic shortening ofthe concrete. At the same time, theself weight of the member is caused toact as the beam cambers upward. It

will be assumed here that all time-dependent losses occur prior tosuperimposed loading, so that thestress is further reduced to theeffective prestress level f pe.

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Cracking moment


5

2

21r

ec

A

P

c

e

h

e

ct

c b

1

1

2

r f

2

e P

cr e M P

Concretecentroid

0

The moment causing cracking may easily be found for a typical beam by writing the equation for the concrete stress at the bottom face, based on thehomogeneous section, and setting it equal to the modulus of rupture:

r

b

cr

c

e f S

M

r

ec

A

P f

2

22 1

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Problem-1


7Calculate the cracking moment and find the

factor of safety against cracking for thesimply supported I-beam shown in crosssection and elevation. The beam has to carrya uniformly distributed servicesuperimposed load totaling 8 kN/m over the12 m span, in addition to its own weight.Normal concrete having density of 24kN/m3 will be used. The beam will bepretensioned using multiple seven-wirestrands; eccentricity is constant and equalto 13.2 cm. The prestressing forceimmediately after transfer (after elasticshortening loss) is 750 kN. Time –

dependent lasses due to shrinkage, creep,and relaxation total 15% of the initialprestressing force. Find the concrete flexuralstresses at mid span and support sectionsunder initial and final conditions.The modulus of rupture of the concrete is2.4 MPa.

P

105

10

15

15

5

10

30

centroidConcrete

centroidSteel

2.13

Dimensions in cm

2.13e

kN/m8 l d ww

m21

P P

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Solution


8

For pretensioned beams using stranded cables, the difference betweensection properties based on the gross and transformed section is usuallysmall. Accordingly, all calculations will be based on properties of thegross concrete section. Average flange thickness will be used.

23

3

92

369

49453

23

232

mm106.4210110

1069.4

mm106.151030

1069.4

mm1069.4cm1069.4351012

1

)2/5.125.17(5.12305.123012

12

mm10110cm11001035)5.1230(2

:PropetiesArea

c

c

t

ct b

c

c

c

A

I r

c

I Z Z

I

I

A10

5

10

15

15

5

10

30

centroidConcrete

centroidSteel

5.12

5.17

5.17

5.12

2.13

60

Dimensions in cm

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Contd.


9

kN.m1.212 N.mm101.2121068.1741044.37

132300

106.42105.637106.154.2

kN5.63775085.085.0

666

336

2

cr

b

ebr cr

ie

M

ec

r P Z f M

P P

14.1144

052.471.2120

isloadlivein theincreaseanrespect towithexpressedcracking,againstfactorsafetyThe

l

Dcr

l

d Dcr cr

M

M M

M

M M M F

kN.m52.47

8

1264.2

8

kN/m2.64241010110kN/m24weightself 22

0

-6330

l w M

Aw

D

c

kN.m1448

128

8

load)livetodueisloaddsupeimposeentirethat the(assumedkN/m8

22

l w M

w

l

l

l

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CE 407-Prestressed Concerte Structures 10

Flexural Strength Analysis…..

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Stress-strain curve for Prestressing steel


11

In the absence of a well-defined yield stressfor prestressing steels of wire and strandtype, the yield stress is defined as the stressat which a total extension of 1% is attained.For alloy bars, the yield stress is taken asequal to the stress producing an extensionof 0.7%.

f pe, ε pe = stress and strain in the steel due to effectiveprestress force P e after all losses.

f py, , ε py = yield stress and yield strain f pu, ε pu=ultimate tensile strength and ultimate strainof the steel f ps, ε ps = stress and strain in the steel when the beamfails.

Prestressing steels do not show a definite yield plateau. Yielding develops graduallyand , in the inelastic range, the stress-strain

curve continues to rise smoothly until thetensile strength is reached.

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Failure of Prestressed Beams


12

For under-reinforced beams, failure is initiated by yieldingof the tensile steel.The associated large tensile strains permit widening of flexuralcracks and upward migration of the neutral axis.

The increased concrete stresses acting on the reducedcompressive area result in a “secondary” compression failure ofthe concrete, even though the failure is initiated by yielding.The stress in steel at failure will be between points A and B.The large steel strains produce visible cracking and considerabledeflection of the member before the failure load is reached. Thisis an important safety consideration.

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Failure of Prestressed Beams (contd.)


13

Over-reinforced beams fail when the compressive strainlimit of the concrete is reached (0.003 according to ACI andSBC), at a load when the steel is still below its yield stress,

between points O and A.This type of failure is accompanied by a downward movementof the neutral axis, because the concrete is stressed into itsnonlinear range although the steel response is still linear. Thistype of failure occurs suddenly with little warning.

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computation of nominal moment resistance, Mn


14

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Minimum reinforcement for flexural members


15

Minimum reinforcement for flexural members

For statically determinate members with a flange of width b in tension, ACI specifies that

bw in the equation giving As,min shall be replaced by b or 2bw whichever is smaller. When

the flange is in compression, bw is sued.

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Flexural Strength EstimationBy Strain-Compatibility

The variation of strain on the cross-section is linear i.e.strains in the concrete and the bonded steel are

calculated on the assumption that plane sectionsremain plane.

Concrete carries no tensile stress, i.e. the tensilestrength of the concrete is ignored.

The stress in the compressive concrete and in the steelreinforcement are obtained from actual or idealizedstress-strain relationships for the respective materials.


16

ASSUMPTIONS

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Notations


17

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Notations


18

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Notations


19

ACI Code Provisions for Tension-Controlled, Transition, and Compression-Controlled

Sections at Increasing Levels of Reinforcement

Sections are tension-controlled when the net tensile strain in the extreme tensionsteel is equal to or greater than 0.005 just as the concrete in compression reachesits assumed strain limit of 0.003

Sections are compression-controlled when the net tensile strain in the extremetensile steel is equal to or less than the 0.002 at the time the concrete incompression reaches its assumed strain limit of 0.003

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Idealized Stress Diagram


20

'85.0 c f

ps f

cu

c

ps

a

b

pd Axis

Neutral

' A

ps f

c1

2/a

C

T

Section Strain Actual stresses Idealized stresses(ACI 318 )

'

c f

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In the above figure, an under-reinforced section at the ultimate moment isshown. The section has a single layer of bonded prestressing steel. At theultimate moment, the extreme fiber compressive strain εcu is taken in ACI318 to be 003.0cu

85.065.0

MPa2856MPafor65.0)28(0.008-0.85

MPa56for65.0MPa....28for85.0

astaken bemayandstrengthconcrete

on thedepends parameterThe.0.85isintensitystressuniformtheand

is blockstressrrectangulasACI318'theof depthThe

1

1

11

1

1

'

c

'

c

'

c

'

c

' c

f f

f f

f

c

'85.0 c f

ps f

cu

c

ps

a

b

pd Axis

Neutral

' A

ps f

c1

2/a

C

T

Section Strain Actual stresses Idealized stresses(ACI 318 )

'

c f

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cb f C

cb f C

C

c

c

1

'

1

'

0.85

)(0.85areahatchedstress

block stressrrectangulatheof VolumeforceecompressivResultant

C will act at the centroid ofthe hatched area A’.

2 :strengthflexuralnominalThe a

d f ATl M p ps pn

tendons. bondedin thestress where ps p ps f A f T

.controlledtensionismemberfor0.9318,ACIIn

factor reductionCapacity;:momentdesignthend

n M M A

'85.0 c f

ps f

cu

c

ps

a

b

pd Axis

Neutral

' A

ps f

c1

2/a

C

T

Section Strain Actual stresses Idealized stresses

'

c f

p A

l

005.0 ps

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2 :strengthflexuralnominalThe a

d f ATl M p ps pn

'85.0 c f

ps f

cu

c

ps

a

b

pd Axis Neutral

' A

ps f

c1

2/a

C

T

Section Strain Actual stresses Idealized stresses

'

c f

p A

l

Assuming sectional and material properties are given, above equation contain

three unknowns, a, and Mn ps f

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Strains and stresses as beam load isincreased to failure


24

Strain distribution (1) results fromapplication of effective prestress force P e, acting alone, after all losses.

At intermediate load stage (2)decompression of the concrete takesplace. Due to bond the increase in steelstrain is the same as the decrease inconcrete strain at that level in the beam.

When the member is overloaded to thefailure stage (3), the neutral axis is at adistance c below the top of the beam.

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Strain in the prestressing tendon at the ultimate loadcondition


25

The strain in the prestressing tendon at the ultimate load condition may beobtained from

c

cd

I e P

A P

E

E A P

E f

p

cu

c

e

c

e

c

p

pe

p

pe

pe

ps

ps

conditionloadultimateatlevelsteelng prestressiat thestrainconcreteThe

1

ed)decompressislevelitsatconcretetheasstrainsteelinincrease(thezeroismoment

appliedexternallywhenlevelsteelng prestressiat theconcretein thestrainThe

/ steelng prestressiin theStrain

ultimateatsteelng prestressiin thestrainTensile

where

3

2

2

1

321


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Strain in the prestressing tendon at the ultimate loadcondition (Contd.)


26

errors.seriousgintroducinwithoutignored beusuallymayand

,oreitherthanlessmuchveryisequationaboveinof magnitudethegeneral,In:1- Note

312

321

ps

.calculated becanultimateatforcetensiletheknown,steelng prestressiof areaWith thesteel.

ng prestressifor thediagramstrain-stresthefromdetermined becanultimateatsteel

ng prestressiin thestresstheknown,isIf .strainecompressivextreme

theandfailureataxisneutraltheof positiontheof in termsdetermined becan:2- Note

31

ps

pscu

ps

ps

f

c

.findhenceand

depth,axisneutralthelocateorder toinsteel)ecompressivd prestresse-nonanyinforce

ecompressivthe(plusforceecompressivconcretewith thesteel)tensiled prestresse-non

anyinforcetensilethe(plustendonsteelin theforcetensiletheequatetonecessary

isitandfailureatknownnotisstresssteelthehowever,general,In:3- Note

psε

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Determination of M n for a singly reinforced section with bonded tendons

1. Select an appropriate trial value of c and determine ε ps (= ε1 + ε2 + ε3 ). Byequating the tensile force in the steel to the compressive force in the concrete,

the stress in the tendon may be determined:

2. Plot the point ε ps and f ps on the graph containing the stress-strain curve for the prestressing steel. If the point falls on the curve, the value of c selected in step 1

is the correct one. If the point is not on the curve, then the stress-strain

relationship for the prestressing steel is not satisfied and the value of c is not

correct.

3. If the point ε ps and f ps obtained in step 2 is not sufficiently close to the stress-

strain curve for the steel, repeat steps 1 and 2 with a new estimate of . A larger

value of c is required if the point plotted in step 2 is below the stress-strain curve

and a smaller value is required if the point is above the curve.


27

p

c psc ps p A

cb f f cb f C f AT 1

'

1

' 85.085.0

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Trial c

(mm)

ε ps f ps(MPa)

Point

plotted

230 0.0120 1918 1

210 0.0128 1751 2

220 0.0124 1835 3

Point 3 lies sufficiently closeto the stress-strain curvefor the tendon and thereforethe correct value for c isclose to 220 mm.

.next trialin theReduce

actual.thanmorealsoisactualthanmoreisIf

85.0 1'

c

c f

A

cb f f

ps

p

c ps

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Problem-2

CE 575: Dr. N. A. Siddiqui

30

Calculate the ultimate flexural strength M n of the rectangular section shown below. The steel tendonconsists of ten 12.7 mm diameter strands ( A p = 1000 mm2) with an effective prestress P e = 1200 kN. The

stress-strain relationship for prestressing steel is also given below and the elastic modulus is E p = 195 ×103 MPa. The concrete properties are f c’ = 35 MPa and E c= 29800 MPa.

(a) Section

500

0.0050

1000

1500

200

0

0.01 0.015 0.020

S t r

e s s ( M P a )

Strain

f py

=1780 MPa

f pu=1910 MPa

350

750

p A

650

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Solution


31Given: A p = 1000 mm

2 ;Effective prestress P e = 1200 kN;

Elastic modulus E p = 195 × 103 MPa; f c’ = 35 MPa and E c= 29800MPa.

(c) Strain atultimate

(d) Concrete stress block at ultimate

'85.0 c f 350

750

ps f

c1

2/a

C

T

(a) Section (b) Straindue to P e

p A

650

003.0cu

c

3 2

80.0

MPa28for65.0)28(0.008-0.85

astaken bemayandstrengthconcreteon thedepends parameterThe

1

1

1

'

c

'

c f f

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Solution (contd.)

c

c ps

650003.000655.0

:ultimateatsteelng prestressiin thestrainTensile

321

00615.0100010195

101200

bygivenis prestresseffectivethetoduetendonsin thestraininitailThe

3

3

1

p p

e

A E

P

00040.0

75035012

1

275101200

350750

101200

29800

11

ed)decompressislevel

itsatconcretetheasstrainsteelinincrease(thezeroismomentapplied

externallywhenlevelsteelng prestressiat theconcretein thestrainThe

3

2332

2

c

e

c

e

c I

e P

A

P

E

c

c

c

cd pcu

650003.0

conditionloadultimateatlevelsteelng prestressiat thestrainconcreteThe3

321:ultimateatsteelng prestressiin thestrainTensile ps

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Solution (contd.)


33

cccb f C

C

c 8340801.03503585.085.0

:forceecompressivresultantof magnitudeThe

1

'

ps p ps f A f T

T

1000

bygivenisforcetensileresultantThe

c f

T C

ps 34.8

henceandthatrequiresmequilibriuHorizontal

shown.assteelfor thecurvestrain-stresson the plottedareand

of valuesingcorrespondtheandselected benowcanof valuesTrial

ps

ps

f

εc

c

c ps

650003.000655.0

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500

0.0050

1000

1500

2000

0.01 0.015 0.020

S t r e s s ( M P a )

Strain

f py=1780 MPa f pu=1910 MPa1

2

3

Trial c(mm)

Pointplotted

230 0.0120 1918 1

210 0.0128 1751 2

220 0.0124 1835 3

ps ps f

Point 3 liessufficiently close tothe stress-straincurve for the tendon

and therefore thecorrect value for c isclose to 220 mm(0.34 c)

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0.9andcontrolledtensionismemberThe

005.000586.0003.0220

220650

cu p

t c

cd

kN.m103510

2

220801.065010001835

2 :momentultimateThe

6

1n

n

p p ps

M

cd A f M

kN.m5.93110359.0:momentdesignThe n M

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Unbonded Tendons

When the prestressing steel is not bonded to the concrete, the stress in thetendon at ultimate, f ps, is significantly less than that predicted for bonded

tendons.

Accurate determination of the ultimate flexural strength is more difficult than

for a section containing bonded tendons. This is because final strain in the

tendon is more difficult to determine accurately.

The ultimate strength of a section containing unbonded tendons may be as low

as 75% of the strength of an equivalent section containing bonded tendons.

Hence, from a strength point of view, bonded construction is to be preferred.


37

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Flexural Strength Analysis Approximate Methods


38

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A i t d i t d d


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Approximate code-oriented procedures bonded tendons


39

'5.01

c

sc st p ps

pu psbdf

A A f A f f

Method 1

21

'

cd f A

bdf

A A f Ad f A M p ps ps

c

sc st p ps

p ps psn

2

85.0

2

' f

f wc p ps pswn

hd hbb f

cd f A M

Rectangle, I or T section with x in fling

I or T section with x out the fling

ps ps

f

wc psf A f

hbb f A '85.0

psf ps psw A A A

ps ps

f

wc psf A f

hbb f A '85.0

3.0)1' c p

ps ps

p f bd

f A

3.0)2' c p

ps ps

p

f bd

f A

2'25.0 pcn bd f M Rectangle, I or T section with x in fling

)5.0()(85.025.0 '2' f f wc pwcn hd hbb f d b f M I or T section with x out the fling

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A i t d i t d d


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40

Method 2 (AASHTO LRFD 2003)

1

211

k k f f pu ps

4.01 k 9.085.0 pu

py

f

f

28.01 k 9.0

pu

py

f

f

'2

c p

sc st pu ps

f d b

A A f Ak

21

'

cd f A

bdf


c

sc st pu ps

p ps psn

285.0

2

' f

f wc p ps pswn

hd hbb f

cd f A M



ps

ps

f

wc psf A f

hbb f A '85.0

psf ps psw A A A

ps ps

f wc psf A f

hbb f A

'85.0

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A i t d i t d d


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41Method 3 (2002 ACI Code)

= 0.28 for ≥ 0.9 [low relaxation]

= 0.40 for ≥ 0.85 [stress relieved]

= 0.55 for ≥ 0.80 [bar]

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A i t d i t d d


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21

'

cd f A

bdf


c

sc st p ps

p ps psn

285.0

2

' f

f wc p ps pswn

hd hbb f

cd f A M



ps

ps

f

wc psf A f

hbb f A '85.0

psf ps psw A A A

ps ps

f

wc psf A f

hbb f A '85.0

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A i t d i t d d


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Approximate code-oriented proceduresunbonded tendons



3.0)1' c p

ps ps

p f bd

f A

21 'c

d f Abdf


c

sc st pu ps

p ps psn

285.0

2

' f

f wc p ps pswn

hd hbb f

cd f A M



ps

ps

f

wc psf A f

hbb f A '85.0

psf ps psw A A A

ps ps

f

wc psf A f

hbb f A '85.0

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Appro imate code oriented procedures


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Approximate code-oriented proceduresunbonded tendons



3.0)2' c p

ps ps

p f bd

f A

2'25.0 pcn bd f M Rectangle, I or T section with x in fling

)5.0()(85.025.0 '2' f f wc pwcn hd hbb f d b f M I or T section with x out the fling

Method 2 (AASHTO LRFD 2003)

105 pe ps f f

21 'c

d f Abdf

A A f A

d f A M p ps psc

sc st pu ps

p ps psn

285.0

2

' f

f wc p ps pswn

hd hbb f

cd f A M



ps

ps

f

wc psf A f

hbb f A '85.0

psf ps psw A A A

ps ps

f

wc psf A f

hbb f A '85.0

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Problem-3


45

Calculate the ultimate flexural strength M n of the rectangular section shown below. The beam is a simplysupported post-tensioned beam which spans 12 m and contains single unbonded cable. The steel tendonconsists of ten 12.7 mm diameter strands ( A p = 1000 mm

2) with an effective prestress P e = 1200 kN. Thestress-strain relationship for prestressing steel is also given below and the elastic modulus is E p = 195 ×103 MPa. The concrete properties are f c’ = 35 MPa and E c= 29800 MPa.

Section

350

750

p A

650

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Solution


46Given: A p = 1000 mm

2 ;Effective prestress P e = 1200 kN;

Elastic modulus E p = 195 × 103 MPa; f c’ = 35 MPa and E c= 29800MPa.

MPa1200

:forceng prestressieffective by thecausedtendonin thestressThe

p

e pe

e

A P f

P

MPa128010001009.6

65035035691200

9.6

69

ultimateattendonunbondedin thestressthe16,toequalratiodepth-to-spanWith the

ps

p

p

'

c

pe ps

f

KA

bd f f f

MPa28for65.0)28(0.008-0.851 '

c

'

c f f

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Contd.


47

mm154801.03503585.0

0012801000

85.0 1'

b f

f A f A f Ac

c

y sc y st ps p

kN.m754102

153801.065010001280

2 :momentultimateThe

6

1

n

p p psn

M

cd A f Tl M

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Further Reading


48

Read more about the ultimate flexural strength of 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 You49

Documents

CE 407-Lecture-3(Flexural Analysis of Prestressed Concrete Beams)-Unprotected