ce 407-lecture-3(flexural analysis of prestressed concrete beams)-unprotected
TRANSCRIPT
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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
2016CE 407-Prestressed Concerte Structures
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
2016CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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.
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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.)
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
14
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Minimum reinforcement for flexural members
CE 407-Prestressed Concerte Structures
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.
CE 407-Prestressed Concerte Structures
16
ASSUMPTIONS
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Notations
CE 407-Prestressed Concerte Structures
17
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Notations
CE 407-Prestressed Concerte Structures
18
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Notations
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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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CE 407-Prestressed Concerte Structures 21
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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CE 407-Prestressed Concerte Structures 22
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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CE 407-Prestressed Concerte Structures 23
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
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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.)
CE 407-Prestressed Concerte Structures
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.
CE 407-Prestressed Concerte Structures
27
p
c psc ps p A
cb f f cb f C f AT 1
'
1
' 85.085.0
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CE 407-Prestressed Concerte Structures 29
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
CE 407-Prestressed Concerte Structures
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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CE 407-Prestressed Concerte Structures 32
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.)
CE 407-Prestressed Concerte Structures
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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CE 407-Prestressed Concerte Structures 34
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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CE 407-Prestressed Concerte Structures 35
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.
CE 407-Prestressed Concerte Structures
37
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Flexural Strength Analysis Approximate Methods
CE 407-Prestressed Concerte Structures
38
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Approximate code-oriented procedures bonded tendons
CE 407-Prestressed Concerte Structures
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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Approximate code-oriented procedures bonded tendons
CE 407-Prestressed Concerte Structures
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
A A f Ad f A M p ps ps
c
sc st pu ps
p ps psn
285.0
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
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Approximate code-oriented procedures bonded tendons
CE 407-Prestressed Concerte Structures
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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Approximate code-oriented procedures bonded tendons
CE 407-Prestressed Concerte Structures
42Method 3 (2002 ACI Code)
21
'
cd f A
bdf
A A f Ad f A M p ps ps
c
sc st p ps
p ps psn
285.0
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
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Approximate code-oriented proceduresunbonded tendons
CE 407-Prestressed Concerte Structures
43Method 1 (2002 ACI Code)
3.0)1' c p
ps ps
p f bd
f A
21 'c
d f Abdf
A A f Ad f A M p ps ps
c
sc st pu ps
p ps psn
285.0
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
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Approximate code-oriented proceduresunbonded tendons
CE 407-Prestressed Concerte Structures
44Method 1 (2002 ACI Code)
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
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
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Problem-3
CE 407-Prestressed Concerte Structures
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
CE 407-Prestressed Concerte Structures
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.
CE 407-Prestressed Concerte Structures
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
2016CE 407-Prestressed Concerte Structures
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