Ce 407-lecture-3(flexural Analysis Of Prestressed Concrete Beams)-unprotected

‫بسم هللا الرحمن الرحيم‬

PRESTRESSED CONCRETE STRUCTURES (CE 407) 1

LECTURE #3

Flexural Analysis of Prestressed Beams-II By

CE 407-Prestressed Concerte Structures

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Contents 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 of

a 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 wellbonded pretensioned beam is shown in a qualitative way. Performance of a grouted post-tensioned beam is similar.

When the jacking force is first applied and the strand is stretched between abutment, the steel stress is fpj. Upon transfer of force to the concrete member, there is an immediate reduction of stress to the initial stress level fpi, due to elastic shortening of the concrete. At the same time, the self weight of the member is caused to act as the beam cambers upward. It will be assumed here that all timedependent losses occur prior to superimposed loading, so that the stress is further reduced to the effective prestress level fpe. CE 407-Prestressed Concerte Structures

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Cracking moment 5

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 the homogeneous section, and setting it equal to the modulus of rupture:

ct

2

h

e

cb

Concrete centroid

1

1 2

fr

f2 

Pe  ec2  M cr  fr 1  2   Ac  r  Sb

CE 407-Prestressed Concerte Structures

0

Pe Pe  M cr

Pe  ec2  1  2  Ac  r 

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Contd. 6

P  ec  M f b  e 1  2b   cr  f r Ac  r  Zb

M cr P  ec   f r  e 1  2b  S2 Ac  r  P Z  ec   M cr   f r Z b  e r b 1  2b  Ac  r 

M cr  total moment at cracking (including moment due to self - weight and superimposed dead and partial live loads) f r  mod ulus of rupture.



Since Z b  I c / cb  Ac r / cb 2

Factor of safety against cracking

It is defined as a live load factor which may be

 r2    M cr   f r Z b  Pe   e   cb 

less than, equal to, or larger than unity. Then

 Zb   M cr   f r Z b  Aps f pe   e   Ac 

 Fcr 

CE 407-Prestressed Concerte Structures

M D  M d  Fcr M l  M cr M cr  M D  M d Ml Alghrafy 2016

Problem-1

30 10

7

Calculate the cracking moment and find the factor of safety against cracking for the 5 simply supported I-beam shown in cross section and elevation. The beam has to carry 15 a uniformly distributed service Concrete centroid superimposed load totaling 8 kN/m over the 13.2 15 12 m span, in addition to its own weight. Steel centroid Normal concrete having density of 24 5 kN/m3 will be used. The beam will be pretensioned using multiple seven-wire 10 strands; eccentricity is constant and equal Dimensions in cm to 13.2 cm. The prestressing force wd  wl  8 kN/m immediately after transfer (after elastic shortening loss) is 750 kN. Time – e  13.2 dependent lasses due to shrinkage, creep, and relaxation total 15% of the initial P prestressing force. Find the concrete flexural stresses at mid span and support sections under initial and final conditions. The modulus of rupture of the concrete is 12 m 2.4 MPa. 10

P P

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Solution 8

For pretensioned beams using stranded cables, the difference between section properties based on the gross and transformed section is usually small. Accordingly, all calculations will be based on properties of the gross concrete section. Average flange thickness will be used. 30 10

Area Propeties : 10

12.5

5 17.5

15

60 17.5

13.2

15

Ac  2  (30 12.5)  35 10  1100 cm 2  110 103 mm 2 1  I c  2    30 12.53  30 12.5  (17.5  12.5 / 2) 2    12  Concrete centroid 1  3 5 4 9 4  10  35   I c  4.69 10 cm  4.69 10 mm  12  Steel centroid

5 12.5

10 Dimensions in cm

CE 407-Prestressed Concerte Structures

I c 4.69 109 Zb  Zt    15.6 106 mm3 ct 30 10 r2 

I c 4.69 109   42.6 103 mm 2 3 Ac 110 10

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Contd. Pe  0.85 Pi  0.85  750  637.5 kN

9

3  r2   6 3  42.6  10 M cr  f r Z b  Pe   e   2.4 15.6 10  637.5 10    132   300   cb   M cr  37.44 106  174.68 106  212.1106 N.mm  212.1 kN.m





w0  self weight  Ac  24 kN/m 3  110 103 10-6  24  2.64 kN/m w0l 2 2.64 12 2 MD    47.52 kN.m 8 8 wl  8 kN/m (assumed that the entire supeimposed load is due to live load) wl l 2 8 12 2 Ml    144 kN.m 8 8

The safety factor against cracking, expressed with respect to an increase in the live load is Fcr 

M cr  M D  M d M cr  M D  0 212.1  47.52  0    1.14 Ml Ml 144

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Flexural Strength Analysis…..

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Stress-strain curve for Prestressing steel 11

Prestressing steels do not show a definite yield plateau. Yielding develops gradually and , in the inelastic range, the stress-strain curve continues to rise smoothly until the tensile strength is reached.

In the absence of a well-defined yield stress for prestressing steels of wire and strand type, the yield stress is defined as the stress at which a total extension of 1% is attained. For alloy bars, the yield stress is taken as equal to the stress producing an extension of 0.7%. CE 407-Prestressed Concerte Structures

fpe, εpe = stress and strain in the steel due to effective prestress force Pe after all losses. fpy, , εpy = yield stress and yield strain fpu, εpu=ultimate tensile strength and ultimate strain of the steel fps, εps = stress and strain in the steel when the beam fails.

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Failure of Prestressed Beams 12

For under-reinforced beams, failure is initiated by yielding of the tensile steel. The associated large tensile strains permit widening of flexural cracks and upward migration of the neutral axis. The increased concrete stresses acting on the reduced compressive area result in a “secondary” compression failure of the 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 considerable deflection of the member before the failure load is reached. This is an important safety consideration. CE 407-Prestressed Concerte Structures

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Failure of Prestressed Beams (contd.) 13

Over-reinforced beams fail when the compressive strain limit of the concrete is reached (0.003 according to ACI and SBC), 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 movement of the neutral axis, because the concrete is stressed into its nonlinear range although the steel response is still linear. This type of failure occurs suddenly with little warning. CE 407-Prestressed Concerte Structures

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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. CE 407-Prestressed Concerte Structures

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Flexural Strength Estimation By Strain-Compatibility 16

ASSUMPTIONS

 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 sections remain plane.  Concrete carries no tensile stress, i.e. the tensile strength of the concrete is ignored.  The stress in the compressive concrete and in the steel reinforcement are obtained from actual or idealized stress-strain relationships for the respective materials. CE 407-Prestressed Concerte Structures

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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 tension steel is equal to or greater than 0.005 just as the concrete in compression reaches its assumed strain limit of 0.003 Sections are compression-controlled when the net tensile strain in the extreme tensile steel is equal to or less than the 0.002 at the time the concrete in compression reaches its assumed strain limit of 0.003

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Idealized Stress Diagram 20

b

 cu

A' a

dp

f

0.85 f c'

' c

1c

c

a/2 C

Neutral Axis

 ps Section

Strain

CE 407-Prestressed Concerte Structures

f ps Actual stresses

f ps

T

Idealized stresses (ACI 318 )

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b

 cu

A' a

dp

f

0.85 f c'

' c

1c

c

a/2

C

Neutral Axis

f ps

 ps

f ps

T

Idealized stresses (ACI 318 ) In the above figure, an under-reinforced section at the ultimate moment is shown. The section has a single layer of bonded prestressing steel. At the ultimate moment, the extreme fiber compressive strain εcu is taken in ACI 318 to be  cu  0.003 The depth of the ACI318' s rectangular stress block is  c Section

Strain

Actual stresses

1

and the uniform stress intensity is 0.85 f . The parameter 1 depends on the ' c

concrete strength and may be taken as

1  0.85 for f c'  28 MPa....1  0.65 for f c'  56 MPa 1  0.85 - 0.008 ( f c'  28)  0.65 for 56MPa  f c'  28 MPa 0.65  1  0.85 CE 407-Prestressed Concerte Structures

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b

 cu

A' a

dp

f

0.85 f c'

' c

a/2

1c

c

C

Neutral Axis

Ap

l

f ps

 ps

Section

Strain

f ps

Actual stresses

T

Idealized stresses

Resultant compressive force C  Volume of the rectangular stress block  C  stress  hatched area  0.85 f c'  b( 1c)  C  0.85 f c'b1c

T  f ps Ap

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

where f ps  stress in the bonded tendons.

a   The nominal flexural strength : M n  Tl  Ap f ps  d p   2  And the design moment : M   M n ;   Capacity reduction factor In ACI 318,   0.9 for member is tension controlled .  ps CE 407-Prestressed Concerte Structures

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 0.005 Alghrafy 2016

b

 cu

A' a

dp

f

0.85 f c'

' c

a/2

1c

c

C

Neutral Axis

Ap

Section

l

f ps

 ps Strain

Actual stresses

f ps

T

Idealized stresses

a   The nominal flexural strength : M n  Tl  Ap f ps  d p   2 

Assuming sectional and material properties are given, above equation contain three unknowns, a, f ps and Mn

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Strains and stresses as beam load is increased to failure 24

Strain distribution (1) results from application of effective prestress force Pe, acting alone, after all losses. At intermediate load stage (2) decompression of the concrete takes place. Due to bond the increase in steel strain is the same as the decrease in concrete strain at that level in the beam. When the member is overloaded to the failure stage (3), the neutral axis is at a distance c below the top of the beam.

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Strain in the prestressing tendon at the ultimate load condition 25

The strain in the prestressing tendon at the ultimate load condition may be obtained from        ps

1

2

3

where  ps  Tensile strain in the prestressing steel at ultimate

 1  Strain in the prestressing steel   pe 

f pe Ep



Pe / Ap Ep

 2  The strain in the concrete at the prestressing steel level when externally applied moment is zero (the increase in steel strain as the concrete at its level is decompressed)  Pe Pe e 2     Ic   Ac  3  The concrete strain at the prestressing steel level at ultimate load condition 1  Ec

 dp c     cu   c  CE 407-Prestressed Concerte Structures

Strain in the prestressing tendon at the ultimate load condition (Contd.) 26

 ps  1   2   3 Note - 1 : In general, the magnitude of  2 in above equation is very much less than either 1 or  3 , and may usually be ignored without introducing serious errors.

 ps   1   3 Note - 2 :  ps can be determined in terms of the position of the neutral axis at failure c and the extreme compressive strain  cu . If  ps is known, the stress in the prestressing steel at ultimate f ps can be determined from the stres - strain diagram for the prestressing steel. With the area of prestressing steel known, the tensile force at ultimate can be calculated.

Note - 3 : In general, however, the steel stress is not known at failure and it is necessary to equate the tensile force in the steel tendon (plus the tensile force in any non - prestressed tensile steel) with the concrete compressive force (plus the compressive force in any non - prestressed compressive steel) in order to locate the neutral axis depth, and hence find ε ps . CE 407-Prestressed Concerte Structures

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Determination of Mn for a singly reinforced section with bonded tendons 27 1.

Select an appropriate trial value of c and determine εps (= ε1 + ε2 + ε3). By equating the tensile force in the steel to the compressive force in the concrete, the stress in the tendon may be determined:

0.85 f c'b1c T  Ap f ps  C  0.85 f b1c  f ps  Ap Plot the point εps and fps 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. If the point εps and fps obtained in step 2 is not sufficiently close to the stressstrain 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. ' c

2.

3.

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Contd.. 28 4. 5.

Interpolate between the plots from steps 2 and 3 to obtain a close estimate for εps and fps and the corresponding value for c. With the correct values of fps and c, determined in step 4, calculate the ultimate nominal moment Mn and find the design strength.

a   M n  Ap f ps  d p   2  where a  1c

Ɛt=Ɛps

Design strength  M n

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0.85 f c'b1c f ps  Ap If f ps is more than actual  c is also more than actual.

Trial c (mm)

εps

fps (MPa)

Point plotted

230

0.0120

1918

1

210

0.0128

1751

2

220

0.0124

1835

3

 Reduce c in the next trial.

Point 3 lies sufficiently close to the stress-strain curve for the tendon and therefore the correct value for c is close to 220 mm.

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Problem-2 30 Calculate the ultimate flexural strength Mn of the rectangular section shown below. The steel tendon consists of ten 12.7 mm diameter strands (Ap = 1000 mm2) with an effective prestress Pe = 1200 kN. The stress-strain relationship for prestressing steel is also given below and the elastic modulus is Ep = 195 × 103 MPa. The concrete properties are fc’ = 35 MPa and Ec= 29800 MPa.

650 750

(a) Section CE 575: Dr. N. A. Siddiqui

Stress (MPa)

350

Ap

fpu=1910 MPa

200 fpy=1780 MPa 0 1500 100 0

500 0

0

0.005

0.01 0.015 Strain

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Solution 31

Given: Ap = 1000 mm2 ;Effective prestress Pe = 1200 kN; modulus Ep =on 195 103 MPa; fc’ = 35 MPa Ec= 29800 TheElastic parameter 1 depends the×concrete strength and mayand be taken as 1 MPa. 0.85 - 0.008 ( f c'  28)  0.65 for f c'  28 MPa

 1  0.80

0.85 f c'

 cu  0.003

350

c

1c

a/2 C

650 750

2

Ap (a) Section

(b) Strain due to Pe

CE 407-Prestressed Concerte Structures

3 (c) Strain at ultimate

f ps

T

(d) Concrete stress block at ultimate Alghrafy 2016

Solution (contd.) Tensile strain in the prestressing steel at ultimate :  ps  1   2   3 The initail strain in the tendons due to the effective prestressis given by Pe 1200  103 1    0.00615 E p Ap 195  103  1000

 2  The strain in the concrete at the prestressing steel level when externally applied moment is zero (the increase in steel strain as the concrete at its level is decompressed)    3 3 2  1  Pe Pe e  1  1200 10 1200 10  275         0.00040  Ec  Ac I c  29800  750  350  1  3   350  750     12    3  The concrete strain at the prestressing steel level at ultimate load condition 2

d c  650  c    0.003   cu  p  c c     Tensile strain in the prestressing steel at ultimate :

 650  c    c 

 ps  1   2   3  0.00655  0.003 CE 407-Prestressed Concerte Structures

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Solution (contd.) 33

 650  c    ps  0.00655  0.003   c 

The magnitude of resultant compressive force C : C  0.85 f c'b1c  0.85  35  350  0.801  c  8340c The resultant tensile force T is given by T  f ps Ap  1000 f ps Horizontal equilibriu m requires that C  T and hence f ps  8.34c

Trial values of c can now be selected and the corresponding values of ε ps and f ps are plotted on the stress - strain curve for the steel as shown. CE 407-Prestressed Concerte Structures

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2000

fpy=1780 MPa

fpu=1910 MPa

1 3 2

Stress (MPa)

1500 1000

500 0

0

0.005

0.01 0.015

0.02

Strain Trial c (mm)

 ps

f ps

Point plotted

230

0.0120

1918

1

210

0.0128

1751

2

220

0.0124

1835

3

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Point 3 lies sufficiently close to the stress-strain curve for the tendon and therefore the correct value for c is close to 220 mm (0.34 c)

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 dp c   650  220   cu    t   0.003  0.00586  0.005  220   c   The member is tension controlled and   0.9 1c    The ultimate moment : M n  f ps Ap  d p   2   0.801  220   6  M n  1835  1000 650    10  1035 kN.m 2    The design moment : M n  0.9 1035  931.5 kN.m

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Sections containing non-prestressed reinforcement and bonded tendons 36

 cu

b d

'

a

Asc

dp

c

f

0.85 f c'

' c

fy

 sc

Cs

1c

Cc

Neutral Axis

ds Ap

lc l s

f ps

 ps

Ast Section

 st

Strain

fy Actual stresses

From T p  Ts  Cc  C s we have : Cc  T p  Ts  C s  0.85 f c'b1c  f ps Ap  f y ( Ast  Asc )  c  f ps 

0.85 f c'b1c  f y ( Ast  Asc )

f ps

Tp

Ts Idealized stresses (ACI318)

Ap f ps  Ast f y  Asc f y 0.85 f c'b1

M n  Cs  ls  Cc  lc  Tp  d s  d p 

Ap

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Unbonded Tendons 37

 When the prestressing steel is not bonded to the concrete, the stress in the

tendon at ultimate, fps, 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.

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38

Flexural Strength Analysis Approximate Methods

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Approximate code-oriented procedures bonded tendons 39

Method 1 1) p 

Aps f ps bd p f c'

 0 .3

 Aps f p   Ast  Asc   Rectangle, I or T section with x in fling c   M n  Aps f ps d p 1   A f d    hf ps ps p  bdf c' 2 Apsf  0.85 f c' b  bw   Aps    f ps hf   c  M n  Apsw f ps  d p    0.85 f c' b  bw h f  d   I or T section with x out the fling 2 2   hf  Apsf  0.85 f c' b  bw   Aps Apsw  Aps  Apsf f ps Aps f p   Ast  Asc     f ps  f pu 1  0.5 ' bdf c  

2) p 

Aps f ps bd p f c'

 0.3

M n  0.25 f c'bd p2

Rectangle, I or T section with x in fling

M n  0.25 f c'bw d p2  0.85 f c' (b  bw )h f (d  0.5h f )

CE 407-Prestressed Concerte Structures

I or T section with x out the fling

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Approximate code-oriented procedures bonded tendons 40

Method 2 (AASHTO LRFD 2003) f ps

 k k   f pu 1  1 2  1  

k2 

Aps  f pu   Ast  Asc 

k1  0.4 k1  0.28

0.85 

f py f pu

f py f pu

 0 .9

 0.9

b  d p  f c'

 Aps f pu   Ast  Asc   Rectangle, I or T section with x in fling c   Aps f ps  d p   M n  Aps f ps d p 1  hf ' bdf c 2 Apsf  0.85 f c' b  bw   Aps    f ps hf   c  M n  Apsw f ps  d p    0.85 f c' b  bw h f  d   I or T section with x out the fling 2 2   hf  Apsf  0.85 f c' b  bw   Aps Apsw  Aps  Apsf f ps

CE 407-Prestressed Concerte Structures

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Approximate code-oriented procedures bonded tendons Method 3 (2002 ACI Code)

41

𝛾𝑝 = 0.28 for 𝑓𝑝𝑦 ≥ 0.9 𝑓𝑝 [low relaxation] 𝛾𝑝 = 0.40 for 𝑓𝑝𝑦 ≥ 0.85 𝑓𝑝 [stress relieved] 𝛾𝑝 = 0.55 for 𝑓𝑝𝑦 ≥ 0.80 𝑓𝑝 [bar] CE 407-Prestressed Concerte Structures

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Approximate code-oriented procedures bonded tendons Method 3 (2002 ACI Code)

42

 Aps f p   Ast  Asc   Rectangle, I or T section with x in fling c   M n  Aps f ps d p 1   A f d    hf ps ps p '  ' bdf 2   A  0 . 85 f b  b  Aps   psf c w c   f ps h  c  f  M n  Apsw f ps  d p    0.85 f c' b  bw h f  d   I or T section with x out the fling 2 2   hf  Apsf  0.85 f c' b  bw   Aps Apsw  Aps  Apsf f ps

CE 407-Prestressed Concerte Structures

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Approximate code-oriented procedures unbonded tendons Method 1 (2002 ACI Code) 1) p 

Aps f ps bd p f c'

43

 0 .3

 Aps f pu   Ast  Asc   Rectangle, I or T section with x in fling c   Aps f ps  d p   M n  Aps f ps d p 1  hf ' bdf c 2 Apsf  0.85 f c' b  bw   Aps    f ps hf   c  M n  Apsw f ps  d p    0.85 f c' b  bw h f  d   I or T section with x out the fling 2 2   hf  Apsf  0.85 f c' b  bw   Aps Apsw  Aps  Apsf f ps CE 407-Prestressed Concerte Structures

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Approximate code-oriented procedures unbonded tendons Method 1 (2002 ACI Code) 2) p 

Aps f ps bd p f c'

44

 0.3

M n  0.25 f c'bd p2

Rectangle, I or T section with x in fling

M n  0.25 f c'bw d p2  0.85 f c' (b  bw )h f (d  0.5h f )

I or T section with x out the fling

Method 2 (AASHTO LRFD 2003)

f ps  f pe  105  Aps f pu   Ast  Asc   Rectangle, I or T section with x in fling c   Aps f ps  d p   M n  Aps f ps d p 1  hf ' bdf c 2 Apsf  0.85 f c' b  bw   Aps    f ps hf   c  M n  Apsw f ps  d p    0.85 f c' b  bw h f  d   I or T section with x out the fling 2 2   hf  Apsf  0.85 f c' b  bw   Aps Apsw  Aps  Apsf f ps CE 407-Prestressed Concerte Structures

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Problem-3 45 Calculate the ultimate flexural strength Mn of the rectangular section shown below. The beam is a simply supported post-tensioned beam which spans 12 m and contains single unbonded cable. The steel tendon consists of ten 12.7 mm diameter strands (Ap = 1000 mm2) with an effective prestress Pe = 1200 kN. The stress-strain relationship for prestressing steel is also given below and the elastic modulus is Ep = 195 × 103 MPa. The concrete properties are fc’ = 35 MPa and Ec= 29800 MPa.

350

650 750

Ap

Section CE 407-Prestressed Concerte Structures

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Solution 46

Given: Ap = 1000 mm2 ;Effective prestress Pe = 1200 kN; Elastic modulus Ep = 195 × 103 MPa; fc’ = 35 MPa and Ec= 29800 MPa.

1  0.85 - 0.008 ( f c'  28)  0.65 for f c'  28 MPa The stress in the tendon caused by the effective prestressing force Pe : f pe 

Pe  1200 MPa Ap

With the span - to - depth ratio equal to 16, the stress in the unbonded tendon at ultimate f ps  f pe  69 

f c' bd p 6.9  KAp

 f ps  1200  69 

35  350  650  1280 MPa 6.9 100 1000

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Contd. 47

c

Ap f ps  Ast f y  Asc f y 0.85 f c'b1



1000 1280  0  0  154 mm 0.85  35  350  0.801

 c   The ultimate moment : M n  Tl  f ps Ap  d p  1  2   0.801153   6  M n  1280 1000 650   10  754 kN.m 2  

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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 You 49

CE 407-Prestressed Concerte Structures

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