1

EXAMPLE OF PRESTRESS LOSSES CALCULATION ACCORDING TO AASHTO-LRFD 2007 REFINED METHOD

By

March 19, 2007

PRESTRESS LOSS 2005 PROVISIONS SIGN CONVENTION Positive terms are:

o Tension in steel o When prestress loss is labeled as such it is assigned a positive sign o When prestress gain is labeled as such it is assigned a positive sign o Compression in concrete o Bending causing bottom fiber tension o Eccentricity of deck relative to composite section if deck is above girder as in common practice o Eccentricity of prestress force if prestress is below section centroid as in common practice

NOTATION Ac = area of section calculated using the gross composite concrete section properties of the

girder and the deck and the deck-to-girder modular ratio (in.2) Ad = area of deck concrete Ag = gross area of precast section (in.2) Ati = area of transformed section calculated using the initial girder concrete modulus of

elasticity Aif = area of transformed section of girder calculated using the girder concrete modulus of

elasticity at service Act = area of transformed composite section calculated using moduli of elasticity of girder

and deck at service Aps = area of prestressing steel (in.2) Ecd = modulus of elasticity of deck concrete (ksi) Ect = modulus of elasticity of concrete at transfer or at time of load application (ksi) Ep = modulus of elasticity of deck concrete ed = eccentricity of deck with respect to the transformed gross composite section, usually

negative as it is above centroid epc = eccentricity of prestressing force with respect to centroid of gross composite section,

positive if below centroid epg = eccentricity of prestressing force with respect to centroid of gross girder section, positive if

below centroid epti = eccentricity of prestressing force with respect to centroid of initial transformed

section of girder eptf = eccentricity of prestressing force with respect to centroid of final transformed section

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of girder eptc = eccentricity of prestressing force with respect to centroid of composite transformed

section of girder and deck fcgp = the concrete stresses at the center of gravity of the prestressing tendons due to the

prestressing force immediately after transfer and the self-weight of the member at the sections of maximum moment (ksi)

f'ci = specified compressive strength of concrete at time of prestressing for pretensioned members and at time of initial loading for nonprestressed members. If concrete age at time of initial loading is unknown at design time, f'ci may be taken as 0.80f'c (ksi)

fpi = prestressing steel stress immediately prior to transfer (ksi) fpt = stress in prestressing steel immediately after transfer (ksi) H = relative humidity (%). In the absence of better information, H may be taken from Figure

5.4.2.3.3-1 Ic = moment of inertia of section calculated using the gross composite concrete section

properties of the girder and the deck and the deck-to-girder modular ratio at service (in.4) Ig = moment of inertia of gross precast section (in.4) Iti = moment of inertia of transformed section calculated using the initial concrete modulus of

elasticity (in.4) Itf = moment of inertia of transformed section calculated using the girder concrete

modulus of elasticity at service Itc = moment of inertia of transformed composite section calculated using concrete moduli

of elasticity at service Kdf = transformed section coefficient that accounts for time-dependent interaction between

concrete and bonded steel in the section being considered for time period between deck placement and final time

Kid = transformed section coefficient that accounts for time-dependent interaction between concrete and bonded steel in the section being considered for time period between transfer and deck placement

kf = factor for the effect of concrete strength khc = humidity factor for creep khs = humidity factor for shrinkage (%) Kid = transformed section coefficient that accounts for time-dependent interaction between

concrete and bonded steel in the section being considered for time period between transfer and deck placement

ktd = time development factor kvs = factor for the effect of the volume-to-surface ratio of the component Mg= midspan moment due to member self weight (kip-in.) Md= midspan moment due deck weight, diaphragms and other loads introduced before

composite action is effected (kip-in.) MSIDL= midspan moment due to dead loads after deck has become composite with girder (kip-in.) MLL= midspan moment due to live load (kip-in.) Pe = effective prestressing, including long term effects due to creep, shrinkage and relaxation

and including elastic stress change (loss or gain) due to initial prestress, girder weight, deck weigh and all other loads introduced before composite action takes effect, but excluding any elastic stress gain due to composite dead loads or due to live loads (kip)

Pi = prestressing force immediately prior to transfer (kip)

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t = maturity of concrete (days), defined as age of concrete between time of loading for creep calculations, or end of curing for shrinkage calculations, and time being considered for analysis of creep or shrinkage effects

td = age at deck placement (days) tf = final age (days) ti = age of concrete at transfer of prestressing (days) V/S = volume-to-surface ratio ybc = eccentricity of bottom fibers with respect to centroid of gross composite section, positive if

below centroid yb = eccentricity of bottom fibers with respect to centroid of gross girder section, positive if

below centroid ybti = eccentricity of bottom fibers with respect to centroid of initial transformed section of

girder ybtf = eccentricity of bottom fibers with respect to centroid of final transformed section of

girder ybtc = eccentricity of bottom fibers with respect to centroid of composite transformed

section of girder and deck sh = shrinkage strain of concrete, positive when shortening, end of member curing and time at

which shrinkage effects are determined, Equation 5.4.2.3.3-1 bdf = shrinkage strain of girder between time of deck placement and final time per Equation

5.4.2.3.3-1 bid = concrete shrinkage strain of girder between the time of transfer and deck placement per

Equation 5.4.2.3.3-1 ddf = shrinkage strain of deck concrete between placement and final time per Equation 5.4.2.3.3-

1 (t,ti) = concrete creep coefficient at time t due to loading applied at time ti per Equation 5.4.2.3.2-1 b(td,ti) = girder creep coefficient at time of deck placement due to loading introduced at transfer per

Equation 5.4.2.3.2-1 b(tf,td) = girder creep coefficient at final time due to loading at deck placement per Equation

5.4.2.3.2-1 b(tf,ti) = girder creep coefficient at final time due to loading introduced at transfer per Equation

5.4.2.3.2-1 d(tf,td) = deck creep coefficient at final time due to loading introduced shortly after deck placement,

per Eq. 5.4.2.3.2-1 fcd = change in concrete stress at centroid of prestressing strands due to long-term losses

between transfer and deck placement, combined with deck weight and superimposed loads fcdf = change in concrete stress at centroid of prestressing strands due to shrinkage of deck

concrete fpCD = prestress loss due to creep of girder concrete between time of deck placement and final

time (ksi) fpCR = prestress loss due to creep of girder concrete between transfer and deck placement (ksi) fpES = sum of all losses or gains due to elastic shorting or extension at the time of application of

prestress and/or external loads (ksi) fpES1 = sum of all losses or gains due to elastic shorting or extension at the time of application of

initial prestress and member self weight, per Equation 5.9.5.2.3a-1 (ksi) fpES2 = Elastic gain in prestressing steel stress, ksi, at the time of application of deck weight

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fpES3 = Elastic gain in prestressing steel stress, ksi, at the time of application of superimposed dead loads (barriers, wearing surface, etc.)

fpES4 = Elastic gain in prestressing steel stress, ksi, at the time of application of live load fpLT = losses due to long-term shrinkage and creep of concrete, and relaxation of the steel (ksi) fpR = total prestress loss due to relaxation of prestressing steel, which may be calculated as the

sum of fpR1 and fpR2 , or estimated as 2.4 ksi for low relaxation strands and 10 ksi for stress relieved strands

fpR1 = prestress loss due to relaxation of prestressing strands between time of transfer and deck placement per Equation 5.9.5.4.2c-1 (ksi)

fpR2 = prestress loss due to relaxation of prestressing strands in concrete section between time of deck placement and final time per Equation 5.9.5.4.3c-1 (ksi)

fpSD = prestress loss due to shrinkage of girder concrete between time of deck placement and final time (ksi)

fpSR = prestress loss due to shrinkage of girder concrete between transfer and deck placement (ksi) fpSS = prestress loss due to shrinkage of deck in composite section (ksi) fpT = total losses (ksi)

h = correction factor for relative humidity of ambient air, per Equation 5.9.5.3-2

st = correction factor for specified concrete strength at time of prestress transfer to the concrete members, Equation 5.9.5.3-3

EXAMPLE This example uses the data of Example 9.4 of the precast/Prestressed Concrete Institute Bridge Design Manual. The prestress losses and concrete stresses are calculated using the Refined Method of Article 5.9.5.4. of the 2005 Interim of the AASHTO LRFD Bridge Design Specifications, Third Edition. The bridge consists of six, 120-ft simple span 72 in. deep AASHTO-PCI bulb-tee girders spaced at 9-ft. The girders are designed to act compositely with the 8-in. cast-in-place concrete deck to resist the superimposed dead loads and live loads. The superimposed dead loads consist of the railing and a 2 in. future wearing surface, both are assumed for calculation of losses and stresses to be introduced immediately after the deck has gained design strength. The top in. of the deck is assumed to be worn out with time. It is included in weight calculation but not in cross section properties. The cast-in-place haunch over the girder top flange is assumed to be 0.5 in. thick and 42 in. wide. The bridge cross-section is shown in Figure 1. The bridge is constructed in a region with relative humidity (%), H = 70. Precast concrete strength at release, 'cif = 5.8 ksi, and at service,

'cf = 6.5 ksi. Cast-in-place concrete strength at 28 days,

'cf =

4.0 ksi. Prestressing steel: 48-0.5-in. diameter, 270-ksi low relaxation strand, with a centroid at 6.92 from bottom girder fibers. Precast gross section properties are: 2in.767=gA , 4in.545894=gI , ..yb in6036= , Volume-to-surface ratio, V/S = 3. Deck V/S ratio = 3.51. Construction schedule allows for the following assumptions: Concrete age at prestress transfer, ti = 1 day; age at deck placement, td = 90 days. Final conditions are assumed to occur at concrete age, tf = 20000 days. Bending moments at the mid-span cross section are as reported in the PCI BDM. They are listed in Table 1.

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Table 1 Moment at mid-span (k-in)

Dead Load Live Load plus Dynamic Load Allowance Non-composite Composite Composite

Girder, Mg Slab, Md Barriers + FWS, MSIDL HL-93, MLL 17,258 19,915 6,480 32,082

3' 3'

51'

5 spaces @ 9'

8" uniform deck thickness

Figure 1 Bridge Cross-Section

MATERIAL PROPERTIES: Modulus of elasticity of concrete:

ksi33000 51 'c.

c fwE = (5.4.2.4-1) Girder at release: ( ) ksi445685

10005614033000

51

=

+= ...E.

c

Girder at final time: ( ) ksi4718561000

561403300051

=

+= ...E.

c

Deck: ( ) ksi360741000

41403300051

=

+=.

c .E

CREEP:

(a) Girder

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Creep coefficient at final time due to loading at transfer, ( )ifb t,t t = tf - ti = 20000 - 1 = 19999 days ( ) ( )( ) 00613130451130451 === ...S/V..kvs ( )( ) 0017000805610080561 ...H..khc ===

( ) 7408515

15 .

.fk

ci'f =+=+=

( )( ) 001199998546119999

461.

.tftk

ci'td =+=

+= ( ) ( )( )( )( )( )( ) 48110017400010619191 11801180 ......tkkkk.t,t ..itdfhcvsifb === Girder creep coefficient, ( )idb t,t , at time of deck placement due to loading introduced at transfer: td = 90 days, and t = tf - ti = 90 - 1 = 89 days.

( )( ) 70.0898.546189

461 '=+=

+= tftk

citd

( ) ( )( ) 0417048191 1180 ...tkkkk.t,t .itdfhcvsidb ===

Girder creep coefficient at final time due to loading at deck placement ti = 90 days ( ) ( )( ) 8709048191 11801180 ..tkkkk.t,t ..itdfhcvsdfb ===

(b) Deck: ( ) ( )( ) 099.051.313.045.1/13.045.1 === SVkvs ( )( ) 00.170008.056.1008.056.1 === Hkhc

( ) 191480015

15 .

).(fk

ci'f =+=+=

Deck creep at final time due to loads introduced shortly after deck placement: ( ) ( )( )( )( )( )( ) 24210011910019909191 11801180 ......tkkkk.t,t ..itdfhcvsdfb ===

SHRINKAGE

(a) Girder

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Shrinkage strain between prestress transfer and final time: ( ) ( )( ) 02.170014.000.214.000.2 === Hkhs ( )( )( )( )( ) 000384000048000174002106110480 3 ......x.kkkk tdfhsvsbif === Girder shrinkage strain between initial time and deck placement time, t=90-1 = 89 days:

( )( ) 70.0898.546189

461 '=+=

+= tftk

citd

( )( ) 0002690000384070010480 3 ...x.kkkk tdfhsvsbid === in./in. Girder shrinkage strain between deck placement and final time:

000115000026900003840 ...bidbifbdf === in./in.

(b) Deck Shrinkage strain between end of deck curing and final time: ( ) ( )( ) 02.170014.000.214.000.2 === Hkhs ( )( )( )( )( ) 000579000048000119102199010480 3 ......x.kkkk tdfhsvsddf === in./in. CROSS SECTION PROPERTIES

(a) Gross Precast Section: Gross precast section properties are given in the problem statement; 2in.767=gA , 4in.545894=gI ,

..yb in6036=

(b) Gross Composite Section To obtain gross composite section, the haunch and deck width are first transformed to precast concrete. Transformed deck width = 578247183607108 ./))((E/Eww cdddtf === in. Similarly, transformed haunch width = (24) (3607)/ (4718) = 32.11 in. The gross composite section properties are then obtained using customary calculations, which are summarized in Table 2

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Compo-nent

Width, b

Thickness

Modulus of

ElasticityModular ratio, n (n)(b)

Area, (in2) yb

(A)*(yb) yb-yNA Icg

Icg+A(yb-yNA)2 (in4)

in. in. ksi in. in2 in in4 in4

1 Girder 4718 1 767 36.6 28072 -17.92 545894 7921302 Deck 108.00 7.50 3607 0.76 82.57 619 76.3 47219 21.73 2903 2953803 Haunch 42.00 0.50 3607 0.76 32.11 16 72.3 1160 17.73 0 5049

Sum 1402 54.52 1092558

Table 2- Gross Composite Section Properties

(c) Initial Transformed Section The initial transformed section consists of the concrete girder with a value of Eci = 4456 ksi, and strands transformed to precast concrete using a modular ratio ni = Es/Eci= 28,500/4456 = 6.40. Steel is transformed using (ni-1) = 5.40. See Table 3.

Compo-nent

Width, b

Thickness

Modulus of

ElasticityModular ratio, n (n)(b)

Area, (in2) yb

(A)*(yb) yb-yNA Icg

Icg+A(yb-yNA)2 (in4)

in. in. ksi in. in2 in in4 in4

1 Girder 4456 1 767 36.6 28072 -1.46 545894 5475252 Strands 28500 5.40 40 6.92 274 28.22 0 31562

Sum 807 35.14 579087

Table 3- Initial Transformed Section Properties

(d) Transformed Precast Section at Deck Placement The transformed precast section at deck placement time is the same as the initial transformed section except that the girder modulus is taken at the value at service Ec = 4,718 ksi. section

Compo-nent

Width, b

Thickness

Modulus of

ElasticityModular ratio, n (n)(b)

Area, (in2) yb

(A)*(yb) yb-yNA Icg

Icg+A(yb-yNA)2 (in4)

in. in. ksi in. in2 in in4 in4

1 Girder 4718 1 767 36.6 28072 -1.37 545894 5473262 Strands 28500 5.04 37 6.92 256 28.31 0 29676

Sum 804 35.23 577003

Table 4- Transformed Precast Section Properties

(e) Transformed Precast Section at Deck Placement composite

The transformed composite section consists of the transformed gross section combined with the transformed strand, Table 5.

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Compo-nent

Width, b

Thickness

Modulus of

ElasticityModular ratio, n (n)(b)

Area, (in2) yb

(A)*(yb) yb-yNA Icg

Icg+A(yb-yNA)2 (in4)

in. in. ksi in. in2 in in4 in4

1 Girder 4718.00 1 767 36.6 28072 -16.69 545894 7596322 Deck 108.00 7.50 3607.00 0.76 82.57 619 76.3 47219 22.96 2903 3292583 Haunch 42.00 0.50 3607.00 0.76 32.11 16 72.3 1160 18.96 0 57704 Strands 28500 5.04 37 6.92 256 -46.37 0 79609

Sum 1439 53.29 1174268

Table 5- Tranformed Composite Section Properties

Table 6 gives a summary of the section properties need for calculation of the prestress loss components. Table 6- Section properties

Section Properties Precast Girder Composite Girder, Deck

Gross Transformed-initial Transformed-

final Gross Transformed

Area (in2) 767 807 804 1402 1439 Bottom fiber eccentricity (in.) 36.60 35.14 35.23 54.52 53.29

Moment of inertia (in4) 545894 579087 577003 1092558 1174268 Prestress eccentricity (in.) 29.68 28.22 28.31 47.60 46.37 Deck eccentricity (in.) 21.73

Note that ed, the eccentricity of the deck center relative to the center of the composite section, ((72+.5+3.75)-54.52) = 21.73 in. is negative because it is above the centroid of the composite section. LONG TERM LOSSES:

(I) TIME OF TRANSFER TO TIME OF DECK PLACEMENT (LRFD 5.9.5.4.2)

(a) SHRINKAGE OF GIRDER CONCRETE, fpSR (5.9.5.4.2a)

idpbidpSR KEf = where 000269.0bid = , ,ksi,E p 50028= and

( )[ ]ifbg

pgg

g

ps

ci

pid

t,t.IeA

AA

EE

K

+

++

=70111

12

( )( ) ( )( )( )78.0

49.17.01545894

68.297671767344.7

4456285001

12

=+

+

+

=idK

10

( )( )( ) 980.578.028500000269.0KEf idpbidpSR === ksi

(b) CREEP OF GIRDER CONCRETE, fpCR (5.9.5.4.2b)

( ) ididbcgpci

ppCR Kt,tfE

Ef =

where cgpf is concrete stress due to initial prestress plus girder weight:

ti

ptig

ti

pti

tiicgp I

eMI

eA

Pf

+=

21

( )( )( )( ) ( ) ( )( ) ksi0483579087

222817258579087

2228807

152021530482

.....f cgp =

+=

( )( )( ) 814.1578.004.1048.34456

28500fpCR =

= ksi

(c) RELAXATION OF PRESTRESSING STRANDS, fpR1 (5.9.5.4.2c) The relaxation loss, fpR1, may be assumed equal to 1.2 ksi for low-relaxation strands.

(d) TOTAL LONG TERM LOSS BETWEEN INITIAL TIME AND DECK PLACEMENT

( )1pRpCRpSR fff ++ = 5.980+15.814+1.200 = 22.994 ksi

(II) TIME OF DECK PLACEMENT TO FINAL TIME

(a) SHRINKAGE OF GIRDER CONCRETE, fpSD (5.9.5.4.3a)

dfpbdfpSD KEf = where bdf = 0.000115, pE = 28,500 ksi, and

( )[ ]ifbc

pcc

c

ps

ci

pdf

t,t.IeA

AA

EE

K

+

++

=70111

12

( )( ) ( )( )( )790

4917011092558

6047140211402

34474456

285001

12

.....

K df =+

+

+

=

( )( )( ) 589.279.028500000115.0fpSD ==

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(b) CREEP OF GIRDER CONCRETE, fpCD (5.9.5.4.3b)

( ) ( )( ) ( ) dfdfbcdc

pdfidbifbcgp

ci

ppCD Kt,tfE

EKt,tt,tf

EE

f += The first term represents loss due to creep caused by initial loads.

( )( )( ) ksi776.679.004.148.1048.34456

28500f 1pCD =

= The second term represents gain due to creep caused by forces introduced beyond the initial loading. These forces are the some of the long term losses between initial and deck placement, plus the deck weight plus the superimposed loads. The corresponding concrete stress increment at steel centroid, fcd, can be calculated using:

( ) ( )tc

ptcSIDL

tf

ptfd

g

2pgg

g

ps1pRpCRpSRcd I

eMIeM

IeA

1AA

ffff

+++=

Mslab = (1659.6) (12) = 19915 kip-in., and MSIDL = 2160 + 4320 = 6480 kip-in. The prestress eccentricity relative to the centroid of transformed precast and transformed composite section are .in31.28eptf = and .in37.46eptc

( ) ( )( ) ( )( ) ( )( )ksi726.1

117426837.466480

57700331.2819915

54589468.297671

767344.7994.22f

2

cd

=

+

=

( )( )( ) ksi166.779.087.0726.14718

28500f 2pCD =

= Net value = gain)net (i.e. ksi390.0166.7776.6fff 2pCD1pCDpCD ==+=

(c) RELAXATION OF PRESTRESSING STRANDS, fpR2 (5.9.5.4.3c)

20.1ff 1pR2pR == ksi

(d) GAIN DUE TO SHRINKAGE OF DECK CONCRETE, fpSS (5.9.5.4.3d)

( )( )dfbdfcdfc

ppSS t,t7.01KfE

Ef +=

12

where ( )( )

+

=c

dpc

cdfd

cddddfcdf I

eeA1

t,t7.01EA

f

Ad = (108) (7.5) + (42) (0.5) = 831 in2 ed = 21.73 in.

( )( )( )( )( )

( )( ) 158.01092558

73.2160.471402

124.27.01

3607831000579.0fcdf =

+= ksi

( )( ) ( )( ) ( )( )( ) 219.188.07.0179.0158.04718

28500t,t7.01KfEE

f dfbdfcdfc

ppSS =+

=+= ksi

(e) TOTAL LONG TERM LOSS BETWEEN DECK PLACEMENT AND FINAL TIME (excluding deck shrinkage effects = ksi3.401.200.392.59)fff( pR2pCRpSR =+=++ Long term loss should be applied as a negative prestress to the corresponding concrete section to obtain the loss of compressive concrete stress. However, an exception is that the stress gain due to concrete shrinkage corresponds to a tensile concrete stress increment which is calculated separately. An exact solution would be to use the net section properties. An acceptable approximation is to use the gross section properties. The precast section should be used with the loss between initial and deck placement. The gross composite section should be used for the second time period.

(III) CONCRETE BOTTOM FIBER STRESSES

Note that no elastic losses are required to be explicitly calculated in order to correctly calculate the concrete stresses, as long as the proper section properties are used.

(a) Concrete stresses while section is still precast only: Concrete stress due to initial prestress plus self weight

=

+=

ti

bg

ti

bpti

tii1cb I

yMI

yeA1Pf

( )( )( )( ) ( )( ) ( )( ) ksi342.3579087

14.3517258579087

14.3522.28807

15.202153.048f 1cb =

+= Concrete stress due to loss between initial time and deck placement, and due to deck weight and superimposed dead load

( )

+++=

g

bpgg

g

ps1pRpCRpSR2cb I

yeA1

AA

ffff

13

( ) ( )( )( ) ksi556.0545894

60.3668.297671767344.7994.22f 2cb =

+

=

Concrete stress due to deck placement

ksi216.1I

y)M(f

tf

btfd3cb ==

(b) Concrete stresses after section becomes composite:

Concrete stress due to loss between deck placement and final

( )

+++=

c

bcpcc

c

pspR2pCDpSDcb4 I

yeA1

AA

ffff

( ) ( )( ) ksi 0.0771,092,558

(54.52)47.601,40211,4027.3443.40fcb4 =

+

=

Due to deck shrinkage

dfc

dbc

cdfd

cddddfcbSS KI

eyA1

)t,t(7.01EA

f

+

=

ksi -0.195 (0.79)558,092,1

)73.21)(52.54(402,11

)24.2)(70.0(1)607,3)(831)(000579.0(fcbSS =

+=

Concrete stress due to superimposed dead load ( )

ksi294.0I

yMf

tc

btcSIDL5cb ==

Concrete stress due to live load

ksi457.11173788

)29.53)(32082(I

yMf

tc

btcLL6cb ===

Net concrete stress at bottom fibers at final time

ksi0.4521.4560.2940.195-0.0771.2160.5563.342ff cbicb ===

(IV) ELASTIC LOSSES AND GAINS TO CALCULATE STEEL STRESS IF NEEDED Elastic losses (or gains) should not be used in concrete stress analysis as they are already included if transformed section properties are used. If elastic losses (or gains) are needed to calculated effective steel stress, for example in the shear design, they are calculated as shown below.

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(a) ELASTIC SHORTENING LOSS AT PRESTRESS TRANSFER, fpES (5.9.5.2.3a)

( ) 495.19048.34456

28500fEE

f cgpci

p1pES =

== ksi

(b) ELASTIC GAIN DUE TO DECK WEIGHT

( )( ) 902.5

57700331.2819915

471828500

IeM

EE

ftf

ptfd

c

p2pES =

== ksi

(c) ELASTIC GAIN DUE TO SUPERIMPOSED DEAD LOAD

( )( ) 5461

117426837466480

471828500

3 ..

IeM

EE

ftc

ptcSIDL

c

ppES =

== ksi

(d) ELASTIC GAIN DUE TO LIVE LOAD:

( )( ) 6537

1174268374632082

471828500

4 ..

IeM

EE

ftc

ptcLL

c

ppES =

== ksi

Effective steel stress = fpi -(fpLT + fpES1 + fPES2 + fPES3 + fpES4)

= 202.5- (25.174 + 19.495 - 5.902 - 1.546 - 7.653) = 172.932 ksi

The following pages show the results of the Excel Spreadsheet Prestress_Loss PCI BDM 9.4 070319, which may be downloaded from the website www.structuresprograms.unomaha.edu , Folder Excel Spreadsheet

15

Prestress Loss and Stresses at Midspan of Pretensioned Concrete Composite GirderRef: See NCHRP Report 496, "Prestress Losses in Pretensioned High-Strength Concrete Bridge Girders," 2003

Project Name: PCI BDM 9.4 Date:Designer: Maher Tadros

A 767.00 in2 H 70 %I 545894 in4 fci 5.800 ksiyb 36.6 in f'c 6.500 ksih 72 in f'cd 4.000 ksiV/S 3 in ti (release) 1 daysAps 7.344 in

2 t (deck pour) 90 daysfpi 202.5 ksi tf (final) 27375 daysEs 28500 ksiypb 6.92 inspan

Deck Girder 17258 kip-inWidth 108 in Deck, haunch, diaphragms 19915 kip-inThickness 7.5 in SI dead ld. 6480 kip-inHaunch width 42.0 in 80% of HL93 plus impact* 32082 kip-inHaunch thickness 0.5 inV/Sd 3.50

19-Mar-07

*The LL moment to be input here is the same used to check bottom fiber stress at final conditions (AASHTO LRFD Service III)

Yellow cells--and only yellow cells are input data! Other cells contain notation, components, and results.

Beam Materials

Service load moments

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Project Name: Date: 3/19/2007Designer:

Material PropertiesModulus of Elasticity E=33000*w^1.5*K1(f'c)^0.5 Beam initial Eci 4456in KSI w=0.140+f'c/1000 at deck placement Ec 4718(5.4.2.4) Deck Ecd 3607Shrinkage =0.00048(5/(1+fci))kskhsktd Beam initial to final bif 0.000381(5.4.2.3.3) ks=(1.45-0.13V/S) initial to deck placam. bid 0.000268

khs=2.0-0.014H deck placem. to final bdf 0.000113ktd=t/(61-4fci+t) Deck ddf 0.000579

Creep Beam initial to final bif 1.479(5.4.2.3.2) khc=(1.56-0.008*H) initial to deck placement bid 1.039

ktd =t/(61-4fci+t) Deck placem. to final bdf 0.870Deck Deck placem. to final ddf 2.247 Gross section

Steel modular ratio ni=Es/Eci 6.3953 Transfromed section factors, K Kid 0.77822 0.78194" (at deck placement) n=Es/Ec 6.0411 K=1/(1+ni*net*Aps/Anet*(1+0.7*bif)) Kdf 0.78620 0.78966" (deck) nd=Ed/Ec 0.7645 (5.9.5.4.2-2, 5.9.5.4.3-2)

Section PropertiesSection Properties Precast Beam

Gross Net (-APS) Tr.-initial Transformed-final Deck Haunch Gross Net Tr.-final A (in.2) 767.00 759.66 806.62 804.02 619.22 16.05 1402.27 1394.93 1439.30yb (in.) 36.6 36.89 35.14 35.23 76.25 72.25 54.52 54.77 53.29I (in.4) 545894 539362 579083 577005 2903 0.33 1092539 1075814 1174254ep (in.) 29.68 29.97 28.22 28.31 47.60 47.85 46.37ed (in.) 21.63 21.38 =1+(A*ep2)/I 2.2377 2.2648 2.1094 2.1170 3.9077 3.9685 3.6358b =1+A*e*yb/I 2.5263 2.5569 2.3815 2.3901 1.0000 4.3305 4.3978 4.0291 =1+A*e*(yb-h)/I -0.4762 -0.4820 -0.4489 -0.4506 1.0000 -0.0681 -0.0691 -0.0633

Composite Bm, DeckTransformed Deck

=1.90kskhc(5/(1+fci))ktd*ti-0.118

Area of deck incl. haunch = 831

PCI BDM 9.4Maher Tadros

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Formulas in this method use gross section properties to approximate net section properties. Otherwise the results are identical to those of NCHRP 18-07

Project NamPCI BDM 9.4 Date:Designer: Maher Tadros

Loading Change Net

202.519.493

Prestress transfer, f cgp = 3.048 ksi 183.05.969 177.0

15.843 161.21.200 160.0

23.012 160.05.904 165.91.546 167.47.449 167.42.548 164.96.764 158.17.161 165.31.200 164.11.187 165.32.165 165.37.654

Total loss and effective prestress including gain due to LL, 29.567 172.9

Only the two highlighted loss values are needed in concrete stress analysis, when transformed section properties are used

Cause Initial Final Cause Initial Final I Final II Final IIIPi (transf. section, release) 4.391 4.391 Pi (transf. section, release) -0.828 -0.828 -0.828 -0.414Mg (transf., release) -1.047 -1.047 Mg (transf., release) 1.098 1.098 1.098 0.549Loss (gross section, precast) -0.557 Loss (net section, precast) 0.105 0.105 0.052deck weight (transf., service) -1.216 deck weight (transf., service) 1.269 1.269 0.634SIDL (transf. composite) -0.294 SIDL (transf. composite) 0.103 0.103 0.052Loss (gross, composite) -0.076 Loss (net, composite) 0.001 0.001 0.000Deck shrinkage (gross composite) -0.195 Deck shrinkage (gross composite) 0.564 0.564 0.282LL (transf., composite) -1.456 LL (transf., composite) 0.639 0.639Net 3.343 -0.451 0.271 2.313 2.952 1.795

Code Limit 3.480 -0.484 -0.578 2.925 3.900 2.600

Bottom Fibers Top Fibers

Prestress Loss Using 2007 LRFD Detailed Method

19-Mar-07

fps

(5) Relaxation between release and deck place (Loss)Total long-term (initial to deck placemnt)id

Total long-term (deck placemnt to final)df

Extreme Fiber Stresses (using transformed/net section properties)

(13) Elastic gain due to LL n*(MLLecomp_tr/Icomp_tr)

(7) Elastic gain due to superimposed DL (on composite section) (MADLecomp-fin/Icomp-fin)*nDeck + SIDL: f cd and f pED

(10) Creep of beam due to deck and SIDL (Eq 5.9.5.4.3b-1) (Gain)

(4) Creep between release and deck place (Eq 5.9.5.4.2b-1) (Loss)

(6) Elastic gain due to deck weight (Mdecketr-fin/Ibm-tr-fin)*n

(11) Relaxation between deck place and final (Loss)(12) Shrinkage of deck (Eq 5.9.5.4.3d-1,2) (Gain)

(8) Shrinkage of beam between deck place and final (Eq 5.9.5.4.3a-1) (Loss)(9) Creep of beam between deck place and final, initial loads (Eq 5.9.5.4.3b-1) (Loss)

(1) Initial prestress just before release

(3) Shrinkage between release and deck place (Eq 5.9.5.4.2a-1) See list of equations below (Loss)

Prestress Loss Components (ksi)

(2) Elastic shortening due to initial prestress plus self weight (Loss)

'cif24.0'cf19.0'cif60.0 'cf40.0'cf45.0 'cf60.0

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