aashto pci bdm loss ex. 9.4 070319
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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
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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
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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
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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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