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7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
1/25Fal l 2012 |PCI Journal8
Anew method or estimating time-dependent loss
o prestress was introduced in the 2005 interim
revisions o the American Association o State
Highway and Transportation OcialsAASHTO LRFD
Bridge Design Specifcations1 ollowing the recommen-
dations o the National Cooperative Highway ResearchPrograms (NCHRP) report 496.2 The primary goal o the
research documented in NCHRP report 496 was to update
the methodology or estimating prestress loss, extending
its applicability to include high-strength concrete gird-
ers. The method that was developed (and has since been
adopted as AASHTO LRFD specications article 5.9.5.4)
is much more rened and rigorous than previous AASHTO
methods. The renements, while in many ways making the
method more technically sound, introduce some nuances
that can be conusing to practitioners more amiliar with
previous approaches. The goal o this paper is to clariy the
AASHTO LRFD specications loss o prestress provisionsby demonstrating their oundation in basic mechanics.
This paper ollows the organization o the prestress loss
provisions currently in AASHTO. Each o the equations
appearing in article 5.9.5.4 will be described in detail,
ollowed by a brie discussion o the shrinkage and creep
models developed or high-strength concrete as recommend-
ed in NCHRP report 496 and currently shown in AASHTO
LRFD specications article 5.4.2. The time-dependent
material property models and the methods or estimating
prestress loss are independent o each other. In other words,
any suitable shrinkage and creep models may be used in the
This paper details the time-dependent analysis method or
determining loss o prestress in pretensioned bridge girders in
the American Association o State Highway and Transportation
OfcialsAASHTO LRFD Bridge Design Specifcations.
This paper aims to make the loss o prestress method more
widely understood and better applied in practice.
This paper clarifes misconceptions related to transormed sec-
tion properties and prestress gains.
AASHTO LRFD Bridge DesignSpecificationsprovisions for
loss of prestress
Brian D. Swartz, Andrew Scanlon, and Andrea J. Schokker
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
2/25 10PCI Journal|Fal l 2012
our components:
riction
anchorage seating
prestressing steel relaxation
elastic shortening
Friction and anchorage seating are primarily o concern or
posttensioned construction, although precasters must be
aware o these two components as part o the pretensioning
process. Precasters must also be aware o the relaxation
losses that occur between jacking and transer. While some
precasters overjack to compensate or riction and seat-
ing losses, many do not overjack to counteract relaxation
losses beore transer. Designers should ensure that the
steel stress assumed just beore transer is realistic given
the precasting environment and relaxation losses that occur
during abrication. Guidelines or estimating relaxation
losses beore transer have historically been part o AAS-
HTO LRFD specications, but they are no longer provided
as o the 2005 interim revisions. Relaxation beore transer
is not insignicant. In act, approximately 1/4 o the total
relaxation loss happens during the rst day that the strand
is held in tension as calculated by the intrinsic relaxation
equation developed by Magura et al.4 Because the equa-
tion or relaxation beore transer is no longer included by
AASHTO, Eq. (5.9.5.4.4b-2) is reproduced rom the 20045
AASHTO LRFD specications here or convenience. It
applies only to low-relaxation strands.
f
t f
ffpR
pj
py
pj=
log( ).
24
400 55 (AASHTO
5.9.5.4.4b-2)
where
t = time in days rom stressing to transer
pj = initial stress in the tendon ater anchorage seating
py = specied yield strength o prestressing steel
Elastic shortening losses occur as the concrete responds
elastically, and instantaneously, to the compressive load
transerred rom the bonded prestressing. The designer can
consider this eect in two dierent ways:
The elastic shortening loss o prestressthe dier-
ence in strand tension just ater transer and just be-
ore transercan be calculated explicitly. The loss o
prestress can be determined iteratively using AASH-
TO LRFD specications Eq. (5.9.5.2.3a-1) or directly
using Eq. (C5.9.5.2.3a-1). Stresses in the concrete are
prestress loss method. Explanations o concepts that apply to
more than one equation are presented in detail ollowing the
comprehensive description o the method. Finally, a discus-
sion is oered to convey the authors perspective on the
current method and recommendations or improvement.
Material property models
Use o the AASHTO LRFD specications prestress loss
provisions requires a method or calculating each o the
ollowing material properties:
concretemodulusofelasticityEc
concretestrainduetoshrinkagesh
concretecreepstraincrviaacreepcoefcient
prestressingsteelmodulusofelasticityEp
prestressingsteelrelaxationfpR
The AASHTO LRFD specications provide guidance on
each o these, with the most signicant recent changes
to the creep and shrinkage model. Those changes were
introduced in the 2005 interim revisions ollowing recom-
mendations rom NCHRP report 496. 2 The changes also
included some slight modications to the calculation o
concrete elastic modulus and steel relaxation.
Detailed inormation related to the AASHTO LRFD speci-
cations material property model or high-strength concrete,
along with a basic denition o the concrete creep coecientused in the prestress loss provisions, is available in appendix
A. Further documentation on the high-strength concrete
model is provided by Tadros et al.2 and Al-Omaishi et al.3
Loss of prestress
In the current AASHTO LRFD specications methodology,
loss o prestress is calculated in three stages:
at transer
between transer and the time o deck placement
between the time o deck placement and nal time
The introduction o time o deck placement in the 2005
interim revisions was a signicant change rom the previ-
ous method. The calculations required or each stage are
discussed in detail in the ollowing paragraphs.
Losses at transfer
Losses relative to the initial jacking stress immediately
ater transer o prestress to the concrete can be split into
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
3/25Fal l 2012 |PCI Journal0
Superposition is applied or creep strains resulting
rom dierent stress increments
Relationships between stress and strain in concrete
and steel are prescribed.
The prestress loss method in the AASHTO LRFD speci-
cations is split into two dierent periods: beore andater deck placement. In both periods, concrete shrinkage,
concrete creep, and prestressing relaxation are considered.
In this paper, the eects o dierential deck shrinkage will
be treated separately to avoid the conusion that may ol-
low i prestress gains due to deck shrinkage are combined
and superimposed incorrectly with prestress losses due to
creep, shrinkage, and relaxation.
For both shrinkage and creep o concrete, the equations or
change in prestress are ounded on Hookes law applied
to the prestressing steel, which is assumed linear elastic
at all times. All equations are the product o the change in
concrete strain at the centroid o the prestressing strands
and the elastic modulus o prestressing steel. An eort was
made to make the undamental Hookes law relationship
readily apparent in this paper by reormatting equations.
Concrete shrinkage before deck placement
The prestress loss due to shrinkagepSR is given by
AASHTO LRFD specications Eq. (5.9.5.4.2a-1).
f E KpSR bid p id= (AASHTO 5.9.5.4.2a-1)
where
bid = shrinkage strain o girder concrete between time o
transer or end o curing and time o deck placement
Kid = transormed section coecient that accounts or
time-dependent interaction between concrete and bonded
steel in section being considered or period between trans-
er and deck placement
The terms in Eq. (5.9.5.4.2a-1) can be regrouped as shown
in Eq. (1).
f E KpSR p bid id= ( ) (1)
Hookes law is clearly evident in Eq. (1). The concrete strain
at the centroid o the prestressing steel is the product bidKid.
The Kidterm is an adjustment to the ree shrinkage strain o
concrete bidduring this time to account or the time-depen-
dent interaction between concrete and bonded steel, which
provides some internal restraint against shrinkage. It arises
rom considerations o strain compatibility and equilibrium
on the cross section. Kidis discussed in detail later in this
paper. In addition, a derivation o the equation is provided in
appendix B. The subscripts b, i, and din the shrinkage strain
then determined by applying the eective prestress
orce ater transer (the prestressing orce beore trans-
er minus the elastic shortening loss) to the net section
concrete properties. The use o gross section properties
is typically an acceptable simplication.
I only the concrete stresses need to be known and a
calculation o eective prestressing orce immediately atertranser is not needed, the use o transormed section prop-
erties may provide a more direct solution. In this case, the
concrete stresses can be ound by applying the prestressing
orce beore transer (not calculating any elastic shortening
losses explicitly) to the transormed section properties. The
elastic shortening losses are accounted or implicitly by use
o transormed section properties.
To calculate eective prestress and concrete stresses ater
transer, a time-dependent analysis that considers shrinkage
and creep o concrete and relaxation o prestressing steel is
required.
Time-dependent loss of prestress
The time-dependent analysis method or estimating loss
o prestress is independent o the material property model
used. No assumptions inherent in the time-dependent
analysis method impose concrete material characteristics.
Thereore, the prestress loss method in the AASHTO
LRFD specications is not specic to high-strength con-
crete. Rather, it is a time-dependent analysis approach or a
concrete girder with bonded prestressing steel that reer-
ences the concrete material property models in the AAS-
HTO LRFD specications developed or high-strengthconcrete. Any other suitable material property model could
be used with the time-dependent analysis approach.
Loss o prestress is calculated by tracing the change in
strain in the prestressing steel that occurs with time due
to concrete strains caused by elastic, creep, and shrink-
age deormations, along with losses due to relaxation. The
analysis is based on strain compatibility and equilibrium in
the cross section at any time along with prescribed stress-
strain relationships or steel and concrete. The time-depen-
dent analysis approach is more straightorward than the
complex equations in the AASHTO LRFD specicationssuggest. The basic assumptions inherent in the time-depen-
dent analysis method are as ollows:
Pure beam behavior; that is, plane sections remain plane.
Strain compatibility; that is, perect bond between the
concrete and prestressing steel. Thereore the change
in strain in the prestressing steel is equal to the change
in strain in the surrounding concrete.
As required or equilibrium, internal orces equal
external orces.
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
4/25 11PCI Journal|Fal l 2012
loss o prestress due to relaxation is small and varies over
a small range. Thereore, the specications recommend
assuming a total relaxation rom transer to nal time o
2.4 ksi (16.5 MPa), with hal o that assumed to occur
beore deck placement pR1.
Girder concrete shrinkage after deck place-
mentThe equation or shrinkage losses ater deckplacement pSD mirrors that or losses beore deck place-
ment and is given by AASHTO LRFD specications
Eq. (5.9.5.4.3a-1).
f E KpSD bdf p df= (AASHTO 5.9.5.4.3a-1)
where
bd = shrinkage strain o girder concrete ater deck place-
ment
Kd = transormed section coecient that accounts or
time-dependent interaction between concrete and
bonded steel in section being considered ater deck
placement
The terms in Eq. (5.9.5.4.3a-1) can be regrouped as shown
in Eq. (3).
f E KpSD p bdf df= ( ) (3)
The concrete strain at the centroid o the prestressing steel
over this period is the product bdKd. Kd is a transormed
section coecient analogous to Kid, except that it is de-
veloped or use with the properties o the ull composite(girder plus deck) cross section. The shrinkage strain rom
ater deck placement bd is calculated most readily when
recognizing that it is the dierence between nal shrinkage
strain and that at the time o deck placement (Eq. [4]).
bdf bif bid = (4)
where
bi= shrinkage strain o girder concrete between time o
transer or end o curing and nal time
Concrete creep after deck placement The
equation or creep losses ater deck placement pCD is
the sum o creep eects due to stresses applied at transer
and stresses applied ater transer. It is given by AASHTO
LRFD specications Eq. (5.9.5.4.3b-1).
fE
Ef t t t t K
E
Ef t t
pCD
p
ci
cgp b f i b d i df
p
c
cd b f d
= ( ) ( )
+ (
, ,
, )) Kdf (AASHTO 5.9.5.4.3b-1)
term represent beam (girder), initial time (transer or start o
curing), and deck placement time td.
Concrete creep before deck placement The pre-
stress loss due to creep pCR is given by AASHTO LRFD
specications Eq. (5.9.5.4.2b-1).
fE
Ef t t KpCR
p
ci
cgp b d i id = ( ) ,(AASHTO
5.9.5.4.2b-1)
where
Eci = modulus o elasticity o concrete at transer
cgp = concrete stress at centroid o prestressing tendons
due to the prestressing orce immediately ater
transer and sel-weight o the member at section
o maximum moment
b(td,ti) = creep coecient or girder concrete at the time td
o deck placement due to loading applied at the
time at transer ti
The terms in Eq. (5.9.5.4.2b-1) can be regrouped as shown
in Eq. (2).
f Ef
Et t KpCR p
cgp
ci
b d i id =
( )
, (2)
Hookes law is again seen in Eq. (2) with the term
f
Et t K
cgp
ci
b d i id
( )
,
as the concrete strain due to creep at the centroid o the
prestressing steel. The term
f
E
cgp
ci
is the elastic strain in the concrete at the prestressing cen-
troid due to stresses applied at transer. The product o thiselastic strain and the girder creep coecient at the time o
deck placement tddue to stresses applied at time ti results in
the creep strain in the concrete at the centroid o prestress-
ing. As with the shrinkage strain, the transormed section co-
ecient Kidis used to represent the internal restraint oered
by the bonded prestressing steel against concrete creep.
Prestressing steel relaxation before deck
placement The AASHTO LRFD specications provide
guidance or a detailed calculation o prestressing steel
relaxation, but when low-relaxation strands are used the
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
5/25Fal l 2012 |PCI Journal2
where
b(t,ti) = creep coecient or girder concrete at nal time
tdue to loading applied at the time ti o transer
cd = change in concrete stress at centroid o pre-
stressing strands due to time-dependent loss o
prestress between transer and deck placementcombined with deck weight and superimposed
loads
b(t,td) = creep coecient or girder concrete at nal time
tdue to loading applied at time tdo deck place-
ment
The terms in Eq. (5.9.5.4.3b-1) can be regrouped as shown
in Eq. (5).
f Ef
Et t t t pCD p
cgp
ci
b f i b d i=
( ) ( )
+
, ,
EEf
Et t Kp
cd
c
b f d df
( )
Kdf
,
(5)
The rst term accounts or the creep eects, due to the
initial prestressing, that continue ater deck placement.
This is calculated as the dierence between the nal
creep eects and those already considered at the time o
deck placement in Eq. (2). The term [b(t,ti) b(td,ti)] is
not equal to b(t,td) because the creep unction is driven
largely by the time when the stress changed. The second
term in Eq. (5) will generally be opposite in sign relative
to the rst. The initial stress at transercgp will typically
be compression, while the changes in stress due to deck
weight, superimposed dead load, and loss o prestress will
typically be tensile stress increments. Figure 1 claries the
Figure 1. Schematic summary o the components aecting prestress losses due to creep. Note: Ec = modulus o elasticity o concrete at 28 days; Eci = modulus o
elasticity o concrete at transer; fcgp = concrete stress at centroid o prestressing tendons due to prestressing orce immediately ater transer and sel-weight o
member at section o maximum moment; td
= age o girder concrete at time o deck placement; tf= age o girder concrete at end o time-dependent analysis; t
i= age
o girder concrete at time o prestress transer; fcd = change in concrete stress at centroid o prestressing strands due to time-dependent loss o prestress between
transer and deck placement, combined with deck weight and superimposed loads;b(td,ti) = creep coecient or girder concrete at time tdo deck placement due to
loading applied at time tio transer; b(tf,td) = creep coecient or girder concrete at nal time tf due to loading applied at time td o deck placement; b(tf,ti) = creep
coecient or girder concrete at nal time tf due to loading applied at time o transer.
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
6/25 11PCI Journal|Fal l 2012
superposition o the eects. Again, the transormed section
coecient Kd is used to model the restraint o creep by
the bonded prestressing steel. The time-dependent loss o
prestress is shown to occur instantaneously or clarity o
the graphic. In reality, it would occur gradually.
Prestressing steel relaxation after deck place-
ment I a total loss o prestress due to relaxation o 2.4 ksi
(16.5 MPa) is assumed, as recommended by the AASHTO
LRFD specications, hal o that value should be taken
between deck placement and nal time pR2.
Shrinkage of the deck concrete
Starting with the 2005 interim revisions, the AASHTO
LRFD specications method or estimating loss o pre-
stress recognized the interaction between a cast-in-place
deck and a precast concrete girder when they are compos-itely connected. The shrinkage dierential exists primarily
because much o the total girder concrete shrinkage occurs
beore the system is made composite. Thereore, while the
girder and deck behave compositely, the deck has greater
potential shrinkage. Strain compatibility at the deck-girder
interace, however, requires that the two elements behave
as one unit; thereore, an internal redistribution o stresses
is necessary.
The eect o dierential shrinkage can be modeled as an
eective orce at the centroid o the deck (Fig.2).
In typical construction, the deck is above the neutral axis
o the composite section and the centroid o the prestress-
ing steel is below the neutral axis at the critical section.
The eective compressive orce due to dierential shrink-
age causes a tensile strain on the opposite ace (bottom)
o the girder. Assuming strain compatibility between the
prestressing steel and the surrounding concrete, the tensile
strain leads to an increase in the eective orce in the pre-
stressing steel. The AASHTO LRFD specications method
calls this a prestressing gain. This prestressing gain,
however, does not increase the compressive stress in the
surrounding concrete. Rather, the concrete experiences a
tensile stress increment, too. A urther discussion o elastic
gains is provided later in this paper.
An explanation o the article 5.9.5.4.3d equations related
to shrinkage o deck concrete is warranted. First, the
prestress gain due to deck shrinkage pSS is calculated by
AASHTO LRFD specications Eq. (5.9.5.4.3d-1).
fE
Ef K t tpSS
p
c
cdf df b f d = + ( ) 1 0 7. , (AASHTO5.9.5.4.3d-1)
where
cd= change in concrete stress at centroid o prestressing
strands due to shrinkage o deck concrete
The terms in AASHTO Eq. (5.9.5.4.3d-1) can be re-
grouped as shown in Eq. (6).
f Ef
E
t t
pSS p
cdf
c
b f d
=
+ ( )
1 0 7. ,
Kdf (6)
The term in the outer parentheses in Eq. (6) is the strain at
the prestressing steel centroid due to deck shrinkage. The
strain is ound through division o the stress change by
the age-adjusted eective modulus (rather than the elastic
modulus o concrete) because the eective orce due to
deck shrinkage builds up over time and is partially relieved
by creep. A detailed description o age-adjusted eective
modulus is provided later in this paper.
The AASHTO LRFD specications recommend calculat-
ing the stress change caused by the eective deck shrink-
age orce at the centroid o the prestressing steel using
Figure 2. Conceptual model o the eect that deck-girder dierential shrinkage has on the composite cross section. Note: Pdeck= eective orce due to deck shrink-
age applied to composite section at centroid o deck.
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
7/25Fal l 2012 |PCI Journal4
Al-Omaishi et al.,6 calculates extreme bottom ber concrete
stress cbSS using the eective orce dened in Eq. (8).
f PA
y e
IKcbSS deck
c
bc d
c
df= +
1(9)
where
ybc = eccentricity o concrete extreme bottom ber with
respect to centroid o composite cross section
Maintaining a consistent sign convention or Eq. (7) and
(9) is a challenge. In typical construction, Pdeck will be an
eective compression orce and cbSS will be a tensile
stress increment in the extreme bottom concrete ber.
The eective orce Pdeck, will be present on the composite
cross section even i the deck concrete is cracked because
the orce will be transerred by reinorcement across the
cracks and will eventually be carried into the girder via the
composite connection.
Transformed sectioncoefficients Kid and Kdf
The transormed section coecients represent the act
that steel restrains the creep and shrinkage o concrete.
Because the two materials are bonded, the dierences in
time-dependent behavior lead to an internal redistribution
o stress. In addition, the dierence in elastic response
between steel and concrete must be considered in the stress
redistribution.
Consider shrinkage as an example. Figure3 shows sh as
the shrinkage expected o a concrete specimen that has no
restraint against shortening (that is, no reinorcement). Such
an idealized type o specimen was used in developing the
model used to predict shrinkage strains. In a prestressed
concrete girder, however, there is restraint against shrink-
age because the prestressing steel bonded to the concrete
does not shrink. Because strain compatibility must be
satised, equilibrium requires a redistribution o stresses to
accommodate the dierences in time-dependent and elastic
behavior o the two materials. In Fig. 3, res is used to denotethe amount by which the ree shrinkage is reduced at the
level o the prestressing centroid by bonded steel. The net
shortening at the centroid o prestressing is net. The trans-
ormed section coecient is derived to quantiy the eect o
the prestressing steels restraint. It can be thought o as the
simple ratio given in Eq. (10).
Kidnet
sh
=
(10)
Kidand Kdrepresent the same phenomenon. Kidis derived
Eq. (5.9.5.4.3d-2), which is split into its components in
Eq. (7) and (8) or clarity.
f PA
e e
Icdf deck
c
pc d
c
=
1(7)
where
Pdeck = eective orce due to deck shrinkage applied to
composite section at centroid o deck; Pdeck is de-
ned or the purposes o this paper only and is not a
variable used in AASHTO LRFD specications
Ac = area o composite cross section
epc = eccentricity o prestressing orce with respect
to centroid o composite section, positive where
centroid o prestressing steel is below centroid o
composite section
ed = eccentricity o deck with respect to gross composite
section, positive where deck is above girder
Ic = moment o inertia o composite cross section
The eective orce due to deck shrinkage can be calculated
as the product o shrinkage strain, elastic modulus, and
eective deck area (Eq. [8]). Because the gradual buildup
o stress will be partially relieved by simultaneous creep
o the deck concrete, an age-adjusted eective modulus is
used in place o the concrete elastic modulus.
PE
t tAdeck ddf
cd
d f d
d=+ ( )
1 0 7. ,(8)
where
dd = shrinkage strain o deck concrete between time
o deck placement or end o deck curing and
nal time
Ecd = modulus o elasticity o deck concrete
d(t,td) = creep coecient or deck concrete at nal time t
due to loading applied at time tdo deck place-
ment
Ad = eective cross-sectional area o composite deck
concrete
The AASHTO LRFD specications guide the user to
calculate a stress change at the centroid o the prestressing
steel and the prestressing gain due to deck shrinkage but stop
short o providing a prescriptive equation to calculate con-
crete stress at the extreme ber. Equation (9), published by
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
8/25 11PCI Journal|Fal l 2012
where
Ec,e = eective modulus o elasticity o concrete used
to describe the response o concrete to an instan-taneous stress increment considering both elastic
and creep eects
(t,ti) = creep coecient or concrete at time tdue to
stresses applied at time o a stress change ti
I the stress change is not instantaneous, the creep response
o concrete is slightly less. Figure 4 shows the dierence
schematically, where the stress change has been split
into three equal increments (rather than a truly gradual
increase) or the purpose o clarity in the graphic. I the
stress builds over time, the total creep strain is less thanwhen the same stress change is instantaneous. Equa-
tion (11) is adjusted slightly to yield Eq. (12) to represent
an age-adjusted eective modulus o concreteEc,AAEM.
EE
t t
E
t tc AAEM
c
i
c
i
,, . ,
=+ ( )
+ ( ) 1 1 0 7
(12)
where
= aging coecient
with respect to the girder only and is used or calculations
beore deck placement, while Kdis derived or the girder-deck
composite system. A detailed derivation is in appendix B.
Age-adjusted effective modulus
The AASHTO LRFD specications account or the creep
o concrete by using an age-adjusted eective modulus in
some instances. The method approximates the principle
o creep superposition attributed to McHenry7 or a time
varying stress history. Figure4 shows the general rela-
tionship between elastic modulus, eective modulus, and
age-adjusted eective modulus. When a change in stress
in concrete c is applied instantaneously, the short-term
strain can be calculated according to the elastic modulusEc
or the concrete. I the stress is maintained, the strain willincrease gradually as concrete creeps (rom point A to B
in Fig. 4). The net behavior o the concrete can be approxi-
mated by dening an eective elastic modulus (Eq. [11])
to describe the cumulative stress-strain behavior rom the
origin to point B.
EE
t tc eff
c
i
,,
=
+ ( )1 (11)
Figure 3. Bonded prestressing steel partially restrains the ree shrinkage strain inherent to the concrete. The eect is represented mathematically by the transormed
section coecient. Note:An = area o net cross section;Ap = area o prestressing steel; e= eccentricity between the centroid o the girder concrete and the centroid
o the prestressing steel; net= net shrinkage strain o section at centroid o prestressing steel considering shrinkage o concrete and restraint against shrinkage rom
bonded prestressing steel; res = portion o ree shrinkage o concrete eectively restrained by bonded prestressing steel at centroid o prestressing steel;
sh = concrete strain due to shrinkage.
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ater jacking. Thereore, the tension is less than at the time
o jacking. This loss o stress in the prestressing steel due
to shortening o the concrete at transer results in less pre-
compression o the concrete. This point will be contrasted
with the idea o an elastic gain.
An analogy to reinorced (nonprestressed) concrete helps
to clariy the concept. When a reinorced concrete beam
is loaded (Fig.5), tension stresses and eventually crack-ing would be expected on the bottom ace o a simply
supported beam. As load is applied to the beam, the steel
undergoes an elongation and thereore takes on a tensile
orce. As a result o the applied load, the steel experiences
a gain in tension. This orce would not be considered to
be precompressing the concrete in this region or acting to
resist the ormation o cracks. The tension gain in the steel
ollows an elongation that is also experienced by the sur-
rounding concrete, rendering a tensile stress increment in
the concrete as well as the steel.
Applied to reinorced concrete, the concept seems quitestraightorward. With respect to prestressed concrete, how-
ever, the term gain is misleading. Based on the terminol-
ogy alone, it seems reasonable to sum all prestress loss and
gain terms to arrive at an eective prestressing orce. I
the intent is to estimate the stress in the prestressing steel,
a summation o all prestress loss and gain terms is appro-
priate. Such an estimate may be desirable when checking
steel stresses against a specied limit. When checking
concrete stresses against a tension limit, however, the ap-
proach is problematic. Prestress gains can be caused by
the application o deck weight, superimposed dead load,
and live load. In addition, prestress gains result rom deck
The aging coecient was rst proposed by Trost8,9 in
1967, then rened by Bazant10 and Dilger.11 The AASHTO
LRFD specications adopt a constant value o 0.7 or . A
description o the method is also available in Collins and
Mitchell.12
Elastic losses and gains
Prestress loss occurs when the strain in the steel decreasesto match the shortening o concrete at the same level as the
steel. The application o load, however, such as the deck
weight at time o deck placement, causes an elongation in
the prestressing steel and a corresponding increase in steel
stress. The application o load also causes an increment o
tensile stress in the concrete at the level o the steel. Use o
the termprestress gain to describe this elastic response to
load causes conusion. The ollowing discussion attempts
to clariy the matter.
The elastic shortening loss at transer is well understood,
and the equation or estimating that loss did not change in2005 with the implementation o a new time-dependent
loss calculation method. This is a loss o prestress (relative
to the stress in the strands just beore transer) that ollows
concretes elastic response to the applied load rom preten-
sioned strands. The concrete and prestressing strands must
reach a point o equilibrium. The concrete girder is holding
the prestressing steel at a length greater than its zero-stress
length. In turn, the concrete experiences compressive stress
and a consequent shortening based on the elastic properties
o the concrete material. Because the concrete girder has
shortened, the prestressing steel bonded to the concrete is
allowed to get closer to its zero-stress length than it was
Figure 4. Schematic denition o an eective modulus and age-adjusted eective modulus as used to account or the creep response o concrete. Note: Ec = modulus
o elasticity o concrete; Ec,AAEM= age-adjusted eective elastic modulus o concrete used to describe response o concrete to gradual stress increment consider-ing both elastic and creep eects; Ec,eff= eective modulus o elasticity o concrete used to describe the response o concrete to an instantaneous stress increment
considering both elastic and creep eects.
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made o only the concrete componentAn, where the
area o prestressing steelAp has been subtracted.
Transormed cross section: A transormed girder cross
section accounts or the dierences in the elastic
response o concrete and steel. I they are assumed to
be perectly bonded, the steel experiences the same
strain as the surrounding concrete. The steel has a
much stier response per unit area because it has a
higher elastic modulus than the concrete. The dier-
ence is accounted or by transorming the steel to an
equivalent area o concrete, ound as the product o
the steel area and the modular ratio between steel and
concrete n, where n is equal to E
E
p
c
.
Gross cross section: A gross girder section disregards
the dierences in elastic response between concreteand steel. The area o the steel is treated no dierently
rom concrete, and the total gross areaAg o the cross
section is used in calculating the properties.
O the three, the use o transormed section properties and
net section properties to calculate stresses are most accu-
rate and, in act, numerically equal. Gross section proper-
ties are used most commonly in practice or simplicity.
The error involved with the use o gross section properties
is typically negligible because steel makes up a small per-
centage o the cross-sectional area.13
First, a proo will be oered to demonstrate that the trans-
ormed and net section properties oer identical results.
Consider the pretensioning arrangement dened in Fig.8.
The theory under inspection is that the use o net section
and transormed section properties will yield exactly the
same stress in the concrete c. Correct application o both
techniques is presented in Eq. (13), ollowed by a proo.
c
n TR
P
A
P
A= =
'
(13)
shrinkage. In each o these cases, the elongation that leads
to a prestress gain (increase in tension) is coupled with an
elongation in the concrete that leads to tensile stress (just
as in the reinorced concrete beam). Thereore, it would be
a gross error to assume that the prestress gains due to the
application o external loads or deck shrinkage act to ur-
ther precompress the surrounding concrete. Such an error
is likely to result i the true eective prestress orce, ound
by summation o all losses and gains, is used to calculate
concrete stress by a traditional combined stress ormula-
tion.
The plot o eective prestress over time in Fig.6 (red line)
was presented in NCHRP report 496 and has been repro-
duced in numerous settings since. The idea is to summarize
the components infuencing the eective prestressing orce.
It is undamentally correct, but possibly misleading, in its
treatment o prestress gains. To clariy the point, plots orthe extreme ber concrete stresses have been added by
the authors. The plots assume a simply supported precast
concrete girder with the centroid o prestressing below the
neutral axis with a cast-in-place deck on top. Stresses are
shown, conceptually, at the midspan section. While Fig. 6
is intended to be realistic, in some cases scale has been
sacriced or clarity.
Transformed section properties:Prestressing effects
The idea o using transormed section properties to cal-culate stresses is now mentioned in the AASHTO LRFD
specications (C5.9.5.2.3a) and in much o the literature
related to the NCHRP report 496 recommendations. Al-
though the issue has been addressed by many others,6,1315
urther clarication may still be necessary. First, the vari-
ous ways o idealizing a cross section will be summarized,
with reerence to a simple, concentrically prestressed cross
section (Fig.7):
Net cross section: A net girder section treats the con-
crete and steel as separate components (though strain
compatibility still applies). The net cross section is
Figure 5. Typical fexural response o a simply supported reinorced concrete beam.
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has been transormed to an equivalent area o con-
crete based on the modular ratio
Equation (13) could be rewritten as Eq. (14).
P
P
A
A
n
TR
'
= (14)
where
P' = eective tensile stress in prestressing strands ater
transer, considering elastic shortening losses
P = eective tensile stress in prestressing strands just
beore transer
ATR = area o transormed section, where the area o steel
Figure 6. Schematic summary o the time-dependent response o a prestressed concrete girder with respect to the eective prestressing orce and the extreme ber
concrete stresses at the top and bottom o the girder. Source: Data rom Tadros, Al-Omaishi, Seguirant, and Gallt (2003).
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(Fig. 7), the right side o Eq. (14) becomes the expression
in Eq. (21).
A
A
A
A nAn
A
A
n
TR
n
n p p
n
=+
=
+
1
1(21)
It is clear that Eq. (20) and (21) are identical, thus provingthe relationships shown in Eq. (13) and (14). Thereore, the
use o net section properties and transormed section prop-
erties produces identical results, but loss o prestress dur-
ing transer due to elastic shortening o concrete pES must
be calculated explicitly to determine the eective prestressing
orce needed to use net section properties [Eq. (22)].
P P A fp pES'
= (22)
A more detailed derivation o the relationship between net
section and transormed section properties, including treat-
Equation (15) was developed as an expression or P'
(Fig. 8).
P E A E AL L
Lp p p p p p
' '
'
= =
(15)
where
p
'= tensile strain in prestressing steel ater transer,
considering eects o elastic shortening
p = tensile strain in prestressing steel beore transer
L = length o concrete member just beore transer
L' = length o concrete member ater transer, considering
elastic shortening due to prestress
The change in length o the concrete component upon orce
transer can be approximated by Hookes law equation or
change in length o an axially loaded member [Eq. (16)].
L LP L
A En c
( ) =''
(16)
Substituting Eq. (16) into Eq. (15) yields Eq. (17).
P A EP
A Ep p p
n c
'
'
=
(17)
Solving or P'yields Eq. (18).
P
E A
E
E
A
A
p p p
p
c
p
n
' =
+
1(18)
From Fig. 8, the orce in the strands beore transer is given
by Eq. (19).
P E Ap p p= (19)
Substituting Eq. (18) and (19) into the the let side o
Eq. (14) yields the expression in Eq. (20).
P
P
E A
E
E
A
A
E A E
E
A
A
p p p
p
c
p
n
p p p p
c
p
n
'
=
+
=
+
1
1
1
=
+
1
1 nA
A
p
n
(20)
Recalling the denition o transormed section properties
Figure 7. Comparison o net section, gross section, and transormed section
representations. Note:Ag = gross area o cross section, including both steel and
concrete;An = area o net concrete section;Ap = area o prestressing steel;ATR
= area o transormed section, where the area o steel has been transormed
to an equivalent area o concrete based on the modular ratio; Ec = modulus
o elasticity o concrete; Ep = modulus o elasticity o prestressing steel; n=modular ratio Ep/Ec.
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ment o eccentric prestressing, is available in Huang.14
For now, the ollowing observations can be made:
The use o net section properties and transormed sec-
tion properties produces identical results or stress in
the concrete.
The use o transormed section properties is the most
direct method or calculating concrete stress because it
does not require an independent calculation o elastic
shortening losses.
A separate calculation o eective prestress would beneeded when transormed section properties are used
i eective prestress must be quantied.
Gross section properties can be used as a direct re-
placement or net section properties with minimal er-
ror when the area o prestressing steel is small relative
to the area o concrete.13
Transformed section
properties: Applied loads
It is also appropriate to use transormed section proper-
ties when calculating stresses caused by externally applied
loads. I perect bond is assumed, then the prestressing
steel will experience the same strain as the surrounding
concrete when the cross section undergoes the combined
eects o curvature and axial strain but will exhibit a
stier response because o its higher elastic modulus. The
additional stiness is accounted or by use o transormed
section properties.
Net section properties can be used with equal accuracybut additional computational eort. This approach would
require calculation o stresses due to the combination o
external load on the net section and orce in the prestress-
ing steel as it resists elongation. A closed orm solution or
this approach is cumbersome. The most practical approach
may be a series o iterations until strain compatibility is
satised.
Again, gross section properties are oten used to simpliy
stress calculations (ignoring the orce in the prestress-
ing steel as it resists elongation mentioned or net section
properties). The use o gross section properties introduces
Figure 8. Typical pretensioning sequence. Note:Ap = area o prestressing steel; Ep = modulus o elasticity o prestressing steel; L= length o concrete member just
beore transer; L'= length o concrete member ater transer, considering elastic shortening due to prestress; P= eective tensile stress in prestressing strands just
beore transer; P'= eective tensile stress in prestressing strands ater transer, considering elastic shortening losses; p = tensile strain in prestressing steel beore
transer; 'ptensile strain in prestressing steel ater transer, considering eects o elastic shortening.
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a technical error, but it is generally negligible. In addition,
the simplication error is oten conservative because gross
section properties underestimate the true stiness o the
cross section.
Discussion
The previous sections have provided a description o theprestress loss calculation method presented in the current
AASHTO LRFD specications, including the bases or the
various steps in the calculations. The increased complexity
in the provisions relative to previous versions has caused
some concern among practitioners and has introduced the
potential or errors in application o the provisions because
o a lack o amiliarity with the methodology. The ollow-
ing sections provide a discussion o some o the issues
involved.
Uncertainty in time-dependentanalysis
The complexity and rigor o the AASHTO LRFD speci-
cations method, relative to previous methods, suggests
an improved renement and precision in the estimate o
prestress losses and the determination o time-dependent
stresses in the concrete. However, time-dependent behav-
ior o prestressed concrete elements is dicult to predict
because o the uncertainty in many dependent variables,
such as the ollowing:
the actual compressive strength o concrete
the elastic modulus, shrinkage strain, and creep straino concrete
initial jacking orce in the prestressing strands, which
is only required to be within 5% o the target orce
according to PCIsManual or Quality Control or
Plants and Production o Structural Precast Concrete
Products16
construction practices and sequence o construction
(that is, time o deck placement)
eective area o the deck behaving compositely withthe girder
magnitude o applied loads
as-built geometry o the nished structure and location
o the prestressing strands
Given the many sources o uncertainty, one must have
reasonable expectations or any time-dependent analysis
method or prestressed concrete. It is possible that the
increased complexity o the AASHTO LRFD specica-
tions method could reduce its accuracy i engineers do not
understand the provisions and apply them incorrectly. This
paper has sought to explain the undamental principles and
subsequently reduce the number o instances where the
provisions are applied in error.
Stages for analysis
The AASHTO LRFD specications method recognizes theplacement o a cast-in-place deck as a signicant action in
the lie o a prestressed girder. The sel-weight o the deck
decompresses the concrete precompression region, and
the action o the composite system changes the way that
the girder responds to loads. In act, the AASHTO method
requires the user to dene a variable td that is the age o the
girder concrete at the time o deck placement. This type o
detail in the construction sequence is beyond the designers
control and in many cases beyond the designers ability
to estimate. Thereore, it is important to understand the
sensitivity o the method to this variable.
Figure 9 was developed to show the sensitivity o the pre-
stress loss calculations to tdover a range o realistic values.
The plot is based on example 9.4 in the PCI Bridge Design
Manual.17 Elastic shortening losses have not been included
in the plot because they are not aected by the time o
deck placement. In addition, the prestress gain due to deck
shrinkage is not included in the plots or prestress loss.
Deck shrinkage is included, however, in the extreme ber
concrete stress results presented later. Figure 9 shows plots
or the total time-dependent prestress loss (sum o shrink-
age, creep, and relaxation eects) and or the division o
those losses beore and ater deck placement. Relaxation
losses are assumed to be divided evenly between the twoperiods, regardless o the time o deck placement, as rec-
ommended by the AASHTO LRFD specications method.
The relaxation losses are small relative to the other compo-
nents, so this assumption does not have a signicant eect
on the plot in Fig. 9.
Figure 9 shows that the sensitivity o total prestress losses
to the time o deck placement is insignicant, especially
compared with the inherent uncertainty in the time-depen-
dent analysis o a prestressed concrete girder. The same
conclusion can also be reached by a more rigorous time-
step analysis method.18 I there are particular reasons tohave an accurate estimate o prestress losses at the time o
deck placement, such as a more reliable estimate o cam-
ber, then the division o the time periods at deck placement
may be necessary. For the design and layout o prestress-
ing, however, stress calculations at transer and at service
are likely to control and the numerical value chosen or the
time o deck placement is o little consequence.
For the same example, the sensitivity o bottom-ber
concrete stresses to the time o deck placement is also o
interest. The bottom-ber concrete stress at service, or
this example, varies over a range o 37 psi (0.26 MPa)
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PE
t tAdeck ddf bdf
cd
d f d
d= ( )+ ( )
1 0 7. ,(23)
Transformed section coefficients
As detailed previously, the transormed section coecients
Kidand Kdrepresent the restraint that the bonded prestress-ing steel oers against shrinkage and creep in the concrete.
There are, however, some inconsistencies between the nal
ormulation o those equations and the behavior they rep-
resent. The equations rom the AASHTO LRFD specica-
tions method are given below.
KE
E
A
A
A e
It
id
p
ci
p
g
g pg
g
b f
=
+
+
+
1
1 1 1 0 7
2
. , tti( )
(AASHTO 5.9.5.4.2a-2)
or deck placement times between 30 days and 365 days,
with the tensile stress increasing as the girder age at deck
placement increases. To put that range into perspective, the
removal o one 1/2 in. (12 mm) diameter prestressing strand
at the centroid o prestressing would change the bottom
ber stress at service by approximately 60 psi (0.4 MPa).
In other words, the time o deck placement is practically
insignicant or nal time estimates o eective prestress
or extreme ber concrete stress.
Deck shrinkage
The AASHTO LRFD specications method appropriately
recognizes the eective orce that develops as a result o
deck shrinkage. The magnitude o that orce, however, is a
unction o the shrinkage dierential between the deck and
the girder, not the total deck shrinkage. Thereore, it would
be more correct to replace Eq. (8) with Eq. (23). The 2011
edition o the PCI Bridge Design Manual19 recommends
using hal the value obtained by Eq. (8) in design due to
the likelihood o deck cracking and reinorcement reducing
this eect.
Figure 9. Plot o the American Association o State Highway and Transportation Ocials prestress loss methods sensitivity to the time o deck placement input vari-able as applied to Example 9.4 in the PCI Bridge Design Manual. Note: 1 ksi = 6.895 MPa.
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sponsorship o the Portland Cement Association (PCA
project index no. 08-04). The contents o this report refect
the views o the authors, who are responsible or the acts
and accuracy o the data presented. The contents do not
necessarily refect the views o PCA.
References
1. AASHTO (American Association o State Highway
and Transportation Ocials). 2012.AASHTO LRFD
Bridge Design Specifcations, 6th Edition. Washing-
ton, DC: AASHTO.
2. Tadros, M. K., N. Al-Omaishi, S. J. Seguirant, and J.
G. Gallt. 2003. Prestress Losses in Pretensioned High-
Strength Concrete Bridge Girders. National Coopera-
tive Highway Research Program report 496. Wash-
ington, DC: Transportation Research Board, National
Academy o Sciences.
3. Al-Omaishi, N., M. K. Tadros, and S. J. Seguirant.
2009. Elasticity Modulus, Shrinkage, and Creep o
High-Strength Concrete as Adopted by AASHTO.
PCI Journal 54 (3): 4463.
4. Magura, D. D., M. A. Sozen, and C. P. Siess. 1964. A
Study o Stress Relaxation in Prestressing Reinorce-
ment. PCI Journal 9 (2): 1357.
5. AASHTO. 2004.AASHTO LRFD Bridge Design
Specifcations.3rd ed. Washington, DC: AASHTO.
6. Al-Omaishi, N., M. K. Tadros, and S. J. Seguirant.2009. Estimating Prestress Loss in Pretensioned,
High-Strength Concrete Members. PCI Journal 54
(4): 132159.
7. McHenry, D. 1943. New Aspect o Creep in Concrete
and Its Application to Design.American Society or
Testing MaterialsProceedings43: 10691084.
8. Trost, H. 1967. Auswirkungen des Superprosition-
springzips au Kriech- und Relaxations-Probleme bei
Beton und Spannbeton. [In German.]Beton- und
Stahlbetonbau 62 (10): 230238.
9. Trost, H. 1967. Auswirkungen des Superprosition-
springzips au Kriech-und Relaxations-Probleme bei
Beton und Spannbeton. [In German.]Beton- und
Stahlbetonbau 62 (11): 261269.
10. Baant, Z. P. 1972. Prediction o Concrete Creep E-
ects Using Age-Adjusted Eective Modulus Meth-
od.ACI Journal 69 (4): 212217.
11. Dilger, W. H. 1982. Creep Analysis o Prestressed
Concrete Structures using Creep Transormed Section
where
epg = eccentricity o prestressing steel centroid with respect
to gross concrete section
Ig = moment o inertia o the gross concrete sectionK
E
E
A
A
A e
It t
df
p
ci
p
c
c pc
c
b f i
=+
+
+
1
1 1 1 0 7
2
. , (( )
(AASHTO 5.9.5.4.3a-2)
The Kidcoecient is intended to represent the concrete-
steel interaction in the girder beore deck placement.
Thereore, the age-adjusted eective modulus used in the
ormulation would be better dened by the creep coe-
cient beore deck placement b(td,ti), rather than the creep
coecient at nal time b(t,ti).
The equation or the transormed section coecient or the
composite section Kdhas a similar inconsistency. Because
Kdrepresents behavior in response to loads applied at the
time o deck placement, the age-adjusted eective modulus
should be dened using the creep coecient or loads ap-
plied at deck placement b(t,td).
Last, the transormed section coecients are relatively
insensitive to the input variables or typical cross sections.
For common bridge girders, values will oten be in the 0.80
to 0.90 range. Given the complexity o the calculation and
the relatively steady value o the result, it may be reason-able to adopt a constant value or standard bridge types.
Conclusion
The AASHTO LRFD specications method or loss
o prestress is a rened approach to the time-dependent
analysis o prestressed girders that remains independent o
any particular material property model. The computational
intensity can be overwhelming or designers exposed to
the method or the rst time, and the apparent complexity
o the equations can be intimidating. However, a thorough
understanding o the undamental concepts involved withthe methods development provides clarity and improves
the designers ability to apply the provisions correctly.
Despite the rigor o the method, one should remain mindul
o the inherent uncertainty involved with time-dependent
analysis o prestressed members. Opportunities to simpliy
the provisions and reduce the complexity o the equations
should be explored urther.
Acknowledgments
The research reported in this paper (PCA R&D SN 3123b)
was conducted by Pennsylvania State University with the
7/29/2019 AASHTO LRFD Bridge Design Specifications provisions for loss of prestress
17/25Fal l 2012 |PCI Journal4
Properties. PCI Journal 27 (1): 89117.
12. Collins, M. P., and D. Mitchell. 1991. Prestressed
Concrete Structures. Englewood Clis, NJ: Prentice-
Hall.
13. Hennessey, S. A., and M. K. Tadros. 2002. Signi-
cance o Transormed Section Properties in Analy-sis or Required Prestressing. PCI Journal 47 (6):
104107.
14. Huang, T. 1972. Estimating Stress or a Prestressed
Concrete Member.Journal o the Precast Concrete
Institute 17 (1): 2934.
15. Walton, S., and T. Bradberry. 2004. Comparison o
Methods or Estimating Prestress Losses or Bridge
Girders.In Proceedings, Texas Section ASCE Fall
Meeting, September 29-August 2, 2004, Houston,
Texas. Austin, TX: Texas Section American Society
o Civil Engineers. Accessed June 19, 2012. http://tp.
dot.state.tx.us/pub/txdot-ino/library/pubs/bus/bridge/
girder_comparison.pd
16. PCI Plant Certication Committee. 1999.Manual or
Quality Control or Plants and Production o Struc-
tural Precast Concrete Products. MNL-116-99. 4th ed.
Chicago, IL: PCI
17. PCI Bridge Design Manual Steering Committee. 1997.
Precast Prestressed Concrete Bridge Design Manual.
MNL-133. 1st ed. Chicago, IL: PCI.
18. Swartz, B. D., A. J. Schokker, and A. Scanlon. 2010.
Examining the Eect o Deck Placement Time on the
Behavior o Composite Pretensioned Concrete Bridge
Girders. 2010 b Congress and PCI Convention
Bridge Conerence Proceedings. Chicago, IL: PCI.
CD- ROM.
19. PCI Bridge Design Manual Steering Committee. 2011.
Bridge Design Manual. MNL-133-11. 3rd ed. Chi-
cago, IL: PCI.
Notation
Ac = area o composite cross section
Ad = eective composite cross-sectional area o deck
concrete
Ag = gross area o cross section, including both steel
and concrete
An = area o net concrete section
Ap = area o prestressing steel
ATR = area o transormed section where the area o
steel has been transormed to an equivalent area
o concrete based on the modular ratio
e = eccentricity between the centroid o the girder
concrete and the centroid o the prestressing steel
ed = eccentricity o deck with respect to gross compos-ite section, positive where deck is above girder
epc = eccentricity o prestressing orce with respect
to centroid o composite section, positive where
centroid o prestressing steel is below centroid o
composite section
epg = eccentricity o prestressing steel centroid with
respect to gross concrete section
epn = eccentricity o prestressing steel centroid with
respect to net concrete section
Ec = modulus o elasticity o concrete at 28 days
Ec,AAEM = age-adjusted eective elastic modulus o con-
crete used to describe response o concrete to
gradual stress increment considering both elastic
and creep eects
Ecd = modulus o elasticity o deck concrete
Ec,e = eective modulus o elasticity o concrete used to
describe the response o concrete to an instanta-
neous stress increment considering both elasticand creep eects
Eci = modulus o elasticity o concrete at transer
Ect = modulus o elasticity o concrete at time tunder
consideration
Ep = modulus o elasticity o prestressing steel
c = stress in the concrete
fc' = specied compressive strength o concrete
cgp = concrete stress at centroid o prestressing tendons
due to prestressing orce immediately ater
transer and sel-weight o member at section o
maximum moment
pj = initial stress in the tendon ater anchorage seating
py = specied yield strength o prestressing steel
Ic = moment o inertia o composite cross section
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Ig = moment o inertia o gross concrete section
In = moment o inertia o net concrete section
k = adjustment actor or specied concrete compres-
sive strength at time o transer or end o curing
khc = adjustment actor or average ambient relativehumidity in creep coecient calculations
khs = adjustment actor or average ambient relative
humidity in shrinkage calculations
ks = adjustment actor or member size, specically
volumetosurace area ratio
ktd = adjustment actor or time development that sets
the rate at which shrinkage strain asymptotically
approaches ultimate value (set equal to 1.0 when
determining nal shrinkage)
K1 = correction actor or source o aggregate to be
taken as 1.0 unless determined by physical test
and as approved by authority o jurisdiction
Kd = transormed section coecient that accounts or
time-dependent interaction between concrete and
bonded steel in section being considered ater
deck placement
Kid = transormed section coecient that accounts or
time-dependent interaction between concrete
and bonded steel in section being considered orperiod between transer and deck placement
L = length o concrete member just beore transer
L' = length o concrete member ater transer, consid-
ering elastic shortening due to prestress
n = ratio o elastic moduli o prestressing steel and
girder concrete
P = eective tensile stress in prestressing strands just
beore transer
P' = eective tensile stress in prestressing strands ater
transer, considering elastic shortening losses
Pc = restraint orce applied to concrete by bonded
prestressing steel
Pdeck = eective orce due to deck shrinkage applied to
composite section at centroid o deck
Pp = eective compression orce applied to prestress-
ing steel by shrinkage o concrete
RH = ambient relative humidity
t = time in days rom stressing to transer or relax-
ation calculations; time o interest ater applica-
tion o stress or creep calculations
td = age o girder concrete at time o deck placement
t = age o girder concrete at end o time-dependent
analysis
ti = age o concrete when load is initially applied
V/S = ratio o volume to surace area
wc = unit weight o concrete
ybc = eccentricity o concrete extreme bottom ber
with respect to centroid o composite cross sec-
tion
n = variable representing 1
2
+
A e
I
n pn
n
= aging coecient
c = change o stress in concrete
cbSS = stress increment at bottom concrete ber due to
dierential shrinkage between precast concrete
girder and cast-in-place composite deck
cd = change in concrete stress at centroid o prestress-
ing strands due to time-dependent loss o prestressbetween transer and deck placement combined
with deck weight and superimposed loads
cd = change in concrete stress at centroid o prestress-
ing strands due to shrinkage o deck concrete
pCD = loss o prestress between time o deck placement
and nal time due to creep o girder concrete
pCR = loss o prestress between time o transer and
deck placement due to creep o girder concrete
pES = loss o prestress during transer due to elastic
shortening o concrete
pR = loss o prestress beore transer due to relaxation
pR1 = loss o prestress between transer and deck place-
ment due to relaxation
pR2 = loss o prestress ater deck placement due to
relaxation
pSD = loss o prestress between time o deck placement
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o transer
b(t,td) = creep coecient or girder concrete at nal time t
due to loading applied at time tdo deck placement
d(t,td) = creep coecient or deck concrete at nal time t
due to loading applied at time tdo deck placement
b(t,ti) = creep coecient or girder concrete at nal time
tdue to loading applied at time o transer
and nal time due to shrinkage o girder concrete
pSR = loss o prestress between time o transer and
deck placement due to shrinkage o girder con-
crete
pSS = increase o eective prestressing orce due to
dierential shrinkage between a precast concretegirder and cast-in-place composite deck
bd = shrinkage strain o girder concrete ater deck
placement
bid = shrinkage strain o girder concrete between time
o transer or end o curing and time o deck
placement
bi = shrinkage strain o girder concrete between time
o transer or end o curing and nal time
c = strain in the concrete
cr = creep strain
dd = shrinkage strain o deck concrete between time
o deck placement or end o deck curing and nal
time
el = elastic strain
net = net shrinkage strain o section at centroid o pre-
stressing steel considering shrinkage o concrete
and restraint against shrinkage rom bondedprestressing steel
p = tensile strain in prestressing steel beore transer
p
' = tensile strain in prestressing steel ater transer,
considering eects o elastic shortening
res = portion o ree shrinkage o concrete eectively
restrained by bonded prestressing steel at centroid
o prestressing steel
sh = shrinkage strain
total = total strain
c = concrete stress
= creep coecient
(t,ti) = creep coecient or concrete at time tdue to
stresses applied at time ti
b(td,ti) = creep coecient or girder concrete at time tdo
deck placement due to loading applied at time ti
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wc = unit weight o concrete
fc' = specied compressive strength o concrete
The model or determining elastic modulus is largely
unchanged rom previous versions o AASHTO, still based
largely on density and compressive strength. The K1 actor
has been added as an adjustment to the elastic modulus oconcrete based on the stiness o the specic coarse aggre-
gate in the mixture. In the absence o data to calibrate K1, a
value o 1.0 is used. FigureA1 is a schematic o the model
used to predict elastic modulus, including a conceptual
representation o its sensitivity to key input variables.
Shrinkage of concrete
Shrinkage o concrete is a decrease in volume primarily
due to the loss o excess water over time. The AASHTO
LRFD specications model is shown in Eq. (5.4.2.3.3-1).
sh s hs f tdk k k k = ( )0 48 10 3. (AASHTO 5.4.2.3.3-1)
where
ks = adjustment actor or member size, specically vol-
umetosurace area ratio
khs = adjustment actor or average ambient relative humid-
ity in shrinkage calculations
k = adjustment actor or specied concrete compressive
strength at time o transer or end o curing
Appendix A: AASHTO
LRFD specifications
model for high-strength
concrete properties
National Cooperative Highway Research Program report
4962 recommended new material property models to better
characterize the behavior o high-strength concrete. The
models were developed empirically by testing represen-
tative concrete mixtures rom our dierent states. New
models were proposed or elastic modulus, shrinkage, and
creep. Detailed inormation related to the development o
the model is available elsewhere,2,3 so only a brie sum-
mary is provided here.
Elastic modulus
American Association o State Highway and Transporta-
tion OcialsAASHTO LRFD Bridge Design Specifca-
tions Eq. (5.4.2.4-1) is used to predict concrete elastic
modulus.
E K w fc c c= 33 000 11 5
,. ' (AASHTO 5.4.2.4-1)
where
K1 = correction actor or source o aggregate to be taken
as 1.0 unless determined by physical test and as ap-
proved by authority o jurisdiction
Figure A1. Schematic o the model or estimating the concrete elastic modulus demonstrating the eects o key variables. Note: K1 = correction actor or source oaggregate to be taken as 1.0 unless determined by physical test and as approved by authority o jurisdiction; wc= unit weight o concrete.
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ktd = adjustment actor or time development that sets the
rate at which shrinkage strain asymptotically ap-
proaches ultimate value (set equal to 1.0 when deter-
mining nal shrinkage)
The equation asymptotically approaches an ultimate
shrinkage value experimentally determined as 0.00048 or
the baseline specimen. Adjustment actors alter the ulti-
mate shrinkage or conditions that dier rom the baselinetest specimen. The time-development actor ktdsets the
rate at which the ultimate shrinkage value is approached.
Figure A2 shows a schematic o the model used to predict
shrinkage strains, including a conceptual representation o
its sensitivity to key input variables.
Creep of concrete
Creep is an increase in strain due to sustained loads on the
concrete (expressed schematically in Fig.A3).
Figure A2. Schematic o the model or estimating the shrinkage strain o concrete demonstrating the eects o key variables. Note: f'c = specied compressive
strength o concrete; RH= ambient relative humidity; V/S= ratio o volume to surace area.
Figure A3. Schematic denition o concrete creep behavior relative to its elastic response. Note: fc = stress in the concrete; c = strain in the concrete
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In the AASHTO LRFD specications method, the creep
strain due to a given stress increment is expressed in terms
o the elastic strain caused by the same stress. The ratio
o the creep strain to the elastic strain is termed the creep
coecient. The creep coecient at time tdue to a stresschange that occurs at time ti is determined by the ollowing
equation.
t t k k k k t i s hc f td i, ..( ) = 1 9 0 118 (AASHTO 5.4.2.3.2-1)
where
khc = adjustment actor or average ambient relative humid-
ity in creep coecient calculations
ti = age o concrete when load is initially applied
Similar to the shrinkage model, the equation or the creep
coecient asymptotically approaches an ultimate value
that is initially set or baseline conditions. A series o cor-
rection actors adjusts the ultimate values or conditions
other than those used in developing the baseline equation.
The creep coecient is also based largely on the age o the
concrete when the load is applied. Stress applied at a later
age will lead to smaller creep strains than the same stress
change at an earlier age. Figure A4 shows a schematic o
the model used to predict creep strains, including a concep-
tual representation o its sensitivity to key input variables.
Although the model was developed using specimens under
sustained compressive stress, AASHTO inherently as-
sumes that the same model can be used to represent time-
dependent strains ollowing a tension stress increment.
Figure A4. Schematic o the model or estimating the creep strain o concrete demonstrating the eects o key variables and dening the creep coecient relative
to the elastic strain. Note: Ec = modulus o elasticity o concrete; fc = stress in the concrete; f'c = specied compressive strength o concrete; RH= ambient relative
humidity; ti = age o girder concrete at time o prestress transer; V/S= ratio o volume to surace area; cr = creep strain; el= elastic strain; total= total strain;
(t,ti) = creep coecient or concrete at time tdue to stresses applied at time ti.
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plied to the concrete rom the prestressing steel.
res ct
c
n
c pn
n
EP
A
P e
I= +
2
(26)
where
Ect = modulus o elasticity o concrete at time tunder
consideration
epn = eccentricity o prestressing steel centroid with respect
to net concrete section
In = moment o inertia o net concrete section
Equation (26) can be solved or the eective orce Pc ap-
plied to the concrete shown in Eq. (27).
P
E A
A e
I
E Ac
res ct n
n pn
n
res ct n
n
=
+
=
1
2(27)
where
n = variable representing 1
2
+
A e
I
n pn
n
The orce Pc will accumulate gradually, so it is appropri-
ate to substitute an age-adjusted eective modulus or the
elastic modulus i creep eects are also being considered.Equation (12) denes the age-adjusted eective modulus.
ReplacingEctin Eq. (27) withEc,AAEMas dened in Eq. (12)
(Ec,AAEM is replaced withEct,AAEMandEc withEct) yields Eq. (28).
PE A
t tc
res ct n
n i
=+ ( ) { }
1 ,(28)
The restraint strain res in Eq. (28) can be represented as
the dierence between the ree shrinkage strain and the net
strain given in Eq. (29).
res sh net = (29)
Requiring equilibrium o orces, the terms in Eq. (28) and
(25) can be set equal to one another. Also, Eq. (29) will be
substituted into Eq. (28) to remove the unknown res. Solv-
ing or the ratio
net
sh
,
which is the denition o the transormed section coe-
cient Kid(Eq. [24]), produces Eq. (30).
Appendix B: Transformed
section coefficient
(derivation)
To aid understanding o the transormed section coe-
cientsK
idandK
d, theK
idequation will be derived. Thederivation oKd is similar but with respect to the com-
posite (girder plus deck) section properties rather than the
girder section properties. The development o the equation
is similar or both shrinkage and creep. For the sake o
consistency throughout the presentation, only shrinkage
will be discussed. The main ideas o this derivation are
documented by Tadros et al.2
The derivation reerences Fig. 3. The shrinkage strain dis-
tribution across the girder section is aected by the pres-
ence o bonded prestressing steel. sh is the ree shrinkage
o concrete that would exist without any internal restraint.
res denotes the reduction in shrinkage at the centroid o the
prestressing caused by the steels restraint. The net change
in strain at the centroid o the prestressing is given by net.
The transormed section coecient is the ratio o the net
strain to the ree strain expressed in Eq. (24).
Kid
net
sh
=
(24)
Three assumptions are made in developing an equation or
Kid:
The shrinkage strain is uniorm over the cross section.
The concrete would undergo a ree shrinkage sh in the
absence o prestressing steel.
Compatibility requires the same strain in the steel as in
the surrounding concrete.
Figure 3 shows that the shrinkage o the surrounding con-
crete imposes a strain neton the prestressing steel. The e-
ective compressive orce imposed on the steel by concrete
shrinkage Pp can be calculated using Eq. (25).
P A Ep p p net= (25)
For equilibrium within the cross section, an equal and
opposite orce must be applied to the concrete. Moreover,
the eective orce on the concrete causes a strain res at
the centroid o the prestressing steel. Equation (26) can
be written recognizing that the stress in the concrete at the
centroid o the prestressing must be equal to the product o
the strain res and the modulus o elasticity o the concrete.
Furthermore, the stress is caused by the combined axial
and eccentricity eects o the eective tensile orce Pc ap-
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KA
A
E
Et t
idnet
sh p
n
p
ct
n i
= =
+
+ ( ) { }
1
1 1 ,
(30)
Equation (30) closely resembles the ormulation given in
the American Association o State Highway and Transpor-
tation OcialsAASHTO LRFD Bridge Design Specifca-
tions. The ollowing assumptions are made to arrive at the
AASHTO LRFD equation:
Gross section properties are substituted or net section
properties.
The time o interest or the concrete elastic modulus is
assumed to be the time o transer. ThereoreEctwill
be replaced withEci.
The aging coecient will be assigned a constant
value o 0.7 as supported by the work o Dilger11 and
explained previously.
The time o interest in the creep coecient used in the
age-adjusted eective modulus term is assumed to be
the nal time t.
The assumptions, when substituted in Eq. (30), produce the
ollowing AASHTO equation or Kid:
KE
E
A
A
A e
I t t
id
p
ci
p
g
g pg
g
f
=
+
+
+
1
1 1 1 0 7
2
. , ii( ) { }
(AASHTO 5.9.5.4.2a-2)
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About the authors
Brian D. Swartz, PhD, PE, is an
assistant proessor o civilengineering at The University o
Hartord in West Hartord, Conn.
Andrew Scanlon, PhD, is a
proessor o civil engineering at
Pennsylvania State University in
University Park, Pa.
Andrea Schokker, PhD, PE, LEED
AP, is the executive vice chancel-
lor at the University o Minnesota
Duluth (UMD) and was the
ounding department head o the
Civil Engineering Department at
UMD in Duluth, Minn.
Abstract
This paper details the American Association o State
Highway and Transportation OcialsAASHTO
LRFD Bridge Design Specifcations time-dependent
analysis method used or determining the loss o pre-
stress in pretensioned bridge girders. The undamental
mechanics on which the method relies are explained
in detail. New concepts introduced as part o the 2005
interim revisions are claried. This paper aims to make
the loss o prestress method more widely understood
and more accurately applied in practice.
Keywords
Creep, LRFD, prestress gain, prestress loss, relaxation,
shrinkage, transormed section.
Review policy
This paper was reviewed in accordance with the
Precast/Prestressed Concrete Institutes peer-review
process.
Reader comments
Please address any reader comments to journal@pci
.org or Precast/Prestressed Concrete Institute, c/o PCI
Journal, 200 W. Adams St., Suite 2100, Chicago, IL60606. J