AASHTO LRFD Bridge Design Specifications provisions for loss of prestress

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


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



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.



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



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.



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.



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



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.



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.



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.



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).



(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.



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.



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)



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.



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



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



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



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



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.



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



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.



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-



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)


25/25

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

Documents

AASHTO LRFD Bridge Design Specifications provisions for loss of prestress