prestressed concrete - 7 estimation of prestress losses

8/12/2019 Prestressed Concrete - 7 Estimation of Prestress Losses

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University of Western AustraliaSchool of Civil and Resource Engineering 2004

7. Prestressed Concrete :

Estimation of prestresslosses

Introduction

Post-tensioning - immediate losses

Post-tensioning - time dependent losses

Pre-tensioning - immediate losses

Pre-tensioning - time dependent losses

Its not all lost -

just some of it, but enough to concern us!


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INTRODUCTION

Jacking and locking-off cause stresses and strains in tendons and

concrete, and these cause the tendon force to diminish; hence the termloss of prestress.

Some of the losses occur during jacking, and/or immediately upon

transfer; these losses are called immediate losses.

Other losses occur progressively with time, as the tendon and concreteage and undergo inelastic deformations; these losses are termed time-dependent losses, or deferred losses.

Individual losses are small, but when added together amount to a

significant decline in the original jacking force: typically 15% to 25%;

hence must be considered by the designer and constructor.

Important initial decisions by designers of prestressed concrete are:

adopt at least medium strength concrete, to minimise creep, and

adopt very high strength tendons, of low relaxation, to minimisepercentage loss of prestress force.


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POST-TENSIONING - IMMEDIATE LOSSES

Immediate losses are comprised of a number of separate, but

sometimes related, causes. These are due to (Note that they

do not always apply):

Elastic deformation of concrete.

(Friction in jack and anchorage - usually minor.)

Friction between tendon and duct wall.

Draw-in losses.

(Other, specific to type of construction - consider.)

Lets consider the major losses . .


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Loss due to elastic deformation of concrete :

BEFORE STRESSING:

AFTER STRESSING:

Extension of

tendon= ( s pi / E p ).L

Shortening of concrete

= ( s ci / E cj ).L

s pi is initial stress

in tendon, ands ci is induced

compressive stress

in concrete.

Note that

Pi = s ci A c= s pi Ap

Ecj is elastic

modulus of

concrete at time of

stressing

There is an important point to note about this . . .

Post-tensioning

Immediate loss:


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We must be sure that the required prestress force is applied. Thisis so important that two separate methods of measuring the force must be

adopted and checked against one another. The methods are

Observe Po from thegaugeon the jack (which must be recentlycalibrated to + 3%), and then

Measure the extension of the tendon, ensuring that a correction is appliedfor the contraction of the concrete, and from this calculate the prestress

force Po.

The forces Po must agree within 10%. Otherwise, we must search for apossible problem, and fix it!

If this is done properly for a single tendon, then there is no loss of

prestress to be accounted for. . . .

. . . But not so for multiple tendons, e.g. slab stressing - itmay be necessary to re-stress, ensuring that all strands are

stressed to the correct force.

Consider this example . . .


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Consider a beam with two tendons, 1 and 2.

Suppose we stress tendon 1 first. The concrete shortens, but we continue

stressing until tendon 1 is at required force.

Now stress tendon 2. The concrete shortens further, so tendon 1 also shortens . .

. . . and loses some of its force! WHOOPS ! !

tendon 1

tendon 2

e1

e2Elastic loss in tendon 1 due to prestress P2 applied to

tendon 2

DP21 = P2 [1/A + e1 e2 / I] Ep / Ecj Ap1

Two options:

Sequential stressing (chasing the tail), or

Overstress tendon 1 by DP21, if possible and safe.


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When a draped tendon is stressed, it bears hard against the duct wall.

As stressing proceeds, the tendon stretches and slides along the duct wall:

LIVE

END

DEAD

END

Lpa

Friction resists this sliding, so the jacking force diminishes towards the dead end.

The diminished force at any position is given by P = P0 e-mq where

m is the coefficient of limiting friction between tendon and duct.

q is the sum of: the total angle of the tendon change a totbetween the subject positionand that at the jack, and

a wobble angle bp.Lpa, allowing for constructional imperfections.So P = P

0

e -m( atot + bp Lpa )

Loss due to friction between tendon and duct wall :Post-tensioning

Immediate loss:


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What is a tot ?

a tot is the total change of angle

between a point at which we

know the force (e.g. at jacking

end) and the point in which we

are interested.

This example shows a tot from theleft hand (live) end, to mid-span in a

simply supported beam.

This is a tot .

a tot

a tot is clearly q 0 -q L/2

q 0 = e0

q L/2 = 0

a tot = e0 - 0 = e0

Post-tensioning

Immediate loss:

So how do we selectmandbp? . . .


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(Approximate only - consult AS3600 and trade literaturefor each case.)

m = 0.20 for galvanised spun duct, and0.14 for polyethelene duct.

bp = 0.015 to 0.025 rads/m

Post-tensioning

Immediate loss:Selection of the coefficients and p:


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After stressing,

before transfer:Bearing plate

Duct

Strands

Loss due to draw-in of tendon: Diagrammatic only

Anchor head

Permanentwedges,

tightlydriven

After transfer:

Draw-in

length dx

So there is some loss of force in the tendon . . .


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. . but the shortening of the tendon is impeded

by reverse friction on the duct wall, so . . .

tendon force

distance fromlive end

Distance x over which draw-in dxmodifies the tendon force.

Modified tendon

force

For a given dx, x can be calculatedfrom:x = { ( Epdx) / (spj K) }0.5

In this formulation, we use the rate

of change of the tendon force just asfor duct friction :

K = m ( a tot + bp Lpa) / Lpa

The loss due to draw-in often doesnot affect the force at mid-span,

except for short span members.

Post-tensioning

Immediate loss:


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So how can we estimate the effect of theseimmediate losses on the tendon force over

the entire beam ?

The easiest method is graphical . . . .


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After jacking but prior to transfer:

force in tendon

length along member0 L

P0

jacking force prior

to transfer

loss of force overfull length of memberdue to duct friction

NOTE: Applies to a parabolically draped tendon in post-tensioned design. For other draping conditions, force

declines towards dead end, but not uniformly with length.

JACK

END

DEAD

END


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Immediately after transfer (Initial prestress) :

Force in tendon

Length along member0 L

P0

Pi (0) Pi (L)

Loss (if any) due to

elastic shortening of

concrete

+ loss (if any)at anchorage.

plus additional loss

(if any) due to draw-in at anchorage.

NOTE: Pi diminishes from the live to the dead end.

Usually our interest is in the mid-span, or mid-spans for

continuous beams or slabs, and dead end .


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POST-TENSIONING -TIME DEPENDENT LOSSES

With the passing of time, and influenced by environmentalfactors, the prestresss force diminishes further. The losses are

additive to those which occur at stressing and transfer. The

separate, but inter-related causes are:

Losses due to shrinkage of concrete.

Losses due to creep of concrete.

Losses due to relaxation of tendon.

We now consider these separately, and their relationshipto one another . . .


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Loss due to shrinkage of concrete :

Concrete shrinks with time, dependent on:

chemical process of hydration.

hypothetical thickness of section th.

moisture changes during the entire life of structure.

restraint offered during hydration and later.

shrinkage strain ecs

time

ecs ( )

DRY ENVIRONMENT

MOISTER

ENVIRONMENT


time

ecs ( )8

Longitudinal rebar (if any) reduces the shrinkage, and so ecs is modified :

ecs

= ecs

(from above) . 1/ ( 1 + 15 As

/ Ag

) . . . . . .


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The tendon(s) in a prestressed beam shorten as the beam shrinks,

and so the prestress force declines. It is not the total shrinkage,

but that which occurs after the time of prestressing T0 , which

concerns us :

time

ecs ( )

ecs (T0 )

T0 T

ecs (T ) Shrinkage which causes loss

of prestress to time T= e cs (T) -ecs (T0)= age atprestressing

Loss of prestress due to shrinkage is given by :

sp (shrinkage) = Ep . [ e cs (T) -ecs (T0) ]where ecs has been modified to allow for long. rebar, if any.


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Loss due to creep of concrete :

Concrete loaded in compression creeps with time, dependent on:

chemical process of hydration.

hypothetical thickness of section th.

moisture changes during entire life of structure.

intensity of prestress, and its age of application T0.

How do we account for the intensity of stress ? . . .


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sustained stress sc

total strain ec

0.5f climit of validity

Ecj

sc

= stress on concrete

ec = strain of concrete

Ecj = elastic modulus of

concrete at time j after

casting - this is typically

less than Ec, which is at28 days.

The elastic strain is easily estimated as sc / Ecj. But how do we estimatethe creep strain, which is additional to the elastic strain ?

Concrete under sustained stress :

elastic strain

total strain at time T

total strain

at infinite time


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limit of validity0.5f c

total strain ec

sustained stress sc

Ecj

Creep strain at time T is proportional to immediate elastic strain :

ecc (T) = fcc (T) . sc / Ecj

Creep Factor cc(T) :


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fcc can be estimated from AS3600 : Loss of prestress due to creep is then

sp(creep) = Ep . ecc in which Ep = elastic modulus of tendon

ecc = fcc sci / Ec

sci = stress on concrete, under prestress and

sustained loading, at the level of the tendon.


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Loss due to Tendon Relaxation:

Under sustained tensile strain, any metallic member relaxes, i.e. loses some of

its load due to creep. For prestressing wire, strand and bars, relaxation is

measured in a standard manner, and adjustments are then made for the realdesign condition. Diagrammatically, the test is (strand shown):

strand1. Apply

0.7fp

2. Measure this, and maintain by adjusting

force, for 1000 hours.

Initial stress = 0.7 fp

Stress after 1000 hours = 0.7 fp - x

Basic relaxation Rb = x / (0.7fp), expressed as % age.

Design relaxation R modifies Rb thus . . . . .

k


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where

k4 is duration factor

k5 is maturity factor

k6 is temperature factor.

R = k4.k5.k6.Rb

So loss of prestress

sp (relaxation) =R/100 spi

0

1

2

0.4 0.5 0.6 0.7 0.8 spi / fp

maximum

permissible

value of spi / fp

10 20 30 40

Annual average

temperature (oC)

1

2k6

0.6

1.0

1.4

1 10 100 1000 10000Time (days)

k4

k5


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So shrinkage and creep of concrete, together with relaxation

of tendon steel, cause long term (deferred) loss of prestress.

Their effects are inter-active. For example, shrinkage and

creep of concrete reduce the prestress force, and thereby the

loss due to tendon relaxation. This can be accounted for by

a modification factor applied to the relaxation loss thus :

% age loss due to relaxation

= R [ 1 - (loss of stress due to shrinkage and creep)/spi) ]The total losses due to deferred effects are then applied overthe entire length of the beam, and summarised graphically

thus . . . .


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After long period of time (Effective prestress):

Force in tendon

0 L

P0

Pi (0) Pi (L)

Pe(0) Pe(L)Combined losses due

to shrinkage and

creep of concrete,and relaxation of

tendon.Length along beam

So the time dependent (deferred) losses have a constant

effect along the length of the beam, AND

we must be concerned with the mid-spans for bending, and

support points, especially dead end, for shear.

L


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PRE-TENSIONING - IMMEDIATE LOSSES

Immediate losses are comprised of

Elastic deformation of concrete - always!

Friction in jack and anchorage.

Other - consider.

Elastic deformation of concrete

It is common to express this problem thus:

The jacking force Po required to achieve initial force Pi is:

Po = Pi [ 1 + (1/A + e2/I) (Ep/Ec) Ap ]


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PRE-TENSIONING - TIME-DEPENDENT LOSSES

Simple - the same as for post-tensioning time-dependent losses:

Shrinkage.

Creep.

Tendon relaxation.But note that pre-tensioning usually

occurs in the very early life of the member.

So fcp, ft, and Ecj are small.

To improve these properties at transfer, it

is common to use either or both of:

High early strength cement,

Steam curing.

See the literature for these topics.

Lets try to summarise all this . . .


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Cause of loss Pre-tensioningPost-tensioning

IMMEDIATE:

LONG-TERM:

Concrete shrinkage

Concrete creep

Tendon relaxation

Elastic deformation

of concrete

One tendon: No

More than one: Yes

Friction in jack

or anchorageFriction in duct

Draw-in

Other

Not if properly done

Yes

Consider

Consider

Yes

Yes

Yes

Yes

Not if properly done -

care at cradles !No

No

Consider

Yes

Yes

Yes

Prestress Losses - Summary


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SUMMARY

Losses always occur, and must be estimated.

Immediate losses occur during jacking and/or transfer.

Long-term losses occur progressively with time.

Each causal factor causes small loss, but sum of theselosses is significant.

Rational methods for estimating losses exist, e.g. in

Section 6 of AS3600 - 2001, which provides guidance

on relevant parameters. With careful planning, prestress losses can be accounted

for, and minimised.

prestressed concrete - 7 estimation of prestress losses

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