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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
shrinkage strain ecs
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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shrinkage strain ecs
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.