1 PRESTRESS LOSSES Reading: Nawy, E.G., Prestressed Concrete – A Fundamental Approach, 3 rd Edition, Chapter 3. SOURCES AND COMPUTATION OF LOSSES There are essentially two types of prestress losses that can take place in prestressed concrete members: __________________________________________ and ___________________ __________________________. These two types of losses can be described in the following. Immediate Losses : These losses depend upon the type of member: pretrensioned or post-tensioned. In a pretensioned member, an immediate loss is that due to ______________________________ of the member. Immediate losses in a post-tensioned member are those due to __________________ and ___________________________________. Post-tensioned members can also be subjected to elastic shortening losses when _______________________ ____________________ is used. Time-Dependent Losses : The losses that depend upon elapsed time after stressing are independent of the member type. These losses are: ______________________________________________ ______________________________________________ _______________________________. There are two methods that can be used to estimate losses in prestressed concrete: (a) lump sum approximations; and (b) refined estimations. One should keep in mind that all estimates for prestress loss are just that – ESTIMATIONS. As we get into the details of the “refined” estimations, be aware of all the assumed behavior that exists in the estimation. Prior to ACI 318-83, lump sum loss calculations were allowed. However, today’s Code deems lump sum estimates obsolete.
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Prestress Losses in wire Strands for Prestressed Concrete
There are two methods that can be used to estimate losses in prestressed concrete: (a) lump sum approximations; and (b) refined estimations. One should keep in mind that all estimates for prestress loss are just that – ESTIMATIONS. As we get into the details of the “refined” estimations, be aware of all the assumed behavior that exists in the estimation.
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PRESTRESS LOSSES Reading: Nawy, E.G., Prestressed Concrete – A Fundamental Approach, 3rd Edition, Chapter 3. SOURCES AND COMPUTATION OF LOSSES
There are essentially two types of prestress losses that can take place in prestressed concrete
members: __________________________________________ and ___________________
__________________________. These two types of losses can be described in the following.
Immediate Losses:
These losses depend upon the type of member: pretrensioned or post-tensioned. In a
pretensioned member, an immediate loss is that due to ______________________________
of the member. Immediate losses in a post-tensioned member are those due to
__________________ and ___________________________________. Post-tensioned
members can also be subjected to elastic shortening losses when _______________________
____________________ is used.
Time-Dependent Losses:
The losses that depend upon elapsed time after stressing are independent of the member type.
These losses are:
______________________________________________
______________________________________________
_______________________________.
There are two methods that can be used to estimate losses in prestressed concrete: (a) lump sum
approximations; and (b) refined estimations. One should keep in mind that all estimates for
prestress loss are just that – ESTIMATIONS. As we get into the details of the “refined” estimations,
be aware of all the assumed behavior that exists in the estimation. Prior to ACI 318-83, lump sum
loss calculations were allowed. However, today’s Code deems lump sum estimates obsolete.
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Prestress losses are most conveniently broken down into components. We will address loss
calculations based upon the member type being considered. The prestress loss can be determined
using the following “formulas” for pretensioned members:
( )( )
( )
0
0
, before transfer
, after transfer
, initial prestress
pT pES pR tr
pR tr s pCR pSH
pi pJ pR tr pES
f f f t t
f t t f f
f f f t t f
∆ = ∆ + ∆
+ ∆ + ∆ + ∆
= −∆ −∆
The following can be used for post-tensioned members:
( )
at jacking
at transfer
, after transfer
initial prestress
pT pF pES
pA
pR tr s pCR pSH
pi pJ pA pF
f f f
f
f t t f f
f f f f
∆ = ∆ + ∆
+ ∆
+ ∆ + ∆ + ∆
= −∆ −∆
The subscripts and times are defined below:
0t = time at jacking;
trt = time at transfer of prestressing force;
st = time at stabilization of losses (i.e. during the service loading stage);
j = jacking;
R = relaxation;
ES = elastic shortening
A = anchorage;
F = friction;
CR = creep;
SH = shrinkage.
The AASHTO-LRFD Specifications allow lump-sum estimates for prestressing losses with the caveat
that the following conditions are met.
1. Members that are post-tensioned must be non-segmental members with spans less than 160 feet and concrete stressed an age of 10-30 days.
2. Members that are pretensioned must be stressed at an age where the concrete strength is no less than 3,500-psi.
3. Members must be made from normal weight concrete.
4. Members cannot be steam-cured, nor moist-cured.
5. The prestressing steel must be normal or low-relaxation.
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6. There must be average exposure conditions at the site.
If these conditions are met, there is a simple table (Table 1 shown below) that can be used for loss
calculations. It should be noted that the table defined PPR as _____________________________
______________, which is basically the ratio prestressed reinforcement to total reinforcement within
the cross-section.
Table 1: AASHTO Lump Sum Approximations.
ELASTIC SHORTENING The loss due to elastic shortening is based upon mechanics of materials approaches. We should all
appreciate that the strain lost due to elastic shortening deformations can be computed using,
ESES Lε ∆
= (1)
Therefore, if we can compute the member deformation due to elastic shortening, ES∆ , we can
determine the strain lost resulting from elastic shortening. Losses due to elastic shortening are
different when pretensioned and post-tensioned members are considered.
Pretensioned Members:
When the member is pretensioned, the computation of loss is straight-forward,
i s ipES s ES cs
c c c
PE nPf E nfA E A
ε∆ = = = = (2)
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where: csf is the stress in the concrete at the level of the prestressing steel; and n is the modular
ratio. Of course, this will vary depending upon the location of the tendon centroid within the
cross-section. The initial prestressing force, iP , that will cause elastic shortening is a little
difficult to estimate if the jacking force, loss due to friction, and loss due to seating are not
known. Therfore, Nawy (1999) has suggested that 90% of the initial prestressing force given be
used.
Post-Tensioned Members:
In the case of post-tensioned members, the computation is a little more difficult. The reason for
this is that when a post-tensioned member is considered, one can jack tendons in sequence rather
than jacking them all at once. The loss due to elastic shortening in this case can be computed as,
( )1
1 N
pES pES jj
f fN =
∆ = ⋅ ∆∑ (3)
where: N is the number of tendons (or groups/pairs) sequentially jacked. Use of equation (3) is
best illustrated via example. It should be noted that the last tendon or group of tendons to be
stressed suffers no elastic shortening, while the first tendon or group of tendons suffers the
highest losses.
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EXAMPLE 1 – COMPUTATION OF ELASTIC SHORTENING LOSSES
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EXAMPLE 2 – COMPUTATION OF ELASTIC SHORTENING LOSS
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CREEP
As in the case of elastic shortening, the loss due to creep all begins with mechanics of materials.
Recall our rheological model for the time-varying loss due to creep. The basic mechanics of
materials approach to creep loss takes the following form,
0.6
0.610
pspCR t cs
c
psu cs
c
Ef C f
EEt C f
t E
∆ = ⋅ ⋅
= ⋅ ⋅ ⋅ +
(4)
where:
uC = the ultimate creep coefficient (usually 2.35 often used); t = the time (in days); cE = the elastic modulus of the concrete; psE = the elastic modulus of the prestressing steel; csf = the compressive stress in the concrete at the level of the prestressing steel centroid.
Creep loss is generally a function of the location along the member where the compressive stress is
analyzed. This results from the tendon centroid (in general) varying along the length of the concrete
member. The average concrete stress between anchorage points can be used for post-tensioned
members. In a prestensioned member, the average along the member length can be used.
There seems to be many procedural recommendations for computing creep in prestressed concrete
members. The first we will consider is ACI Committee 423. This committee’s recommendation is
given below,
( )pspCR CR cs csd
c
Ef K f f
E∆ = ⋅ ⋅ − (5)
where:
CRK = is a creep coefficient (reduce by 20% for lightweight concrete) = 2.00 for pretensioned members = 1.60 for post-tensioned members csf = the stress in the concrete at the level of the prestressing steel centroid immediately
after transfer, csdf = the stress in the concrete at the level of the prestressing steel due to all
superimposed dead loads applied after transfer.
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A second recommendation for computation of losses comes from the AASHTO LRFD
Specifications. This computation is slightly simpler than equation (5), but more complicated in other
respects. The loss due to creep according to AASHTO-LRFD is,
12.0 7.0 0.0pCR cgp cdpf f f∆ = ⋅ − ⋅∆ ≥ (6)
where:
cgpf = is the stress a the center of gravity of the prestressing steel centroid at transfer (ksi);
cdpf∆ = the change in concrete stress at the center of gravity of the prestressing steel due to permanent loads (with the exception of the load acting at the time the prestressing steel is applied). Values should be calculated a the same section (or sections) at which cgpf is computed (ksi).
SHRINKAGE Recalling our discussion of the factors that affect shrinkage, any relationship used for shrinkage loss
estimation should include consideration of ________________________________, _____________
__________________________, and member ________________________________. It is
assumed that shrinkage begins at the end of the curing period (e.g. 7-days). If one would like to
compute the shrinkage strain that occurs from 28-days to 1-year, a subtraction procedure should be
employed.
The loss of prestress resulting from shrinkage strain can be computed using mechanics of materials
relationships,
pSH SH psf Eε∆ =
where: SHε is the shrinkage strain. ACI Committee 209 suggests the following computation for the
shrinkage strain at any time, t,
( )6, 780 10SH t SH
tt
ε γα
− = ⋅ × ⋅ + (7)
where:
t = is the time in days, α = 35 if moist cured for 7-days, = 55 is steam cured for 1-3 days, SHγ = is a correction factor that accounts for conditions other than standard conditions.
The correction factor accounts for relative humidity, volume-to-surface ratio, concrete composition, etc.
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The general form of the ACI Committee 423 recommendation for prestress losses due to shrinkage
takes the following form,
( ) ( )68.2 10 1 0.06 100pSH SH psVf K RH ES
− ∆ = × ⋅ ⋅ − ⋅ − ⋅ (8)
where: SHK is 1.0 for pretensioned members and is taken from Table 2 for post-tensioned members.
Table 2: Shrinkage Factor for Post-Tensioned Members
As one might expect, there are also AASHTO-LRFD recommendations for prestress loss
computation. These are also broken down into pretensioned and post-tensioned members. For
pretensioned members,
17.0 0.150pSHf H ksi∆ = − (9)
and for post-tensioned members,
13.5 0.123pSHf H ksi∆ = − (10)
where: H is the relative humidity (%) obtained from local statistics or a map.
A relative humidity map can be found in the PCI Design Handbook and the AASHTO – LRFD
Specifications. Such a map is shown in Figure 1.
STEEL RELAXATION An empirical relationship for steel relaxation loss can be developed using mechanics of materials.
As is usually the case, we will also look at ACI Committee 423 and AASHTO-LRFD
recommendations. If we know the initial prestress, pif ′ , and an empirical relationship describing the
relaxation over time, we can write the loss is prestress as,
2 1log log 0.55pipR pi
py
ft tf ffα
′− ′∆ = ⋅ ⋅ − (11)
where:
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α = 10 for stress-relieved strands, = 45 for low-relaxation strands, 1t = initial time (hours) for interval under consideration, 2t = final time (hours) for interval under consideration, pif ′ = initial stress in prestressing steel at the beginning of the interval considered, pyf = yield stress of the prestressing steel.
It should be noted that the initial prestress to yield stress should be greater than 0.55.
Figure 1: Annual Average Ambient Relative Humidities (AASHTO 2001).
The ACI Committee 423 recommendation includes losses due to other sources. The loss in prestress
resulting from relaxation using ACI 423 recommendations is,
( )pR RE pES pCR pSHf K J f f f C ∆ = − ⋅ ∆ + ∆ + ∆ ⋅ (12)
where the loss due to elastic shortening, creep, and shrinkage should be computed using previous
ACI 423 recommendations for these losses. The relaxation loss constants, and REK J are taken from
Table 3 and the constant C is taken from Table 4.
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Table 3: Relaxation Loss Constants, and REK J Used in ACI 423 Recommendation.
Table 4: Relaxation Constant, C Used in ACI 423 Recommendation.
The AASHTO – LRFD Specifications also contain a recommendation that is a little bit simpler than
that implied in equation (12). The procedure is a two-level procedure where relaxation loss is
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computed during two stages: (a) at transfer and (b) after transfer. The loss that occurs before
transfer of prestress is computed using,
( )1
log 240.55pJ
pR pJpy
ftf f
fα ⋅
∆ = ⋅ − ⋅
(13)
where: t is the time (days) from initial stressing to transfer; pJf is the stress in the tendons at the end
of the jacking sequence; α is a constant which is 10 for stress-relieved strands and 40 for low-
relaxation strands. The loss that occurs after transfer for stress-relieved strands and pretensioned
members is computed using,
( )2 20 0.4
0.2pR pES
pSH pCR
f f
f f
∆ = − ∆
− ⋅ ∆ + ∆ (14)
and the loss after transfer for stress-relieved strands and post-tensioned members is computed
using,
( )2 20 0.3 0.4
0.2pR pF pES
pSH pCR
f f f
f f
∆ = − ∆ − ∆
− ⋅ ∆ + ∆ (15)
If low-relaxation strands are used, the loss after transfer can be taken as 30% of the values obtained
using equations (14) and (15).
FRICTION As a prestressing tendon is pulled, its lengthening will be resisted by frictional forces along the
tendon. This is especially important in post-tensioned members. In general, loss due to friction is