Prestress Loss
Introduction
In prestressed concrete applications, most important variable is the
prestress.
Prestress does not remain constant (reduces) with time.
Even during prestressing of tendons, and transfer of prestress,
there is a drop of prestress from the initially applied stress.
Reduction of prestress is nothing but the loss in prestress.
Early attempts to produce prestressed concrete was not successful due to loss of
prestress transferred to concrete after few years.
Prestress loss is nothing but the reduction of initial applied prestress to an
effective value.
In other words, loss in prestress is the difference between initial prestress and
the effective prestress that remains in a member.
Loss of prestress is a great concern since it affects the strength of member and
also significantly affects the members serviceability including Stresses in
Concrete, Cracking, Camber and Deflection.
Prestress Loss
Loss of prestress is classified into two types:
1. Short-Term or Immediate Losses
immediate losses occur during prestressing of tendons, and
transfer of prestress to concrete member.
2. Long-Term or Time Dependent Losses
Time dependent losses occur during service life of structure.
1. Immediate Losses include
i. Elastic Shortening of Concrete
ii. Slip at anchorages immediately after prestressing and
iii. Friction between tendon and tendon duct, and wobble Effect
2. Time Dependent Losses include
i. Creep and Shrinkage of concrete and
ii. Relaxation of prestressing steel
Prestress Losses
ImmediateTime
Dependent
Relaxation
Anchorage
Slip
Elastic
ShorteningFriction
ShrinkageCreep
Losses in Various Prestressing Systems
Type of Loss Pre-tensioning Post-tensioning
1. Elastic Shortening Yes
i. No, if all the cables are
simultaneously tensioned.
ii. If the wires are tensioned in
stages loss will exist.
2. Anchorage Slip No Yes
3. Friction Loss No Yes
4. Creep and Shrinkage
of ConcreteYes Yes
5. Relaxation of Steel Yes Yes
Immediate Losses
Elastic Shortening of Concrete
In pre-tensioned concrete, when the prestress is transferred to
concrete, the member shortens and the prestressing steel also
shortens in it. Hence there is a loss of prestress.
In case of post-tensioning, if all the cables are tensioned
simultaneously there is no loss since the applied stress is recorded
after the elastic shortening has completely occurred.
If the cables are tensioned sequentially, there is loss in a tendon
during subsequent stretching of other tendons.
Loss of prestress mainly depends on modular ratio and average
stress in concrete at the level of steel.
Loss due to elastic shortening is quantified by drop in prestress
(fs) in a tendon due to change in strain in tendon (s).
The change in strain in tendon is equal to the strain in concrete
(c) at the level of tendon due to prestressing force.
This assumption is due to strain compatibility between concrete
and steel.
Strain in concrete at the level of tendon is calculated from the
stress in concrete (fc) at the same level due to prestressing force.
Strain compatibility
Loss due to elastic shortening is quantified by the drop in
prestress (fs) in a tendon due to change in strain in tendon
(s).
Change in strain in tendon is equal to strain in concrete (c) at
the level of tendon due to prestressing force, which is called
strain compatibility between concrete and steel.
Strain in concrete at the level of tendon is calculated from the
stress in concrete (fc) at the same level due to the prestressing
force.
A linear elastic relationship is used to calculate the strain from
the stress.
Elastic Shortening
1. Pre-tensioned Members: When the tendons are cut and
the prestressing force is transferred to the member,
concrete undergoes immediate shortening due to
prestress.
2. Tendon also shortens by same amount, which leads to
the loss of prestress.
Elastic Shortening
1. Post-tensioned Members: If there is only one tendon,
there is no loss because the applied prestress is recorded
after the elastic shortening of the member.
2. For more than one tendon, if the tendons are stretched
sequentially, there is loss in a tendon during subsequent
stretching of the other tendons.
Pre-tensioned Members: operation of pre-tensioning through
various stages by animation.
Pre-tensioning of a member
Prestressing bed
Elastic Shortening
Casting bed
Duct
jackAnchorage
Post-tensioning of a member
Post-tensioned Members: complete operation of post-tensioningthrough various stages by animation
Elastic Shortening
Linear elastic relationship is used to calculate the strain from the
stress.
Quantification of the losses is explained below.
fs= Ess
= Esc
= Es(fc/Ec)
fs= nfc
For simplicity, the loss in all the tendons can be calculated based
on the stress in concrete at the level of CGS.
This simplification cannot be used when tendons are stretched
sequentially in a post-tensioned member.
In most Post-tensioning systems when the tendon force is
transferred from the jack to the anchoring ends, the friction
wedges slip over a small distance.
Anchorage block also moves before it settles on concrete.
Loss of prestress is due to the consequent reduction in the
length of the tendon.
Certain quantity of prestress is released due to this slip of wire
through the anchorages.
Amount of slip depends on type of wedge and stress in the wire.
Anchorage Slip
The magnitude of slip can be known from the tests or from the
patents of the anchorage system.
Loss of stress is caused by a definite total amount of
shortening.
Percentage loss is higher for shorter members.
Due to setting of anchorage block, as the tendon shortens,
there develops a reverse friction.
Effect of anchorage slip is present up to a certain length,
called the setting length lset.
Anchorage loss can be accounted for at the site by over-
extending the tendon during prestressing operation by the
amount of draw-in before anchoring.
Loss of prestress due to slip can be calculated:
s
, = Slip of anchorage
L= Length of cable
A= Cross-sectional area of the cable
E = Modulus of Elasticity of steel
P = Prestressing Force in the cab
sP E
A L
where
le.
Frictional Loss
In Post-tensioned members, tendons are housed in ducts or
sheaths.
If the profile of cable is linear, the loss will be due to
straightening or stretching of the cables called Wobble Effect.
If the profile is curved, there will be loss in stress due to friction
between tendon and the duct or between the tendons themselves.
A typical continuous post-tensioned member
(Courtesy: VSL International Ltd.)
Friction
Post-tensioned Members
Friction is generated due to curvature of tendon, and vertical
component of the prestressing force.
5
Variation of prestressing force after stretching
Px
Friction
Post-tensioned Members
P0
The magnitude of prestressing force, Px at any distance, x from
the tensioning end follows an exponential function of the type,
o
, P = Prestressing force at the jacking end
= Coeficient of friction between cable and the duct
umulative angle in radian throug
kxx oP P e
where
C
h which
the tangent to the cable profile has turned
between any two points under consideration
k = Friction coefficient
Creep of Concrete
Time-dependent increase of deformation under sustained load.
Due to creep, the prestress in tendons decreases with time.
Factors affecting creep and shrinkage of concrete
Age
Applied Stress level
Density of concrete
Cement Content in concrete
Water-Cement Ratio
Relative Humidity and
Temperature
Time Dependent Losses
For stress in concrete less than one-third of the characteristic
strength, the ultimate creep strain (cr,ult) is found to be
proportional to the elastic strain (el).
The ratio of the ultimate creep strain to the elastic strain is
defined as the ultimate creep coefficient or simply creep
coefficient, Cc.
Cc =cr ultel
The loss in prestress (fp ) due to creep is given as follows.
fs = Es cr, ult =Es Cc el
Since cr,ult = Cc el
Es is the modulus of the prestressing steel
Curing the concrete adequately and delaying the application of
load provide long-term benefits with regards to durability, loss of
prestress and deflection.
In special situations detailed calculations may be necessary to
monitor creep strain with time.
Specialized literature or standard codes can provide guidelines
for such calculations.
Following are applicable for calculating the loss of prestress
due to creep.
Creep is due to sustained (permanent) loads only. Temporary
loads are not considered in calculation of creep.
Since the prestress may vary along the length of the member,
an average value of the prestress is considered.
Prestress changes due to creep, which is related to the
instantaneous prestress.
To consider this interaction, the calculation of creep can be
iterated over small time steps.
Shrinkage of Concrete
Time-dependent strain measured in an unloaded and
unrestrained specimen at constant temperature.
Loss of prestress (fs ) due to shrinkage is as follows.
fs = Es sh
where Es is the modulus of prestressing steel.
The factors responsible for creep of concrete will have influence
on shrinkage of concrete also except the loading conditions.
The approximate value of shrinkage strain for design shall be
assumed as follows:
For pre-tensioning = 0.0003
For post-tensioning =
Where t = age of concrete at transfer in days.
10
0.002
( 2)Log t
Relaxation
Relaxation is the reduction in stress with time at constant
strain.
decrease in the stress is due to the fact that some of the
initial elastic strain is transformed in to inelastic strain
under constant strain.
stress decreases according to the remaining elastic strain.
Factors effecting Relaxation :
Time
Initial stress
Temperature and
Type of steel.
Relaxation loss can be calculated according to the any code.
Losses in Prestress
Notation
Geometric Properties
1. Commonly used Notations in prestressed member are
Ac = Area of concrete section
= Net area of concrete excluding the area of prestressing steel.
As = Area of prestressing steel = Total area of tendons.
A = Area of prestressed member
= Gross area of prestressed member = Ac + As
At = Transformed area of prestressed member
= Area of member when steel area is replaced by an equivalent area
of concrete = Ac + nAs = A + (n 1)As
Here,
n = the modular ratio = Es/Ec
Ec = short-term elastic modulus of concrete
Es = elastic modulus of steel.
Areas for prestressed members
CGC, CGS and eccentricity of typical prestressed members
CGC = Centroid of concrete = Centroid of gravity of section, may lie outside concrete
CGS = Centroid of prestressing steel = Centroid of the tendons.
CGS may lie outside the tendons or the concrete
I = Moment of inertia of PC member = Second moment of area of gross section about
CGC.
It = Moment of inertia of transformed section = Second moment of area of the
transformed section about the centroid of the transformed section.
e = Eccentricity of CGS with respect to CGC = Vertical distance between CGC and
CGS. If CGS lies below CGC, e will be considered positive and vice versa
Load Variables
Pi = Initial prestressing force = force applied to tendons by jack.
P0= Prestressing force after immediate losses = Reduced value of prestressing force
after elastic shortening, anchorage slip and loss due to friction.
Pe = Effective prestressing force after time-dependent losses = Final prestressing
force after the occurrence of creep, shrinkage and relaxation.
Strain compatibility
Loss due to elastic shortening is quantified by the drop in prestress (fs) in a
tendon due to change in strain in tendon (s).
Change in strain in tendon is equal to strain in concrete (c) at the level of
tendon due to prestressing force, which is called strain compatibility between
concrete and steel.
Strain in concrete at the level of tendon is calculated from the stress in
concrete (fc) at the same level due to the prestressing force.
A linear elastic relationship is used to calculate the strain from the stress.
The quantification of the losses is explained below
fs = Es s
= Es c
= EsfcEc
= n
For simplicity, the loss in all the tendons can be calculated based
on the stress in concrete at the level of CGS.
This simplification cannot be used when tendons are stretched
sequentially in a post-tensioned member.
Pre-tensioned Axial Members
Original length of member at transfer of prestress
Length after elastic shortening
Pi
P0
Elastic Shortening
Elastic shortening of a pre-tensioned axial member
The stress in concrete due to prestressing force after immediate
losses (P0/Ac) can be equated to the stress in transformed section
due to the initial prestress (Pi /At).
The transformed area At of the prestressed member can be
approximated to the gross area A.
The strain in concrete due to elastic shortening (c) is the
difference between the initial strain in steel (si) and the residual
strain in steel (s0).
Elastic Shortening
Pre-tensioned Axial Members
Elastic Shortening
Pi
P0
Length of tendon before stretching
si
s0 c
Elastic shortening of a pre-tensioned axial member 25
The following equation relates the strain variables.
c = si - s0
The strains can be expressed in terms of the prestressing forces.
c =
si =
s0 =
Substituting the expressions of the strains
=
(
+
) =
(
+
) =
(
+ ) = or
=
+or
=
Thus, the stress in concrete due to the prestressing force after
immediate losses (P0/Ac) can be equated to the stress in the
transformed section due to the initial prestress (Pi /At).
Problem
1. A prestressed concrete sleeper produced by pre-tensioning
method has a rectangular cross-section of 300mm 250 mm
(b h). It is prestressed with 9 numbers of straight 7mm
diameter wires at 0.8 times the ultimate strength of 1570
MPa. Estimate the percentage loss of stress due to elastic
shortening of concrete. Consider n = 6.
Solution
a)Approximate solution considering gross section
The sectional properties are.
Area of a single wire, Aw = /4 72 = 38.48 mm2
Area of total prestressing steel, As = 9 38.48 = 346.32 mm2
Area of concrete section, Ac = 300 250 = 75 103 mm2
Moment of inertia of section, I = 300 2503/12 = 3.91 108 mm4
Distance of centroid of steel area (CGS) from the soffit,
Prestressing force, Pi = 0.8 1570 346.32 N = 435 kN
Eccentricity of prestressing force, e = (250/2) 115.5 = 9.5 mm
The stress diagrams due to Pi are shown.
Since the wires are distributed above and below the CGC, the
losses are calculated for the top and bottom wires separately.
Stress at level of top wires (y = yt = 125 40)
Stress at level of bottom wires (y = yb = 125 40),
Loss of prestress in top wires = nfcAs (in terms of force)= 6 4.9 (4 38.48)
= 4525.25 N
Loss of prestress in bottom wires = 6 6.7 (5 38.48)
= 7734.48 N
Total loss of prestress = 4525 + 7735
= 12259.73 N 12.3 kN
Percentage loss = (12.3 / 435) 100% = 2.83%
b) Accurate solution considering transformed section.
Transformed area of top steel,
A1 = (6 1) 4 38.48 = 769.6 mm2
Transformed area of bottom steel,
A2 = (6 1) 5 38.48 = 962.0 mm2
Total area of transformed section,
AT = A + A1 + A2 = 75000.0 + 769.6 + 962.0
= 76731.6 mm2
Centroid of the section (CGC)
= 124.8 mm from soffit of beam
Moment of inertia of transformed section,
IT = Ig + A(0.2)2 + A1(210 124.8)2 + A2(124.8 40)2
= 4.02 108mm4
Eccentricity of prestressing force,
e = 124.8 115.5
= 9.3 mm
Stress at the level of bottom wires,
Stress at the level of top wires,
Loss of prestress in top wires = 6 4.81 (4 38.48)
= 4442 N
Loss of prestress in bottom wires = 6 6.52 (5 38.48)
= 7527 N
Total loss = 4442 + 7527 = 11969 N
12 kN
Percentage loss = (12 / 435) 100% = 2.75 %
It can be observed that the accurate and approximate solutions
are close. Hence, the simpler calculations based on A and I is
acceptable.
Pre-tensioned Bending Members
Changes in length and the prestressing force due to elastic
shortening of a pre-tensioned bending member.
Due to the effect of self-weight, the stress in concrete varies
along length.
To have a conservative estimate of the loss, the maximum stress
at the level of CGS at the mid-span is considered.