Loss of PC

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.