Prestressed Concrete Structure - Amlan-Menson

8/12/2019 Prestressed Concrete Structure - Amlan-Menson

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For high strength concrete (say M100 grade of concrete and above) under uniaxial

compression, the ascending and descending branches are steep.

0 c

f c f ck

E s E ci

0 c

f c f ck

E s E ci

Figure 1-6.3 Stress-strain curves for high strength concrete under compression

The equation proposed by Thorenfeldt, Tomaxzewicz and Jensen is appropriate for high

strength concrete.

c

c ck nk

c

n

f = f

n - +

0

0

1

(1-6.4)

The variables in the previous equation are as follows.

f c = compressive stress

f ck = characteristic compressive strength of cubes in N/mm 2 c = compressive strain

0 = strain corresponding to f ck

k = 1 for c 0

= 0.67 + ( f ck / 77.5) for c > 0. The value of k should be greater than 1.

n = E ci / ( E ci E s)

E ci = initial modulus

E s = secant modulus at f ck = f ck / 0.

The previous equation is applicable for both the ascending and descending branches of

the curve. Also, the parameter k models the slope of the descending branch, which

increases with the characteristic strength f ck . To be precise, the value of 0 can be

considered to vary with the compressive strength of concrete.


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Curve under uniaxial tension

The stress versus strain behaviour of concrete under uniaxial tension is linear elastic

initially. Close to cracking nonlinear behaviour is observed.

f c

c

f c f c

c

f c

c

f c f c

(a) (b)

Figure 1-6.4 a) Concrete panel under tension, b) Stress-strain curve for concrete

under tension

In calculation of deflections of flexural members at service loads, the nonlinearity is

neglected and a linear elastic behaviour f c = E c c is assumed. In the analysis of ultimate

strength, the tensile strength of concrete is usually neglected.

Creep of Concrete Creep of concrete is defined as the increase in deformation with time under constant

load. Due to the creep of concrete, the prestress in the tendon is reduced with time.

Hence, the study of creep is important in prestressed concrete to calculate the loss in

prestress.

The creep occurs due to two causes.

1. Rearrangement of hydrated cement paste (especially the layered products)

2. Expulsion of water from voids under load

If a concrete specimen is subjected to slow compressive loading, the stress versus

strain curve is elongated along the strain axis as compared to the curve for fast loading.

This can be explained in terms of creep. If the load is sustained at a level, the increase

in strain due to creep will lead to a shift from the fast loading curve to the slow loading

curve (Figure 1-6.5).


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c

f c Fast loading

Slow loading

Effect of creep

c

f c Fast loading

Slow loading

Effect of creep

Figure 1-6.5 Stress-strain curves for concrete under compression

Creep is quantified in terms of the strain that occurs in addition to the elastic strain due

to the applied loads. If the applied loads are close to the service loads, the creep strain

increases at a decreasing rate with time. The ultimate creep strain is found to be

proportional to the elastic strain. The ratio of the ultimate creep strain to the elastic

strain is called the creep coefficient .

For stress in concrete less than about one-third of the characteristic strength, the

ultimate creep strain is given as follows.

cr,ult el = (1-6.5)

The variation of strain with time, under constant axial compressive stress, is

represented in the following figure.

s t r a

i n

Time (linear scale)

cr, ult = ultimate creep strain

el = elastic strain s

t r a

i n

Time (linear scale)

cr, ult = ultimate creep strain

el = elastic strain

Figure 1-6.6 Variation of strain with time for concrete under compression

If the load is removed, the elastic strain is immediately recovered. However the

recovered elastic strain is less than the initial elastic strain, as the elastic modulus

increases with age.

There is reduction of strain due to creep recovery which is less than the creep strain.

There is some residual strain which cannot be recovered (Figure 1-6.7).


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s t r a

i n

Time (linear scale)

Residual strain

Creep recoveryElastic recovery

Unloading s t r a

i n

Time (linear scale)

Residual strain

Creep recoveryElastic recovery

Unloading

Figure 1-6.7 Variation of strain with time showing the effect of unloading

The creep strain depends on several factors. It increases with the increase in the

following variables.

1) Cement content (cement paste to aggregate ratio)

2) Water-to-cement ratio

3) Air entrainment4) Ambient temperature.

The creep strain decreases with the increase in the following variables.

1) Age of concrete at the time of loading.

2) Relative humidity

3) Volume to surface area ratio.

The creep strain also depends on the type of aggregate.

IS:1343 - 1980 gives guidelines to estimate the ultimate creep strain in Section 5.2.5 . It

is a simplified estimate where only one factor has been considered. The factor is age of

loading of the prestressed concrete structure. The creep coefficient is provided for

three values of age of loading.

Table 1-6.1 Creep coefficient for three values of age of loading

Age of Loading Creep Coefficient

7 days 2.2

28 days 1.6

1 year 1.1


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It can be observed that if the structure is loaded at 7 days, the creep coefficient is 2.2.

This means that the creep strain is 2.2 times the elastic strain. Thus, the total strain is

more than thrice the elastic strain. Hence, it is necessary to study the effect of creep in

the loss of prestress and deflection of prestressed flexural members. Even if the

structure is loaded at 28 days, the creep strain is substantial. This implies higher loss of

prestress and higher deflection.

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. Specialised literature or international codes can provide guidelines for such

calculations.

Shrinkage of Concrete

Shrinkage of concrete is defined as the contraction due to loss of moisture. The study of

shrinkage is also important in prestressed concrete to calculate the loss in prestress.

The shrinkage occurs due to two causes.

1. Loss of water from voids

2. Reduction of volume during carbonation

The following figure shows the variation of shrinkage strain with time. Here, t 0 is the time

at commencement of drying. The shrinkage strain increases at a decreasing rate with

time. The ultimate shrinkage strain ( sh ) is estimated to calculate the loss in prestress.

S h

r i n k

a g e s

t r a

i n

t 0 Time (linear scale)

sh

S h

r i n k

a g e s

t r a

i n

t 0 Time (linear scale)

sh

Figure 1-6.8 Variation of shrinkage strain with time


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Like creep, shrinkage also depends on several factors. The shrinkage strain increases

with the increase in the following variables.

1) Ambient temperature

2) Temperature gradient in the members

3) Water-to-cement ratio

4) Cement content.

The shrinkage strain decreases with the increase in the following variables.

1) Age of concrete at commencement of drying

2) Relative humidity

3) Volume to surface area ratio.

The shrinkage strain also depends on the type of aggregate.

IS:1343 - 1980 gives guidelines to estimate the shrinkage strain in Section 5.2.4 . It is a

simplified estimate of the ultimate shrinkage strain ( sh ).

For pre-tension

sh = 0.0003 (1-6.6)

For post-tension

(1-6.7)( )sh

=log t +10

0.00022

Here, t is the age at transfer in days. Note that for post-tension, t is the age at transfer

in days which approximates the curing time.

It can be observed that with increasing age at transfer, the shrinkage strain reduces. As

mentioned before, curing the concrete adequately and delaying the application of load

provide long term benefits with regards to durability and loss of prestress.

In special situations detailed calculations may be necessary to monitor shrinkage strain

with time. Specialised literature or international codes can provide guidelines for such

calculations.


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1.6.2 Properties of Grout

Grout is a mixture of water, cement and optional materials like sand, water-reducing

admixtures, expansion agent and pozzolans. The water-to-cement ratio is around 0.5.

Fine sand is used to avoid segregation.

The desirable properties of grout are as follows.

1) Fluidity

2) Minimum bleeding and segregation

3) Low shrinkage

4) Adequate strength after hardening

5) No detrimental compounds

6) Durable.

IS:1343 - 1980 specifies the properties of grout in Sections 12.3.1 and Section 12.3.2 .

The following specifications are important.

1) The sand should pass 150 m Indian Standard sieve.

2) The compressive strength of 100 mm cubes of the grout shall not be less than 17

N/mm 2 at 7 days.


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1.6.5 Codal Provisions of Concrete

The following topics are covered in IS:1343 - 1980 under the respective sections. These

provisions are not duplicated here.

Table 1-6.2 Topics and sectionsWorkability of concrete Section 6

Concrete mix proportioning Section 8

Production and control of concrete Section 9

Formwork Section 10

Transporting, placing, compacting Section 13

Concrete under special conditions Section 14

Sampling and strength test of concrete Section 15

Acceptance criteria Section 16

Inspection and testing of structures Section 17


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1.7 Prestressing Steel This section covers the following topics.

Forms of Prestressing Steel

Types of Prestressing Steel

Properties of Prestressing Steel

Codal Provisions of Steel

1.7.1 Forms of Prestressing Steel

The development of prestressed concrete was influenced by the invention of high

strength steel. It is an alloy of iron, carbon, manganese and optional materials. The

following material describes the types and properties of prestressing steel.

In addition to prestressing steel, conventional non-prestressed reinforcement is used for

flexural capacity (optional), shear capacity, temperature and shrinkage requirements.

The properties of steel for non-prestressed reinforcement are not covered in this section.

It is expected that the student of this course is familiar with the conventional

reinforcement.

Wires A prestressing wire is a single unit made of steel. The nominal diameters of the wires

are 2.5, 3.0, 4.0, 5.0, 7.0 and 8.0 mm. The different types of wires are as follows.

1) Plain wire: No indentations on the surface.

2) Indented wire: There are circular or elliptical indentations on the surface.

Strands

A few wires are spun together in a helical form to form a prestressing strand. The

different types of strands are as follows.

1) Two-wire strand: Two wires are spun together to form the strand.

2) Three-wire strand: Three wires are spun together to form the strand.

3) Seven-wire strand: In this type of strand, six wires are spun around a central wire.

The central wire is larger than the other wires.
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1.7.2 Types of Prestressing Steel

The steel is treated to achieve the desired properties. The following are the treatment

processes.

Cold working (cold drawing)The cold working is done by rolling the bars through a series of dyes. It re-aligns the

crystals and increases the strength.

Stress relieving

The stress relieving is done by heating the strand to about 350 C and cooling slowly.

This reduces the plastic deformation of the steel after the onset of yielding.

Strain tempering for low relaxation

This process is done by heating the strand to about 350 C while it is under tension.

This also improves the stress-strain behaviour of the steel by reducing the plastic

deformation after the onset of yielding. In addition, the relaxation is reduced. The

relaxation is described later.

IS:1343 - 1980 specifies the material properties of steel in Section 4.5 . The following

types of steel are allowed.1) Plain cold drawn stress relieved wire conforming to IS:1785, Part 1 , Specification

for Plain Hard Drawn Steel Wire for Prestressed Concrete, Part I Cold Drawn

Stress Relieved Wire .

2) Plain as-drawn wire conforming to IS:1785, Part 2 , Specification for Plain Hard

Drawn Steel Wire for Prestressed Concrete, Part II As Drawn Wire.

3) Indented cold drawn wire conforming to IS:6003 , Specification for Indented Wire

for Prestressed Concrete. 4) High tensile steel bar conforming to IS:2090 , Specification for High Tensile Steel

Bars used in Prestressed Concrete.

5) Uncoated stress relieved strand conforming to IS:6006 . Specification for

Uncoated Stress Relieved Strand for Prestressed Concrete.


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1.7.3 Properties o f Prestressing Steel

The steel in prestressed applications has to be of good quality. It requires the following

attributes.

1) High strength

2) Adequate ductility3) Bendability, which is required at the harping points and near the anchorage

4) High bond, required for pre-tensioned members

5) Low relaxation to reduce losses

6) Minimum corrosion.

Strength of Prestressing Steel

The tensile strength of prestressing steel is given in terms of the characteristic tensile

strength ( f pk ).

The characteristic strength is defined as the ultimate tensile strength of the coupon

specimens below which not more than 5% of the test results are expected to fall.

The ultimate tensile strength of a coupon specimen is determined by a testing machine

according to IS:1521 - 1972 , Method for Tensile Testing of Steel Wire . The following

figure shows a test setup.

Extensometer

Wedge grips

Coupon specimen

Extensometer

Wedge grips

Coupon specimen

(a) Test set-up


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(b) Failure of a strand

Figure 1-7.3 Testing of tensile strength of prestressing strand

The minimum tensile strengths for different types of wires as specified by the codes are

reproduced.

Table 1-7.1 Cold Drawn Stress-Relieved Wires ( IS: 1785 Part 1 )

Nominal Diameter (mm) 2.50 3.00 4.00 5.00 7.00 8.00

Minimum Tensile Strength f pk

(N/mm 2)

2010 1865 1715 1570 1470 1375

The proof stress (defined later) should not be less than 85% of the specified tensile

strength.

Table 1-7.2 As-Drawn wire ( IS: 1785 Part 2 )Nominal Diameter (mm) 3.00 4.00 5.00

Minimum Tensile Strength f pk (N/mm 2) 1765 1715 1570

The proof stress should not be less than 75% of the specified tensile strength.

Table 1-7.3 Indented wire ( IS: 6003 )

Nominal Diameter (mm) 3.00 4.00 5.00

Minimum Tensile Strength f pk (N/mm2) 1865 1715 1570

The proof stress should not be less than 85% of the specified tensile strength.

For high tensile steel bars ( IS: 2090 ), the minimum tensile strength is 980 N/mm 2. The

proof stress should not be less than 80% of the specified tensile strength.


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Stiffness of Prestressing Steel

The stiffness of prestressing steel is given by the initial modulus of elasticity. The

modulus of elasticity depends on the form of prestressing steel (wires or strands or

bars).

IS:1343 - 1980 provides the following guidelines which can be used in absence of testdata.

Table 1-7.4 Modulus of elasticity ( IS: 1343 - 1980 )

Type of steel Modulus of elasticity

Cold-drawn wires 210 kN/mm 2

High tensile steel bars 200 kN/mm 2

Strands 195 kN/mm 2

Al lowable Str ess in Prest ress ing Steel

As per Clause 18.5.1 , the maximum tensile stress during prestressing ( f pi ) shall not

exceed 80% of the characteristic strength.

pi pf 0.8 k f

(1-7.1)

There is no upper limit for the stress at transfer (after short term losses) or for the

effective prestress (after long term losses).

Stress-Strain Curves for Prestressing Steel The stress versus strain behaviour of prestressing steel under uniaxial tension is initially

linear (stress is proportional to strain) and elastic (strain is recovered at unloading).

Beyond about 70% of the ultimate strength the behaviour becomes nonlinear and

inelastic. There is no defined yield point.

The yield point is defined in terms of the proof stress or a specified yield strain. IS:1343

- 1980 recommends the yield point at 0.2% proof stress. This stress corresponds to an

inelastic strain of 0.002. This is shown in the following figure.


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0.002

Proof stress

p

f p

0.002

Proof stress

p

f p

Figure 1-7.4 Proof stress corresponds to inelastic strain of 0.002

The characteristic stress-strain curves are given in Figure 5 of IS:1343 - 1980 . The

stress corresponding to a strain can be found out by using these curves as shown next.

0.002 0.005

0.95 f pk

0.9 f pk

p

f p

0.002 0.005

0.95 f pk 0.85 f

pk

p

f p

Stress relieved wires,strands and bars

As-drawn wires

0.002 0.005

0.95 f pk

0.9 f pk

p

f p

0.002 0.005

0.95 f pk

0.9 f pk

p

f p

0.002 0.005

0.95 f pk 0.85 f

pk

p

f p

0.002 0.005

0.95 f pk 0.85 f

pk

p

f p

Stress relieved wires,strands and bars

As-drawn wires

Figure 1-7.5 Characteristic stress-strain curves for prestressing steel

(Figure 5, IS:1343 - 1980 )

The stress-strain curves are influenced by the treatment processes. The following figure

shows the variation in the 0.2% proof stress for wires under different treatment

processes.

low relaxation

stress relieved

as-drawn

p

f p low relaxation

stress relieved

as-drawn

p

f p

Figure 1-7.6 Variation in the 0.2% proof stress for wires under different treatment

processes


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The design stress-strain curves are calculated by dividing the stress beyond 0.8 f pk by a

material safety factor m =1.15. The following figure shows the characteristic and design

stress-strain curves.

0.8 f pk

p

f p Characteristic curve

Design curve0.8 f

pk

p

f p Characteristic curve

Design curve

Figure 1-7.7 Characteristic and design stress-strain curves for

prestressing steel

Relaxation of SteelRelaxation of steel is defined as the decrease in stress with time under constant strain.

Due to the relaxation of steel, the prestress in the tendon is reduced with time. Hence,

the study of relaxation is important in prestressed concrete to calculate the loss in

prestress.

The relaxation depends on the type of steel, initial prestress and the temperature. The

following figure shows the effect of relaxation due to different types of loading conditions.

p

f p

Fast loading

With sustained loadingEffect of relaxation

p

f p

Fast loading

With sustained loadingEffect of relaxation

Figure 1-7.8 Effect of relaxation due to different types of loading conditions

The following figure shows the variation of stress with time for different levels of

prestressing. Here, the instantaneous stress ( f p) is normalised with respect to the initial

prestressing ( f pi ) in the ordinate. The curves are for different values of f pi /f py , where f py is

the yield stress.


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10090

80

70

60

5010 100 1000 10,000 100,000Time (hours)

f p f pi

p i

p y

f =

f 0.60.70.80.9

10090

80

70

60

5010 100 1000 10,000 100,000Time (hours)

f p f pi

p i

p y

f =

f 0.60.70.80.9

Figure 1-7.9 Variation of stress with time for different levels of prestressing

It can be observed that there is significant relaxation loss when the applied stress is

more than 70% of the yield stress.

The following photos show the test set-up for relaxation test.

Load cell

Specimen

Load cell

Specimen

(a) Test of a single wire strand


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SpecimenSpecimen

(b) Test of a seven-wire strand

Figure 1-7.10 Set-up for relaxation test

The upper limits of relaxation loss are specified as follows.

Table 1-7.5 Relaxation losses at 1000 hours ( IS:1785, IS:6003, IS:6006, IS:2090 )

Cold drawn stress-relieved wires 5% of initial prestress

Indented wires 5% of initial prestress

Stress-relieved strand 5% of initial prestress

Bars 49 N/mm2

In absence of test data, IS:1343 - 1980 recommends the following estimates of

relaxation losses.

Table 1-7.6 Relaxation losses at 1000 hours at 27C

Initial Stress Relaxation Loss (N/mm 2)

0.5 f pk 0

0.6 f pk 350.7 f pk 70

0.8 f pk 90

Fatigue

Under repeated dynamic loads the strength of a member may reduce with the number

of cycles of applied load. The reduction in strength is referred to as fatigue.


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In prestressed applications, the fatigue is negligible in members that do not crack under

service loads. If a member cracks, fatigue may be a concern due to high stress in the

steel at the location of cracks.

Specimens are tested under 2 x 10 6 cycles of load to observe the fatigue. For steel,

fatigue tests are conducted to develop the stress versus number of cycles for failure (S-N) diagram. Under a limiting value of stress, the specimen can withstand infinite number

of cycles. This limit is known as the endurance limit.

The prestressed member is designed such that the stress in the steel due to service

loads remains under the endurance limit. The following photo shows a set-up for

fatigue testing of strands.

Figure 1-7.11 S et-up for fatigue testing of strands

Durability

Prestressing steel is susceptible to stress corrosion and hydrogen embrittlement inaggressive environments. Hence, prestressing steel needs to be adequately protected.

For bonded tendons, the alkaline environment of the grout provides adequate protection.

For unbonded tendons, corrosion protection is provided by one or more of the following

methods.


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1) Epoxy coating

2) Mastic wrap (grease impregnated tape)

3) Galvanized bars

4) Encasing in tubes.

1.7.4 Codal Provisions of Steel

The following topics are covered in IS:1343 - 1980 under the respective sections. These

provisions are not duplicated here.

Table 1-7.7 Topics and sections

Assembly of prestressing and reinforcing steel Section 11

Prestressing Section 12


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2.1 Losses in Prestress (Part I)This section covers the following topics.

Introduction

Elastic Shortening

The relevant notations are explained first.

Notations

Geometric Properties

The commonly used geometric properties of a prestressed member are defined as

follows.

Ac = Area of concrete section

= Net cross-sectional area of concrete excluding the area of

prestressing steel. A p = Area of prestressing steel

= Total cross-sectional area of the tendons.

A = Area of prestressed member

= Gross cross-sectional area of prestressed member.

= Ac + A p

A t = Transformed area of prestressed member

= Area of the member when steel is substituted by an equivalent

area of concrete.

= Ac + mA p

= A + ( m 1) A p

Here,

m = the modular ratio = E p /E c

E c = short-term elastic modulus of concrete

E p = elastic modulus of steel.

The following figure shows the commonly used areas of the prestressed members.
http://elastic%20shortening/http://elastic%20shortening/


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= +

A A c A p A t

= +

A A c A p A t Figure 2-1.1 Areas for prestressed members

CGC = Centroid of concrete

= Centroid of the gross section. The CGC may lie outside the

concrete (Figure 2-1.2).

CGS = Centroid of prestressing steel

= Centroid of the tendons. The CGS may lie outside the tendons or

the concrete (Figure 2-1.2).I = Moment of inertia of prestressed member

= Second moment of area of the gross section about the CGC.

I t = 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 (Figure 2-1.2).

CGSCGCe

CGS

CGCe CGS

CGCe CGSCGCCGSCGCe

CGS

CGCe

CGS

CGC

CGS

CGC

CGS

CGCe

Figure 2-1.2 CGC, CGS and eccentricity of typical prestressed members

Load Variables

P i = Initial prestressing force

= The force which is applied to the tendons by the jack.


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P 0 = Prestressing force after immediate losses

= The reduced value of prestressing force after elastic shortening,

anchorage slip and loss due to friction.

P e = Effective prestressing force after time-dependent losses

= The final value of prestressing force after the occurrence of creep,

shrinkage and relaxation.

2.1.1 Introduction

In prestressed concrete applications, the most important variable is the prestressing

force. In the early days, it was observed that the prestressing force does not stay

constant, but reduces with time. Even during prestressing of the tendons and the

transfer of prestress to the concrete member, there is a drop of the prestressing force

from the recorded value in the jack gauge. The various reductions of the prestressing

force are termed as the losses in prestress.

The losses are broadly classified into two groups, immediate and time-dependent. The

immediate losses occur during prestressing of the tendons and the transfer of prestress

to the concrete member. The time-dependent losses occur during the service life of the

prestressed member. The losses due to elastic shortening of the member, friction at the

tendon-concrete interface and slip of the anchorage are the immediate losses. Thelosses due to the shrinkage and creep of the concrete and relaxation of the steel are the

time-dependent losses. The causes of the various losses in prestress are shown in the

following chart.

Losses

Immediate Time dependent

Elasticshortening

Friction Anchorageslip

Creep Shrinkage Relaxation

Figure 2-1.3 Causes of the various losses in prestress


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2.1.2 Elastic Shortening

Pre-tensioned Members

When the tendons are cut and the prestressing force is transferred to the member, the

concrete undergoes immediate shortening due to the prestress. The tendon also

shortens by the same amount, which leads to the loss of prestress.

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. For more than one tendon, if the tendons

are stretched sequentially, there is loss in a tendon during subsequent stretching of the

other tendons.

The elastic shortening loss is quantified by the drop in prestress ( f p) in a tendon due to

the change in strain in the tendon ( p). It is assumed that the change in strain in the

tendon is equal to the strain in concrete ( c ) at the level of the tendon due to the

prestressing force. This assumption is called strain compatibility between concrete

and steel. The strain in concrete at the level of the tendon is calculated from the stress

in concrete ( f c ) 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.

p p p

p c

c p

c

p c

f = E

= E

f = E

E f = mf (2-1.1)

For simplicity, the loss in all the tendons can be calculated based on the stress inconcrete at the level of CGS. This simplification cannot be used when tendons are

stretched sequentially in a post-tensioned member. The calculation is illustrated for the

following types of members separately.

Pre-tensioned Axial Members

Pre-tensioned Bending Members

Post-tensioned Axial Members


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Post-tensioned Bending Members

Pre-tensioned Axial Members

The following figure shows the changes in length and the prestressing force due to

elastic shortening of a pre-tensioned axial member.

Original length of member at transfer of prestress

Length after elastic shortening

P i

P 0

Original length of member at transfer of prestress

Length after elastic shortening

P i

P 0

Figure 2-1.4 Elastic shortening of a pre-tensioned axial member

The loss can be calculated as per Eqn. (2-1.1) by expressing the stress in concrete in

terms of the prestressing force and area of the section as follows.

(2-1.2)

p c

c

i i p

t

f = mf

P = m

AP P f = m m

A A

0

Note that the stress in concrete due to the prestressing force after immediate losses

(P 0/ A c ) can be equated to the stress in the transformed section due to the initial

prestress ( P i / A t ). This is derived below. Further, the transformed area A t of the

prestressed member can be approximated to the gross area A.

The following figure shows that the strain in concrete due to elastic shortening ( c ) is the

difference between the initial strain in steel ( pi ) and the residual strain in steel ( p0).


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P i

P 0

Length of tendon before stretching pi

p 0

c

P i

P 0

Length of tendon before stretching pi

p 0

c

Figure 2-1.5 Strain variables in elastic shortening

The following equation relates the strain variables.

c = pi - p0 (2-1.3)

The strains can be expressed in terms of the prestressing forces as follows.

c c c

P = A E

0 (2-1.4)

i pi

p p

P = A E

(2-1.5)

p p p

P = A E

00

(2-1.6)

Substituting the expressions of the strains in Eqn. (2-1.3)

i

c c p p p p

i

c c p p p p

i

c p p

i

c p c

P P P = -

A E A E A E

P , P + =

A E A E A E

P m 1 P + =

A A A

P P =

A mA + A

0 0

0

0

0

1 1or

or,

or,

0or i c t

P P =

A A

(2-1.7)

Thus, the stress in concrete due to the prestressing force after immediate losses ( P 0/ Ac )

can be equated to the stress in the transformed section due to the initial prestress ( P i

/ A t ).


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The following problem illustrates the calculation of loss due to elastic shortening in an

idealised pre-tensioned railway sleeper.

Example 2-1.1

A prest res sed conc rete sl eeper pr oduced by pr e-tension ing method has arectangular cross-section of 300mm 250 mm ( b h ). It is prestressed with 9

numbers of s traight 7mm diameter wires at 0.8 times the ultimate strength of 1570

N/mm 2. Estimate the percentage loss of stress due to elastic shor tening of

concrete. Consider m = 6.

250

40

300

40

Solution

a) Approximate solution considering gross section

The sectional properties are calculated as follows.

Area of a single wire, Aw = /4 7 2

= 38.48 mm 2

Area of total prestressing steel, A p = 9 38.48

= 346.32 mm 2

Area of concrete section, A = 300 250

= 75 10 3 mm 2

Moment of inertia of section, I = 300 250 3/12

= 3.91 10 8 mm 4

Distance of centroid of steel area (CGS) from the soffit,


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( )438.48 250- 40 +538.4840y =938.48

= 115.5 mm

Prestressing force, P i = 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 P i 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 = y t = 125 40)

115.5

e

=+

i P - A

i i P P .e- y A I

i P .e y I

( )

( )3 3

3 8

2

435 10 435 10 9.5 = - + 125 - 40

7510 3.9110 = -5.8+ 0.9

= -4.9 N/mm

i i c t t

P P .ef = - + y A I


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Stress at level of bottom wires ( y = y b = 125 40),

( )

( )3 3

3 8

2

435 10 435 10 9.5 = - - 125 - 40

7510 3.9110 = -5.8- 0.9

= -6.7 N/mm

i i c bb

P P .ef = - - y

A I

Loss of prestress in top wires = mf c A p

(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 mm 2

Transformed area of bottom steel,

A2 = (6 1) 5 38.48

= 962.0 mm 2

Total area of transformed section,

AT = A + A 1 + A2

= 75000.0 + 769.6 + 962.0

= 76731.6 mm 2


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Centroid of the section (CGC)

A + A + A y =

A1 2125 (250 - 40) 40

= 124.8 mm from soffit of beam

Moment of inertia of transformed section,I T = I g + A(0.2) 2 + A1(210 124.8) 2 + A2(124.8 40) 2

= 4.02 10 8mm 4

Eccentricity of prestressing force,

e = 124.8 115.5

= 9.3 mm

Stress at the level of bottom wires,3 3

3 8

2

43510 (43510 9.3)84.8= - -

76.7310 4.0210= -5.67 - 0.85

= -6.52 N/mm

c b(f )

Stress at the level of top wires,3 3

3 8

2

435 10 (435 10 9.3)85.2= - +

76.7310 4.0210= -5.67+ 0.86

= -4.81 N/mm

c t (f )

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 %


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be calculated in progressive sequence. Else, an approximation can be used to

calculate the losses.

The loss in the first tendon is evaluated precisely and half of that value is used as an

average loss for all the tendons.

(2-1.9)

p p

c

n i,j

j=

f = f

mf

P = m

A

1

1

2

1

21

=212

Here,

P i,j = initial prestressing force in tendon j

n = number of tendons

The eccentricity of individual tendon is neglected.

Post-tensioned Bending Members

The calculation of loss for tendons stretched sequentially, is similar to post-tensioned

axial members. For curved profiles, the eccentricity of the CGS and hence, the stress in

concrete at the level of CGS vary along the length. An average stress in concrete can

be considered.

For a parabolic tendon, the average stress ( f c,avg ) is given by the following equation.

( )=c,avg c c c f f + f - f 1 223 1

(2-1.10)

Here,

f c 1 = stress in concrete at the end of the member

f c 2 = stress in concrete at the mid-span of the member.

A more rigorous analysis of the loss can be done by evaluating the strain in concrete at

the level of the CGS accurately from the definition of strain. This is demonstrated for a

beam with two parabolic tendons post-tensioned sequentially. In Figure 2-1.7, Tendon

B is stretched after Tendon A. The loss in Tendon A due to elastic shortening during

tensioning of Tendon B is given as follows.

[ ] p p c

p c c

f = E

= E + 1 2 (2-1.11)


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Here, c is the strain at the level of Tendon A. The component of c due to pure

compression is represented as c 1. The component of c due to bending is represented

as c 2. The two components are calculated as follows.

Bc

c

c

LB B A

c

LB

B Ac

P AE

L L

P .e (x).e (x) = dx

L IE

P e (x).e (x) dx

E LI

1

2

0

0

=

=

1

=

(2-1.12)

Here,

A = cross-sectional area of beam

P B = prestressing force in Tendon B

E c = modulus of concrete

L = length of beam

e A( x ), e B( x ) = eccentricities of Tendons A and B, respectively, at distance x

from left end

I = moment of inertia of beam

L = change in length of beam

The variations of the eccentricities of the tendons can be expressed as follows.

(2-1.13) + +

A A A

B B B

x x e (x) = e e

L L x x

e (x) = e eL L

1

1

4 1

4 1

(2-1.14)

2 1

2 1

Where A A A

B B B

, e = e e

e = e e

e A1, e A2 = eccentricities of Tendon A at 1 (end) and 2 (centre), respectively.

e B1, e B2 = eccentricities of Tendon B at 1 and 2, respectively.


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Substituting the expressions of the eccentricities in Eqn. (2-1.12), the second

component of the strain is given as follows.

(2-1.15) ( ) + + + B

A B A B A B A Bc

P = e e e e e e e e

E I 1 1 1 2 2 1 2 21 2 85 15 15


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2.2 Losses in Prestress (Part II)This section covers the following topics

Friction

Anchorage Slip

Force Variation Diagram

2.2.1 Friction

The friction generated at the interface of concrete and steel during the stretching of a

curved tendon in a post-tensioned member, leads to a drop in the prestress along the

member from the stretching end. The loss due to friction does not occur in pre-

tensioned members because there is no concrete during the stretching of the tendons.

The friction is generated due to the curvature of the tendon and the vertical component

of the prestressing force. The following figure shows a typical profile (laying pattern) of

the tendon in a continuous beam.

Figure 2-2.1 A typical continuous post-tensioned member

(Reference: VSL International Ltd.)

In addition to friction, the stretching has to overcome the wobble of the tendon. The

wobble refers to the change in position of the tendon along the duct. The losses due to

friction and wobble are grouped together under friction.

The formulation of the loss due to friction is similar to the problem of belt friction. The

sketch below (Figure 2-2.2) shows the forces acting on the tendon of infinitesimal length

dx .
http://frictional%20loss/http://loss%20due%20to%20anchorage%20slip/http://force%20variation%20diagram/http://frictional%20loss/http://frictional%20loss/http://force%20variation%20diagram/http://loss%20due%20to%20anchorage%20slip/http://frictional%20loss/


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R

N P + dP

P

d

dx

P + dP

N

P d /2

Force triangle

R

N P + dP

P

d

dx

R

N P + dP

P

d

dx

P + dP

N

P d /2

Force triangle

P + dP

N

P d /2

P + dP

N

P d /2

Force triangle

Figure 2-2.2 Force acting in a tendon of infinitesimal length

In the above sketch,

P = prestressing force at a distance x from the stretching end

R = radius of curvature

d = subtended angle.

The derivation of the expression of P is based on a circular profile. Although a cable

profile is parabolic based on the bending moment diagram, the error induced is

insignificant.

The friction is proportional to the following variables.

Coefficient of friction ( ) between concrete and steel. The resultant of the vertical reaction from the concrete on the tendon ( N )

generated due to curvature.

From the equilibrium of forces in the force triangle, N is given as follows.

d N = Psin

d P = Pd

22

2

2(2-2.1)

The friction over the length dx is equal to N = Pd .

Thus the friction ( dP ) depends on the following variables.

Coefficient of friction ( )

Curvature of the tendon ( d )

The amount of prestressing force ( P )


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The wobble in the tendon is effected by the following variables.

Rigidity of sheathing

Diameter of sheathing

Spacing of sheath supports

Type of tendon

Type of construction

The friction due to wobble is assumed to be proportional to the following.

Length of the tendon

Prestressing force

For a tendon of length dx , the friction due to wobble is expressed as kPdx , where k is

the wobble coefficient or coefficient for wave effect.

Based on the equilibrium of forces in the tendon for the horizontal direction, the

following equation can be written.

P = P + dP + (Pd + kPdx)

or , dP = (Pd + kPdx ) (2-2.2)

Thus, the total drop in prestress ( dP ) over length dx is equal to ( Pd + kPdx ). The

above differential equation can be solved to express P in terms of x .

( )

( )

( )

x

x

P x

P

P

P

x

- +kx x

dP = - d + k dx

P

lnP = - + kx

P ln = - + kx

P

P = P e

0

0

0 0

0

0

or,

or,

or,

(2-2.3)

Here,

P 0 = the prestress at the stretching end after any loss due to elastic shortening.

For small values of + kx , the above expression can be simplified by the Taylor series

expansion.

P x = P 0 (1 kx ) (2-2.4)

Thus, for a tendon with single curvature, the variation of the prestressing force is linear

with the distance from the stretching end. The following figure shows the variation of


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prestressing force after stretching. The left side is the stretching end and the right side

is the anchored end.

P x P 0 P x P 0

Figure 2-2.3 Variation of prestressing force after stretching

In the absence of test data, IS:1343 - 1980 provides guidelines for the values of and k .

Table 2-2.1 Values of coefficient of friction

Type of interface

For steel moving on smooth concrete 0.55.

For steel moving on steel fixed to duct 0.30.

For steel moving on lead 0.25.

The value of k varies from 0.0015 to 0.0050 per meter length of the tendon depending

on the type of tendon. The following problem illustrates the calculation of the loss due

to friction in a post-tensioned beam.

Example 2-2.1

A post -tensioned beam 100 mm 300 mm ( b h ) spanning over 10 m is str essed

by successive tensioning and anchoring of 3 cables A, B, and C respectively as

shown in figure. Each cable has cross section area of 200 mm 2 and has initial

stress of 1200 MPa. If the cables are tensioned from one end, estimate the

percentage loss in each cable due to friction at the anchored end. Assume =

0.35, k = 0.0015 / m.


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Cable A Cable B

Cable C

CL

5050 CGC

Cable A Cable B

Cable C

CL

5050 CGC

Solution

Prestress in each tendon at stretching end = 1200 200

= 240 kN.

To know the value of (L), the equation for a parabolic profile is required.

md y y = ( L -d x L 2

4 2 x )

y m

y

L

x

(L)

y m

y

L

x

(L)

Here,

y m = displacement of the CGS at the centre of the beam from the endsL = length of the beam

x = distance from the stretching end

y = displacement of the CGS at distance x from the ends.

An expression of ( x ) can be derived from the change in slope of the profile. The slope

of the profile is given as follows.

md y y

= ( L -d x L 24

2 x )

At x = 0, the slope dy /dx = 4 y m/L. The change in slope ( x ) is proportional to x .

The expression of ( x ) can be written in terms of x as ( x ) = .x ,

where, = 8y m /L2. The variation is shown in the following sketch.


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8 y m /L

4 y m /L

0 L/2 L

8 y m /L

4 y m /L

0 L/2 L

The total subtended angle over the length L is 8 y m/L.

The prestressing force P x at a distance x is given by

P x = P 0e ( + kx ) = P 0e x

where, x = + kx

For cable A, y m = 0.1 m.For cable B, y m = 0.05 m.

For cable C, y m = 0.0 m.

For all the cables, L = 10 m.

Substituting the values of y m and L

0.0043x for cable A= 0.0029x for cable B

0.0015x for cable C x

The maximum loss for all the cables is at x = L = 10, the anchored end.

0.958 for cable A= 0.971 for cable B

0.985 for cable C

-Le

Percentage loss due to friction = (1 e L) 100%

4.2% for cable A= 2.9% for cable B

1.5% for cable C


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Cable A Cable BCable C

CL

CGC

240 kN

Cable A Cable BCable C

CL

CGC

240 kN

Variation of prestressing forces

The loss due to friction can be considerable for long tendons in continuous beams with

changes in curvature. The drop in the prestress is higher around the intermediatesupports where the curvature is high. The remedy to reduce the loss is to apply the

stretching force from both ends of the member in stages.

2-2.2 Anchorage Slip

In a post-tensioned member, when the prestress is transferred to the concrete, the

wedges slip through a little distance before they get properly seated in the conical space.The anchorage block also moves before it settles on the concrete. There is loss of

prestress due to the consequent reduction in the length of the tendon.

The total anchorage slip depends on the type of anchorage system. In absence of

manufacturers data, the following typical values for some systems can be used.

Table 2-2.2 Typical values of anchorage slip

Anchorage System Anchorage Slip ( s )

Freyssinet system

12 - 5mm strands

12 - 8mm strands

4 mm

6 mm

Magnel system 8 mm

Dywidag system 1 mm

(Reference: Rajagopalan, N., Prestressed Concrete )


106/504

Due to the setting of the anchorage block, as the tendon shortens, there is a reverse

friction. Hence, the effect of anchorage slip is present up to a certain length (Figure 2-

2.4). Beyond this setting length , the effect is absent. This length is denoted as l set .

P x

P 0

P x

P 0

Figure 2-2.4 Variation of prestressing force after anchorage slip

2.2.3 Force Variation Diagram

The magnitude of the prestressing force varies along the length of a post-tensioned

member due to friction losses and setting of the anchorage block. The diagram

representing the variation of prestressing force is called the force variation diagram.

Considering the effect of friction, the magnitude of the prestressing force at a distance x

from the stretching end is given as follows.

(2-2.5) - x x P = P e0

Here, x = + kx denotes the total effect of friction and wobble. The plot of P x gives

the force variation diagram.

The initial part of the force variation diagram, up to length l set is influenced by the setting

of the anchorage block. Let the drop in the prestressing force at the stretching end be

P . The determination of P and l set are necessary to plot the force variation diagram

including the effect of the setting of the anchorage block.

Considering the drop in the prestressing force and the effect of reverse friction, the

magnitude of the prestressing force at a distance x from the stretching end is given as

follows.

(2-2.6) ( )' 'x x P = P - P e0


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Here, for reverse friction is analogous to for friction and wobble.

At the end of the setting length ( x = l set ), P x = P x

P

P 0

P x P x

l set x

P x

P x after stretchingP x after settingP x beyond l set

P

P 0

P x P x

l set x

P x

P

P 0

P x P x

l set x

P x

P x after stretchingP x after settingP x beyond l set

Figure 2-2.5 Force variation diagram near the stretching end

Substituting the expressions of P x and P x for x = l set

Since it is difficult to measure separately, is taken equal to . The expression of

P simplifies to the following.

( )( )

( )

( )

set set

set

-l 'l

- +' l

set

set set

P e = P - P e

P e = P - P

P - + ' l = P - P

' P = P + ' l = P l +

0 0

0 0

0 0

0 0

1

1

s set p p

set s set

p p

P = l

A E

l ' = P l +

A E 0

12

11

2

p pset s

s p p

A E l =

' P +

A E = ' =

P

2

0

0

2

1

for

(2-2.7)

P = 2 P 0l set (2-2.8)

The following equation relates l set with the anchorage slip s .

(2-2.9)

Transposing the terms,


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Therefore,

s p pset

A E l =

P 0(2-2.10)

The term P 0 represents the loss of prestress per unit length due to friction.

The force variation diagram is used when stretching is done from both the ends. The

tendons are overstressed to counter the drop due to anchorage slip. The stretching from

both the ends can be done simultaneously or in stages. The final force variation is more

uniform than the first stretching.

The following sketch explains the change in the force variation diagram due to

stretching from both the ends in stages.

a) After stretching from right end

b) After anchorage slip at right end

a) After stretching from right end

b) After anchorage slip at right end

c) After stretching from left end

d) After anchorage slip at left end


d) After anchorage slip at left end


d) After anchorage slip at left end Figure 2-2.6 Force variation diagrams for stretching in stages


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The force variation diagrams for the various stages are explained.

a) The initial tension at the right end is high to compensate for the anchorage

slip. It corresponds to about 0.8 f pk initial prestress. The force variation

diagram (FVD) is linear.

b) After the anchorage slip, the FVD drops near the right end till the length l set .

c) The initial tension at the left end also corresponds to about 0.8 f pk initial prestress.The FVD is linear up to the centre line of the beam.

d) After the anchorage slip, the FVD drops near the left end till the length l set . It is

observed that after two stages, the variation of the prestressing force over the length

of the beam is less than after the first stage.

Example 2-2.2

A four span cont inuous br idge gi rder is post -tensioned with a tendon cons is ting

of twenty strands with f pk = 1860 MPa. Half of the girder is shown in the figu re

below. The symmetrical tendon is simu ltaneously stressed up to 75% f pk from

both ends and then anchored. The tendon properties are A p = 2800 mm 2, E p =

195,000 MPa, = 0.20, K = 0.0020/m. The anchorage sl ip s = 6 mm.

Calculate

a) The expected elongation of the tendon after str etching,b) The force variation diagrams along the tendon before and after anchorage.

13.7 13.7 3 3.7 15.2 15.2 3.7

0.76 0.6 0.76

All dimensions are in metres

0.6CL

Inflection points


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111/504

L

For the two parabolic segments joined at the inflection point as shown in the sketch

above, the slope at the inflection point = 2( e 1 + e 2)/ L.

Here,

e1, e

2 = eccentricities of the CGS at the span and support respectively

L = length of the span

L = fractional length between the points of maximum eccentricity

The change in slope between a point of maximum eccentricity and inflection point is

also equal to .

The change in slope ( ) for each segment of the tendon is calculated using the above

expressions. Next the value of + kx for each segment is calculated using the given

values of , k and x , the horizontal length of the segment. Since the loss in prestress

accrues with each segment, the force at a certain segment is given as follows.

The summation is for the segments from the stretching end up to the point in the

segment under consideration. Hence, the value of ( + kx ) at the end of each

segment is calculated to evaluate the prestressing force at that point ( P x , where x

denotes the point).

e 2

e1

L

- ( +kx) x P = P e0


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0.163 0.111 0.163 0.144 0.1440.144 0.144

+kx 0.0390.050 0.060 0.059 0.059 0.0360.036

The force variation diagram before anchorage can be plotted with the above values of

P x . A linear variation of the force can be assumed for each segment. Since the

stretching is done at both the ends simultaneously, the diagram is symmetric about the

central line.

a) The expected elongation of the tendon after stretching

First the product of the average force and the length of each segment is summed up to

the centre line.

0.050 ( +kx ) 0.149

0.110 0.185 0.244 0.303 0.339

e - + x

0.7380.712

0.7830.9521.000 0.8960.861

0.831

P x (kN)3906 3718 3500

33633246 3058 2883

2781

[ ] [ ]

[ ] [ ]

[ ] [ ]

[ ]

av1 1

P L = 3906 + 3718 13.7 + 3718 + 3500 13.72 21 1

+ 3500 +3363 3+ 3363 + 3246 3.72 21 1

+ 3246 + 3058 15.2 + 3058 + 2883 15.22 21+ 2883 + 2718 3.72

= 227612.2 kN


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The elongation ( ) at each stretching end is calculated as follows.

322761210=

2800195000= 0.417 m

av

P P

P L =

A E

b) The force variation diagrams along the tendon before and after anchorage

After anchorage, the effect of anchorage slip is present up to the setting length l set . The

value of l set due to an anchorage slip s = 6 mm is calculated as follows.

62800195000=13.7

=15.46 m

s P P set

0

A E l =

P

The quantity P 0 is calculated from the loss of prestress per unit length in the first

segment. P 0 = (3906 3718) kN /13.7 m = 13.7 N/mm. The drop in the prestressing

force ( p) at each stretching end is calculated as follows.

02= 213.715464= 423.7 kN

p set = P l

Thus the value of the prestressing force at each stretching end after anchorage slip is

3906 424 = 3482 kN. The force variation diagram for l set = 15.46 m is altered to show

the drop due to anchorage slip.

The force variation diagrams before and after anchorage are shown below. Note that

the drop of force per unit length is more over the supports due to change in curvature

over a small distance.


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2500

3000

3500

4000

0 20 40 60 8

Distance from end (m)

P r e s

t r e s s

i n g

f o r c e

( k N )

00

After anchorage Before anchorage

2500

3000

3500

4000

0 20 40 60 8

Distance from end (m)

P r e s

t r e s s

i n g

f o r c e

( k N )

After anchorage Before anchorage


115/504

2.3 Losses in Prestress (Part III)This section covers the following topics.

Creep of Concrete

Shrinkage of Concrete

Relaxation of Steel

Total Time Dependent Losses

2.3.1 Creep of Concrete

Creep of concrete is defined as the increase in deformation with time under constant

load. Due to the creep of concrete, the prestress in the tendon is reduced with time.

The creep of concrete is explained in Section 1.6, Concrete (Part II). Here, theinformation is summarised. 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 .

The ultimate creep strain is then given as follows.

(2-3.1)cr,ult el

=

IS:1343 - 1980 gives guidelines to estimate the ultimate creep strain in Section 5.2.5 . It

is a simplified estimate where only one factor has been considered. The factor is age of

loading of the prestressed concrete structure. The creep coefficient is provided for

three values of age of loading.

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. Specialised literature or international codes can provide guidelines for such

calculations.

The loss in prestress ( f p ) due to creep is given as follows.

f p = E p cr, ult (2-3.2)


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Here, E p is the modulus of the prestressing steel.

The following considerations are applicable for calculating the loss of prestress due to

creep.

1) The creep is due to the sustained (permanently applied) loads only.

Temporary loads are not considered in the calculation of creep.2) Since the prestress may vary along the length of the member, an average value

of the prestress can be considered.

3) The prestress changes due to creep and the creep is related to the

instantaneous prestress. To consider this interaction, the calculation of creep can

be iterated over small time steps.

2.3.2 Shrinkage of Concrete

Shrinkage of concrete is defined as the contraction due to loss of moisture. Due to the

shrinkage of concrete, the prestress in the tendon is reduced with time.

The shrinkage of concrete was explained in details in the Section 1.6, Concrete (Part II).

IS:1343 - 1980 gives guidelines to estimate the shrinkage strain in Section 5.2.4 . It is a

simplified estimate of the ultimate shrinkage strain ( sh ). Curing the concrete adequately

and delaying the application of load provide long term benefits with regards to durabilityand loss of prestress. In special situations detailed calculations may be necessary to

monitor shrinkage strain with time. Specialised literature or international codes can

provide guidelines for such calculations.

The loss in prestress ( f p ) due to shrinkage is given as follows.

f p = E p sh (2-3.3)

Here, E p is the modulus of the prestressing steel.

2.3.3 Relaxation of Steel

Relaxation of steel is defined as the decrease in stress with time under constant strain.

Due to the relaxation of steel, the prestress in the tendon is reduced with time. The

relaxation depends on the type of steel, initial prestress ( f pi ) and the temperature. To


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calculate the drop (or loss) in prestress ( f p), the recommendations of IS:1343 - 1980

can be followed in absence of test data.

Example 2-3.1

A conc rete beam of dimension 100 mm 300 mm is post-tens ioned with 5straight wires of 7mm diameter. The average prestress after short-term losses is

0.7 f pk = 1200 N/mm 2 and the age of loading is given as 28 days. Given that E p =

200 103 MPa, E c = 35000 MPa, find out the losses of prestress due to creep,

shrinkage and relaxation. Neglect the weight of the beam in the computation of

the stresses.

300

100

50 CGS

300

100

50 CGS

Solution

Area of concrete A = 100 300

= 30000 mm 2

Moment of inertia of beam section

I = 100 300 3 / 12

= 225 10 6 mm 4

Area of prestressing wires A p = 5 ( /4) 7 2

= 192.42 mm 2

Prestressing force after short-term losses

P 0 = A p .f p0

= 192.4 1200

= 230880 N


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Modular ratio m = E p / E c

= 2 10 5 / 35 10 3

= 5.71

Stress in concrete at the level of CGS

0 0

24 6

230880 230880= - - 50

310 22510

c P P e

f = - - e A I

= 7.69 2.56

= 10.25 N/mm 2

Loss of prestress due to creep

( f p)cr = E p cr, ult

= E p el

= E p (f c /E c )

= m f c

= 5.71 10.25 1.6

= 93.64 N / mm 2

Here, = 1.6 for loading at 28 days, from Table 2c -1 (Clause 5.2.5.1, IS:1343 - 1980).

Shrinkage strain from Claus e 5.2.4.1, IS:1343 - 1980

sh = 0.0002 / log 10 (t + 2)

= 0.0002 / log 10 (28 + 2)

= 1.354 10 -4

Loss of prestress due to shrinkage

( f p)sh = E p sh

= 2 10 5 1.354 10 -4

= 27.08 N/mm 2


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From Table 2c-2 (Table 4, IS:1343 - 1980)

Loss of prestress due to relaxation

( f p)rl = 70.0 N/mm 2

Loss of prestressing force = f p A p

Therefore,

Loss of prestressing force due to creep = 93.64 192.42

= 18018 N

Loss of prestressing force due to shrinkage

= 27.08 192.42

= 5211 N

Loss of prestressing force due to relaxation

= 70 192.42

= 13469 N

Total long-term loss of prestressing force (neglecting the interaction of the losses and

prestressing force)

= 18018 + 5211 + 13469

= 36698 N

Percentage loss of prestress = 36698 / 230880 100%

= 15.9 %

2.3.4 Total Time-dependent Loss

The losses of prestress due to creep and shrinkage of concrete and the relaxation of the

steel are all time-dependent and inter-related to each other. If the losses are calculated

separately and added, the calculated total time-dependent loss is over-estimated. To

consider the inter-relationship of the cause and effect, the calculation can be done for

discrete time steps. The results at the end of each time step are used for the next time

step. This step-by-step procedure was suggested by the Precast / Prestressed


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Concrete Institute (PCI) committee and is called the General method (Reference: PCI

Committee, Recommendations for Estimating Prestress Losses, PCI Journal, PCI, Vol.

20, No. 4, July-August 1975, pp. 43-75 ).

In the PCI step-by-step procedure, a minimum of four time steps are considered in the

service life of a prestressed member. The following table provides the definitions of thetime steps (Table 2-3.3).

Table 2-3.3 Time steps in the step-by-step procedure

Step Beginning End

1 Pre-tension: Anchorage of steel

Post-tension: End of curing

Age of prestressing

2 End of Step 1 30 days after prestressing or

when subjected to superimposed

load

3 End of Step 2 1 year of service

4 End of Step 3 End of service life

The step-by-step procedure can be implemented by a computer program, where the

number of time steps can be increased.

There are also approximate methods to calculate lump sum estimates of the total loss.

Since these estimates are not given in IS:1343 - 1980 , they are not mentioned here.


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3.1 Analysis of Members under Axial LoadThis section covers the following topics.

Introduction

Analysis at Transfer

Analysis at Service Loads

Analysis of Ultimate Strength Analysis of Behaviour

Notations

Geometric Properties

A prestressed axial member may also have non-prestressed reinforcement to carry the

axial force. This type of members is called partially prestressed members. The

commonly used geometric properties of a prestressed member with non-prestressed

reinforcement are defined as follows.

A = gross cross-sectional area

A c = area of concrete

A s = area of non-prestressed reinforcement

A p = area of prestressing tendons

A t = transformed area of the section

= A c + ( E s / E c ) A s + ( E p / E c ) A p

The following figure shows the commonly used areas of a prestressed member with

non-prestressed reinforcement.
http://analysis%20at%20transfer/http://analysis%20at%20service%20loads/http://analysis%20of%20ultimate%20strength/http://analysis%20of%20behaviour/http://analysis%20of%20behaviour/http://analysis%20of%20ultimate%20strength/http://analysis%20at%20service%20loads/http://analysis%20at%20transfer/


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= +

A A c A p A s

+= +

A A c A p A s

+

A

A t A

A t Figure 3-1.1 Areas for a prestressed member with non-prestressed reinforcement

3.1.1 Introduction

The study of members under axial load gives an insight of the behaviour of a

prestressed member as compared to an equivalent non-prestressed reinforced concrete

member. Prestressed members under axial loads only, are uncommon. Members such

as hangers and ties are subjected to axial tension. Members such as piles may have

bending moment along with axial compression or tension. In this section, no

eccentricity of the CGS with respect to CGC is considered. The definitions of CGS and

CGC are provided in Section 2.1, Losses in Prestress (Part I). The following figure

shows members under axial loads.


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Hangers PilesHangers Piles Figure 3-1.2 Members under axial load

The analysis of members refers to the evaluation of the following.

1) Permissible prestress based on allowable stresses at transfer .

2) Stresses under service loads . These are compared with allowable stresses

under service conditions.

3) Ultimate strength. This is compared with the demand under factored loads .

4) The entire axial load versus deformation behaviour.

The stages for loading are explained in Section 1.2, Advantages and Types of

Prestressing

3.1.2 Analysis at Transfer

The stress in the concrete ( f c ) in a member without non-prestressed reinforcement can

be calculated as follows.

c c

P f = - A

0

(3-1.1)

Here,

P 0 = prestress at transfer after short-term losses.

In presence of non-prestressed reinforcement, the stress in the concrete can be

calculated as follows.


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c c s c

P f = -

A + (E /E )A0

(3-1.2) s

The permissible prestress is determined based on f c to be within the allowable stress at

transfer.

3.1.3 Analysis at Service Loads

The stresses in concrete in a member without non-prestressed reinforcement can be

calculated as follows.

(3-1.3) ec

c t

P P f = -

A A

Here,

P = external axial force (In the equation, + for tensile force and vice

versa.)

P e = effective prestress.

If there is non-prestressed reinforcement, A c is to be substituted by ( A c + ( E s /E c ) A s) and

A t is to be calculated including A s .

The value of f c should be within the allowable stress under service conditions.

3.1.4 Analysis of Ultimate Strength

The ultimate tensile strength of a section ( P uR ) can be calculated as per Clause 22.3,

IS:1343 - 1980 .

In absence of non-prestressed reinforcement,(3-1.4a)

uR Pk pP = f A0.87

In presence of non-prestressed reinforcement,

(3-1.4b) uR y s Pk pP = f A + f A0.87 0.87


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In the previous equations,

f y = characteristic yield stress for non-prestressed reinforcement with mild

steel bars

= characteristic 0.2% proof stress for non-prestressed reinforcement

with high yield strength deformed bars.

f pk = characteristic tensile strength of prestressing tendons.

The ultimate tensile strength should be greater than the demand due to factored loads.

The ultimate compressive strength of a section ( P uR ) can be calculated in presence of

moments by the use of interaction diagrams . For a member under compression with

minimum eccentricity, the ultimate strength is given as follows. Here, the contribution of

prestressing steel is neglected.

P uR = 0.4 f ck A c + 0.67 f y A s (3-1.5)

3.1.5 Analysis of Behaviour

The analysis of behaviour refers to the determination of the complete axial load versus

deformation behaviour. The analyses at transfer, under service loads and for ultimate

strength correspond to three instants in the above behaviour.

The analysis involves three principles of mechanics (Reference: Collins, M. P. andMitchell, D., Prestressed Concrete Structures , Prentice-Hall, Inc., 1991).

1) Equilibrium of internal forces with the external loads at any point of the load

versus deformation behaviour. The internal forces in concrete and steel are

evaluated based on the respective strains, cross-sectional areas and the

constitutive relationships.

2) Compatibility of the strains in concrete and in steel for bonded tendons. This

assumes a perfect bond between the two materials. For unbonded tendons, thecompatibility is in terms of total deformation.

3) Constitutive relationships relating the stresses and the strains in the materials.

The relationships are developed based on the material properties.


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Equilibrium Equation

At any instant, the equilibrium is given by the following equation.

P = A c f c + A sf s + A pf p (3-1.6)

Here,

f c = stress in concrete

f s = stress in non-prestressed reinforcementf p = stress in prestressed tendons

P = axial force.

Compatibility Equations

For non-prestressed reinforcement

s = c (3-1.7)

For prestressed tendons

p = c + p (3-1.8)

Here,

c = strain in concrete at the level of the steel

s = strain in non-prestressed reinforcement

p = strain in prestressed tendons

p = strain difference in prestressed tendons with adjacent concrete

The strain difference ( p) is the strain in the prestressed tendons when the concrete

has zero strain ( c = 0). This occurs when the strain due to the external tensile axial

load balances the compressive strain due to prestress. At any load stage,

p = pe ce (3-1.9)

Here,

pe = strain in tendons due to P e , the prestress at service

ce = strain in concrete due to P e .

The strain difference is further explained in Section 3.4, Analysis of Member under

Flexure (Part III).

Constitutive Relationships

The constitutive relationships can be expressed in the following forms based on the

material stress-strain curves shown in Section 1.6, Concrete (Part II), and Section 1.7,

Prestressing Steel.


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For concrete under compression

f c = F 1 ( c ) (3-1.10)

For prestressing steel

f p = F 2 ( p) (3-1.11)

For reinforcing steel

f s = F 3 ( s) (3-1.12)

The stress versus strain curve for concrete is shown below. The first and third

quadrants represent the behaviour under tension and compression, respectively.

c

f c

c

f c

Figure 3-1.3 Stress versus strain for concrete

The stress versus strain curve for prestressing steel is as shown below.

p

f p

p

f p

Figure 3-1.4 Stress versus strain for prestressing steel

The following stress versus strain curve is for reinforcing steel.


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s

f s

s

f s

Figure 3-1.5 Stress versus strain for reinforcing steel

The equilibrium and compatibility equations and the constitutive relationships can be

solved to develop the axial force versus deformation curve. The deformation can be

calculated as c L, where L is the length of the member.

The following plot shows the axial force versus deformation curves for prestressed and

non-prestressed sections. The two sections are equivalent in their ultimate tensile

strengths.

Deformation

Axial force

Cracking Tensile strengths

Compressive strengths

Prestressed section

Non-prestressed section

Deformation

Axial force



Deformation

Axial force



Prestressed section

Non-prestressed section Figure 3-1.6 Axial force versus deformation curves

From the previous plot, the following can be inferred.

1) Prestressing increases the cracking load.

2) Prestressing shifts the curve from the origin.

For the prestressed member, there is a compressive deformation in absence

of external axial force.

A certain amount of external force is required to decompress the member.


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3) For a given tensile load, the deformation of the prestressed member is

smaller.

Prestressing reduces deformation at service loads.

4) For a given compressive load, the deformation of the prestressed member is

larger.

Prestressing is detrimental for the response under compression.

5) The compressive strength of the prestressed member is lower.

Prestressing is detrimental for the compressive strength.

6) For a partially prestressed section with the same ultimate strength, the axial load

versus deformation curve will lie in between the curves for prestressed and non-

prestressed sections.

The above conclusions are generic for prestressed members.


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3) Constitutive relationships relating the stresses and the strains in the materials.

Variation of Internal Forces

In reinforced concrete members under flexure, the values of compression in concrete

(C ) and tension in the steel ( T ) increase with increasing external load. The change in

the lever arm ( z ) is not large.

In prestressed concrete members under flexure, at transfer of prestress C is located

close to T . The couple of C and T balance only the self weight. At service loads, C

shifts up and the lever arm ( z ) gets large. The variation of C or T is not appreciable.

The following figure explains this difference schematically for a simply supported beam

under uniform load.

Reinforced concreteC 2 > C 1 , z 2 z 1

Prestressed concreteC 2 C 1, z 2 > z 1

w 2 > w 1 w 2

z 2C 2T 2

C 1

w 1

z 1T 1

w 2

z 2C 2T 2

w 1

C 1T 1

z 1

Reinforced concreteC 2 >

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Prestressed Concrete Structure - Amlan-Menson