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The Indian Concrete Journal June 201748

POINT OF VIEW

Long term losses in pre-stressed concrete member as per IS 1343:2012 and IS 1343:1980

P. Markandeya Raju and T. Raghuram Sandeep

In prestressed concrete structures, Creep and Shrinkage of concrete and Relaxation of prestressing steel are long term phenomena and cause gradual loss of compression in concrete and tension in prestressing steel. Their inaccurate estimation leads to serviceability problems like excessive camber and cracking of concrete. While revising IS 1343:1980, many modifications were incorporated in time dependent losses. There is a need to understand the background of these changes before applying and this article is a beginning in this direction. After discussing various parameters affecting Long term losses, the calculations involved and inferences on results were presented. A typical example explaining calculation of losses based on both codes is also presented.

1. InTRoducTIonLosses in prestressed concrete structures can be categorised as shown in Table 1.

In properly designed and manufactured members, the loss of stress due to Creep, Shrinkage and Relaxation of

steel account for major portion of the total loss. So their magnitudes are vital in arriving at the residual prestress. A reasonably accurate prediction of these losses is important to ensure satisfactory performance of structures in service. If prestress losses are underestimated, the tensile strength of concrete can exceed under service loads, causing cracking and unexpected excessive deflection. On the other hand, overestimating prestress losses can lead to excessive camber and an uneconomical design. To determine with precision the extent of the losses from them is a challenging task. The scientific model presented in [1] the new code (IS 1343:2012) provides an accurate mathematical model that is in line with Model Code 90 [2] (CEB MC-90).

The objective of this paper is to discuss in detail, each parameter of long term loss as mentioned in the new code (IS 1343: 2012) and to explain them with a typical example based on assumed data. The results are compared with those obtained based on old code [3] (IS 1343:1980).

Table 1. Categorisation of lossesInstantaneous Losses or Initial Losses of Prestress Time Dependent Losses or Final Losses of Prestress

1. Loss of prestress due to Elastic deformation of beam 1. Loss of prestress due to Bending of the beam

2. Loss of prestress due Anchorage slip 2. Loss of prestress due to Relaxation or creep of prestressing steel

3. Loss of prestress due to Friction between tendon and duct(a) Curvature effect(b) Wave effect or Length effect

3. Loss of prestress due to Shrinkage in the beam

4. Losses of prestress due to Controlled prestressing force4. Loss of prestress due to Temperature changes

5. Loss of prestress due to Elongation of the tendon.

The Indian Concrete Journal June 2017 49

POINT OF VIEW

2.cReeP In concReTeCreep in concrete is associated with time, level of applied stress, density of concrete, cement content, water cement ratio, thickness of the elements and atmospheric conditions like humidity and temperature. Creep is particularly important in prestressed concrete as the continued long term shortening of concrete in compression leads to reduction in prestressing force. The creep strain is proportional to elastic strain at lower stress levels in concrete. Hence most of the statutory codes express creep strain as a factor times elastic strain and the factor is called ‘’Creep co-efficient”. Several factors influence creep of concrete. Some of the most important factors influencing creep are presented, with an emphasis on those factors that are most important for evaluating residual prestress.

2.1 Age at loading

The strength of concrete increases with time due to the hydration of the cement i.e., creep decreases with age at loading. Creep is inversely proportional to the degree of hydration, and hence the type of cement will influence the creep. For instance, at the same age of loading, the use of slow-hardening cement will increase creep compared with the use of standard Portland cement.

At higher prestress levels, creep stress is not proportional to Elastic stress and the rate of change of creep with Elastic stress increases and the variation is non-linear as shown in Figure 1. In addition, other factors which influence the hydration of the cement, such as water/cement ratio,

moisture conditions and temperature, will also affect the creep at a given age of loading. Assuming a constant stress state in the concrete, the gain in strength with time reduces the creep since the stress-strength ratio will decrease. The age at loading will also decrease the contribution from the drying creep on the total strains since most of the drying of the concrete would have already occurred without affecting the creep behaviour.

2.2 Untensioned reinforcement

The untensioned reinforcement in concrete has a restraining effect on the creep strain as some of the concrete stress will be transferred to reinforcement as the long-term strains due to creep and shrinkage develop. They will reduce the prestress loss in the tendons but not the loss of stress in concrete. Oh et al. (1995) studied the effect of different reinforcement ratios on creep of high-strength concrete and observed that the reduction of creep strain in concrete with reinforcement ratios of 0.64% and 1.78% was 15% and 33% respectively [5,6] (Figure 2).

2.3 Size

The main influence of size on creep is in drying state (after curing period). The size of the structure affects the drying rate and thus the drying creep rate. Creep decreases as the volume to surface ratio increases. A possible explanation for the size effect on creep is that since the drying is much slower for larger structures, the hydration in the inner parts of the structure will continue and thus relatively higher concrete strength is achieved, which reduces creep, when the drying process initiates [6].


POINT OF VIEW

2.4 Properties of concrete

The properties of concrete that have significant influence on creep are those which affect the strength development in concrete i.e., water-cement ratio and type and fineness of cement. Considering the same age at loading and applied stress, the use of different types of cement will influence creep. Slow-hardening cements will exhibit the largest creep strains and rapid- hardening cements, the lowest. Another property worth mentioning is the fineness of cement which also influences the development of concrete strength. The finer the cement, the higher the specific surface area and faster is the rate of development of strength. Similarly the aggregate content of the concrete will have a restraining effect on the creep deformations since the aggregate does not undergo creep. The most frequently used aggregates, such as granite and gneiss, have very low volume change and significantly high modulus of elasticity and thus higher restraining capacity than the cement paste. This lowers the creep in concretes with higher aggregate content.

2.5 Relative humidity

Relative humidity is defined as the ratio of partial pressure of water vapour to the equilibrium or saturated vapour pressure. A concrete member is said to be in moisture equilibrium if the moisture in the surrounding air is same as that in the member. The ambient relative humidity that is in moisture equilibrium with the surrounding air will have very low influence on creep in concrete. The higher the relative humidity, lesser the drying and hence resulting in lower creep values. However, the moisture content of the concrete specimen will influence creep. The lower the moisture content the lower the creep. Results from several studies indicate that completely dry specimens exhibit significantly lower or no creep than those containing small amounts of moisture. Further as temperature increases, equilibrium vapour pressure increases and hence relative humidity decreases.

In IS 1343:1980, ultimate creep coefficients are presented for different ages of loading. These recommendations are only for structures where losses need not be evaluated at various stages. They are not valid for evaluation of residual prestress or deflection or camber in structures at various stages of stressing/loading/measuring. IS 1343:1980 has no mention of the effect of relative humidity and element thickness on loss of prestress. Although IS 1343:2012 calculates creep co-efficient for a given relative humidity and element thickness, accurate prediction of relative humidity on the day of loading during design calculations is not practically possible. Further there is a need to incorporate the effect of untensioned reinforcement in the model for the determination of creep

co-efficient. Majority of the short comings of IS 1343:1980 are addressed in IS 1343:2012 with a scientific mathematical model that incorporates various parameters. However, this model is valid only if stress in concrete does not exceed one-third of characteristic compressive strength of concrete and concrete should be of normal concrete ranging from M30 to M60. These models are not applicable for special concretes [1].

3. ShRInkage In concReTeConcrete starts to lose moisture and undergoes a change in volume (due of chemical reaction between cement and water) towards the end of curing period. This phenomenon, known as concrete shrinkage, starts to develop rapidly after the end of the curing period. Excess water in concrete evaporates and cement matrix around aggregate contracts. Shrinkage is basically divided into two components namely Autogenous shrinkage and drying shrinkage. Autogenous shrinkage occurs during early hydration and is caused by the internal consumption of water during hydration as the hydration products occupy less volume than the unhydrated cement and water [7]. Drying shrinkage is caused by loss of water from concrete to the atmosphere [7]. Generally this loss of water is from the cement paste, but with a few types of aggregates (with high water absorption), the main loss of water contributing to the drying shrinkage of concrete is from aggregate. Drying shrinkage is relatively slow and the stress it induces when restrained is partially relieved by tensile creep. The rate of drying shrinkage is dependent upon the relative humidity of the surrounding air and the element geometry. The drying shrinkage is partially reversible, i.e. upon rewetting; the swelling strains will be less than the preceding shrinkage strains [6]. Similar to creep, shrinkage also depends on various factors that are presented below.

3.1 Properties of concrete

Generally, higher the water-cement ratio, greater is the shrinkage. The water content in concrete has maximum influence on shrinkage as it is proportional to the amount of water that can leave the pore system of concrete. The increase in cement content at constant water-cement ratio also increases shrinkage. This is because, the hydrated cement occupies less volume than cement paste in concrete. Another factor which influences the shrinkage is aggregate content. Since the aggregate is minimally affected by moisture changes in concrete, it does not shrink and thus has a restraining effect on concrete.

3.2 Untensioned reinforcement

Similar to the effect of aggregate, untensioned reinforcement has a restraining effect on shrinkage of concrete. In prestressed


POINT OF VIEW

concrete, shrinkage is unrestrained as untensioned steel is not considered for evaluating losses. There may be a reduction in shrinkage loss if the effect of untensioned reinforcement is considered. Oh et al. (1995) studied the effect of different reinforcement ratios on shrinkage of concrete and observed that the reduction of shrinkage strain in concrete with reinforcement ratios of 0.64 % and 1.78% was 14% and 30% respectively [5,6]

3.3 Size

The size of a concrete member mainly influences the drying rate and there by the rate of shrinkage significantly. It also has an effect on the final shrinkage strain. The influence of size on shrinkage of a concrete member is proportional to the volume to surface ratio, i.e. ratio of volume of member to the surface in contact with surrounding air. The lower the ratio, faster is the development of shrinkage. But the final shrinkage strain decreases with increase in volume to surface ratio, which means that final shrinkage strain is size dependent [6].

3.4 Relative humidity

The moisture content of the concrete specimen if not in moisture equilibrium with ambient relative humidity will have influence on drying shrinkage. The lower the moisture content the lower is the shrinkage because the rate of drying is faster.

IS 1343:1980 defines the values of ultimate shrinkage strain depending on age of loading alone and states that it has to be increased by 50% under dry atmospheric conditions for post-tensioned members. Whereas IS 1343:2012 has incorporated various parameters affecting shrinkage as discussed above. However, the new model as per revised code does not factor in, the age of loading which is true to the practical situation.

4. RelaxaTIon of STeelUnder sustained loading of prestressing force, the strand steel gradually relaxes. The resulting reduction in prestress is called Relaxation loss. Relaxation loss increases with prestress and temperature. The relaxation losses of low-relaxation strands are considerably less than the loss in normal-relaxation strand. Relaxation of a prestressing strand depends on the stress level in the strand. However, because of other prestress losses, there is a continuous reduction of the strand stress, which causes a reduction in relaxation [6].

To understand the process of calculation of long-term losses a typical example is considered for study.

5. exaMPle PRobleMA post-tensioned concrete beam shown in Figure 3 is stressed on 7th day.

Span of the beam = 20.0 m Diameter of strand = 9.5 mm Number of strands =10 Grade of concrete = M35 Curing period = 5 days Relative humidity (RH) = 80%. Loss due elastic shortening = 5% Nominal cover to steel = 75 mm. Live load =10 kN/m

Age at which live load is subjected on the beam = 45 days.

The following are evaluated

1. Residual prestress on 28, 45, 70, 90, 25550 days when all strands are stressed on 7th day. (Single stage stressing)

2. Residual prestress on 28, 45, 70, 90, 25550 days when 5 strands are stressed on 7th day and remaining 5 strands on 28th day. (Multistage stressing)

5.1 Creep

As per Clause 6.2.5 of IS 1343:2012, creep loss is evaluated based on creep co-efficient method. The final creep co-efficient given in the Table of Clause 6.2.5 can also be arrived by equations given in the same clause and corresponding sub-clauses. The final creep co-efficient given in the table corresponds to grade of concrete ranging from M30 to M60, subject to the condition that the compressive stress does


POINT OF VIEW

not exceed 0.36fck. These values can be used where the end results are not sensitive to precise values.

The creep co-efficient is given by

( ) ( )

( ) = Creep strain at time t>to.

( ) = Initial strain at loading.

= Initial time of loading.

The creep coefficient ( ) is given by

( ) ( )

Where

= Notional creep co-efficient to which the creep co-efficient reaches asymptotically with time (this value can be taken as value reached in 70 years).

( ) = Co-efficient describing development with time

The notional creep co-efficient is given by

( ) ( )

= a factor to allow for the effect of relative humidity on the notional creep coefficient.

⁄ √

⁄ √

RH = Relative humidity of the ambient environment in percentage

ho= notional factor (Approximately the distance travelled by water molecule from the centre point of the cross-section to the surface of the concrete) it is given by

= Area of cross-sectional area (mm2) = The perimeter of the member in contact with the atmosphere or exposed to drying (mm).

( ) = a factor to allow for the effect of concrete strength on the notional creep coefficient

√

√

( ) = a factor to allow for the effect of concrete age at loading on the notional creep coefficient

( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

= Creep strain at time t>to.

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

= Initial strain at loading.

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )


The creep coefficient

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

is given by

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

where

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )


( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

= Co-efficient describing development with time

The notional creep co-efficient

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

is given by

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )




( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

= Area of cross-section (mm2)

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

= The perimeter of the member in contact with the atmosphere or exposed to drying (mm).

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

= a factor to allow for the effect of concrete strength on the notional creep coefficient

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

= a factor to allow for the effect of concrete age at loading on the notional creep coefficient

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( ) = 2.28 (from 7th day to infinity)

Similarly the ultimate or notional creep co-efficients are evaluated for 28,45,70,90, 365 and infinity (70 years i.e. 365 70 = 25550) days and presented in the Table 2. Creep co-efficient is directly proportional to creep loss. The values from the table clearly indicate that the members which are stressed at the early age will have more loss due to creep and loss decreases with the increase in the age of loading.

Creep co-efficient as per IS 1343:1980 is given in Clause 5.2.5.1 for age of loading on 7th, 28th and 365th day. Co-efficients for remaining days are interpolated and presented in Table 2.

For this case study it can be observed that the old code underestimates the creep co-efficient values for age of

Table 2. Notional creepco-efficient

t, days

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

(IS 1343:2012)

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

(IS 1343:1980)

7 0.64 2.28 2.2028 0.49 1.75 1.6045 0.45 1.60 1.5770 0.41 1.47 1.5490 0.39 1.40 1.51365 0.30 1.07 1.10

25550 0.13 0.47 -


POINT OF VIEW

loading up to 45 days and over estimates after then. The development of creep with time is given in Clause 6.2.5.2 of IS 1343:2012 as follows.

= 2.28 (from 7th day to infinity) Similarly the ultimate or notional creep co-efficients are evaluated for 28,45,70,90, 365 and infinity (70 years i.e. 365 70 = 25550) days and presented in the Table 2. Creep co-efficient is directly proportional to creep loss. The values from the table clearly indicate that the members which are stressed at the early age will have more loss due to creep and loss decreases with the increase in the age of loading. Creep co-efficient as per IS 1343:1980 is given in Clause 5.2.5.1 for age of loading on 7th, 28th and 365th day. Co-efficients for remaining days are interpolated and presented in Table 2. For this case study it can be observed that the old code underestimates the creep co-efficient values for age of loading up to 45 days and over estimates after then. The development of creep with time is given in Clause 6.2.5.2 of IS 1343:2012 as follows.

( ) [

( )]

Where

t = age of concrete in days at the moment considered,

to= age of concrete at loading in days,

= coefficient depending on relative humidity (RH in percentage) and notional member size (h0 in mm). RH = relative humidity expressed as percentage. RH0 = 100 (that is, 100 percentage of relative humidity)

[ (

) ]

[ ( ) ]

In the present case, member is stressed on 7th day and losses are evaluated on 28th day.

( ) [ ( )]

The co-efficient ‘0.38’ is a fraction of the ultimate creep co-efficient that has occurred from 7th day to 28th day i.e. creep co-efficient for 7 to 28 days is . Similarly creep co-efficient for respective days are evaluated and presented in Table 3. Creep co-efficient cannot be estimated in respective intervals with respect to IS 1343:1980 as there are no guidelines to arrive at creep co-efficient for evaluating residual prestressing force at various stages. 5.2 Shrinkage

As per IS 1343:2012 Clause 6.2.4. Total shrinkage strain is given by

Where = total shrinkage train = drying shrinkage strain

where




( ) [

( )]

Where




[ (

) ]

[ ( ) ]


( ) [ ( )]




= coefficient depending on relative humidity (RH in percentage) and notional member size (h0 in mm).

RH = relative humidity expressed as percentage.

RH0 = 100 (i.e. 100% of the relative humidity)

RH = relative humidity expressed as percentage.



( ) [

( )]

Where




[ (

) ]

[ ( ) ]


( ) [ ( )]




The co-efficient ‘0.38’ is a fraction of the ultimate creep co-efficient that has occurred from 7th day to 28th day i.e. creep co-efficient for 7 to 28 days is 0.38 x 2.28 =0.85. Similarly creep co-efficient for respective days are evaluated and presented in Table 3.

Table 3. Creep co-efficient for various intervals

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

(As per IS 1343:2012)

t, days Stressed on 7th day Stressed on 28th day


( ) [

( )]

Where




[ (

) ]

[ ( ) ]


( ) [ ( )]




( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )


( ) [

( )]

Where




[ (

) ]

[ ( ) ]


( ) [ ( )]




( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

7 0 0 - -28 0.38 0.85 0 045 0.44 1.01 0.35 0.6270 0.51 1.16 0.46 0.8090 0.55 1.25 0.51 0.89365 0.76 1.73 0.75 1.32

25550 1.00 2.28 1.00 1.25

Creep co-efficient cannot be estimated in respective intervals with respect to IS 1343:1980 as there are no guidelines to arrive at creep co-efficient for evaluating residual prestressing force at various stages.

5.2 Shrinkage



( ) [

( )]

Where




[ (

) ]

[ ( ) ]


( ) [ ( )]




where


( ) [

( )]

Where




[ (

) ]

[ ( ) ]


( ) [ ( )]




= total shrinkage strain


( ) [

( )]

Where




[ (

) ]

[ ( ) ]


( ) [ ( )]



Where = total shrinkage train = drying shrinkage strain = drying shrinkage strain

= autogeneous shrinkage strain Table mentioned in the Clause 6.2.4.2 of IS 1343:2012 gives the values of autogeneous shrinkage strain with respect to grade of concrete. These values have been multiplied by 106 and hence it has to be divided by 106 to get the actual values. This strain is the ultimate strain or total autogeneous shrinkage strain that a member will experience it its life time. Whereas the development of autogeneous shrinkage with time is given by equation mentioned in Clause 6.2.4.4 which is as follows.

( ) ( )

( ) ( √ ) Where, t = time in days ( ) = Total strain occurred at a given point of time. ( ) = co-efficient describing autogeneous shrinkage with time. = Autogeneous shrinkage strain that can occur in a member in its life time. In the present case, for M35 grade concrete, the autogeneous shrinkage strain from the table in Clause 6.2.4.2 is

The autogeneous shrinkage strain is computed for different time intervals and presented in Table 4. The co-efficient ( ) gives the fraction of autogeneous shrinkage that has occurred upto‘t’ days. For example co-efficient for 7 days is ‘0.41’ which indicates that 41% of the total autogeneous shrinkage ( ) has occurred upto 7th day. Remaining 59% will occur from 7th day to 25550th day. The summation of column 3 in Table 4 gives the total percentage of autogeneous strain that has occurred from 0 days to 25550 days which is always 100% i.e. factor 1.00, and summation of column 4 gives the total autogeneous shrinkage strain that occurred from 0 days to 25550 days i.e. 1.00 . 5.2.1 Drying Shrinkage

Drying shrinkage generally begins at the end of curing period. Drying shrinkage is given by Clause 6.2.4.3 of IS 1343:2012 as follows

Where, = Total drying shrinkage strain that a member will undergo in its life time i.e. for 70 years = coefficient depending on notional size ho and the values of kh for corresponding hoare given in table of Clause 6.2.4.3 of IS 1343:2012. ho = notion factor = 125.217 mm From table of Clause 6.2.4.3, after interpolation kh= 0.987 = unrestrained drying shrinkage which depends on grade of concrete and relative humidity (RH) expressed in terms of percentage (50 and 80). These values are given in the table of Clause 6.2.4.3 of IS 1343:2012. IS 1343:2012 is not clear on source from which RH has to be considered. Hence in general 50% can be considered for internal members of the building or structure which are not exposed to atmosphere and 80% for components which are exposed to atmosphere.

= autogeneous shrinkage strain

Table mentioned in the Clause 6.2.4.2 of IS 1343:2012 gives the values of autogeneous shrinkage strain with respect to grade of concrete. These values have been multiplied by 106 and hence it has to be divided by 106 to get the actual values. This strain is the ultimate strain or total autogeneous shrinkage strain that a member will experience it its life time. Whereas the development of autogeneous shrinkage with time is given by equation mentioned in Clause 6.2.4.4 which is as follows.


( ) ( )






( ) ( )





where,

t = time in days


( ) ( )





= Total strain occurred at a given point of time.


( ) ( )





= co-efficient describing autogeneous shrinkage with time.


( ) ( )





= Autogeneous shrinkage strain that can occur in a member in its life time.

In the present case, for M35 grade concrete, the autogeneous shrinkage strain from the table in Clause 6.2.4.2 is


( ) ( )





The autogeneous shrinkage strain is computed for different time intervals and presented in Table 4.

The co-efficient


( ) ( )





gives the fraction of autogeneous shrinkage that has occurred upto‘t’ days. For example co-


POINT OF VIEW

efficient for 7 days is ‘0.41’ which indicates that 41% of the total autogeneous shrinkage


( ) ( )





has occurred upto 7th day. Remaining 59% will occur from 7th day to 25550th day. The summation of column 3 in Table 4 gives the total percentage of autogeneous shrinkage strain that has occurred from 0 days to 25550 days which is always 100% i.e. factor 1.00, and summation of column 4 gives the total autogeneous shrinkage strain that occurred from 0 days to 25550 days i.e. 1.00 45 10-6

5.2.1 Drying ShrinkageDrying shrinkage generally begins at the end of curing period. Drying shrinkage is given by Clause 6.2.4.3 of IS 1343:2012 as follows


( ) ( )





where,


( ) ( )





= Total drying shrinkage strain that a member will undergo in its life time i.e. for 70 years


( ) ( )





= coefficient depending on notional size ho and the values of kh for corresponding hoare given in table of Clause 6.2.4.3 of IS 1343:2012.

ho = notion factor = 125.217 mm

From table of Clause 6.2.4.3, after interpolation kh= 0.987


( ) ( )





= unrestrained drying shrinkage which depends on grade of concrete and relative humidity (RH) expressed in terms of

percentage (50 and 80). These values are given in the table of Clause 6.2.4.3 of IS 1343:2012.

IS 1343:2012 is not clear on source from which RH has to be considered. Hence in general 50% can be considered for internal members of the building or structure which are not exposed to atmosphere and 80% for components which are exposed to atmosphere.

In the present case study M35 grade concrete and RH = 80% are considered. From the table in Clause 6.4.2 for M35 grade of concrete,


( ) ( )





is obtained as 292.000


( ) ( )





10-6after interpolation.

Drying shrinkage strain at infinity (70 years = 25550 days) is given by

In the present case study M35 grade concrete and RH = 80% are considered. From the table in Clause 6.4.2 for M35 grade of concrete, is obtained as 292.000 10-6after interpolation. Drying shrinkage strain at infinity (70 years = 25550 days) is given by 6

= 288.318 10-6

The development of drying shrinkage strain with time is given by Clause 6.2.4.5 as follows

( ) Where, = total drying shrinkage strain occurred at a given point of time ‘t’ in days. ( ) = co-efficient describing the drying shrinkage with time. It is given by the equation mentioned in Clause 6.2.4.4 of IS 11343:2012 as follows

( ) ( )

( ) √

= age of concrete at the beginning of drying shrinkage i.e. no of days curing has been done = 5 days (assumed in the present problem)

( ) ( )

( ) √

Drying shrinkage strain on 7th day is

Similarly drying shrinkage strains are computed for different time intervals and presented in the Table 5. The co-efficient ( ) gives the fraction of drying shrinkage that has occurred upto‘t’ days. For example, co-efficient for 28 days is ‘0.29’ which indicates that 29% of total drying shrinkage ( ) has occurred upto 28 days. Remaining 71% will occur from 28th day to 25550th day. The summation of column 4 in Table 5 gives the total percentage of drying strain that has occurred from 5th day (Curing period) to 25550th day which is always 100% i.e. factor 1.00, and summation column 5 gives the total drying shrinkage strain that has occurred during 5th day to 25550th day i.e. 1.00 . Shrinkage strain as per IS 1343:1980 is given by Clause 5.2.4.1. For post tensioned members ultimate shrinkage strain is given by

( )

Where t = age of concrete at transfer in days (7th day) For member loaded on 7th day

( )

= 0.987 292.00 10-6


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )



= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

where,


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

= total drying shrinkage strain occurred at a given point of time ‘t’ in days.


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

= co-efficient describing the drying shrinkage with time.

Table 4. Autogeneous shrinkage straint, days Coeff.Describingautogeneous shrinkage

strain with time (

( ) ( )





( ) ( )

Where




( ) ( )


⁄ √

⁄ √





√

√


( ) ( )

as (t ))Coeff. of autogeneous

shrinkage strain occurred during the interval

Autogeneous shrinkage strain occurred during the interval = Coeff. of autogeneous shrinkage

strain occurred during the interval X Eca ( 45 10-6 )

7 0.41 0.41 (Up to 7th day) 18.49 X 10-6 (On 7th day)

28 0.65 0.24 10.89 X 10-6

45 0.74 0.09 3.85 X 10-6

70 0.81 0.07 3.32 X 10-6

90 0.85 0.04 1.69 X 10-6

25550 1.00 0.15 6.75 X 10-6

Total 1.00 45 X 10-6


POINT OF VIEW

It is given by the equation mentioned in Clause 6.2.4.4 of IS 1343:2012 as follows


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

= age of concrete at the beginning of drying shrinkage i.e. no of days curing has been done

= 5 days (assumed in the present problem)


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )



= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

Similarly drying shrinkage strains are computed for different time intervals and presented in the Table 5.

The co-efficient


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

gives the fraction of drying shrinkage that has occurred upto ‘t’ days. For example, co-efficient for 28 days is ‘0.29’ which indicates that 29% of total drying shrinkage


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

has occurred upto 28 days. Remaining 71% will occur from 28th day to 25550th day. The summation of column 4 in Table 5 gives the total percentage of drying strain that has occurred from 5th day (Curing period) to 25550th day which is always 100% i.e. factor 1.00, and summation column 5 gives the total drying shrinkage strain that has occurred during 5th day to 25550th day i.e. 1.00 .


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

Shrinkage strain as per IS 1343:1980 is given by Clause 5.2.4.1. For post tensioned members ultimate shrinkage strain is given by


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

where

t = age of concrete at transfer in days (7th day)

For member loaded on 7th day


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

This code is silent on estimation of shrinkage strain in respective intervals and has no special consideration for multi stage prestressing.

The autogeneous shrinkage strain and drying shrinkage strain are combined together to arrive at total shrinkage strain which are presented in Table 6.

Table 5. Drying shrinkage straint, days


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

Coeff.describing drying shrinkage strain with time

(


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

)

Coeff. Of drying shrinkage strain occurred during the

interval

Drying shrinkage strain occurred during the interval = Coeff. Of autogeneous shrinkage

strain occurred during the interval


= 288.318 10-6



( ) ( )

( ) √


( ) ( )

( ) √



( )


( )

5 0 - - -

7 2 0.03 0.03 9.93 X 10-6 (On 7th day)

28 23 0.29 0.26 73.96 X 10-6

45 40 0.42 0.13 36.18 X 10-6

70 65 0.54 0.12 34.75 X 10-6

90 85 0.60 0.07 18.93 X 10-6

25550 (

Creep loss ( ) .

Total shrinkage strain in the period 70-90 days is obtained from Table 6 as 20.62 10-6

Shrinkage loss = .

Residual prestressing force on 90thday .

5.4.5 Residual prestressing force at infinity (25550 days)

Strands stressed on 7th day will have a residual prestressing force of 657.605 kN on 90th day with a loss % of 0.432 in the period 70-90 days. Bending stress or net stress in the beam at mid span is.

( )

From Table 3, Creep co-efficient on 90th and 25550th day are 1.25 and 2.28 respectively.

Creep loss ( ) .

Total shrinkage strain in the period 90 - days is obtained from Table 6 as 121.32 10-6

Shrinkage loss =

As per Clause 19.5.2.3 of IS 1343:2012, for long term relaxation loss values given in Table 6 of code should be multiplied by 3. Loss between 90 days to infinity is

Residual prestressing force on 25550thday

5.4.6 Losses as per IS 1343:1980

Creep co-efficient for a member loaded on 7th day, from Table 2 is 2.2. Total loss due to creep is

Shrinkage strain for a member loaded on 7th day is obtained from Table 6 as 209.59 10-6. Total loss due to shrinkage is

Relaxation loss for initial stress of 0.7 fpis 70 MPa. Total relaxation loss is

) 1.00 0.40 114.57 X 10-6

Total drying shrinkage strain 1.00 288.318 X 10-6

Table 6. Comparison of total shrinkage strainPeriod As per IS 1343:2012 Strain

as per IS 1343:1980Autogenous

strainDrying strain

Total Shrinkage

Strain

7-28 10.89 X10-6 73.96 X10-6 84.85 X10-6

209.59 X 10-6

28-45 3.85 X10-6 36.18 X10-6 40.04 X10-6

45-70 3.32 X10-6 34.75 X 10-6 38.07 X10-6

70-90 1.69 X10-6 18.93 X10-6 20.62 X10-6

90-

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







6.75 X10-6 114.57 X10-6 121.32 X10-6

Total strain 26.51X10-6 278.38 X10-6 304.89 X10-6


POINT OF VIEW

5.3 Relaxation in steel

Relaxation loss depends on the initial jacking force. The maximum initial jacking force as per Clause 19.5.1 of IS 1343:2012 is 76% of ultimate tensile strength of wire or bar or strand. Assuming that the average stress to be 95% of 76% and considering 5% loss for elastic shortening, the average stress in the strand after anchorage will be as follows.

This code is silent on estimation of shrinkage strain in respective intervals and has no special consideration for multi stage prestressing. The autogeneous shrinkage strain and drying shrinkage strain are combined together to arrive at total shrinkage strain which are presented in Table 6. 5.3 Relaxation in steel


From Table 6 of IS 1343:2012, for low relaxation strands, loss is 2.5%. For long-term relaxation losses, the values given in Table 6 of IS 1343:2012 should be multiplied by 3. The revised code has no mention of time limit for which the long term relaxation losses have to be evaluated.However it is understood that the long term relaxation loss (multiplying values of Table 6 of IS 1343:2012 by 3) have to be considered while evaluated losses at infinity. The loss 2.5% is the total loss due to relaxation i.e. at 1000 hours at 20 ± 2°C. Code did not specify any values for calculating losses up to 1000 hours and temperature greater than 20°C to evaluate relaxation loss in intervals. In other words the break up for total loss (2.5%) for different intervals upto 1000 hours is not available in the revised code. IRC 112:2011 considers the values up to 1000 hours and above and for the early age relaxation in case of initial temperatures higher than 40° C, asin case of steam curing [8, 9]. From Table 4 of IS 1343:1980 for the initial stress of 0.7fp, relaxation loss is 70 MPa. 5.4 Residualprestressing force for single stage stressing

When all the strands are stressed in single stage, the total prestressing force in the beam is calculated as follows.

Ultimate tensile strength of strand is 102.3 kN from Table 1 of IS 14268:1995. Maximum force allowed is only 76%. Assuming the average stress to be 95% of 76% and considering 5% loss for elastic shortening, the average prestressing force in the strand after anchorage will be as follows.

5.4.1 Residual prestressing force on 28th day

Strands stressed on 7th day will have a prestressing force of 716 kN located at 0.075 m from the bottom of girder. Bending stress in the beam at mid span due to prestressing is

( )

Self-weight of the beam

Bending moment due to self-weight of beam is

Bending tensile at the soffit of the beam at mid span due to self-weight is

From Table 6 of IS 1343:2012, for low relaxation strands, loss is 2.5%. For long-term relaxation losses, the values given in Table 6 of IS 1343:2012 should be multiplied by 3. The revised code has no mention of time limit for which the long term relaxation losses have to be evaluated.However it is understood that the long term relaxation loss (multiplying values of Table 6 of IS 1343:2012 by 3) have to be considered while evaluated losses at infinity. The loss 2.5% is the total loss due to relaxation i.e. at 1000 hours at 20 ± 2°C. Code did not specify any values for calculating losses up to 1000 hours and temperature greater than 20°C to evaluate relaxation loss in intervals. In other words the break up for total loss (2.5%) for different intervals upto 1000 hours is not available in the revised code. IRC 112:2011 considers the values up to 1000 hours and above and for the early age relaxation in case of initial temperatures higher than 40° C, asin case of steam curing [8, 9]. From Table 4 of IS 1343:1980 for the initial stress of 0.7fp, relaxation loss is 70 MPa.

5.4 Residual prestressing force for single stage stressing










( )













( )












( )












( )



Bending tensile at the soffit of the beam at mid span due to self-weight is Bending tensile at the soffit of the beam at mid span due to self-weight is

Bending compressive stress or net stress in concrete at the level of steel is

From Table 3, Creep co-efficient on 28th day is 0.85. Loss of prestressing force due to creep from 7th day to 28th day is

Where

m = modular ratio, expressed as ratio of modulus of elasticity of steel to concrete

fc = Stress in concrete at the level of steel

As= Total area of prestressing steel = 10 54.8 mm2


Shrinkage loss = .

Loss due to relaxation of steel is 2.5 % i.e.

Residual prestressing force on 28th day


Strands stressed on 7th day will have a residual prestressing force of 672.734 kNon 28th day with a loss % of 6.043 in the period 7-28 days. Bending stress in the beam at mid span due to prestressing is

( )

Bending compressive stress or net stress in concrete at the level of steel up to 45th day i.e. just before the beam is subjected to live load

From Table 3, Creep co-efficients on 28th and 45th day are 0.85 and 1.01 respectively.

Creep loss ( )




Where





Shrinkage loss = .





( )



Creep loss ( )




Where





Shrinkage loss = .





( )



Creep loss ( )

Where




Where





Shrinkage loss = .





( )



Creep loss ( )


As= Total area of prestressing steel = 10



Where





Shrinkage loss = .





( )



Creep loss ( )

54.8 mm2



Where





Shrinkage loss = .





( )



Creep loss ( )

Total shrinkage strain in the period 7-28 days is obtained from Table 6 as 84.85



Where





Shrinkage loss = .





( )



Creep loss ( )

10-6

Shrinkage loss =



Where





Shrinkage loss = .





( )



Creep loss ( )




Where





Shrinkage loss = .





( )



Creep loss ( )




Where





Shrinkage loss = .





( )



Creep loss ( )



Where





Shrinkage loss = .





( )



Creep loss ( )


POINT OF VIEW


Strands stressed on 7th day will have a residual prestressing force of 672.734 kN on 28th day with a loss % of 6.043 in the period 7-28 days. Bending stress in the beam at mid span due to prestressing is



Where





Shrinkage loss = .





( )



Creep loss ( )




Where





Shrinkage loss = .





( )



Creep loss ( )


Creep loss =



Where





Shrinkage loss = .





( )



Creep loss ( )

Total shrinkage strain in the period 28-45 days is obtained from Table 6 as 40.04 x 10-6Total shrinkage strain in the period 28-45 days is obtained from Table 6 as 40.04 10-6

Shrinkage loss = .


Live load of 10 kN/m is subjected on the beam on 45th day. Bending moment due to this live load is

Bending tensile stress at the soffit of the beam at mid span the due to live load is

Bending compressive stress or net stress on 45th day after the application of live load is

( )


Strands stressed on 7th day will have a residual prestress force of 665.672 kN on 45th day with a loss % of 1.050 in the period 28-45days. Bending stress or net stress in the beam at mid span is.


Creep loss ( )


Shrinkage loss = .



Strands stressed on 7th day will have a residual prestressing force of 660.460 kN on 70th day with a loss % of 0.783 in the period 45-70 days. Bending stress in the beam at mid span due to prestressing is.

( )



Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )


Residual prestressing force on 45thday.


Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )



Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )




Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )




Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )




Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )



Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )


5.4.3 Residual prestressing force on 70th day Strands stressed on 7th day will have a residual prestress force of 665.672 kN on 45th day with a loss % of 1.050 in the period 28-45 days. Bending stress or net stress in the beam at mid span is.


Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )



Creep loss =


Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )




Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )


10-6

Shrinkage loss =


Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )


Residual prestressing force on 70thday =


Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )



Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )


5.4.4 Residual prestressing force on 90th day Strands stressed on 7th day will have a residual prestressing force of 660.460 kN on 70th day with a loss % of 0.783 in the period 45-70 days. Bending stress in the beam at mid span due to prestressing is.


Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )



Shrinkage loss = .





( )




Creep loss ( )


Shrinkage loss = .




( )



Creep loss =Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







Total shrinkage strain in the period 70-90 days is obtained from Table 6 as 20.62 x 10-6

Shrinkage loss =

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







Residual prestressing force on 90thday =

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =








POINT OF VIEW

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







5.4.5 Residual prestressing force at infinity (25550 days) Strands stressed on 7th day will have a residual prestressing force of 657.605 kN on 90th day with a loss % of 0.432 in the period 70-90 days. Bending stress or net stress in the beam at mid span is.

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =








Creep loss =

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







Total shrinkage strain in the period 90 -

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







days is obtained from Table 6 as 121.32

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







10-6

Shrinkage loss =

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =








Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =








Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







5.4.6 Losses as per IS 1343:1980Creep co-efficient for a member loaded on 7th day, from Table 2 is 2.2. Total loss due to creep is

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =







Shrinkage strain for a member loaded on 7th day is obtained from Table 6 as 209.59 x 10-6. Total loss due to shrinkage is

Creep loss ( ) .


Shrinkage loss = .




( )


Creep loss ( ) .


Shrinkage loss =








Total loss evaluated from both the codes is presented in Table 7.

5.5 Residual prestressing force for multi stage stressing

In the present problem strands are stressed in two stages. In first stage, 5 strands are stressed on 7th day and in second stage remaining 5 strands are stressed on 28th day. Total force on the beam due to prestressing at first stage is


Strands stressed on 7th day will have a prstressing force of 360 kN located at 0.075 m from soffit of the beam. Bending stress in the beam at mid span due to first stage stressing is

( )

Tensile stress due to self-weight of beam

Bending compressive stress or net stress in concrete at the level of steel up to 28th day

From Table 3, Creep co-efficient for 28th day corresponding to strands stressed on 7th day is 0.85.

Creep loss .

Total shrinkage strain in the period 7-28 is obtained from Table 6 as 84.85 10-6.

Shrinkage loss = .


Residual prestressing force on 28th day just before stressing second stage strands

.

Prestressing force in the beam due to second stage stressing is 360 kN. Bending stress in the beam at the soffit of the beam due to second stage stressing is










( )




Creep loss .


Shrinkage loss = .



.



5.5.1 Residual prestressing force on 28th dayStrands stressed on 7th day will have a prstressing force of 360 kN located at 0.075 m from soffit of the beam. Bending stress in the beam at mid span due to first stage stressing is






( )




Creep loss .


Shrinkage loss = .



.









( )




Creep loss .


Shrinkage loss = .



.









( )




Creep loss .


Shrinkage loss = .



.




Creep loss






( )




Creep loss .


Shrinkage loss = .



.



Total shrinkage strain in the period 7-28 is obtained from Table 6 as 84.85






( )




Creep loss .


Shrinkage loss = .



.



10-6.

Table 7. Comparison of total long term lossType of loss As per IS 1343:2012

(kN)As per IS 1343:1980

(kN)Creep 27.756 42.309

Shrinkage 32.581 22.397Relaxation 47.492 38.360Total loss 107.829 103.066


POINT OF VIEW

Shrinkage loss =






( )




Creep loss .


Shrinkage loss = .



.









( )




Creep loss .


Shrinkage loss = .



.









( )




Creep loss .


Shrinkage loss = .



.








( )




Creep loss .


Shrinkage loss = .



.









( )




Creep loss .


Shrinkage loss = .



.


5.5.2 Residual prestressing force on 45th day 5.5.2 Residual prestressing force on 45th dayStrands stressed on 7th day will have a residual prestressing force of 343.118 kN on 28th day with a loss of 4.689% in the period 7-28 days. Bending stress in the beam at mid span due to first stage stressing is

Strands stressed on 7th day will have a residual prestressing force of 343.118 kN on 28th day with a loss of 4.689% in the period 7-28 days. Bending stress in the beam at mid span due to first stage stressing is

( )

Bending compressive stress or net stress in concrete at the level of steel up to 45th day i.e. just before the beam is subjected to live load is

From Table 3, Creepco-efficients on 28th day and 45th day corresponding to strands stressed on 7th day are 0.85 and 1.01 respectively.

Creep loss ( ) .

Creep loss in first stage strands due stressing of second stage strands i.e. strands which are stressed on 28th day.

From Table 3, Creep co-efficient on 45th day corresponding to strands stressed on 28th day is 0.62.

Creep loss .

Total shrinkage strain in the period 28-45 days is obtained from Table 6as 40.04 10-6.

Shrinkage loss = .

Residual prestressing force on 45th day in first stage strands is

.

Loss in second stage strands from 28th day to 45th day

Creep loss .

Shrinkage loss = .

Residual prestressing force on 45th day in second stage strands is

.



( )



( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )

From Table 3, Creep co-efficients on 28th day and 45th day corresponding to strands stressed on 7th day are 0.85 and 1.01 respectively.

Creep loss =


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )

Creep loss in first stage strands due to stressing of second stage strands i.e. strands which are stressed on 28th day.


Creep loss =


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )



( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )

10-6.

Shrinkage loss =


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )



( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )


Creep loss =


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )

Shrinkage loss =


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )



( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )

5.5.3 Residual prestressing force on 70th dayStrands stressed on 7th day will have a residual prestressing force of 336.889 kN on 45th day with a loss of 1.815% in the period 28-45 days. Bending stress in the beam at mid span due to first stage stressing is


( )



Creep loss ( ) .



Creep loss .


Shrinkage loss = .


.


Creep loss .

Shrinkage loss = .


.



( )

Bending compressive stress in concrete at the level of steel up to 45th day i.e. just before the beam is subjected to live load is



Creep loss ( ) .


From Table 3, Creep co-efficients on 45th day and 70th day corresponding to strands stressed on 28th day is 0.80 and 0.62. Bending compressive stress in concrete at the level of steel after beam is subjected to live load on 45th day is

( )

Creep loss ( ) .

Total shrinkage strain in the period 45-70 days is obtained from Table 3 as 38.07 10-6.

Shrinkage loss = .


.


Strands stressed on 28th day will have a residual prestressing force of 345.359 kN on 45th day with a loss of 4.067% in the period 28-45 days. Bending compressive stress in concrete at the level of steel after beam is subjected to live load on 45th day is

Creep loss ( ) .

Shrinkage loss = .


.


Strands stressed on 7th day will have a residual prestressing force of 334.233 kN on 70th day with a loss of 0.788% in the period 45-70 days. Bending stress at mid span due to due to first stage stressing is



POINT OF VIEW

Creep loss =



Creep loss ( ) .



( )

Creep loss ( ) .


Shrinkage loss = .


.



Creep loss ( ) .

Shrinkage loss = .


.







Creep loss ( ) .



( )

Creep loss ( ) .


Shrinkage loss = .


.



Creep loss ( ) .

Shrinkage loss = .


.



Creep loss =



Creep loss ( ) .



( )

Creep loss ( ) .


Shrinkage loss = .


.



Creep loss ( ) .

Shrinkage loss = .


.






Creep loss ( ) .



( )

Creep loss ( ) .


Shrinkage loss = .


.



Creep loss ( ) .

Shrinkage loss = .


.



10-6.

Shrinkage loss =



Creep loss ( ) .



( )

Creep loss ( ) .


Shrinkage loss = .


.



Creep loss ( ) .

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5.5.4 Residual prestressing force on 90th dayStrands stressed on 7th day will have a residual prestressing force of 334.233 kN on 70th day with a loss of 0.788% in the period 45-70 days. Bending stress at mid span due to due to first stage stressing is

( )



Creep loss ( )


From Table 3, Creepco-efficients on 70th day and 90th day corresponding to strands stressed on 28th day is 0.80 and 0.89. Bending compressive stress in concrete at the level of steel after beam is subjected to live load on 45th day is

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5.5.5 Residual prestressing force at infinity (25550 days) 5.5.5 Residual prestressing force at infinity (25550 days)Strands stressed on 7th day will have a residual prestressing force of 332.773 kN on 90th day with a loss of 0.437% in the period 70-90 days. Bending stress in the beam at mid span due to first stage stressing is


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Compressive stress or net stress in concrete at the level of steel up to 70th day


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As per Clause 19.5.2.3 of IS 1343:2012, for long term relaxation loss values given in Table 6 of code should be multiplied by 3. Loss between 90 - days is

Residual prestressing force at infinity in first stage strands is

.

Loss in second stage strands from 90th day to infinity days

Strands stressed on 28th day will have a residual prestressing force of 342.101kN on 90th day with a loss of 0.333% in the period 70-90 days. Bending compressive stress in concrete at the level of steel after beam is subjected to live load on 45th day is

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Residual prestressing force at infinity in second stage strands is

.

Residual prestressing force in the beam after all the losses is

= 307.546 + 320.071 = 627.617kN.

6. Conclusions

The calculations of long term losses play a major role in performance of prestressed concrete structures [10]. The revised IS 1343 has incorporated many factors into the model for calculating long term losses. A typical case study has been considered to elaborate the long term loss calculation procedure based on old and revised code. The following general observationscan be drawn from this study.

With reference to Creep

In IS: 1343:1980, ultimate creep coefficients were given for different ages of loading. These recommendations are only for structures where losses need not be evaluated at various stages. Creep co-efficient cannot be estimated in respective intervals with respect to IS 1343:1980 as there are no guidelines to arrive creep co-efficient for evaluating residual stresses at various stages.They are not valid for evaluation of residual prestress or deflection or camber in structures at various stages of stressing/loading/measuring. These short comings are addressed in new code with a scientific mathematical model that incorporates various parameters like member size, relative humidity, and age of loading. However, this model is valid only if stress in concrete does not exceed one-third characteristic compressive strength of concrete and concrete should be of normal concrete ranging from M30 to M60. These models are not applicable for special concretes.

With reference to Shrinkage loss

Shrinkage strain as per IS 1343:1980 is given by the Clause 5.2.4.1. For post tensioned members ultimate shrinkage strain is given by the equation which is dependent on age of concrete at the time of transfer of prestress. Shrinkage strain cannot be estimated in respective intervals and for multistage case with respect to IS 1343:1980 as there are no guidelines for evaluating residual stresses at various stages. Whereas IS1343:2012 has incorporated various parameters affecting the shrinkage like Relative humidity, member size and grade of concrete and can obtain shrinkage loss for multistage prestressing. IS 1343:2012 is not clear on source from which RH has to be considered.

With reference to Relaxation loss

From Table 6 of IS 1343:2012 for low relaxation strands loss percentage is based on initial prestress. For long-term relaxation losses, the values given in this table should be multiplied by 3. The %loss from this



.


= 307.546 + 320.071 = 627.617kN.

6. Conclusions










POINT OF VIEW



.


= 307.546 + 320.071 = 627.617kN.

6. Conclusions










.


= 307.546 + 320.071 = 627.617kN.

6. Conclusions









= 307.546 + 320.071 = 627.617kN.

6. concluSIonSThe calculations of long term losses play a major role in performance of prestressed concrete structures [10]. The revised IS 1343 has incorporated many factors into the model for calculating long term losses. A typical case study has been considered to elaborate the long term loss calculation procedure based on old and revised code. The following general observationscan be drawn from this study.






From Table 6 of IS 1343:2012 for low relaxation strands loss percentage is based on initial prestress. For long-term relaxation losses, the values given in this table should be multiplied by 3. The %loss from this table is the total loss due to relaxation i.e. at 1000 hours at 20 ± 2°C. Code is silent on losses up to 1000 hours and for temperature greater than 20°C and on evaluation of relaxation loss in intervals. IRC 112:2011 code considers the values up to 1000 hours and above and for temperatures greater than 20°C. As per Table 4 of IS 1343:1980, the relaxation loss is dependent on the initial prestress only.

With reference to the results from case study

Although the variation of total loss as calculated for the case considered from both the codes is not significant, it may definitely vary with the problem statement.

References

1. ______Indian standard code of practice for prestressed concrete (second revision), IS 1343: 2012.Bureau of Indian Standards, New Delhi.

2. Comite euro-international du-beton, CEB-FIP model code 1990.

3. ______Indian standard code of practice for prestressedconcrete, IS 1343: 1980.Bureau of Indian Standards, New Delhi.

4. Beeby A.W. and Narayanan R. S., Designers’ guide to Eurocode 2: design of concrete structures, Thomas Telford Publication.

5. Oh B. H., Cha S. W., Um J. Y. and Lim D. H., Effects of reinforcement and humidity onthe creep and shrinkage behaviour of high-strength concrete members, Creep andShrinkage of Concrete, RILEM Symposium Proceedings of the Fifth International, 1995, pp. 517-522.

6. Peter L., Assessment of long-term losses in prestressedconcrete structure, Thesis submitted to Lund University, for Ph.D, Lund University, 2012.

7. American concrete institute guide for modelling and calculating shrinkage and creep in hardened concrete, ACI 209.2R-08.

8. ______Indian road congress code of practise for concrete bridges, IRC 112:2011. Indian Road Congress, New Delhi.

9. Viswanathan T., Calculation of time dependent losses in prestressed concrete as per IRC : 112 and IRC :18, Journal of Indian Road Congress, April - June 2014, Vol. 74, No. 4, pp. 146-161.

10. Gilbert R.I., Time effects in concrete structures, Elsevier Science publishing company, New york, 1988.


POINT OF VIEW

Dr. P. Markandeya Raju holds a B.Tech (Civil Engineering) from Nagarjuna University; M.E (Structural Engineering) from Andhra University; PhD from JNTU, Hyderabad. He is a Professor of Civil Engineering at MVGR College of Engineering (Autonomous), Vizianagaram, Andhra Pradesh and has 15 years of teaching experience. He has more than 45 papers to his credit in various national and international conferences and journals. His areas of interest are prestressed steel structures, computer applications in structural engineering and durability studies on special concretes.

T. Raghuram Sandeep holds a B.Tech. (Civil Engineering) from JNTU, Hyderabad and M.E (Structural Engineering) from Andhra University, Visakhapatnam. He is a Technical Officer at Civil Engineering Division of BARC, Visakhapatnam, Andhra Pradesh. He published three technical papers in reputed International journals. His research interests are partial prestressing, prestressing in concrete and steel-concrete composite structures.

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