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PRESTRESS LOSSES Reading:

Nawy, E.G., Prestressed Concrete – A Fundamental Approach, 3rd Edition, Chapter 3.

SOURCES AND COMPUTATION OF LOSSES There are essentially two types of prestress losses that can take place in prestressed concrete members: __________________________________________ and ___________________ __________________________. These two types of losses can be described in the following. Immediate Losses: These losses depend upon the type of member: pretrensioned or post-tensioned. In a pretensioned member, an immediate loss is that due to ______________________________ of the member. Immediate losses in a post-tensioned member are those due to __________________ and ___________________________________. Post-tensioned members can also be subjected to elastic shortening losses when _______________________ ____________________ is used. Time-Dependent Losses: The losses that depend upon elapsed time after stressing are independent of the member type. These losses are: ______________________________________________ ______________________________________________ _______________________________. There are two methods that can be used to estimate losses in prestressed concrete: (a) lump sum approximations; and (b) refined estimations. One should keep in mind that all estimates for prestress loss are just that – ESTIMATIONS. As we get into the details of the “refined” estimations, be aware of all the assumed behavior that exists in the estimation. Prior to ACI 318-83, lump sum loss calculations were allowed. However, today’s Code deems lump sum estimates obsolete.

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Prestress losses are most conveniently broken down into components. We will address loss calculations based upon the member type being considered. The prestress loss can be determined using the following “formulas” for pretensioned members: ∆f pT = ∆f pES + ∆f pR ( t0 , ttr ) + ∆f pR ( ttr , ts ) + ∆f pCR + ∆f pSH f pi = f pJ − ∆f pR ( t0 , ttr ) − ∆f pES

before transfer after transfer initial prestress

The following can be used for post-tensioned members:

∆f pT = ∆f pF + ∆f pES

at jacking

+ ∆f pA

at transfer

+ ∆f pR ( ttr , ts ) + ∆f pCR + ∆f pSH

after transfer

f pi = f pJ − ∆f pA − ∆f pF

initial prestress

The subscripts and times are defined below: t0 = time at jacking; ttr = time at transfer of prestressing force; ts = time at stabilization of losses (i.e. during the service loading stage); j

= jacking;

R = relaxation; ES = elastic shortening A = anchorage; F = friction; CR = creep; SH = shrinkage. The AASHTO-LRFD Specifications allow lump-sum estimates for prestressing losses with the caveat that the following conditions are met. 1. Members that are post-tensioned must be non-segmental members with spans less than 160 feet and concrete stressed an age of 10-30 days. 2. Members that are pretensioned must be stressed at an age where the concrete strength is no less than 3,500-psi. 3. Members must be made from normal weight concrete. 4. Members cannot be steam-cured, nor moist-cured. 5. The prestressing steel must be normal or low-relaxation.

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6. There must be average exposure conditions at the site. If these conditions are met, there is a simple table (Table 1 shown below) that can be used for loss calculations. It should be noted that the table defined PPR as _____________________________ ______________, which is basically the ratio prestressed reinforcement to total reinforcement within the cross-section.

Table 1: AASHTO Lump Sum Approximations.

ELASTIC SHORTENING The loss due to elastic shortening is based upon mechanics of materials approaches. We should all appreciate that the strain lost due to elastic shortening deformations can be computed using,

ε ES =

∆ ES L

(1)

Therefore, if we can compute the member deformation due to elastic shortening, ∆ ES , we can determine the strain lost resulting from elastic shortening. Losses due to elastic shortening are different when pretensioned and post-tensioned members are considered. Pretensioned Members: When the member is pretensioned, the computation of loss is straight-forward,

∆f pES = Esε ES =

PE nP i s = i = nf cs Ac Ec Ac

(2)

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where: f cs is the stress in the concrete at the level of the prestressing steel; and n is the modular ratio. Of course, this will vary depending upon the location of the tendon centroid within the cross-section. The initial prestressing force, Pi , that will cause elastic shortening is a little difficult to estimate if the jacking force, loss due to friction, and loss due to seating are not known. Therfore, Nawy (1999) has suggested that 90% of the initial prestressing force given be used. Post-Tensioned Members: In the case of post-tensioned members, the computation is a little more difficult. The reason for this is that when a post-tensioned member is considered, one can jack tendons in sequence rather than jacking them all at once. The loss due to elastic shortening in this case can be computed as,

∆f pES =

1 ⋅ N

N

∑ ( ∆f ) pES

j =1

j

(3)

where: N is the number of tendons (or groups/pairs) sequentially jacked. Use of equation (3) is best illustrated via example. It should be noted that the last tendon or group of tendons to be stressed suffers no elastic shortening, while the first tendon or group of tendons suffers the highest losses.

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EXAMPLE 1 – COMPUTATION OF ELASTIC SHORTENING LOSSES

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EXAMPLE 2 – COMPUTATION OF ELASTIC SHORTENING LOSS

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CREEP As in the case of elastic shortening, the loss due to creep all begins with mechanics of materials. Recall our rheological model for the time-varying loss due to creep. The basic mechanics of materials approach to creep loss takes the following form,

∆f pCR = Ct ⋅

E ps Ec

⋅ f cs

E t 0.6 = ⋅ Cu ⋅ ps ⋅ f cs 0.6 Ec 10 + t

(4)

where: Cu = the ultimate creep coefficient (usually 2.35 often used); t = the time (in days); Ec = the elastic modulus of the concrete;

E ps = the elastic modulus of the prestressing steel; f cs = the compressive stress in the concrete at the level of the prestressing steel centroid. Creep loss is generally a function of the location along the member where the compressive stress is analyzed. This results from the tendon centroid (in general) varying along the length of the concrete member. The average concrete stress between anchorage points can be used for post-tensioned members. In a prestensioned member, the average along the member length can be used. There seems to be many procedural recommendations for computing creep in prestressed concrete members. The first we will consider is ACI Committee 423. This committee’s recommendation is given below, ∆f pCR = K CR ⋅

E ps Ec

⋅ ( f cs − f csd )

(5)

where: K CR

f cs f csd

= = = =

is a creep coefficient (reduce by 20% for lightweight concrete) 2.00 for pretensioned members 1.60 for post-tensioned members the stress in the concrete at the level of the prestressing steel centroid immediately after transfer, = the stress in the concrete at the level of the prestressing steel due to all superimposed dead loads applied after transfer.

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A second recommendation for computation of losses comes from the AASHTO LRFD Specifications. This computation is slightly simpler than equation (5), but more complicated in other respects. The loss due to creep according to AASHTO-LRFD is,

∆f pCR = 12.0 ⋅ f cgp − 7.0 ⋅ ∆f cdp ≥ 0.0

(6)

where:

f cgp

= is the stress a the center of gravity of the prestressing steel centroid at transfer

∆f cdp

(ksi); = the change in concrete stress at the center of gravity of the prestressing steel due to permanent loads (with the exception of the load acting at the time the prestressing steel is applied). Values should be calculated a the same section (or sections) at which f cgp is computed (ksi).

SHRINKAGE Recalling our discussion of the factors that affect shrinkage, any relationship used for shrinkage loss estimation should include consideration of ________________________________, _____________ __________________________, and member ________________________________. It is assumed that shrinkage begins at the end of the curing period (e.g. 7-days). If one would like to compute the shrinkage strain that occurs from 28-days to 1-year, a subtraction procedure should be employed. The loss of prestress resulting from shrinkage strain can be computed using mechanics of materials relationships,

∆f pSH = ε SH E ps where: ε SH is the shrinkage strain. ACI Committee 209 suggests the following computation for the shrinkage strain at any time, t,

t

⋅ ( 780 ×10−6 ) ⋅ γ SH ε SH ,t = t + α where: t

= α = = γ SH =

is the time in days, 35 if moist cured for 7-days, 55 is steam cured for 1-3 days, is a correction factor that accounts for conditions other than standard conditions. The correction factor accounts for relative humidity, volume-to-surface ratio, concrete composition, etc.

(7)

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The general form of the ACI Committee 423 recommendation for prestress losses due to shrinkage takes the following form, ∆f pSH = 8.2 × 10−6 ⋅ K SH

(

)

V ⋅ 1 − 0.06 S

⋅ (100 − RH ) ⋅ E ps

(8)

where: K SH is 1.0 for pretensioned members and is taken from Table 2 for post-tensioned members.

Table 2: Shrinkage Factor for Post-Tensioned Members

As one might expect, there are also AASHTO-LRFD recommendations for prestress loss computation. These are also broken down into pretensioned and post-tensioned members. For

pretensioned members,

∆f pSH = 17.0 − 0.150 H

ksi

(9)

and for post-tensioned members,

∆f pSH = 13.5 − 0.123H

ksi

(10)

where: H is the relative humidity (%) obtained from local statistics or a map. A relative humidity map can be found in the PCI Design Handbook and the AASHTO – LRFD Specifications. Such a map is shown in Figure 1.

STEEL RELAXATION An empirical relationship for steel relaxation loss can be developed using mechanics of materials. As is usually the case, we will also look at ACI Committee 423 and AASHTO-LRFD recommendations. If we know the initial prestress, f pi′ , and an empirical relationship describing the relaxation over time, we can write the loss is prestress as,

log t2 − log t1 f pi′ 0.55 ∆f pR = f pi′ ⋅ ⋅ − f α py where:

(11)

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α = 10 for stress-relieved strands, = t1 = t2 = f pi′ =

45 for low-relaxation strands, initial time (hours) for interval under consideration, final time (hours) for interval under consideration, initial stress in prestressing steel at the beginning of the interval considered,

f py = yield stress of the prestressing steel. It should be noted that the initial prestress to yield stress should be greater than 0.55.

Figure 1: Annual Average Ambient Relative Humidities (AASHTO 2001). The ACI Committee 423 recommendation includes losses due to other sources. The loss in prestress resulting from relaxation using ACI 423 recommendations is, ∆f pR = K RE − J ⋅ ( ∆f pES + ∆f pCR + ∆f pSH ) ⋅ C

(12)

where the loss due to elastic shortening, creep, and shrinkage should be computed using previous ACI 423 recommendations for these losses. The relaxation loss constants, K RE and J are taken from Table 3 and the constant C is taken from Table 4.

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Table 3: Relaxation Loss Constants, K RE and J Used in ACI 423 Recommendation.

Table 4: Relaxation Constant, C Used in ACI 423 Recommendation.

The AASHTO – LRFD Specifications also contain a recommendation that is a little bit simpler than that implied in equation (12). The procedure is a two-level procedure where relaxation loss is

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computed during two stages: (a) at transfer and (b) after transfer. The loss that occurs before

transfer of prestress is computed using,

∆f pR1 =

log ( 24 ⋅ t ) f pJ ⋅ − 0.55 ⋅ f pJ α f py

(13)

where: t is the time (days) from initial stressing to transfer; f pJ is the stress in the tendons at the end of the jacking sequence; α is a constant which is 10 for stress-relieved strands and 40 for lowrelaxation strands. The loss that occurs after transfer for stress-relieved strands and pretensioned

members is computed using, ∆f pR 2 = 20 − 0.4∆f pES − 0.2 ⋅ ( ∆f pSH + ∆f pCR )

(14)

and the loss after transfer for stress-relieved strands and post-tensioned members is computed using,

∆f pR 2 = 20 − 0.3∆f pF − 0.4∆f pES − 0.2 ⋅ ( ∆f pSH + ∆f pCR )

(15)

If low-relaxation strands are used, the loss after transfer can be taken as 30% of the values obtained using equations (14) and (15).

FRICTION As a prestressing tendon is pulled, its lengthening will be resisted by frictional forces along the tendon. This is especially important in post-tensioned members. In general, loss due to friction is broken down into loss from the following sources; ____________________________________________________ ____________________________________________________ The length effect accounts for frictional sources that are encountered when the tendon is intended to be straight. In essence, a “straight” duct is not really straight and it will wobble along the length of the member. The vibration and placement of the concrete can displace the tendon ducts. Therefore, as the tendon(s) are pulled, they will encounter the sides of the tendon duct and therefore, frictional force will be developed. The curvature effect causes friction because the tendon “wants” to assume

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a straight configuration within the duct. This straightening out is prevented by the curved tendon duct. This then causes friction as well. There are a couple of methods that are available to “overcome” the frictional losses; 1. The first is tendon overtensioning. In this procedure, the tendon is “pulled on” with a magnitude of force sufficient to overcome the frictional losses. This procedure results in a variation in stress along the tendon. Portions of the tendon may be stresses to a higher level during jacking than other portions. 2. The second is jacking from both ends. This method tends to require more field coordination and effort. This method is often used with the tendon lengths become very long, or the angles of the tendon bend are large. Jacking from both ends does not help the simple beam, but continuous beams receive significant benefit.

Effect of Overtensioning The effect of overtensioning when jacking from one end can be graphically seen in the figure below.

Figure 2: Graphical Depiction of Tendon Stress and Friction Loss. Figure 2 illustrates the stress in the tendon when it is “over-jacked”. As one can see as jacking commences, the stress in the tendon is highest at the anchorage end. At the end opposite from the jacking end, the tendon stress is the least. When the jack is released and the prestressing force is transferred to the beam, the anchorage seating causes the associated loss in prestress. The “kink” in the curve indicates that frictional forces are capable of keeping a level of stress in the tendon. If

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frictional loss is high, there is a sharp “kink”. When frictional loss is low, there is a very shallow “kink”. Furthermore, when the frictional forces are low, there is a much more uniform state of stress in the tendon between the jacking end and the opposite end.

Friction Loss Due to Curvature Effect The loss in prestress due to curvature is the first frictional loss that we will consider. Consider a free-body-diagram of a segment of curved tendon shown below.

(a)

(b)

Figure 3: Forces Present on Tendon Segments During Jacking. The change in angle, dα , that occurs over the segment and the force normal to the tendon length are given by, dα = dx

R

and N = F ⋅ dα = F ⋅ dx

R

The frictional loss over this infinitely small segment is then given by,

dF = − µ N = − µ ( Fdα ) Rearranging terms in the equation above gives,

dF = − µ dα F Integrating both sides of the equation gives,

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F2

∫

F1

α

1 dF = − µ dα F 0

∫

where: F1 is the force at the jacking end; and F2 is the force at the “other” end. Carrying out the integration gives, ln F2 − ln F1 = − µα µL

− F2 = e − µα = e R F1

Solving for the force at the jacking end gives, F2 = F1e − µα

(16)

which can be used to compute frictional losses due to curvature effect.

Friction Loss Due to Wobble Effect A similar plan of attack can be used to address the losses due to wobble (length) effect. The change in force over a small length of tendon can be written as, dF = − K ⋅ F ⋅ dL where: K is a wobble coefficient. The wobble coefficient is a frictional coefficient that gives indication of the magnitude of the frictional force which results from wobble of the tendon within the duct. As one can see in the equation above, the wobble coefficient allows calculation of the change in force over the small length as a fraction of the force in the tendon. We can integrate both sides of the above equation giving, F2

∫

F1

L

1 dF = − KdL F 0

∫

Carrying out the integration gives an expression that can be used to compute the loss in prestressing force that results from wobble effect, F2 = F1e − KL

(17)

Combined Effects of Curvature and Wobble The combined effect of the curvature effect and wobble effect can be written as a simple summation of equations (16) and (17),

(

F2 = F1 e − µα + e − KL

)

18

If one recognizes the similarities in derivation (i.e. the integration) a different form of the combined effect can be written (more like the ACI form of the loss expression). Recalling the derivation of equations (16) and (17) the integration of forces with the combined effects can be written as, F2

∫

F1

α

L

1 dF = − µ dα − KdL F 0 0

∫

∫

ln F2 − ln F1 = − µα − KL Performing a little algebra results in, F2 = F1e − µα − KL

(18)

Equation (18) forms the basis of the ACI approach to computing frictional losses. The frictional loss can be expressed as the change in force between the jacking end and the “other” end. Therefore,

∆f pF = F1 − F2 Plugging in equation (18) gives,

(

∆f pF = F1 − F1e − µα − KL = F1 1 − e

−( µα + KL )

)

If µα + KL is small, we can rewrite the frictional loss as follows, e

−( µα + KL )

= 1 − ( µα + KL )

∆f pF = FJ ( µα + KL ) The ACI 318-99 equations for computing frictional losses evolve from the equation above. Equation (18) can be written in “reverse” fashion by dividing both sides by e

−( µα + KL )

. Thus, the force at the

jacking end can be computed using the force at “some distance” from the jacking end,

Ps = Px e µα + KL

Ps = Px (1 + µα + KL ) where: Px is the force at a distance x from the jacking end; and Ps is the force at the jacking end. Equations (19) form the basis for the ACI 318-99 friction loss calculation procedure. One item left in computing the loss for friction is to determine values for the wobble coefficient and typical friction coefficients. These values are given in the Code and are reproduced in Table 5 on the next sheet.

(19)

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Table 5: Wobble Coefficients and Friction Coefficients for Typical Members

Comments on Computing Angles The angle, α , is the sweep as one moves from one end of the tendon to the other. If we have parabolic tendon with known radius, R, the angle sweep can be computed using Figure 4.

Figure 4: Parabolic Tendon Profile and Angle Sweep. Assuming that the eccentricity at the center and ends is known, the angle sweep can be computed as,

α=

8y x

(20)

When considering harped tendons, compute the angle sweeps for each segment and add them together. Furthermore, if a tendon profile varies over any given pulling length, the engineer must add up frictional losses over each of these lengths.

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ANCHORAGE SEATING The prestress loss resulting from seating of the anchorage mechanisms can be computed using mechanics of materials. Unfortunately, we will need to know how much “slop” is typically present in the anchorage mechanism to determine this loss. Using mechanics of materials relations as a starting point, the loss in prestress resulting from anchorage seating for a mechanism with known wedge slip, ∆ A , can be computed using,

∆f pA =

∆A ⋅ E ps L

The wedge slip should be taken from manufacturer’s literature or recommendations.

(21)

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EXAMPLE 3 – STEP-BY-STEP PRESTRESS LOSS CALCULATION

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EXAMPLE 4 – STEP-BY-STEP LOSS COMPUTATION

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PRESTRESS LOSSES Reading:

Nawy, E.G., Prestressed Concrete – A Fundamental Approach, 3rd Edition, Chapter 3.

SOURCES AND COMPUTATION OF LOSSES There are essentially two types of prestress losses that can take place in prestressed concrete members: __________________________________________ and ___________________ __________________________. These two types of losses can be described in the following. Immediate Losses: These losses depend upon the type of member: pretrensioned or post-tensioned. In a pretensioned member, an immediate loss is that due to ______________________________ of the member. Immediate losses in a post-tensioned member are those due to __________________ and ___________________________________. Post-tensioned members can also be subjected to elastic shortening losses when _______________________ ____________________ is used. Time-Dependent Losses: The losses that depend upon elapsed time after stressing are independent of the member type. These losses are: ______________________________________________ ______________________________________________ _______________________________. There are two methods that can be used to estimate losses in prestressed concrete: (a) lump sum approximations; and (b) refined estimations. One should keep in mind that all estimates for prestress loss are just that – ESTIMATIONS. As we get into the details of the “refined” estimations, be aware of all the assumed behavior that exists in the estimation. Prior to ACI 318-83, lump sum loss calculations were allowed. However, today’s Code deems lump sum estimates obsolete.

2

Prestress losses are most conveniently broken down into components. We will address loss calculations based upon the member type being considered. The prestress loss can be determined using the following “formulas” for pretensioned members: ∆f pT = ∆f pES + ∆f pR ( t0 , ttr ) + ∆f pR ( ttr , ts ) + ∆f pCR + ∆f pSH f pi = f pJ − ∆f pR ( t0 , ttr ) − ∆f pES

before transfer after transfer initial prestress

The following can be used for post-tensioned members:

∆f pT = ∆f pF + ∆f pES

at jacking

+ ∆f pA

at transfer

+ ∆f pR ( ttr , ts ) + ∆f pCR + ∆f pSH

after transfer

f pi = f pJ − ∆f pA − ∆f pF

initial prestress

The subscripts and times are defined below: t0 = time at jacking; ttr = time at transfer of prestressing force; ts = time at stabilization of losses (i.e. during the service loading stage); j

= jacking;

R = relaxation; ES = elastic shortening A = anchorage; F = friction; CR = creep; SH = shrinkage. The AASHTO-LRFD Specifications allow lump-sum estimates for prestressing losses with the caveat that the following conditions are met. 1. Members that are post-tensioned must be non-segmental members with spans less than 160 feet and concrete stressed an age of 10-30 days. 2. Members that are pretensioned must be stressed at an age where the concrete strength is no less than 3,500-psi. 3. Members must be made from normal weight concrete. 4. Members cannot be steam-cured, nor moist-cured. 5. The prestressing steel must be normal or low-relaxation.

3

6. There must be average exposure conditions at the site. If these conditions are met, there is a simple table (Table 1 shown below) that can be used for loss calculations. It should be noted that the table defined PPR as _____________________________ ______________, which is basically the ratio prestressed reinforcement to total reinforcement within the cross-section.

Table 1: AASHTO Lump Sum Approximations.

ELASTIC SHORTENING The loss due to elastic shortening is based upon mechanics of materials approaches. We should all appreciate that the strain lost due to elastic shortening deformations can be computed using,

ε ES =

∆ ES L

(1)

Therefore, if we can compute the member deformation due to elastic shortening, ∆ ES , we can determine the strain lost resulting from elastic shortening. Losses due to elastic shortening are different when pretensioned and post-tensioned members are considered. Pretensioned Members: When the member is pretensioned, the computation of loss is straight-forward,

∆f pES = Esε ES =

PE nP i s = i = nf cs Ac Ec Ac

(2)

4

where: f cs is the stress in the concrete at the level of the prestressing steel; and n is the modular ratio. Of course, this will vary depending upon the location of the tendon centroid within the cross-section. The initial prestressing force, Pi , that will cause elastic shortening is a little difficult to estimate if the jacking force, loss due to friction, and loss due to seating are not known. Therfore, Nawy (1999) has suggested that 90% of the initial prestressing force given be used. Post-Tensioned Members: In the case of post-tensioned members, the computation is a little more difficult. The reason for this is that when a post-tensioned member is considered, one can jack tendons in sequence rather than jacking them all at once. The loss due to elastic shortening in this case can be computed as,

∆f pES =

1 ⋅ N

N

∑ ( ∆f ) pES

j =1

j

(3)

where: N is the number of tendons (or groups/pairs) sequentially jacked. Use of equation (3) is best illustrated via example. It should be noted that the last tendon or group of tendons to be stressed suffers no elastic shortening, while the first tendon or group of tendons suffers the highest losses.

5

EXAMPLE 1 – COMPUTATION OF ELASTIC SHORTENING LOSSES

6

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EXAMPLE 2 – COMPUTATION OF ELASTIC SHORTENING LOSS

8

9

CREEP As in the case of elastic shortening, the loss due to creep all begins with mechanics of materials. Recall our rheological model for the time-varying loss due to creep. The basic mechanics of materials approach to creep loss takes the following form,

∆f pCR = Ct ⋅

E ps Ec

⋅ f cs

E t 0.6 = ⋅ Cu ⋅ ps ⋅ f cs 0.6 Ec 10 + t

(4)

where: Cu = the ultimate creep coefficient (usually 2.35 often used); t = the time (in days); Ec = the elastic modulus of the concrete;

E ps = the elastic modulus of the prestressing steel; f cs = the compressive stress in the concrete at the level of the prestressing steel centroid. Creep loss is generally a function of the location along the member where the compressive stress is analyzed. This results from the tendon centroid (in general) varying along the length of the concrete member. The average concrete stress between anchorage points can be used for post-tensioned members. In a prestensioned member, the average along the member length can be used. There seems to be many procedural recommendations for computing creep in prestressed concrete members. The first we will consider is ACI Committee 423. This committee’s recommendation is given below, ∆f pCR = K CR ⋅

E ps Ec

⋅ ( f cs − f csd )

(5)

where: K CR

f cs f csd

= = = =

is a creep coefficient (reduce by 20% for lightweight concrete) 2.00 for pretensioned members 1.60 for post-tensioned members the stress in the concrete at the level of the prestressing steel centroid immediately after transfer, = the stress in the concrete at the level of the prestressing steel due to all superimposed dead loads applied after transfer.

10

A second recommendation for computation of losses comes from the AASHTO LRFD Specifications. This computation is slightly simpler than equation (5), but more complicated in other respects. The loss due to creep according to AASHTO-LRFD is,

∆f pCR = 12.0 ⋅ f cgp − 7.0 ⋅ ∆f cdp ≥ 0.0

(6)

where:

f cgp

= is the stress a the center of gravity of the prestressing steel centroid at transfer

∆f cdp

(ksi); = the change in concrete stress at the center of gravity of the prestressing steel due to permanent loads (with the exception of the load acting at the time the prestressing steel is applied). Values should be calculated a the same section (or sections) at which f cgp is computed (ksi).

SHRINKAGE Recalling our discussion of the factors that affect shrinkage, any relationship used for shrinkage loss estimation should include consideration of ________________________________, _____________ __________________________, and member ________________________________. It is assumed that shrinkage begins at the end of the curing period (e.g. 7-days). If one would like to compute the shrinkage strain that occurs from 28-days to 1-year, a subtraction procedure should be employed. The loss of prestress resulting from shrinkage strain can be computed using mechanics of materials relationships,

∆f pSH = ε SH E ps where: ε SH is the shrinkage strain. ACI Committee 209 suggests the following computation for the shrinkage strain at any time, t,

t

⋅ ( 780 ×10−6 ) ⋅ γ SH ε SH ,t = t + α where: t

= α = = γ SH =

is the time in days, 35 if moist cured for 7-days, 55 is steam cured for 1-3 days, is a correction factor that accounts for conditions other than standard conditions. The correction factor accounts for relative humidity, volume-to-surface ratio, concrete composition, etc.

(7)

11

The general form of the ACI Committee 423 recommendation for prestress losses due to shrinkage takes the following form, ∆f pSH = 8.2 × 10−6 ⋅ K SH

(

)

V ⋅ 1 − 0.06 S

⋅ (100 − RH ) ⋅ E ps

(8)

where: K SH is 1.0 for pretensioned members and is taken from Table 2 for post-tensioned members.

Table 2: Shrinkage Factor for Post-Tensioned Members

As one might expect, there are also AASHTO-LRFD recommendations for prestress loss computation. These are also broken down into pretensioned and post-tensioned members. For

pretensioned members,

∆f pSH = 17.0 − 0.150 H

ksi

(9)

and for post-tensioned members,

∆f pSH = 13.5 − 0.123H

ksi

(10)

where: H is the relative humidity (%) obtained from local statistics or a map. A relative humidity map can be found in the PCI Design Handbook and the AASHTO – LRFD Specifications. Such a map is shown in Figure 1.

STEEL RELAXATION An empirical relationship for steel relaxation loss can be developed using mechanics of materials. As is usually the case, we will also look at ACI Committee 423 and AASHTO-LRFD recommendations. If we know the initial prestress, f pi′ , and an empirical relationship describing the relaxation over time, we can write the loss is prestress as,

log t2 − log t1 f pi′ 0.55 ∆f pR = f pi′ ⋅ ⋅ − f α py where:

(11)

12

α = 10 for stress-relieved strands, = t1 = t2 = f pi′ =

45 for low-relaxation strands, initial time (hours) for interval under consideration, final time (hours) for interval under consideration, initial stress in prestressing steel at the beginning of the interval considered,

f py = yield stress of the prestressing steel. It should be noted that the initial prestress to yield stress should be greater than 0.55.

Figure 1: Annual Average Ambient Relative Humidities (AASHTO 2001). The ACI Committee 423 recommendation includes losses due to other sources. The loss in prestress resulting from relaxation using ACI 423 recommendations is, ∆f pR = K RE − J ⋅ ( ∆f pES + ∆f pCR + ∆f pSH ) ⋅ C

(12)

where the loss due to elastic shortening, creep, and shrinkage should be computed using previous ACI 423 recommendations for these losses. The relaxation loss constants, K RE and J are taken from Table 3 and the constant C is taken from Table 4.

13

Table 3: Relaxation Loss Constants, K RE and J Used in ACI 423 Recommendation.

Table 4: Relaxation Constant, C Used in ACI 423 Recommendation.

The AASHTO – LRFD Specifications also contain a recommendation that is a little bit simpler than that implied in equation (12). The procedure is a two-level procedure where relaxation loss is

14

computed during two stages: (a) at transfer and (b) after transfer. The loss that occurs before

transfer of prestress is computed using,

∆f pR1 =

log ( 24 ⋅ t ) f pJ ⋅ − 0.55 ⋅ f pJ α f py

(13)

where: t is the time (days) from initial stressing to transfer; f pJ is the stress in the tendons at the end of the jacking sequence; α is a constant which is 10 for stress-relieved strands and 40 for lowrelaxation strands. The loss that occurs after transfer for stress-relieved strands and pretensioned

members is computed using, ∆f pR 2 = 20 − 0.4∆f pES − 0.2 ⋅ ( ∆f pSH + ∆f pCR )

(14)

and the loss after transfer for stress-relieved strands and post-tensioned members is computed using,

∆f pR 2 = 20 − 0.3∆f pF − 0.4∆f pES − 0.2 ⋅ ( ∆f pSH + ∆f pCR )

(15)

If low-relaxation strands are used, the loss after transfer can be taken as 30% of the values obtained using equations (14) and (15).

FRICTION As a prestressing tendon is pulled, its lengthening will be resisted by frictional forces along the tendon. This is especially important in post-tensioned members. In general, loss due to friction is broken down into loss from the following sources; ____________________________________________________ ____________________________________________________ The length effect accounts for frictional sources that are encountered when the tendon is intended to be straight. In essence, a “straight” duct is not really straight and it will wobble along the length of the member. The vibration and placement of the concrete can displace the tendon ducts. Therefore, as the tendon(s) are pulled, they will encounter the sides of the tendon duct and therefore, frictional force will be developed. The curvature effect causes friction because the tendon “wants” to assume

15

a straight configuration within the duct. This straightening out is prevented by the curved tendon duct. This then causes friction as well. There are a couple of methods that are available to “overcome” the frictional losses; 1. The first is tendon overtensioning. In this procedure, the tendon is “pulled on” with a magnitude of force sufficient to overcome the frictional losses. This procedure results in a variation in stress along the tendon. Portions of the tendon may be stresses to a higher level during jacking than other portions. 2. The second is jacking from both ends. This method tends to require more field coordination and effort. This method is often used with the tendon lengths become very long, or the angles of the tendon bend are large. Jacking from both ends does not help the simple beam, but continuous beams receive significant benefit.

Effect of Overtensioning The effect of overtensioning when jacking from one end can be graphically seen in the figure below.

Figure 2: Graphical Depiction of Tendon Stress and Friction Loss. Figure 2 illustrates the stress in the tendon when it is “over-jacked”. As one can see as jacking commences, the stress in the tendon is highest at the anchorage end. At the end opposite from the jacking end, the tendon stress is the least. When the jack is released and the prestressing force is transferred to the beam, the anchorage seating causes the associated loss in prestress. The “kink” in the curve indicates that frictional forces are capable of keeping a level of stress in the tendon. If

16

frictional loss is high, there is a sharp “kink”. When frictional loss is low, there is a very shallow “kink”. Furthermore, when the frictional forces are low, there is a much more uniform state of stress in the tendon between the jacking end and the opposite end.

Friction Loss Due to Curvature Effect The loss in prestress due to curvature is the first frictional loss that we will consider. Consider a free-body-diagram of a segment of curved tendon shown below.

(a)

(b)

Figure 3: Forces Present on Tendon Segments During Jacking. The change in angle, dα , that occurs over the segment and the force normal to the tendon length are given by, dα = dx

R

and N = F ⋅ dα = F ⋅ dx

R

The frictional loss over this infinitely small segment is then given by,

dF = − µ N = − µ ( Fdα ) Rearranging terms in the equation above gives,

dF = − µ dα F Integrating both sides of the equation gives,

17

F2

∫

F1

α

1 dF = − µ dα F 0

∫

where: F1 is the force at the jacking end; and F2 is the force at the “other” end. Carrying out the integration gives, ln F2 − ln F1 = − µα µL

− F2 = e − µα = e R F1

Solving for the force at the jacking end gives, F2 = F1e − µα

(16)

which can be used to compute frictional losses due to curvature effect.

Friction Loss Due to Wobble Effect A similar plan of attack can be used to address the losses due to wobble (length) effect. The change in force over a small length of tendon can be written as, dF = − K ⋅ F ⋅ dL where: K is a wobble coefficient. The wobble coefficient is a frictional coefficient that gives indication of the magnitude of the frictional force which results from wobble of the tendon within the duct. As one can see in the equation above, the wobble coefficient allows calculation of the change in force over the small length as a fraction of the force in the tendon. We can integrate both sides of the above equation giving, F2

∫

F1

L

1 dF = − KdL F 0

∫

Carrying out the integration gives an expression that can be used to compute the loss in prestressing force that results from wobble effect, F2 = F1e − KL

(17)

Combined Effects of Curvature and Wobble The combined effect of the curvature effect and wobble effect can be written as a simple summation of equations (16) and (17),

(

F2 = F1 e − µα + e − KL

)

18

If one recognizes the similarities in derivation (i.e. the integration) a different form of the combined effect can be written (more like the ACI form of the loss expression). Recalling the derivation of equations (16) and (17) the integration of forces with the combined effects can be written as, F2

∫

F1

α

L

1 dF = − µ dα − KdL F 0 0

∫

∫

ln F2 − ln F1 = − µα − KL Performing a little algebra results in, F2 = F1e − µα − KL

(18)

Equation (18) forms the basis of the ACI approach to computing frictional losses. The frictional loss can be expressed as the change in force between the jacking end and the “other” end. Therefore,

∆f pF = F1 − F2 Plugging in equation (18) gives,

(

∆f pF = F1 − F1e − µα − KL = F1 1 − e

−( µα + KL )

)

If µα + KL is small, we can rewrite the frictional loss as follows, e

−( µα + KL )

= 1 − ( µα + KL )

∆f pF = FJ ( µα + KL ) The ACI 318-99 equations for computing frictional losses evolve from the equation above. Equation (18) can be written in “reverse” fashion by dividing both sides by e

−( µα + KL )

. Thus, the force at the

jacking end can be computed using the force at “some distance” from the jacking end,

Ps = Px e µα + KL

Ps = Px (1 + µα + KL ) where: Px is the force at a distance x from the jacking end; and Ps is the force at the jacking end. Equations (19) form the basis for the ACI 318-99 friction loss calculation procedure. One item left in computing the loss for friction is to determine values for the wobble coefficient and typical friction coefficients. These values are given in the Code and are reproduced in Table 5 on the next sheet.

(19)

19

Table 5: Wobble Coefficients and Friction Coefficients for Typical Members

Comments on Computing Angles The angle, α , is the sweep as one moves from one end of the tendon to the other. If we have parabolic tendon with known radius, R, the angle sweep can be computed using Figure 4.

Figure 4: Parabolic Tendon Profile and Angle Sweep. Assuming that the eccentricity at the center and ends is known, the angle sweep can be computed as,

α=

8y x

(20)

When considering harped tendons, compute the angle sweeps for each segment and add them together. Furthermore, if a tendon profile varies over any given pulling length, the engineer must add up frictional losses over each of these lengths.

20

ANCHORAGE SEATING The prestress loss resulting from seating of the anchorage mechanisms can be computed using mechanics of materials. Unfortunately, we will need to know how much “slop” is typically present in the anchorage mechanism to determine this loss. Using mechanics of materials relations as a starting point, the loss in prestress resulting from anchorage seating for a mechanism with known wedge slip, ∆ A , can be computed using,

∆f pA =

∆A ⋅ E ps L

The wedge slip should be taken from manufacturer’s literature or recommendations.

(21)

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EXAMPLE 3 – STEP-BY-STEP PRESTRESS LOSS CALCULATION

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EXAMPLE 4 – STEP-BY-STEP LOSS COMPUTATION

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