Prestress concrete

Helwan University Universit y Faculty of Engineering-Mataria Engineering-Matari a

Civil Engineering Department Theory & Design of Prestressed Concrete Hatem M. Seliem, Ph.D.

C H AP T E R 2 L O SS SS O F P R E S T R E S S

1 CLASSIFICATION CLASSIFICATION OF LOSSES

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Helwan University Universit y Faculty of Engineering-Mataria Engineering-Matari a


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2 GENERAL 

Initial prestressing force applied to the concrete undergoes a progressive process of reduction over a period of approximately five years.

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Early failures of prestressed concrete structures were due to the inability to accurately predict the losses over time.

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In general, losses of prestressing force may be grouped into two categories:



o

Immediate during construction process

o

Time-dependent losses occurring over an extended period

The prestressing jacking force P j (the largest force applied to a tendon) is immediately reduced by losses due to friction, anchorage slip and elastic shortening of the compressed concrete to what is known initial Pi.

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Helwan University Faculty of Engineering-Mataria

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As time passes, the force is further gradually reduced, rapidly at first but then more slowly, because of length changes due to shrinkage and creep of concrete, and due to the relaxation of the highly stressed steel.

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After many years, the prestressing force stabilizes to what is known as effective force Pe.

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For pre-tensioned members, P j never acts on the concrete, but only on the anchorage of the casting bed. The tension is reduced by the time it is applied to the concrete.

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For post-tensioned members, the jacking force is fully applied to the concrete only at the jacking end. Elsewhere, it is diminished by other losses.

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The initial prestress Pi is of primary importance in design, together with the effective prestress Pe.

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An exact determination of prestress losses (especially the time dependent ones) is not feasible because of numerous inter-related factors. In most practical design cases, detailed calculation of losses is unnecessary.

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It is possible to use reasonably accurate lump sum loss estimates.

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For cases where greater accuracy is needed, it is necessary to estimate separate losses, taking care of member geometry, material properties and construction methods. Accuracy of loss calculations may be improved by considering the inter-dependence of time-dependent losses, using discrete time intervals.

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Actual losses affect service load behavior such as deflection, cracking and crack width.

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Overestimation of prestress loss may lead to too much prestressing force, resulting in excessive camber and tensile stresses. Underestimation, on the other hand, will lead to too little prestressing.

Both cases require

accurate calculation of the prestress losses.

3 LUMP SUM ESTIMATE OF LOSSES 

The bases for loss calculations were first introduced in the ACI Code in 1963.

Many

thousands

of

prestressed

concrete

structures

were

satisfactorily built using the approach. The current ACI Code does not have suggestions for lump sum estimates of losses. 

The current AASHTO Specs contain a table for suggested lump sum losses.



Post-Tensioning Institute has also published such tables.

AASHTO Time-dependent Lump-sum Losses

PPR = Partial Prestress Ratio

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Approximate Prestress Loss Values For Post-Tensioning (Nawy, 2003) Table 3.2 Prestress loss, psi

Post-tensioning tendon material

Slabs

Beams and joists

30,000

35,000

Bar

20,000

25,000

Low-relaxation 270K strand

15,000

20,000

Stress-relieved 270K strand and stress-relieved 240K wire

Note: This table of approximate prestress losses was developed to provide a common post-tensioning industry basis for determining tendon requirements on projects in which the designer does not specify the magnitude of prestress losses. These loss values are based on use of normal-weight concrete and on average values of concrete strength, prestress level, and exposure conditions. Actual values of losses may vary significantly above or below the table values where the concrete is stressed at low strengths, where the concrete is highly prestressed, or in very dry or very wet exposure conditions. The table values do not include losses due to friction.

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4 DETAILED ESTIMATION OF LOSSES 

For cases where lump sum losses are inadequate, it is necessary to estimate each of the losses separately, using either assumed data, or for major works, using data developed for the particular job. The separate contributions are then summed to obtain total losses.

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The detailed calculation is complicated because rate of loss from one effect is continuously being changed by the loss resulting from other effects.

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The calculations are further complicated by uncertainties in predicting load history and environmental conditions during the entire service life.

4.1 Anchorage Slip Losses (∆f AS) 

In post-tensioned members, a small amount of the force is lost at the anchorages upon transfer because of the anchorage fitting and movement of the wedges.



The magnitude of the slip ( ∆L) is based on the anchorage system used and shall be specified by the manufacturer.

Δ f AS =

Δ L L

E P

Where ∆L is the magnitude of the slip L is the length of the tendon EP is the modulus of elasticity of the prestressing steel Page 6 of 20





This type of losses could be significant for short beams since the losses are inversely proportional to the length of the cable.

4.2 Elastic Shortening Losses (∆f ES) 

As concrete is compressed, it shortens and the prestressing steel is also shortened due to bonding resulting loss of prestress.



In post-tensioned members, for single tendon, there is no need to calculate elastic shortening loss because it is compensated in jacking (not for several tendons jacked sequentially).

Tendon

(a)

P i

P i L

(b)

ΔES

Elastic shortening (a) Unstressed beam, (b) Longitudinally shortened beam

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The strain in the concrete due to elastic shortening (εES)

ε ES =

Δ ES L

Since the concrete and prestressing steel are bonded, the strain in the concrete and the steel are equal (compatibility), therefore,

Δ f ES = E p ε ES = E p =

Pi Ac E c

E p E c

. f c

∴Δ f ES = n ⋅ f c Where n is the modular ratio (Ep / Ec) f c is the inducted stresses due to prestressing, for strands with eccentricity “e”

f c =

Pi A

−

Pi..e.e M ow .e I

+

I

* For pre-tensioned members, the elastic shortening losses ( ∆f ES) is taken as shown above. * For post-tensioned with single tendon or all tendons tensioned at once (simultaneously), the elastic shortening losses ( ∆f ES) is equal zero. * For post-tensioned with tendons tensioned sequentially, the elastic shortening losses ( ∆f ES) is equal half the value shown above. Pre-tensioned

Δ f ES = n ⋅ f c

Post-tensioned (sequentially) 1

Δ f ES = .n ⋅ f c 2

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Friction Losses (∆f WF) 

For post-tensioned members, the tendons are usually anchored at one end and stretched from the other end. As the steel slides in the duct, frictional losses take place, making the tension at the anchored end less than at the jacking end.



The total friction losses is the sum of: Wobble friction, due to unintended misalignment, which is

o

unavoidable due to workmanship. Curvature friction, due to intended curvature.

o



Although, friction losses vary along the span, the maximum value is typically used.

α Tendon

F1

(a)

Pf = F1 F

F

dα

F1 d

F dF (b

F1

F2 = F1 – F1 (c) Curvature friction losses (a) Tendon alignment.

(b) Forces on infinitesimal length where F 1 is at the jacking end. (c) Polygon of forces assuming F 1 = F 2 over the infinitesimal length in (b).

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4.2.1 Wobble Friction Losses (∆f W) According to the ECP 203, the force at any distance x, can be calculated as follows:

P x = Po .e

− kx

Where Po is the prestressing force at the tensioning end of the cable X is the distance measured from the tensioning en of the cable, and it should be in meters K is the coefficient of friction between the tendon and the surrounding due to wobble effect. K equal to 0.0033 for ordinary cables K equal to 0.0017 for fixed ducts Hence, the loss due to wobble friction is equal to:

Δ f W =

Po − P x A ps

4.2.2 Curvature Friction Losses (∆f F) 

The curvature friction losses is function of the curvature o the tendon and the roughness of the surrounding material.

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The ECP 203 gives the following formula to estimate the force at any distance x produced by jacking force, Po as follows:

P x =

⎛ − μ . x ⎞ ⎜ ⎟ ⎜ r ps ⎟ ⎠ Po .e⎝

Where r ps is the radius of the ducts as given below μ is the coefficient of friction and be taken as follows” μ = 0.55 for friction between steel and concrete μ = 0.30 for friction between steel and steel μ = 0.25 for friction between steel and lead

⎛ μ . x ⎞ ⎟ ≤ 0.20 , ECP 203 allows the use of a simplified expression which is For ⎜ ⎜ r ps ⎟ ⎝ given by:

⎛ μ . x ⎞ ⎟ P x = Po ⎜1 − ⎜ r ps ⎟ ⎝ ⎠ Hence, the loss due to curvature friction is equal to:

Δ f F =

Po − P x A ps

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4.2.3 Simplified Friction Losses (∆f WF)

Given that

⎛ ⎞ ⎜ k . x + μ x ⎟ ≤ 0.20 ⎜ r ps ⎟ ⎝

the ECP 203 allows the use of the following

expression for estimating the total friction losses:

⎡ ⎛ ⎞⎤ x μ ⎟⎥ ; P x = Po ⎢1 − ⎜ kx + ⎜ r ps ⎠⎟⎥ ⎢⎣ ⎝ ⎦ ⎛ μ x ⎞⎟ ⎜ ∴ Po − P x = Po kx + ⎜ ⎟ r ps ⎝

Hence, the total friction loss is equal to:

Δ f WF =

Po − P x A ps

4.2.4 Calculating the Radius of Curvature (r ps) Equation of tendon profile (for a parabolic curve) e = a.x2 + b.x + c the constants (a, b, and c) to be determined from the boundary condtions: at x = 0; e = 0 at x = L/2; e = m/2 at x = L; e = 0

Therefore:

e=4

emax 2

L

. x.( L − x ) Page 12 of 20


α/2


α/2

y

m

α x

x

Since the ratio of the depth of the beam to its span is small, α/2 is a small angle and therefore tan(α/2) = α/2 and the length of the arc is equal to the span of the beam (L).

∴ L ≈ r ps .α ∴ r ps ≈

L

α

≈

L2

8.emax

4.3 Shrinkage Losses (∆f SH) 

Normal concrete mixes contain more water than is required for cement hydration. The free water evaporates with time. The rate depends on humidity, temperature, and size/shape of member. Drying is accompanied by reduction in volume, the change occurring at a higher rate initially. Approximately 80% of shrinkage occurs in the first year.

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The shrinkage losses are calculated as follows:

Δ f SH = ε sh .E P

4.3.1 ECP 203 Estimation of εsh When the relative humidity is known, the shrinkage strain (εsh) can be determined according to Table 2.8.A of the ECP 203 (shown below) using the size/shape factor (B).

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Where B = 2Ac / Pc Ac is the area of the concrete section, mm2 Pc is the perimeter of the concrete section, mm

4.3.2 Alternative ECP 203 Method for Estimation of εsh When the environmental factors are not known, ECP 203 allows the shrinkage strain (εsh) to be taken as follows (Table 10-4 of ECP 203): Prestressing System

Shrinkage Strain (εsh)

Pre-tensioned member (3-5 days after casting)

300x10-

Post-tensioned members (7-14 days after casting)

200x10-6

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For stage construction, ECP 203 allows the assumption that 50% of the shrinkage occurs in the first month and 75% occurs during the first six months.

4.3.3 PCI Method Estimation of εsh Alternatively, Using the PCI Method (PCI Design Handbook, section 4.7)

ε SH = 8.2 × 10

−6

V

K SH (1 − 0.06 )(100 − RH ) S

RH = relative humidity

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KSH = factor related to time from the end of moist curing to application of prestress, days Post-tensioned:

Pretension:

Day

1

7

30

60

KSH

0.92

0.77

0.58

0.45

KSH = 1.0

4.4 Creep Losses (∆f CR) 

The continuous deformation of concrete over extended periods of time is known as creep.



The rate of strain increase is rapid at first, but decreases with time until, after many months, a constant value is approached asymptotically.

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Creep strains have been found to depend on applied sustained load, mix ratio, curing conditions, environmental conditions, and the age of concrete when first loaded.

The creep losses are calculated as follows:

Δ f CR = ε cr .E P

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4.4.1 ECP 203 Estimation of εcr The creep strain (εcr ) can be determined according to clause 2-3-3-5 of ECP 203 as follows:

ε cr = ε o (1 + φ )

=

f o E ct

(1 + φ )

f o = stress in concrete at loading Ect = modulus of elasticity at loading Φ = creep coefficient, to be determined from Table 2.8.B of ECP 203 (shown below) using relative humidity and size/shape factor (B).

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4.4.2 Alternative ECP 203 Method for Estimation of εcr When environmental factors are not known, ECP 203 allows the creep strain ( εcr ) to be taken as follows (Table 10-5 of ECP 203): εcr for every N/mm of the working stress Prestressing System

Pre-tensioned beams

Concrete Stress at the time of prestressing, f ci (N/mm2) f ci > 40

f ci ≤ 40

48x10-6

48x10-6 (40/ f ci)

36x10-6

36x10-6 (40/ f ci)

(3-5 days after casting) Post-tensioned beams (3-5 days after casting)



If the working concrete stress at service loads is greater than 33% of the concrete strength, f cu, the creep strain given in the table above (Table 10-4 of ECP 203) should be increased by the factor α determined from Figure 10-7 of ECP 203.

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4.4.3 Bonded Prestressing According to ECP 203, another formula for determining creep losses for bonded prestressed members can be used as follows:

E P

Δ f CR = φ

E c

f cs

Where * f cs = f cs* − f csd

f *cs = the stress in concrete at the level of centroid of the prestressing steel immediately after transfer f *csd = the stress in concrete at the level of centroid of the prestressing steel due to sustained loads at transfer Φ = 2.0 for pre-tensioned and 1.6 for post-tensioned

4.5 Steel Relaxation Losses (∆f R) 

Prestressing tendons undergo relaxation under constant length, depending on steel stress and time interval. The loss magnitude depends on the duration of the sustained prestressing force, and the ratio of f pi / f py.

The steel relaxation losses can be calculated as follows:

Δ f R =

f pi (log t ) ⎛ f pi k 1

⎞ ⎜ − 0.55 ⎟ ⎜ f py ⎟ ⎝ ⎠

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Where f pi = initial stress after immediate losses and before time dependent losses t = time elapsed after jacking, in hours (max 1000 hours) k1 = coefficient depends on the steel type and is taken as follows: = 10 for normal relaxation stress relived strands = 45 for low relaxation stress relived strands.

For step by step loss analysis:

Δ f R =

f pi (log t 2 − log t 1 ) ⎛ f pi k 1

⎞ ⎜ − 0.55 ⎟ ⎜ f py ⎟ ⎝ ⎠

4.6 Total Losses For pre-tensioned members:

Δ f PT = Δ f ES + Δ f SH + Δ f CR + Δf R For post-tensioned members:

Δ f PT = Δ f AS + Δ f ES + Δ f WF + Δ f SH + Δ f CR + Δf R

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Prestress concrete

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