Chapter 2 Design For Serviceability.pdf

EAS458 PRESTRESSED CONCRETE DESIGN

2018/2019

CHAPTER 2 DESIGN FOR SERVICEABILITY 2.1 INTRODUCTION

The fundamental aim of prestressed concrete is to limit tensile stresses, and hence flexural cracking, in the concrete under working conditions. Design is therefore based initially on the requirements of the serviceability limit state. Subsequently considered are ultimate limit state criteria for bending and shear. In addition to the concrete stresses under working loads, deflection must be checked, and attention must also be paid to the construction stage when the prestress force is first applied to the immature concrete. This stage is known as the transfer condition.

The design sequence of prestressed concrete may therefore be summarised as: 1. Design for serviceability – cracking 2. Check stresses at transfer 3. Check deflections 4. Check ultimate limit state – bending 5. Design shear reinforcement for ultimate limit state The detail illustration by the flow chart is in Fig. 2.1.

The level of prestress and the layout of tendons in a member are usually determined from the serviceability requirements for that member. For example, if a water-tight and crack-free slab is required, tension in the slab must be eliminated or limited to some appropriately low value.

For the serviceability requirements to be satisfied in each region of a member at all times after first loading, a reasonably accurate estimate of the magnitude of the prestress is needed in design. This requires reliable procedures for the determination of both the immediate and time-dependent losses of prestress.

Immediate losses of prestress occur during the stressing and anchoring operation and include elastic shortening of concrete, the short-term relaxation of the tendon, friction along a post-tensioned cable and slip at the anchorages. The time-dependent losses of prestress are caused by creep and shrinkage of the concrete and relaxation of steel.

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EC2 Section 2.3.1

Calculate moment variation, Mv = MT – ΩM0

5.10.2

Stress limit

Structure usage Concrete class

Min. section moduli,

Trial section

Serviceability Limit State

2.3.1

Shape, depth, cover, loss allowances, etc

Self-weight + permanent action moment Total moment Draw Magnel diagram for the critical section Select prestress force and eccentricity Determine tendon profile

5.10.4 – 5.10.9

Calculate losses Check final stresses & stresses under quasipermanent loads Check deflections

8.10.3

Design end-block

6.1, 5.10.8

Ultimate moment of resistance

Prestress system

Untensioned reinforcement

Ultimate moment

6.2

Shear reinforcement design

Ultimate shear force

8.10.3

Check end-block (unbonded)

Ultimate Limit State

7.4

FINISH

Figure 2.1 PSC design flow chart Prepared by: Dr. Noorhazlinda Abd Rahman

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There are two critical stages in the design of PSC for serviceability. The first stage is known as transfer, immediately after the prestress is transferred to the concrete, i.e. when the member is subjected to the maximum prestress and the external load is usually at a minimum. At this stage (at transfer), immediate losses have taken place but no time-dependent losses have yet occurred. The prestressing force immediately after transfer is designated in EN 1992-1-1 as Pm0. At transfer, the concrete is usually young and the concrete strength may be relatively low.

The second critical stage is after time-dependent losses have taken place and the full service load is applied. This stage is known as service. Service stage is at time t when the prestressing force is at a minimum and the external service load is at a maximum. The prestressing force at service stage is designated in EN 1992-1-1 as Pm,t and is often referred to as the effective prestress. To determine the in-service behavior of a member, it is therefore necessary to establish the extent of cracking, if any, by checking the magnitude of the elastic tensile stresses. If a member remains uncracked (i.e. the max. tensile stress at all stages is less than the tensile strength of the concrete), the properties of the uncracked section may be used in all deflection and camber calculations. If cracking occurs, a cracked section analysis may be performed to determine the properties of the cracked section and the postcracking behavior of the member.

2.2 ANALYSIS OF CONCRETE SECTION UNDER WORKING LOADS

The primary analysis of PSC is based on service condition, and on the assumption that stresses in the concrete are limited to values which will corresponds to elastic behaviour. The following assumptions are made in the analysis of PSC design: i.

Plane sections remain plane.

ii. Stress-strain relationships are linear (obeys Hooke’s law). iii. Bending occurs about a principal axis. iv. The prestressing force is the value remaining after all losses have occurred. v. Changes in tendon stress due to applied loads on the member have negligible effect on the behaviour of the member. vi. Section properties are generally based on the gross concrete cross-section.

The stress in the steel is unimportant in the analysis of the concrete section under working conditions. But, the force provided by the steel that is considered in the analysis. Prepared by: Dr. Noorhazlinda Abd Rahman

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EAS458 PRESTRESSED CONCRETE DESIGN 2.2.1

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Section properties, notations and sign convention

For elastic analysis and design, a prestressed concrete section may be characterised by a number of variables and geometric parameters. The sign conventions and notations used for the analysis are indicated in Figure 2.2.

Figure 2.2 Sign convention and notation

Referring to Fig. 2.2, the following notations will be used:

I e

Pm0 Pm,t

: : : : : : : : : : : : : : : : : : :

Area of concrete cross-section Area of prestressing steel Moment of inertia of the section Eccentricity of prestressing force Distance of the extreme top fibres from the C.A Distance of the extreme bottom fibres from the C.A Section modulus of the concrete top fibre Section modulus of the concrete bottom fibre Concrete stress at the top fibre at transfer Concrete stress at the bottom fibre at transfer Concrete stress at the top fibre at service Concrete stress at the bottom fibre at service Allowable concrete compressive stress at transfer Allowable concrete tensile stress at transfer Allowable concrete compressive stress at service Allowable concrete tensile stress at service The prestressing force immediately after transfer The prestressing force at service stage (effective prestress) 1 – losses (%) Prepared by: Dr. Noorhazlinda Abd Rahman

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It should be noted that PSC section may contain ungrouted duct, reinforcing steel and unbonded tendon as shown in Fig. 2.3. The voids and the presence of different materials (which can be transformed to equivalent concrete) should be duly accounted for properties calculated on the basis of gross concrete area provide acceptable analysis results.

Figure 2.3

For the purpose of flexural analysis the sign convention used are as follows (Fig. 2.4): i.

Positive moment produces tension in the bottom fibres of the member;

ii. Negative moment produces tension in the top fibres of the member; iii. Compressive stresses are considered positive; iv. Tensile stresses are considered negative.

Sagging moment

Hogging moment Figure 2.4

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EAS458 PRESTRESSED CONCRETE DESIGN 2.2.2

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Basic theory

Prestressing is artificially induced compressive stress in a structure before it is loaded so that any tensile stress which might be caused by the external loads is automatically cancelled, and failure is eliminated. Following example will clarify the basic theory of prestressing.

B

9.6 N/mm2

+

–

B

9.6 N/mm2

Cross-section

Stress distribution B-B

Figure 2.5

From the simple beam theory, it is known that the pattern of stress distribution at any section in a beam will be compression in the top and tension in the bottom both of equal value. From Fig. 2.5, compressive and tensile stresses at the mid-span will be 9.6 N/mm2 of equal value.

Now by having the above condition (Fig. 2.5), we are aiming to put an initial compression into the beam so that the tension in the beam will be canceled out by this initial compression. There are two ways can be considered as follows:

CASE 1: If the compressive force is applied along the line of CoG (centroidal axial/ concentric prestress), this will give a uniform compression on the beam section (Fig. 2.6). To cancel out the tension in the beam, at least 9.6 N/mm2 is needed. Hence, the applied compressive force will be .

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w = 10 kN/m concentric prestress force, P

9.6 N/mm2

19.2 N/mm2

9.6 N/mm2

+ +

+

=

+

– 9.6 N/mm2

Bending

0

Prestress

Final

Figure 2.6

CASE 2: On the other hand, if, instead of a concentric force, an eccentric force, P is applied at a distance e below the centroidal axis, this is equivalent to applying a concentric force P and a moment Pe.

Let assume

⁄ , (Z for the top and bottom beam section is the same), hence, to cancel out the tensile

stress of 9.6 N/mm2 in Fig. 2.7, a force P of 4.8A = 480 kN need to be applied at an eccentricity of Z/A. And, the compressive stress at the top will be 9.6 N/mm2.

When compare with CASE 1, it can be seen that the applied prestressing force is halved in CASE 2 and so is the compressive stress at the top fibre of the beam. So by introducing of an eccentricity, much better use of the concrete can be made since the stresses at the top and bottom vary over the full range of permissible stresses for the two extreme loading conditions.

Alternatively, if the prestressing force used in both cases is the same, i.e. 960 kN (if consider the above example), then the applied load (w) can be doubled (20 kN/m in CASE 2) and the final stress is the same in both cases. Therefore by introducing the eccentricity, the moment-carrying capacity of a beam is increasing.

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w = 10 kN/m

9.6 N/mm2

P/A N/mm2

Pe /Z N/mm2

–

+ +

+

+

=

+

+

–

Pe /Z N/mm2

9.6 N/mm2

Bending

9.6 + (P/A – Pe /Z)

Prestress

–9.6 + (P/A + Pe /Z) = 0

Eccentric (moment)

Final

Figure 2.7 Until to this stage, two basic governing equations can be concluded: At the top fibre:

At the bottom fibre:

Example 2.1 A rectangular beam

is simply supported over 4.0 m span and supports a live load of

10 kN/m. If a straight tendon is provided at an eccentricity of 65 mm below the centroid of the beam section, determine: (a). the min. presstresing force necessary for no tension under live load at mid-span. (b). the corresponding stresses under self-weight only at the mid-span and at the ends of the member. Prepared by: Dr. Noorhazlinda Abd Rahman

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Tutorial

1. A rectangular concrete beam, 100 mm wide by 250 mm deep, spanning over 8 m is prestressed by a straight cable carrying an effective prestressing force of 250 kN located at an eccentricity of 40 mm. The beam supports a live load of 1.2 kN/m. (a) Calculate the resultant stress distribution for the central cross section of the beam. The density of concrete is 25 kN/m3. (b) Find the magnitude of the prestressing force with an eccentricity of 40 mm which can balance the stresses due to dead and live loads at the bottom fiber of the central section of the beam. 2. A prestressed concrete beam supports a live load of 4 kN/m over a simply supported span of 8 m. The beam is a rectangular section with an overall depth of 400 mm and a width of 200mm. The beam is to be prestressed by an effective presstressing force of 235 kN at a suitable eccentricity such that the resultant stress at the soffit of the beam at the central of the span is zero. (a) Find the eccentricity required for the force. (b) If the tendon is concentric, what should be the magnitude of the prestressing force for the resultant stress to be zero at the bottom fiber of the central span section. 3. A prestressed concrete beam, 200 mm wide and 300 mm deep, is used over an effective span of 6 m to support an imposed load of 4 kN/m. The density of concrete is 25 kN/m3. At the quarter-span section of the beam, find the magnitude of: (a) The concentric prestressing force necessary for zero fiber-stress at the soffit when the beam is fully loaded; (b) The eccentric prestressing force located 100 mm from the bottom of the beam which would nullify the bottom fiber stress due to loading. 4. A concrete beam of rectangular section spanning 8 m has a width of 200 mm and the overall depth of 400 mm. The beam is prestressed by a parabolic cable with an eccentricity of 15 mm at the centre and zero at the supports with an effective prestressing force of 100 kN. The live load on the beam is 2kN/m. Draw the stress distribution diagram at the central section for: (a) Prestress + self-weight (density of concrete = 25 kN/m3); and (b) Prestress + self-weight + live load.

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2.3 DESIGN FOR SERVICEABILITY LIMIT STATE

The design of a prestressed concrete member is based on maintaining the concrete stresses within specified limit at all stages of the life of the member. Hence, the primary design is based on the SLS, with the concrete stress limits based on the acceptable degree of flexural cracking, the necessity to prevent excessive creep and the need to ensure that excessive compression does not result in longitudinal and micro cracking.

The codes of practice limit the allowable stresses in prestressed concrete. Most of the work of PSC design involves ensuring that the stresses in the concrete are within the permissible limits. The stresses produced by the prestress force must be considered in conjunction with the stresses caused by max. and min. values of applied moment. Hence, for the SLS case, at any section in a member, there are two conditions of checking required:

At Transfer (initial condition) This is when the concrete first feels the prestress, immediately after the prestress has been applied. The concrete, at this stage, is usually relatively immature and not at full strength. Hence, transfer is a critical stage which should be considered carefully. The stresses are only due to prestress and self-weight (will consider

).

At Service The stresses induced by the SLS loading, in addition to the prestress and self-weight, must be checked (will consider

). At service stage, the concrete has its full strength but losses will have occurred and so

the prestress force is reduced.

The ultimate capacity at ULS of the PSC section (as for RC) must also be checked. If there is insufficient capacity, the addition of non-prestressed reinforcements is allowed.

2.3.1

Limitation of concrete stresses.

Depending on the serviceability requirements for a particular structure, a designer may set limits on the tensile and compressive stresses in concrete both at transfer and under the full service loads. The concrete stress limits specified in EN 1992-1-1 Clause 5.10.2.2 are as follows (Tab. 2.1):

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Table 2.1 Stresses Symbol Compressive Tensile Notes: ( )

(*)

Loading stage Transfer Value/Equation Symbol ( )

Service Value/Equation

(*)

0

= The characteristic compressive strength of the concrete at time t when it is subjected to the prestressing force. = The characteristic compressive cylinder strength of concrete at 28 days (Tab. 3.1 EN 19921-1). = Mean value of axial tensile strength of concrete (Tab. 3.1, EN 1992-1-1). may be taken equal to – 1 N/mm2 if the sections are designed not to be in tension in service.

If the tensile stress in the concrete is limited to the value of f ctm then all stresses can be calculated on the assumption that the section is uncracked and the gross concrete section is resisting bending. (Similar to Class 1 of prestressed concrete defined in BS code)

If cracked is allowed for certain degree, then calculations may have to be based on a cracked section. Limited cracking is permissible depending on whether the beam is pre- or post-tensioned and the appropriate exposure class.

The choice of whether to permit cracking to take place or not will depend on a number of factors which include conditions of exposure and the nature of loading. If a member consists of precast segments with mortar joints, or if it is essential that cracking should not occur, then it will be designed to be in compression under all load conditions.

2.3.2

Governing inequalities

The design of prestressing requirements is based on the manipulation of the four basic expressions describing the stress distribution across the concrete section. These are used in conjunction with the permissible stresses appropriate to the type of member and covering the following conditions: 1. Initial transfer of prestress force with the association loading (often just the beam’s self-weight); 2. At service, after prestress losses, with minimum and maximum characteristics loading.

For a single-span, simply supported beam it is usually the min. moment at transfer and the max. moment at service that will govern as shown in Figure 2.8.

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Figure 2.8 Prestress beam at transfer and service

From the figure and Eqs. 2.1 to 2.2, the governing equations for a single-span beam are:

At transfer:

At service:

Please be noted that: 1. M0 is referring to the min. applied moment – depend on the self-weight only; 2. MT is referring to the max. applied moment – depend on both the imposed load and self-weight; 3. Pm,t is a prestressing force after considering losses effects (%) = ΩPm0.

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EAS458 PRESTRESSED CONCRETE DESIGN 2.3.3

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Minimum section moduli

The 1st step in PSC design is to calculate the minimum section moduli for the expected moments. To do this, the governing inequalities (Eqs. 2.3 – 2.6) will be utilized. By comparing conditions (2.3) and (2.5), and condition (2.4) and (2.6), the expression for Zt and Zb are as follows:

Since we are dealing with single span simply supported members, it will be assumed that M T and M0 occur at the mid-span. The above equations are the minimum section moduli required for preliminary sizing of the sections. Note that in the above equations, MT and M0 are the functions of self-weight of the section which is unknown at this point. Hence, a trial section or a reasonable self-weight must be assumed initially, and then check once the section has been decided upon giving the actual Zt and Zb values. This min. values of section moduli must be satisfied by the chosen section in order that a prestress force and eccentricity exist which will permit the stress limits to be met; but to ensure that practical considerations are met the chosen section must have a margin above the min. values calculated above. The equations of min. moduli depend on the difference between max. and min. values of moment.

The following formula of the ratio of acceptable span to depth for a prestressed beam may be used as a guide and will generally produce reasonably conservative designs for post-tensioned members.

Example 2.2 Select a rectangular section for a post-tensioned beam to carry, in addition to its self-weight, a uniformly distributed load of 3 kN/m over a simply supported span of 10 m. The member is to be designed with a concrete strength class C40/50 and is restrained against torsion at the ends and at the mid-span. Assume 20% loss of pre-stress.

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Example 2.3 The single-span, simply-supported beam shown below carried the loads as shown. Taking the losses to be 25%, determine an appropriate rectangular section for the member by taking the density of prestressed concrete to be 25 kN/m3.

Tutorial A post-tensioned prestressed beam of a rectangular section 300 mm wide is to be designed for an imposed load of 6.6 kN/m uniformly distributed on a span of 15 m. The member is to be designed with a concrete strength class C40/50. Taking the density of prestressed concrete to be 25 kN/m3 and the characteristic compressive strength of the concrete at transfer is 27 MPa, propose a minimum possible depth of the beam if the loses is 15%.

2.3.4

Prestressing force and eccentricity

The inequalities of equations 2.3 to 2.6 may be rearranged to give expressions for the min. required prestress force for a given eccentricity.

At transfer: ( ( ⁄

) )

(

) (

⁄

)

( ⁄

) )

(

) )

(

)

(

(

)

)

(

)

At service: ( ( ⁄

( (

) )

(

)

(

)

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EAS458 PRESTRESSED CONCRETE DESIGN Note: In Eq. 2.9 and 2.11, it is possible that the denominator term ( ⁄ ⁄

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), might be negative if

. In this case, the sense of the inequality would have to change as the effect of dividing an

inequality by a negative number is to change its sense.

Those equations give a range within which the prestress force must lie to ensure that the allowable stress conditions are met at all stages in the life of the member. In this case of a simply supported beam, the design prestress force will generally be based on the minimum value which satisfies those equations at the critical section for bending in the member.

It is necessary to consider the effect of limiting the eccentricity to a maximum practical value for the section under consideration. Such limits will include consideration of the required minimum cover to the prestressing tendons which will depend on the exposure and structural class assumed for the design. The effect of this limitation will be most severe when considering the maximum moments acting on the section, that is, the inequalities of equations 2.5 and 2.6.

If the limiting value for maximum eccentricity emax, depends on cover requirements, Eq. 2.5 (service, top) becomes: (

)

And Eq. 2.6 (service, bottom) becomes: (

)

This represents linear relationships between MT and Pm0. For the case of a beam subject to sagging moments emax will generally be positive in value, thus equation 2.14 is of positive slope and represents a lower limit to Pm0. It can also be shown that for most practical cases [(Zt/Ac) – emax] < 0, thus equation 2.13 is similarly a lower limit of positive, though smaller slope.

Figure 2.9 represents the general form of those expressions, and it can be seen clearly that providing a prestress force in axcess of Y’ produces only small benefits of additional moment capacity. The value of Y’ is given by the intersection of Eqns. 2.13 and 2.14.

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MT

Pm0 Figure 2.9 Max. moment and prestress force relationship

Thus, (

)

The value of prestress force Pm0 = Y’ may be conveniently considered as a maximum economic value beyond which any increase in prestress force would be matched by a diminishing rate of increase in moment-carrying capacity. If a force larger than this limit is required for a given section it may be more economical to increase the size of this section.

2.4 MAGNEL DIAGRAM

2.4.1

The principles behind the Magnel Diagram

Magnel diagram is a graphical method for the analysis of a pre-stressed concrete beam and for the determination of safe pre-stressing force and eccentricity.

The four lines associated with the limits on stress (the governing inequalities) can be used to determine a range of possible values of pre-stress force for a given or assumed eccentricity by plotting a graph [1/ vs. ].

The relationship between 1/P and e are linear and if plotted graphically, they provide a useful means of determining appropriate values of P and e. Prepared by: Dr. Noorhazlinda Abd Rahman

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When these four equations are plotted, a feasible region is found in which points of 1/P and e simultaneously satisfies all four stress limits. Any such point then satisfies all four equations.

2.4.2

The construction of Magnel Diagram

Equations 2.9 to 2.12 can be used to determine a range of possible values of prestress force for a given or assumed eccentricity. For different assumed values of eccentricity further limits on the prestress force can be determined in an identical manner although the calculations would be tedious and repetitive. In addition, it is possible to assume values of eccentricity for which there is no solution for the prestress force as the upper and lower limit could overlap.

A much more useful approach to design can be developed if the equations are treated graphically as follows. Equations 2.9 to 2.12 can be rearranged into the following form:

At transfer: ( ⁄

⁄ ) ⁄

(

( ⁄

⁄ ) ⁄

(

)

( )

)

(

)

At service: ( ⁄

⁄ ) ⁄

(

( ⁄

⁄ ) ⁄

(

) )

(

)

(

)

The above equations express linear relationships between ⁄

and e.

Note that in eq. 2.16 the sense of the inequality has been reversed to account for the fact that the denominator is negative. The relationship can be plotted as shown in Fig. 2.10(a) and (b) and the area of the graph to one side of each line, as defined by the inequality, can be eliminated, resulting in an area of graph within which any combination of force and eccentricity will simultaneously satisfy all four inequalities and hence will provide a satisfactory design. This form of construction is known as a Magnel Diagram. The additional line (A) shown on the diagram corresponds to a possible physical limitation of the max. eccentricity allowing for the overall depth of section, cover to the prestressing tendons, provision of shear links and so on. Prepared by: Dr. Noorhazlinda Abd Rahman

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Two separate figures are shown as it is possible for line eq. 2.18, derived from eq. 2.11, to have either a ⁄ . positive or a negative slope depending on whether fcc,t is greater or less than

(a)

(b) Figure 2.10

The Magnel Diagram is a powerful design tool as it covers all possible solutions of the inequality equations and enables a range of prestress force and eccentricity values to be investigated. Values of min. and max. prestress force can be readily read from the diagram as can intermediate values where the range of possible eccentricities for a chosen force can be easily determined. The diagram also shows that the min. prestress force (largest value of 1/P m0) corresponds to the max. eccentricity, and as the eccentricity is reduced the prestress force must be increased to compensate. Example 2.5 Draw a Magnel diagram for the beam given in Example 2.2 and determine the min. and max. possible values of prestress force. Assume a max. possible eccentricity of 125 mm allowing for a cover etc. to the tendons.

Example 2.6 A post-tensioned prestressed beam of a rectangular section 250 mm wide is to be designed for an imposed load of 12 kN/m uniformly distributed on a span of 12 m. The member is to be designed with a concrete strength class C40/50. The losses of prestress assumed to be 15%. (a) Determine the minimum possible depth of beam (Ans:

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(b) For the section provided, determine the minimum prestressing force and corresponding eccentricity at the mid-span. (Ans: Under service condition: Pm0,min. = 2383 kN and emin. = 30.1 mm; Under transfer condition: Pm0,min. = 1131 kN and emin. = 194 mm) (c) Check the results by Magnel’s graphical method.

2.5 DESIGN OF TENDON PROFILE

Having obtained a value of prestress force which will permit all stress conditions to be satisfied at the critical section (at the mid-span), it is necessary to determine the eccentricity at which this force must be provided, not only at the critical section but also throughout the length of the member.

At any section along the member, e is the only unknown term in the four equations 2.3 to 2.6 and these will yield two upper and lower limits which must all be simultaneously satisfied. This requirement must be met at all sections throughout the member and will reflect both variations of moment, prestress force and section properties along the member.

The design expressions can be rewritten as: At transfer: [

(

]

[

]

)

(

(

)

)

(

)

(

)

(

)

At service: [

[

(

]

]

(

)

)

Eqs. 2.20 to 2.23 can be evaluated at any section to determine the range of eccentricities within which the resultant force Pm0 must lie. The moments MT and M0 are those relating to the section being considered. As we wish to limit possible tensile stresses, we only examine Eqs. 2.20 and 2.23 corresponding to tension on the top at transfer and on the bottom in service.

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For a member of constant cross-section, if minor changes in prestress force along the length are neglected, the terms in brackets in the above expressions are constant. Therefore the zone within which the centroid must lie is governed by the shape of the bending moment envelopes, as shown in Fig. 2.11.

Figure 2.11

In the case of uniform loading the bending moment envelopes are parabolic, hence the usual practice is to provide parabolic tendon profiles if a straight profile will not fit within the zone. At the critical section, the zone is generally narrow and reduces to zero if the value of the prestress force is taken as the min. value from the Magnel diagram. At sections away from the critical section, the zone becomes increasingly greater than the min. required.

Example 2.7 Determine the cable zone limits at mid-span and ends of the member designed in Examples 2.2, 2.4 and 2.5 for a constant initial prestress force of 700 kN. Data for this question are given in the said examples. Sketch the upper and lower eccentricities.

2.6 LOSSES

In prestressed concrete applications, the most important variable is the prestressing force. In the early days, it was observed that the prestressing force does not stay constant, but reduces with time. Even during prestressing of the tendons and the transfer of prestress to the concrete member, there is a drop of the prestressing force from the recorded value in the jack gauge. The various reductions of the prestressing force are termed as the losses in prestress.

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For the serviceability requirements to be satisfied in each region of a member at all times after first loading, a reasonably accurate estimate of the magnitude of prestress is needed in the design. This requires reliable procedures for the determination of both the immediate and time-dependent losses of prestress (Fig. 2.12).

Immediate losses of prestress occur during the stressing (and anchoring) operation and include elastic shortening of concrete, the short-term relaxation of the tendon, friction along a post-tensioned cable and the slip at the anchorages. Meanwhile, the time-dependent losses of prestress are caused by creep and shrinkage of the concrete, and relaxation of steel.

Figure 2.12 Causes of various losses in prestress

2.6.1

Definitions

The losses of prestress that occur in a tendon are categorised as either immediate losses or time-dependent losses (as illustrated in Fig. 2.13) 𝑃 Jacking force

𝑃

𝑃 Prestressing force immediately after transfer

Final or effective prestressing force

Figure 2.13 Losses of prestress in the tendons

Immediate losses occur when the prestress is transferred to the concrete at time t0 and may vary along the length of the tendon. Immediate losses are the difference between the force imposed on the tendon by the hydraulic prestressing jack Pmax (=Pj) and the force in the tendon immediately after transfer at a distance x from the active end of the tendon Pm0(x) and can be expressed as: Prepared by: Dr. Noorhazlinda Abd Rahman

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( )

Immediate loss =

Time-dependent losses are the gradual losses of prestress that occur with time over the life of the structure. If Pm,t(x) is the force in the prestressing tendon at x from the active end of the tendon after all losses, then,

Time-dependent loss =

( )

( )

Both immediate and time-dependent losses are made up of several components. Immediate losses depend on the method and equipment used to prestress the concrete and include losses due to elastic shortening of concrete, wedge draw-in at the prestressing anchorage, friction in the jack and along the tendon, deformation of the forms for precast members, deformation in the joints between elements of precast structures, temperature changes that may occur during this period and the relaxation of the tendon in a pretension member between the time of tensioning the wires before the concrete is cast and the time of transfer (particularly significant when the concrete is cured at elevated temperatures prior transfer).

Time-dependent losses are the gradual losses of prestress that occur with time over the life of the structure. These include losses caused by the gradual shortening of concrete at the steel level due to creep and shrinkage, relaxation of the tendon after transfer and time-dependent deformation that may occur within the joints in segmental construction.

2.6.2

Immediate losses

The magnitude of immediate losses is taken as the sum of the losses caused by each relevant phenomenon. Where appropriate, the effects of one type of immediate loss on the magnitude of other immediate losses should be considered. For example, in a pretensioned member, the loss caused by relaxation of the tendon prior to transfer will affect the magnitude of the immediate loss caused by elastic deformation of concrete.

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EAS458 PRESTRESSED CONCRETE DESIGN

2018/2019

Elastic deformation losses 1. Pretensioned member When the tendons are cut and the prestressing force is transferred to the member, the concrete undergoes immediate shortening due to the prestress. The tendon also shortens by the same amount, which leads to the loss of prestress. The elastic shortening loss is quantified by the drop in prestress (ΔPel) in a tendon due to the change in strain in the tendon (Δεp,0). It is assumed that the change in strain in the tendon, Δεp,0 is equal to the strain in concrete (εcp,0) at the level of the tendon due to the prestressing force. This assumption is called strain compatibility between concrete and steel. The strain in concrete at the level of the tendon is calculated from the stress in concrete (σcp,0) at the same level due to the prestressing force. A linear elastic relationship is used to calculate the strain from the stress.

The quantification of the losses due to elastic shortening of the concrete is derived mathematically below: i). The elastic shortening loss = the loss in prestress force,

ii). The change in strain in a tendon (

) = the strain in concrete (

) at the level of the tendon:

Substitute into 2.26:

iii). The elastic shortening loss = The drop in prestress (ΔPel) = [the change in stress of the tendon] x [cross sectional area of prestressing steel].

(

)

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EAS458 PRESTRESSED CONCRETE DESIGN

2018/2019

2. Post-tensioned member If there is only one tendon, or with two or more tendons stressed simultaneously, the elastic deformation of the concrete occurs during the stressing operation before the tendons are anchored, hence in this case, the elastic shortening losses are zero.

In a member containing more than one tendon and where the tendons are stressed sequentially, stressing of a tendon causes an elastic shortening loss in all previously stresses and anchored tendons. Consequently, the 1st tendon to be stressed suffers the largest elastic shortening loss and the last tendon to be stressed suffers no elastic shortening loss at all. Elastic shortening losses in the tendons stressed early in the prestressing sequence can be reduced by re-stressing the tendons (prior to grouting of the prestressing ducts). It is relatively simple to calculate the elastic shortening losses in an individual tendon of a posttensioned member, provided the stressing sequence is known. For most cases, it is sufficient to determine the average loss of stress as follows:

In post-tensioned members, the tendons are not bonded to the concrete until grouting of the duct occurs after the stressing sequence is completed. It is the shortening of the member between the anchorage plates that leads to elastic shortening, and not the strain at the steel level, as in the case for pretensioned members.

For simplicity, the loss in all the tendons can be calculated based on the stress in concrete at the level of centroid of prestressing steel. This simplication cannot be used when tendons are stretched sequentially in a post-tensioned member.

3. The simplified formula If the transfer force is Pm0, and the force after elastic shortening loss is P’ (the remaining force), then:

(

)

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EAS458 PRESTRESSED CONCRETE DESIGN

2018/2019

From 2.32: (

)

(

)

Substitute into 2.34: (

(

)

)

**Note that Eq. 2.36 can only be applied for pretensioned member.

For post-tensioned member, it is normally adequate to assume 50% of the above (Eq. 2.36) losses. In this case, the remaining prestress force is: (

)

Example 2.8 A prestressed concrete sleeper produced by pretensioning method has a rectangular cross section of 300 mm wide and 250 mm deep. It is prestressed with 9 nos. of straight 7 mm diameter wires at 80% of the ultimate strength of 1570 N/mm2 as shown in Fig. E.g. 2.8. Estimate the percentage loss of stress due to elastic shortening of concrete. Given Ep = 195,000 MPa and Ecm,0 = 30 GPa.

Figure E.g. 2.8 Prepared by: Dr. Noorhazlinda Abd Rahman

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EAS458 PRESTRESSED CONCRETE DESIGN

2018/2019

Solution using Eq. 2.32: (

)

The calculation of losses will be performed separately for top and bottom wires.

Beam’s properties:

Parameters due to prestresing force: ̅

Pretressing force: ⁄

Stresses due to prestressing force: ⁄ ⁄

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EAS458 PRESTRESSED CONCRETE DESIGN

2018/2019

Elastic shortening (the drop in prestress): ⁄ ⁄

Loss in percentage:

Solution using Eq. 2.36: (

(

)

)

Losses in prestressing (%):

Conclusion: The results are not too much different.

Friction in ducts (friction along the tendon) The friction generated at the interface of concrete and steel during the stretching of a curved tendon in a post-tensioned member, leads to a drop in the prestress along the member from the stretching end. The loss due to friction does not occur in pretensioned members because there is no concrete during the stretching of the tendons.

The friction is generated due to the curvature of the tendon and the vertical component of the prestressing force. In addition to friction, the stretching has to overcome the wobble of the tendon. The wobble refers to the change in position of the tendon along the duct. The losses due to friction and wobble are grouped together under friction.

The magnitude of the friction loss depends on the tendon length, x, and the total angular change of the tendon over that length, as well as the size and type of the duct containing the tendon. An estimation of the loss of force in the tendon due to friction at any distance x from the jacking end may be made using:

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EAS458 PRESTRESSED CONCRETE DESIGN ( )

(

(

)

2018/2019

)

where: : The sum in rad. of the absolute values of successive angular deviations of the tendon over the length x. : The coefficient of friction between the tendon and its duct and depends on the surface characteristics of the tendon and the duct, the presence of rust on the surface of the tendon and the elongation of the tendon. Table 5.1 Cl. 5.10.5.2 EN1992-1-1 (0.19 for strand; 0.17 for cold drawn wire) : The estimate of the unintentional angular deviation (in rad./m) due to wobble effects in the straight or curved parts of internal tendons and depends on rigidity of sheaths, the spacing and fixing of their supports, the care taken in placing the prestressing tendons, the clearance of tendons in the duct, the stiffness of the tendons and the precautions taken during concreting. Typical value is between 0.005 to 0.01 rad./m Example 2.9 A post-tensioned beam shown in Fig. E.g. 2.9 is stressed by two tendons with a parabolic profile and having a total cross-sectional area

. The total initial prestressing force, Pm0 = 10,500 kN

and the total characteristics strength is 14,000 kN. Given the parabolic curve is y = 2.844x10-6x2. Evaluate the prestress loss at mid-span due to elastic shortening of the concrete and friction. Use the following data:

, Ecm(transfer) = 32 kN/mm2, Es = 205 kN/mm2. Take the average

,

e=0

c.a

ɵ

1800 mm

eccentricity for the parabolic tendon as 2/3ec.

ec = 640

30 m

1200 mm

Cross-sectional area A = 1.05 m2 Second moment of area I = 0.36 m4

Cross-section at mid-span

Figure E.g. 2.9 Solution: Loss due to elastic shortening: From Eq. 2.37:

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EAS458 PRESTRESSED CONCRETE DESIGN

(

⁄

2018/2019

)

⁄

(

)

(

)

Loss due to friction: From Eq. 2.38: ( )

(

(

( )

)

)

(

(

)

)

( )

Creep of concrete The sustained compressive stress on the concrete will cause a long-term shortening due to creep, which will similarly reduce the prestress force. As in the previous elastic shortening, the stress in the concrete at the level of the steel is important. Hence; (

)

And,

Therefore, ( (

(

)

) )⁄

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EAS458 PRESTRESSED CONCRETE DESIGN (

)(

(

2018/2019

) )

Note that P’ is the prestressing force after considering all short-term losses

Shrinkage of concrete This is based on empirical figures for shrinkage/unit length of concrete (ε cs) for particular curing conditions and transfer maturity. Typical values range from

(outdoor exposure) to

(indoor exposure). , hence,

Example 2.10 Use info. in Example 2.9, estimate the losses due to creep and shrinkage. Given: creep coefficient (

)

and shrinkage strain,

.

Solution: Loss due to creep: From Eq. 2.41: (

)(

(

) )

P’ is the prestressing force after considering all short-term losses: Short-term loss due to elastic shortening = Short-term loss due to friction = 459 kN Total short-term loss = 814 kN (

)

(

)(

)

Loss due to shrinkage: From Eq. 2.42:

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