Structural analysis
5.10. Prestressed members and structures
5.10.1. General
2-2/clause 5.10 covers specific rules for prestressed concrete members and structures and covers both pre-tensioned as well as post-tensioned bridges. It deals with maximum permis-sible prestressing forces, prestress losses and the treatment of prestress in section design and global analysis. It does not cover the design of anchorage zones, which is covered by 2-2/
clause 8.10. The rules in this section are very much geared towards post-tensioning and some interpretation is needed for pre-tensioned beams made composite with a deck slab, as noted in the text and examples below.
2-1-1/clause 5.10.1(2)allows the effects of prestressing to be considered as an action or as part of the resistance, but individual clauses usually make it clear as to which approach is to be used so there is little real choice. In general, prestress is treated as an action (2-1-1/clause 5.10.1(3)) and is included in the combinations in EN 1990 as such. For example, prestress is treated as an applied force in the design of end blocks (see section 8.10) and, where elastic analysis is used, in member serviceability design and in the flexural design of unbonded or externally post-tensioned members. The effects of prestress are usually split into axial force and moment components.
For the bending resistance of members with bonded prestressing, the prestressing is most conveniently treated as part of the resistance. 2-1-1/clause 5.10.1(4) requires that the contribution of the prestressing tendons to the section resistance should be limited to their additional strength beyond prestressing. This is intended to prevent double-counting of the design prestressing force, which would occur if it was included both on the loading side (as a primary prestress moment and axial force) and in the section resistance calculation.
This requirement is most simply achieved by treating the secondary effects of prestress as an applied action on the loading side, but omitting the primary effects of the design prestressing force. The design prestressing force is then taken into account in the section bending resis-tance by shifting the origin of the design stress–strain diagram for the prestressing tendons by an amount corresponding to the design prestress. The initial strain in the prestress
2-1-1/clause
corresponding to this prestressing force is called the prestrain. This method is discussed further in section 6.1 of this guide.
2-1-1/clause 5.10.1(5)P requires that brittle failure of a prestressed member, caused by sudden failure of prestressing tendons upon cracking of the concrete in flexure, is avoided.
2-2/clause 5.10.1(106) requires this to be achieved using one of the methods in 2-2/clause 6.1(109), which are discussed in section 6.1 of this guide. It is a new codified check for UK designers. The requirement is analogous to that for minimum reinforcement in reinforced concrete elements.
5.10.2. Prestressing force during tensioning 5.10.2.1. Maximum stressing force
EC2 defines maximum permissible limits to the stressing force both during and after tension-ing in order to reduce the risk of tendon failure, to avoid stresstension-ing into the non-linear portion of the prestressing cable’s stress–strain curve and to ensure that excessive relaxation of the stress in the tendons will not occur. The limit during tensioning is covered in this clause while the limit after tensioning is covered in 2-1-1/clause 5.10.3.
The maximum force applied to a tendon, Pmax, should not exceed the value given in 2-1-1/
clause 5.10.2.1(1)Pas follows:
Pmax¼ App;max 2-1-1/(5.41)
where Apis the cross-sectional area of the tendon and p;maxis the maximum allowable stress for the tendon, defined as the minimum of k1fpk or k2fp0:1k where fpk and fp0:1k are the characteristic stress and 0.1% proof stress respectively, as discussed in section 3.3. The values of k1and k2 may be given in the National Annex and are recommended by EC2 to be taken as 0.8 and 0.9 respectively. It is worth noting that, for these recommended values (and assuming a minimum fpk¼ 1:1fp0:1k, as discussed in section 3.3.4), the maximum per-missible tendon jacking stress is slightly greater than the design yield strength (0.88fp0:1k or 0.90fp0:1kcompared to design yield of 0.87fp0:1k if s¼ 1:15). For prestressing strand to EN 10138-3, typically fp0:1k¼ 0:86fpkwhereupon the first limit equates to the even higher value of 0.93fp0:1k. A similar situation arose in BS 5400 Part 4.9
2-1-1/clause 5.10.2.1(2) allows higher stressing, to k3fpk, if the force in the jack can be measured to an accuracy of 5% of the final value of prestressing force. The value of k3 may be given in the National Annex and is recommended by EC2 to be taken as 0.95.
This value should never be assumed during design. Its use is intended for overcoming shortfalls in prestressing force caused by unforeseen problems during construction, such as unexpectedly high friction and wobble losses in ducted tendons. (2-1-1/clause 5.10.3 requires that prestress force should be checked on site by measuring both force and tendon extensions as is normal good practice.) The decision to prestress to this elevated stress must be made in conjunction with the prestressing supplier as it carries an increased risk of strand failure during stressing.
5.10.2.2. Limitation of concrete stress
2-1-1/clause 5.10.2.2 defines several rules for prestressed concrete members to ensure that crushing or splitting of the concrete is avoided during the prestresssing operations and throughout the life of the structure. At anchorages, such effects are generally critical at initial application of the prestress due to the long-term reductions in prestress force and the strength gain of concrete with time, although the prestress force in unbonded and exter-nally prestressed members can potentially increase to a higher value under ultimate load con-ditions – see section 5.10.8.
In the design of post-tensioned members, prestressing forces are applied directly to the member ends as concentrated forces from relatively small anchorages. These forces must then spread out over the cross-section of the member resulting in high local concrete stresses in this zone. The design and detailing of such end blocks is discussed further in section 8.10 of this guide. Transmission lengths in pre-tensioned members are also discussed in section 8.10.
2-1-1/clause 5.10.1(5)P 2-2/clause 5.10.1(106)
2-1-1/clause 5.10.2.1(1)P
2-1-1/clause 5.10.2.1(2)
Although rules for checks on bursting of concrete around anchorages are covered in EC2-2, no explicit check is given on crushing and splitting of the concrete directly in front of the anchor plate in post-tensioned beams. 2-1-1/clause 5.10.2.2(1)P requires such local damage to be avoided. As in previous UK practice, it is therefore necessary to ensure that the concrete has achieved the required minimum strength quoted in the prestressing suppliers’ system specification, referred to in 2-1-1/clause 5.10.2.2(2) as the ‘European Technical Approval’ (ETA).
Where a tendon is loaded in stepped increments or is not stressed to the maximum force assumed in the ETA, 2-1-1/clause 5.10.2.2(4) allows the required minimum concrete strength at transfer to be reduced, subject to an absolute minimum value. This value can be specified in the National Annex and is recommended by EC2 to be taken as 50% of the required minimum concrete strength for full prestressing according to the ETA. 2-1-1/
clause 5.10.2.2(4) recommends that the required concrete strength can be obtained by linear interpolation such that a prestressing force of 30% of the maximum requires 50%
of the concrete strength for full prestress and full prestress requires the full concrete strength specified in the ETA.
The compressive stress in the concrete away from anchorages should also be limited to prevent longitudinal cracking, which is undesirable from durability considerations. 2-1-1/
clause 5.10.2.2(5)defines a limit of 0.6fckðtÞ, where fckðtÞ is the characteristic compressive strength of the concrete at the time of prestressing. This limit corresponds to that provided in 2-2/clause 7.2, which is used to prevent longitudinal cracking where the element is in an aggressive environment. This limit can be increased (to a recommended value of 0.7fckðtÞ, which may be varied in the National Annex) for pre-tensioned elements where tests or experience show that longitudinal cracking will not occur. If the compressive stress in the concrete exceeds 0.45fckðtÞ under the quasi-permanent combination of actions, 2-1-1/
clause 5.10.2.2(5) requires non-linear creep to be considered as discussed in section 3.1.4.
No limits are given for concrete tensile stresses at transfer so it must be assumed that the serviceability limit state crack width limits of 2-2/clause 7.3 apply. The decompression check required by 2-2/Table 7.101N need only be applied at 100 mm from the strands so is inappropriate for the beam top fibre, where this is remote from the strands. Crack widths could, however, be checked and limited to 0.2 mm in accordance with 2-2/Table 7.101N.
Alternatively, the National Annex may modify 2-2/Table 7.101N to give further guidance.
Possibilities would be to redefine the decompression check so that it applies to the extreme fibres for checks at transfer, or to specify a limit of 1 MPa of tension as was permitted in BS 5400 Part 4.9
5.10.3. Prestress force
At any given time, t, and distance, x, from the stressing end of the tendon, the mean prestress force, Pm;tðxÞ, is equal to the maximum force applied at the jacking end ðPmaxÞ minus the immediate losses, PiðxÞ, and time-dependent losses, Pcþ s þ r:
Pm;tðxÞ ¼ Pmax PiðxÞ Pcþ s þ r ðD5.10-1)
This definition is provided in both 2-1-1/clause 5.10.3(1)P and 2-1-1/clause 5.10.3(4).
Care is needed in applying the above, as EC2 presents equations for short-term losses,
PiðxÞ, for a single tendon, whereas the long-term losses, Pcþ s þ r, are presented for the whole group of tendons.
Whatever the initial jacking load, 2-1-1/clause 5.10.3 defines maximum permissible stresses in the tendon immediately after anchoring (post-tensioning) or transfer (pre-tensioning) including the effects of immediate losses only (i.e. at a time of t¼ t0Þ. The prestressing force after stressing and immediate losses is given by:
Pm0ðxÞ ¼ Pmax PiðxÞ ðD5.10-2)
This must nowhere exceed the limit given in 2-1-1/clause 5.10.3(2):
Pm0ðxÞ ¼ Appm0ðxÞ 2-1-1/(5.43)
where pm0ðxÞ is the stress in the tendon at point x immediately after tensioning or transfer.
2-1-1/clause 5.10.3(2) limits pm0ðxÞ to the minimum of k7fpkor k8fp0:1k, where fpkand fp0:1k are the characteristic stress and 0.1% proof stress respectively. The values of k7and k8may be given in the National Annex and are recommended by EC2 to be taken as 0.75 and 0.85 respectively. These are typically a little higher than used in previous UK bridge design practice. For prestressing strand to EN 10138-3, typically fp0:1k¼ 0:86fpk so the second limit governs, giving an allowable force of 73.1% of the characteristic tensile strength.
The limit on force after tensioning was 70% of the characteristic tensile strength of the tendon in BS 5400 Part 4.9
2-1-1/clause 5.10.3(3)requires the following to be considered in determining the immedi-ate losses, PiðxÞ:
. losses due to the elastic deformation of concrete, Pel;
. losses due to short-term relaxation, Pr (only affecting pre-tensioned members where there is a delay between stressing and transfer to the concrete);
. losses due to friction, P ðxÞ;
. losses due to anchorage slip (or wedge draw-in), Psl.
These losses are discussed in sections 5.10.4 and 5.10.5. The time-dependent losses of prestress are designated Pcþ s þ rand result from creep and shrinkage of the concrete and the long-term relaxation of the prestressing steel. Time-dependent losses are discussed in section 5.10.6.
5.10.4. Immediate losses of prestress for pre-tensioning
2-1-1/clause 5.10.4(1) requires the following losses to be considered for pre-tensioned members:
(1) Loss due to friction at the bends (for curved wires or strands) during the stressing process – calculation of friction losses is analogous to that for externally post-tensioned bridges discussed in section 5.10.5.
(2) Loss due to wedge draw-in of the anchorage devices. This depends on the construction process and it is not normally considered by the designer, although the losses can be calculated in the same way as for post-tensioned members discussed in section 5.10.5, if the draw-in is known.
(3) Loss due to the relaxation of the pre-tensioning tendons during the period which elapses between the tensioning of the tendons and the prestressing of the concrete. This is calculated according to 2-1-1/clause 3.3.2.
(4) Loss due to the elastic deformation of the concrete as the result of the action of the pre-tensioned tendons when they are released from the anchorages. The loss of force in each tendon of area Apvaries along its length and can be approximated from:
PelðxÞ ¼ Ap
Ep
EcmðtÞcðxÞ ðD5.10-3)
where cðxÞ is the stress in the concrete adjacent to the tendon at transfer. Ep=EcmðtÞ is the modular ratio, with the modulus for concrete based on its age at transfer. This loss will typically be a greater percentage than that for post-tensioned members for the reasons discussed in the next section. It is possible to refine this equation to allow for the change in concrete stress during transfer by adding a denominator similar to that in 2-1-1/Expression (5.46), but with a zero creep factor, , as follows:
PelðxÞ ¼
Definitions of Ac and zcp are given with the comments on 2-1-1/Expression (5.46).
In equation (D5.10-4), Ap can be based on either one tendon or on a group of 2-1-1/clause
5.10.3(3)
2-1-1/clause 5.10.4(1)
tendons as an approximation. In the latter case, cðxÞ then relates to the cable group centroid. Care is needed to be consistent with definitions.
5.10.5. Immediate losses of prestress for post-tensioning 5.10.5.1. Losses due to elastic deformation of the concrete
To perform a rigorous calculation of elastic loss in a sequentially stressed tendon group, each tendon has to be considered individually and the loss in each determined from the progres-sive stressing of each subsequent tendon. The change in stress induced in each tendon is determined from the change in strain induced in the adjacent concrete, averaged over the tendon’s length, from stressing of each subsequent tendon. The need to use an average concrete strain arises because, where individual tendons are unbonded prior to stressing subsequent tendons, the strain changes in tendon and adjacent concrete are not constrained to be equal and the loss of force in the tendon will be uniform along its length (neglecting the effects of friction). It is therefore usual to calculate an average elastic loss for the entire length of tendon. If an individual tendon is bonded prior to stressing subsequent tendons, the loss of force in it will vary throughout its length as the change in steel strain is constrained to be the same as the change in strain of the concrete immediately adjacent. (This is the case for pre-tensioned beams as discussed in section 5.10.4.)
As a simpler alternative to separate consideration of tendons, 2-1-1/clause 5.10.5.1(2) allows an entire group of tendons to be treated together (acting at their centroid) and the stress from tensioning the complete group used to determine an average loss. This is a common approximation, albeit sometimes slightly unconservative, which has been used in previous UK practice and which is also used in the calculation of long-term losses in 2-1-1/clause 5.10.6. An approximate formula is provided for this average loss when tendons are ‘identical’, by which it is meant that they are the same size and have the same initial stres-sing force. 2-1-1/Expression (5.44) gives the mean loss per tendon, wherein Apis the area of one tendon. The progressive loss of prestress with sequential stressing is accounted for by way of the factor, j:
Pel¼ ApEpX jcðtÞ EcmðtÞ
2-1-1/(5.44) where:
Ap is the cross-sectional area of a tendon
Ep is the modulus of elasticity of prestressing steel
EcmðtÞ is the modulus of elasticity of concrete at time t (see section 3.1.3)
cðtÞ is the variation of stress at the centroid of the tendon group applied at time t. It will include contributions from the prestress force together with any simulta-neous change in other permanent actions, such as the gradual development of self-weight forces developed as a beam lifts from its formwork during stressing.
As discussed above, cðtÞ will be an average value along the tendon group centroid where all tendons are unbonded prior to completion of stressing. It will also include stresses from variations of permanent actions applied after prestressing (e.g. removal of supports, a jacking operation or addition of super-imposed dead load), but these need to be considered separately as they will have a different ‘j’ value
j¼ ðn 1Þ=2n, where n is the number of ‘identical’ tendons successively prestressed. As an approximation, it can be taken as 0.5. Where the stress varies in the tendons due to variations of permanent actions applied after prestressing, j¼ 1 as all tendons are affected similarly.
This is the reason for the ‘P
’ sign in 2-1-1/Expression (5.44). 2-1-1/Expression (5.44) can be applied to determine the total loss in a group of tendons directly if Ap is adjusted accordingly (as is done in Worked examples 5.10-1 and 5.10-3).
One final point to note is that if already installed bonded tendons are included in the beam section properties for subsequent stressing operations, the ‘elastic loss’ in these bonded
2-1-1/clause 5.10.5.1(2)
tendons from stressing subsequent tendons will be included directly in the calculation of concrete cross-sectional stresses. It is then not necessary to apply 2-1-1/Expression (5.44) to these tendons. This approach is followed by some software.
5.10.5.2. Losses due to friction
In post-tensioned systems, prestress losses occur due to friction in the duct as a cable turns through an angle. These losses are caused both by the intended angular deviations forming the cable profile and by unintentional variations in the tendon profile, often referred to as wobble, which arise from tolerances in setting out, from sag in the ducts between duct sup-ports and from movement of ducts during concreting. With external prestressing, friction is concentrated at the points of angular deviation.
2-1-1/clause 5.10.5.2(1)gives the following expression, from which the loss due to friction,
P ðxÞ, in post-tensioned tendons may be estimated:
P ðxÞ ¼ Pmaxð1 e ðþkxÞÞ 2-1-1/(5.45)
where:
is the sum of the angular deviations over a distance x (irrespective of direction or sign) is the coefficient of friction between the tendon and its duct. Values for depend on both tendon and duct type and are best determined from manufacturer’s data for the particular prestressing system to be used. In the absence of such data, 2-1-1/
Table 5.1 defines values which may be assumed for , reproduced here as Table 5.10-1 for convenience
k is the ‘wobble’ factor to account for unintentional angular deviation of the tendon (per unit length). Again, values are best determined from manufacturer’s data. 2-1-1/clause 5.10.5.2(3) recommends values within the range 0:005 < k < 0:01 (per metre) if no such data is available. For external prestressing tendons, 2-1-1/clause 5.10.5.2(4) allows the wobble loss due to unintentional angles to be ignored, although strictly some allowance should be made if it is possible to have any significant angular error in setting out deviation angles on site. Usually such unintentional angles are a small fraction of the intended ones and can therefore be ignored
x is the distance along the length of the tendon from the stressing end (i.e. the point where the stress in the tendon is equal to Pmax)
2-1-1/Expression (5.45) can be rearranged such that the tendon force at x, PðxÞ is given by:
PðxÞ=Pmax¼ e ð þ kxÞ ðD5.10-5)
For small values of ð þ kxÞ, equation (D5.10-5) may be written as:
PðxÞ=Pmax¼ 1 kx ðD5.10-6)
It can be seen that equation (D5.10-6) gives a linear reduction in force along the tendon where there is either no intentional deviation of the tendon (i.e. wobble loss only), or where the angular deviation per metre is linear (parabolic profile).
2-1-1/clause
Table 5.10-1. Coefficients of friction recommended for post-tensioned tendons Internal tendons
Cold drawn wire 0.17 0.25 0.14 0.18 0.12
Strand 0.19 0.24 0.12 0.16 0.10
Deformed bar 0.65 – – – –
Smooth round bar 0.33 – – – –
Note HDPE¼ high-density polyethylene.
It should be noted that the definition and value of the wobble factor, k, used above is not
It should be noted that the definition and value of the wobble factor, k, used above is not