SITUACIONES QUE NO CONTRAINDICAN LA LACTANCIA MATERNA

Consider a pretensioned member with an eccentric prestress force Po transferred to it

as shown in Fig. 4.1. At the level of the prestressing tendons, the strain in the concrete must equal the change in the strain of the steel.

Thus:

σcg/Ecm=∆σp/Es;

∴∆σp=mσcg,

(4.1)

where m=Es/Ecm, the modular ratio, σcg is the stress in the concrete at the level of the

tendons, ∆σp is the reduction in stress in the tendons due to elastic shortening of the

concrete to which they are bonded, and Es and Ecm are the moduli of elasticity of the

steel and concrete respectively. The stress in the concrete is given by

(4.2)

where Pe is the effective prestress force after elastic shortening, Ac and Ic are the

section respectively, and r is the radius of gyration, given by r2=Ic/Ac.

Also,

Pe=Ap(σpo−∆σp),

(4.3)

where σpo is the initial stress in the tendons and Ap is their cross-sectional area.

Although, strictly speaking, the right-hand side of Equation 4.3 is the force in the tendon, for no applied axial force on the section this must equal the force in the concrete. Combining Equations 4.1, 4.2 and 4.3 gives

(4.4)

If the tendons are closely grouped in the tensile zone, the loss due to elastic shortening may be found with sufficient accuracy by taking σcg as the stress in the concrete at the

level of the centroid of the tendons. If the tendons are widely distributed throughout the section, then the above approximation is no longer valid. In this case the influences of the tendons, or groups of tendons, should be determined separately and then superimposed to give the total effective prestress force.

For a post-tensioned member the change in strain in the tendons just after transfer can be assumed to be equal to the strain in the concrete at the same level, even though the ducts have not been grouted and there is no bond between the steel and concrete. The loss of stress in the tendon is therefore still given by Equation 4.1. In practice, the force in post-tensioned members at transfer is not constant owing to friction. However, it is sufficiently accurate to base the elastic shortening loss on the initial prestress force Po, assumed constant along the member.

The value of σcg in Equation 4.4 should reflect the fact that, in general, a member

deflects away from its formwork during tensioning and the stress at any section is modified by the self weight of the member. The additional tensile stress at the level of the tendon is equal to Moe/Ic, so that the total value of σcg is given by

(4.5)

The value of σcg will vary along a member, since generally both e and Mo will vary. In

this case an average value of σcg should be assumed.

tendons tensioned simultaneously, there is no elastic shortening loss, since jacking will proceed until the desired prestress force is reached. In the more usual, and more economical, case where the tendons are tensioned sequentially, after the first tendon the tensioning of any subsequent tendon will reduce the force in those already anchored, with the exception of the last tendon, which will suffer no loss.

While it is possible to determine the resulting forces in a group of tendons for a given sequence of tensioning, the amount of work involved may be large. An acceptable approximation is to assume that the loss in each tendon is equal to the average loss in all the tendons. The loss for the first tendon is approximately equal to mσcg (in practice it is always less but approaches this value as the number of tendons

increases), and the loss for the last tendon is zero, so that the average loss is mσcg/2.

In the case of pretensioned tendons, it is usually assumed that the total force is transferred to the member at one time and that the elastic shortening loss is mσcg.

Example 4.1 ■■

Determine the loss of prestress force due to elastic shortening of the beam shown in

Fig. 4.2. Assume that σpo=1239 N/mm2, Ap=2850 mm2 and m=7.5 for the concrete at

transfer. Section properties: wo=9.97 kN/m; Ac=4.23×105 mm2; Ic=9.36×1010 mm4; r=471 mm. At midspan: At the supports: Thus, in Equation 4.1: ∆σp=1/2×7.5×(14.97+7.95)/2=43 N/mm2,

which represents a loss of 3.5% of the initial stress. ■■

Figure 4.2

For pretensioned members, and for post-tensioned members once the ducts have been grouted, the short-term prestress force is effectively held constant. Any bending moment at a section will induce extra stresses in the steel and concrete due to composite action between the two materials (see Section 5.3), but the prestress force, as measured by the actual force transmitted to the ends of the member via the tendons, remains unaltered. For unbonded members, the prestress force will vary with the loading on the member, but in practice this effect is ignored.

4.3 FRICTION

In post-tensioned members there is friction between the prestressing tendons and the inside of the ducts during tensioning. The magnitude of this friction depends on the type of duct-former used and the type of tendon. There are two basic mechanisms which produce friction. One is the curvature of the tendons to achieve a desired profile, and the other is the inevitable, and unintentional, deviation between the centrelines of the tendons and the ducts.

A small, but finite, portion of a steel cable partly wrapped around a pulley is shown in Fig. 4.3(a). Since there is friction between the cable and the pulley, the forces in the cable at the two ends of the section are not equal. The frictional force is equal to µN, where µ is the coefficient of friction between cable and pulley. The triangle of forces for the short length of cable ∆s is shown in Fig. 4.3(b); for the small angle ∆α, N=T∆α. Thus, considering the equilibrium of the length of cable ∆s:

Tcos (∆α/2)+F=(T−∆T) cos (∆α/2). For the small angle ∆α, cos (∆α/2) ≈ 1.

∴ T+F =T−∆T; ∴ µT ∆α=−∆T.

Thus, in the limit as ∆s → 0: dT/dα=−µT.

Figure 4.3 Friction in a cable.

The solution of this is

T(α)=e−µα ≡ exp (−µα)

or

Tf=To exp (−µαo),

(4.6)

where To and Tf represent the initial and final cable tensions respectively for a length

of cable undergoing an angle change αo.

The variation in tension in a tendon inside a duct undergoing several changes of curvature, as shown in Fig. 4.4, may be described using Equation 4.6. For the first portion of the curve, with radius of curvature rps1, the force in the tendon at point 2 is

given by

P2=P1 exp (−µα1)

=P1 exp (−µs1/rps1),

where s1 is the length of the tendon to point 2. The force in the tendon has been

denoted by P since it is the force in the concrete that is used in design. As noted previously, for no applied axial load the forces in the tendon and concrete must be equal. For most tendon profiles, s may be taken as the horizontal projection of the tendon, so that

P2=P1 exp (−µL1/rps1).

For the portion of the tendon 2–3, the initial force is P2, and the final force P3 is given

by

P3=P2 exp [−µ(L2/rps2)]

=P1 exp [−µ(L1/rps1+L2/rps2)].

This process can be repeated for all the changes in curvature along the length of the tendon. The force P(x) in a curved tendon at an intermediate point along the curved length is given by

P(x)=Po exp (−µx/rps),

where x is the distance from the start of the curve and Po is the tendon force at the

beginning of the curve.

Only variations of curvature in the vertical plane have so far been considered, but in many large bridge decks tendons curve in the horizontal plane as well, and the friction losses for these curvatures must also be taken into account.

The variation between the actual centrelines of the tendon and duct is known as the ‘wobble’ effect (Fig. 4.5). This is generally treated by considering it as additional angular friction, so that the expression for the force in a tendon due to both angular friction and wobble is given by

P(x)=Po exp [−µ(x/rps+kx)],

(4.8)

where k is a profile coefficient with units of rad./m. The value of k depends on the type of duct used, the roughness of its inside surface and how securely it is held in position during concreting.

If µ (x/rps+kx) < 0.2 then Equation 4.8 may be simplified to

P(x)=Po [1−µ (x/rps+kx)].

Values of k should be taken from technical literature relating to the particular duct being used and are generally in the range 50–100×10−4 _{rad/m. For greased strands}

wrapped in plastic sleeves, as used in slabs, k may be taken as 600×10−4 rad/m.

Figure 4.4 Tendon with several curvature changes.

Typical values of µ for wires and strands against different surfaces for tendons which fill approximately 50% of the duct are shown in Table 4.2.

Example 4.2 ■■

For the beam in Example 4.1 determine the prestress loss due to friction at the centre and the right-hand end if the prestress force is applied at the left-hand end. Assume µ=0.19 and k=50×10−4 rad./m.

The total angular deviation in a parabolic curve may be conveniently determined using the properties of the parabola shown in Fig. 4.6.

Thus, for the tendon profile in Fig. 4.2: α=2 tan−1(4dr/L)

=2 tan−1(4×560/20000) =0.223 rad.

The radius of curvature is given by

rps=(d2y/dx2)−1=L2/8dr

=202/(8×0.560) =89.29m.

Table 4.2 Coefficients of friction for different tendon types

Type of wire/strand Bonded Grouted duct Unbonded Steel tube Unbonded HDPE tube Lubricated:

Cold drawn wire 0.16 0.10

Strand

0.18 0.12

Non-lubricated:

Cold drawn wire 0.17 0.24 0.12

Strand 0.19 0.25 0.14

Greased strand 0.05

Po=2850×1239 ×10−3

=3531.2 kN.

Thus, using Equation 4.8:

P(x)=3531.2 exp [−0.19(x/89.29+50×10−4x)].

At midspan:

P(x=10)=3531.2 exp [−0.19(10/89.29+50×10−4_×10)]

=3424.4 kN.

Thus the loss is 106.8 kN, which is 3.0% of the initial force. At the right hand end:

P(x=20)=3531.2 exp [−0.19(0.223+50×10−4×20)]

=3321.6 kN.

The loss is now 209.6 kN, that is 5.9% of the initial force. ■■

The friction losses in the relatively shallow tendon in Example 4.2 are small, but in members with tendons of large curvature the losses may be so large that the member must be tensioned from both ends to achieve an acceptable value of prestress force at the centre. In members with many tendons, it is the usual practice to tension half the number of tendons from one end and the remainder from the opposite end, resulting in the same net prestress force at midspan but a more even distribution of prestress force along the member than if all the tendons had been tensioned from the same end.

Example 4.3 ■■

For the beam in Figure 4.7, determine the minimum effective prestress force if an initial prestress force of 3000 kN is applied (i) at the left-hand end only; (ii) at both ends. Assume the same values of µ and k as in Example 4.2.

(i) The total angular change for the full length of the tendon is given by

The minimum prestress force occurs at the right-hand end of the beam:

P(x=50)=3000 exp [−0.19(0.469+50×10−4×50)]

Figure 4.7

Thus the loss is 383.1 kN, which is 12.8% of the initial force.

(ii) If the beam is tensioned from both ends, the minimum prestress force is at the centre of the beam. Then:

The loss is now 198.3 kN, i.e. 6.6% of the initial force.

The frictional losses in the right-hand span have been greatly reduced by tensioning from both ends, although the prestress force at the centre support is the same in both cases.

■■

There are two additional frictional effects which occur. The first takes place as the tendons pass through the anchorages. This effect is small, however, of the order of 2%, and is usually covered by the calculated duct friction losses, which tend to be conservative. There is also a small amount of friction within the jack itself, between the piston and the jack casing, which causes the load applied to the tendon to be smaller than indicated by the hydraulic pressure within the jack. This is usually determined by the jack manufacturer and compensation made in the pressure gauge readings.

Although friction is a cause of loss of prestress force principally in post-tensioned members, in pretensioned members there is some loss if the tendons are tensioned against deflectors, caused by friction between

the tendon and the deflector. The magnitude of this loss will depend upon the details of the deflector, and will usually be determined from tests on the particular deflection system being used.

Many modern bridges now employ external post-tensioned tendons. Where these pass over deflectors or through diaphragms there is some loss of prestress. However, mid-length friction losses using such tendons are small.

Further information on friction during tensioning may be found in a report of the Construction Industry Research and Information Association (1978).

In document Guía de Lactancia Materna (página 63-69)