Strut And Tie Models
Parabolic tendon profile eccentric at ends of beam
If the prestrcss force does not lie at the centroid of the section at the ends of the beam, but at an eccentricity <?o as shown in figure 11.17, the expression for deflection must be modified. It can he shown that the deflection is the same as that caused by a force P acting at a constant eccentricity c0 throughout the length of the member, plus a force P following a parabolic profile with midspan eccentricity e[ as shown in figure 11.17.
The midspan deflection thus becomes v =
8 El
48 El
Deflections due to more complex tendon profiles are most conveniently estimated on the basis of coefficients which can be evaluated for commonly occurring arrangements. These are on the basis y = (KI?)/EI where K incorporates the variations of curvature due to prestress along the member length.
There are three principal stages in the life of a prestressed member at which deflections may be critical and may need to be assessed.
1. At transfer  a check of actual deflection at transfer for comparison with estimated values is a useful guide that a prestressed beam has been correctly constructed.
2. Under dead load, before application of finishes  deflections must be evaluated to permit subsequent movement and possible damage to be estimated.
3. Longterm under full quasipermanent actions  deflections are required, both to determine the subsequent movement and also to assess the appearance of the final structure.
Shortterm deflections will be based on materials properties associated with characteristic strengths (7n, = 1) and with actual loading ("yf = 1). Longterm assessment however must not only take into account loss in prestress force, but also the effects of creep both on the applied loading and the prestress loading components of the deflection. Creep is allowed for by using an effective modulus of elasticity for the concrete, as discussed in section 6.3.2. Thus if Ec(to) is the instantaneous value, the effective value after creep is given by
where the value of (f>(x,tn), the creep coefficient can be obtained from table 6.12
It can be shown in some instances that when net upward deflections occur, ihese often increase because of creep, thus the most critical downward deflection may well be before creep losses occur, while the most critical upward deflection may be longterm. This further complicates a procedure which already has many uncertainties as discussed in chapter 6: thus deflections must always be regarded as estimates only.
Calculation of deflection
Estimate transfer and longterm deflections for a 200 x 350 mm beam of 10 m span. The prestressing tendon has a parabolic profile with midspan eccentricity 75 mm and the end eccentricity = 0 at both ends. The initial prestress force at transfer. Pq, is 560 kN and there are 20 per cent losses. The imposed load consists of 2.0 kN/m finishes and l.OkN/m variable load. Ecm = 35kN/mm2 and the creep factor 0(<x>Jo) = 2.0.
(a) At transfer
Deflection ya =
384 Ecml 48 Ecml
38435 x 103 x 715 x 10" 48 35 x I03 x 715 x 106 = 9.1  17.5 = 8 mm (upwards)
(b) At application of finishes
Assume that only a small proportion of prestress losses have occurred:
Weight of finishes = 2.0 kN/m therefore
5 x 2.0 x 104 x 1012 'Vb ~ •Va " 384 x 35 x I01 x 715 x 10" 8 + 10 mm = 2 mm (downwards)
(c) In (he long term due to the quasipermanent action plus prestress force after losses Assuming 30 per cent of the variable load contributes to the quasipermanent action:
Quasipermanent action selfweight 4 finishes I 0.3 x variable load
= 1.75 4 2.0 4 0.3 x 1.0 = 4.05 kN/m Prestress forces after losses = 0.SP<> = 0.8 x 560 448 kN
5 4.05 x 104 x 1012 5 448 x 103 x 75 x 102 x 106 'Vc ~ 38411.7 x 101 x 715 x 10A ~ 48 11.3 x 101 x 715 x 10s 63.0  43.3 = 20mm (downwards) < span/250 = 40 mm
Therefore satisfactory.
(d) Movement after application of finishes
^_yA = yc  yb = 20  2 = 18 mm < span/500 = 20 mm (satisfactory)._^
11.4.9 End blocks
In pretcnsioned members, the prestress force is transferred to the concrete by bond over a definite length at each end of the member. The transfer of stress to the concrete is thus gradual. In posttensioned members however, the force is concentrated over a small area at the end faces of the member, and this leads to hightensile forces at right angles to the direction of the compression force. This effect will extend some distance from the end of the member until the compression has distributed itself across the full concrete crosssection. This region is known as the 'end block' and must be heavily reinforced by steel to resist the bursting tension forces. End block reinforcement will generally consist of closed links which surround the anchorages, and the quantities provided arc usually obtained from empirical methods.
Typical 'flow lines' of compressive stress are shown in figure 11.18. from which it can be seen that whatever type of anchorage is used, the required distribution can be expected to have been attained at a distance from the loaded face equal to the lateral dimension of the member. This is relatively independent of the anchorage type.
In designing the end block it is necessary to check that the bearing stress behind the anchorage plate due to the prestressing force does not exceed the limiting stress, /kju, given by
Am is the loaded area of the anchorage plate
/lLi is the maximum area, having the same shape as which can be inscribed in the total area as shown in figure 11.19(a)
The lateral tensile bursting forces can be established by the use of a statically determinate strut and tie model where it is assumed that the load is carried by a truss consisting of concrete struts and links of reinforcement acting as steel ties. In carrying out these calculations a partial factor of safety of yp 1.2 is applied to the prestressing
Flat plate anchorage
Conical anchorage
Flat plate anchorage
Figure 11.18
Stress distribution In end blocks
Conical anchorage
(a) Anchorage zone (end view)
(b) Strut and tie model of load dispersion
(a) Anchorage zone (end view)
(b) Strut and tie model of load dispersion force. EC2 suggests that in determining the geometry of this truss the prestressing force can be assumed to disperse at an angle of 33.7 to the longitudinal axis of the beam as shown in figure 11.19(b). The compressive stresses in the assumed struts should not exceed 0.4 ^ i  and the reinforcement is designed to act at a design strength of
0.87/yk. However if the stress in the reinforcement is limited to 300N/mnr then no checks on crack widths are necessary. This reinforcement, in the form of closed links, is then distributed over a length of the endblock equal to the greater lateral dimension of the block, this length being the length over which it is assumed that the lateral tensile stresses are acting.
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