Capítulo 2. Materiales y métodos
2.1. Área de estudio
unstressed reinforcement in the tension zone, in square millimetres; Amdt 1, Apr. 1994
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b is the width of section in tension zone, in millimetres (For sections with a flange in tension zone, b is the width of equivalent tension zone area, assuming a neutral axis depth of h/3.);
and
h is the height of section, in millimetres.
For deflection and cracking of class 3 elements, see the methods described in annex A.
Table 31 - Depth factors for tensile stresses for class 3 elements
1 2
5.3.2.3 Stress limitations at transfer for beams (SLS) 5.3.2.3.1 Compressive stresses
Design compressive stresses in the concrete at transfer should not exceed 0,45fci at the extreme fibre (in the case of triangular or near triangular distribution of prestress) or 0,3fci for near uniform distribution of prestress, where fci is the concrete strength at transfer.
5.3.2.3.2 Tensile stresses in flexure
Design tensile stresses in flexure in the concrete at transfer should not exceed the values given below:
a) class 1 elements: ensure that at transfer, the tensile stress does not exceed the value of 1 MPa;
b) class 2 elements: ensure that the tensile stress does not exceed the value appropriate to the concrete strength at transfer given in table 30; ensure that elements with pre-tensioned tendons have some tendons or unstressed reinforcement well distributed throughout the tensile zone of the section and elements with post-tensioned tendons have unstressed reinforcement located near the tension face of the element;
c) class 3 elements (see also annex A): the tensile stress should, in general, not exceed the appropriate value for a class 2 element; where this stress is exceeded, regard the section in design as cracked.
5.3.3 Ultimate limit state for beams in flexure 5.3.3.1 Section analysis
When analysing sections under maximum design loads, make the following assumptions:
a) the strain distribution in the concrete in compression is derived from the assumption that plane sections remain plane;
b) the stresses in the concrete in compression are either derived from the stress strain curve given in figure 1, with a γm of 1,5, or are taken as equal to 0,45fcu over the whole compression zone (see figure 4); in both cases, the strain at the outermost compression fibre is taken as 0,0035;
c) the tensile strength of the concrete is ignored;
d) the strains in bonded prestressing tendons and in any unstressed reinforcement, whether in tension or in compression, are derived from the assumption that plane sections remain plane;
e) the stresses in bonded prestressing tendons, whether initially tensioned or untensioned, and in unstressed reinforcement are derived from the appropriate stress/strain curves;
NOTE - The stress/strain curves for prestressing reinforcement are given in figure 3 and those for reinforcement are given in figure 2. An empirical approach towards obtaining the stress in the tendons at failure is given in 5.3.3.2.
f) in post-tensioned elements where the tendons are unbonded, the stress in the tendons does not exceed the values given in table 33 unless a higher stress can be justified on the basis of tests.
Table 32 - Conditions at the ultimate limit state for rectangular beams with pre-tensioned tendons or with post-tensioned tendons
having an effective bond axis to that of the centroid
of the tendons in the
5.3.3.2.1 In the absence of an analysis based on the assumptions given in 5.3.3.1, the moment of resistance of any shape of beam may be obtained from the following equation:
Mu = fpbAps(d-dn) where
Mu is the design moment of resistance of beam;
fpb is the design tensile stress in tendons at failure;
d is the effective depth to centroid of steel area Aps; dn is the depth to centre of compression zone; and Aps is the area of prestressing tendons in tension zone.
5.3.3.2.2 For rectangular beams, and for flanged beams in which the compression block lies within the flange, dn = 0,45x, where x is the neutral axis depth.
5.3.3.2.3 Values for fpb and x may be derived from table 32 for pre-tensioned elements and for post-tensioned elements with effective bond between the concrete and tendons. The effective prestress after all losses shall be at least 0,45fpu. Ignore prestressing tendons and unstressed reinforcement in the compression zone in strength calculations when using this method.
5.3.3.2.4 For rectangular beams, and for flanged beams in which the neutral axis lies within the flange, the stress in the tendons at failure may be derived from table 33 for unbonded tendons.
Table 33 - Conditions at the ultimate limit state for post-tensioned rectangular beams having unbonded tendons
Ratio of depth of neutral axis to that of the centroid of the tendons in the tension zone x/d
for values of
5.3.3.2.5 In table 32, the following assumptions have been made:
a) the effective prestress after all losses have occurred (fpe) does not exceed 0,6fpu; b) the compression block is rectangular with a uniform stress of 0,45fcu;
c) either the tendons are in ducts or, if they are free (as in hollow beams), diaphragms are provided to prevent a reduction of the effective depth; and
d) the effective depth is determined by assuming that the tendons are in contact with the top of the duct or with the soffit of the diaphragms.
5.3.3.2.6 In addition, for unbonded tendons, values of f pb and x may be obtained from equations (15) and (16). (The value of fpb should not be taken as exceeding 0,7fpu.)
fpb = fpe + 7 000 (15) le/d 1 & 1,7 fpu Aps
fcu bd
x = 2,47 fpu Aps (16)
fcu bd fpb fpu d
where
fpb, Aps and d are as in 5.3.3.2.1;
fpe is the design effective prestress in tendons after all losses have occurred;
fpu is the characteristic strength of tendons (see 5.1.5);
fcu is the characteristic strength of concrete (see 5.1.5);
b is the width or effective width of the section or flange in compression zone; and le is the length (see following paragraph).
Equation (15) has been derived by taking the length of the zone of inelasticity within the concrete as 10x. The length le should normally be taken as the length of the tendons between end anchorages. In the case of continuous multispan beams, this length may be determined as in figure 26.
Figure 26 — Determination of le
5.3.3.3 Non-rectangular beams
Non-rectangular sections may be analysed using the assumptions given in 5.3.3.1 or the design formulae given in 5.3.3.2.
5.3.3.4 Unstressed reinforcement in the tension zone
In the absence of a rigorous analysis, the area of reinforcement As may be replaced by an equivalent area of prestressing tendons Asfy /fpu.
5.3.4 Shear resistance of beams
Calculation for shear resistance is only required for the ultimate limit state. The provisions of this subclause apply to class 1, class 2, and class 3 prestressed concrete elements. Consider the ultimate shear resistance of the concrete alone, Vc, at both sections, uncracked (see 5.3.4.2) and cracked (see 5.3.4.3) in flexure. Take the lower value and, if necessary, provide shear reinforcement (see 5.3.4.4).
5.3.4.1 Maximum shear stress
Under no circumstances should the maximum design shear stress v exceed the lesser of 0,75 fcu or 4,75 MPa (this includes an allowance for a γm of 1,40).
5.3.4.2 Sections uncracked in flexure
5.3.4.2.1 The ultimate shear resistance of a section uncracked in flexure, Vco, corresponds to the occurrence of a maximum design principal tensile stress at the centroidal axis of the section
ft = 0,23 fcu
5.3.4.2.2 In the calculation of Vco, take the value of prestress at the centroidal axis as 0,8fcp. The value of Vco is given by
Vco = 0,67 bh ft² 0,8 fcp ft (17)
where
ft = 0,23 fcu , taken as positive;
fcp is the design compressive stress at the centroidal axis due to prestress, taken as positive;
b is the width of beam, which, for T-beams, I-beams and L-beams, is replaced by the width of rib, bw; and
h is the overall depth of beam.
Table 34 gives values of Vco /bh obtained from equation (17) for different concrete grades and applicable values of fcp.
Table 34 Values of Vco /bh
1 2 3 4 5
fcp
MPa
Vco /bh MPa Concrete grade
30 40 50 60
2 1,27 1,41 1,54 1,64
4 1,59 1,74 1,90 2,00
6 1,85 2,02 2,17 2,12
8 2,08 2,26 2,42 2,56
10 2,29 2,48 2,65 2,80
14 2,65 2,87 3,06 3,22
5.3.4.2.3 In flanged beams where the centroidal axis occurs in the flange, limit the principal tensile stress ft to at the intersection of the flange and web. When calculating Vco, use 0,8 of the
0,23 fcu
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stress due to prestress at this intersection. Amdt 1, Apr. 1994
5.3.4.2.4 For a section uncracked in flexure and with inclined tendons or compression zones, the component of prestressing force or that of compression force normal to the longitudinal axis of the beam may be added to Vco.
5.3.4.2.5 In a pre-tensioned beam, the critical section should be taken at a distance from the edge of the bearing equal to the height of the centroid of the section above the soffit. Where this section occurs within the prestressed development length, the compressive stress at the centroidal axis due to prestress to be used in equation 17 may be calculated from the following relationship:
fcpx = fcp
x
lp 2 & x lp where
fcp is the design stress at the end of the prestress development length lp.
The prestress development length lp should be taken as the greater of the transmission length (see 5.8.4) or the overall depth of the element.
5.3.4.3 Sections cracked in flexure
5.3.4.3.1 Calculate the design ultimate shear resistance Vcr of a section cracked in flexure, using the following equation:
(18) Vc r ' 1 & 0,55 fp e
fpu vc bd % Mo V M where
d is the distance from extreme compression fibre to centroid of steel area (Aps + As) in tension zone;
Mo is the moment necessary to produce zero stress in concrete at the extreme tension fibre;
and
Mo is equal to ;0,8 fpt Ι Yt where
Yt is the distance from the centroid of the concrete section to the extreme tension fibre;
fpt is the stress at the extreme tension fibre due to prestress only;
I is the second moment of area; and
fpe is the design effective prestress in tendons after all losses have occurred (should not be taken as exceeding 0,6 fpu).
NOTE ) Where the steel area in the tension zone consists of tendons and reinforcement, fpe may be taken as the value obtained by dividing the effective prestressing force by an equivalent area of tendons equal to
Aps % As fy fpu
where
fpu is the characteristic strength of tendons (see 5.3.3.2 or figure 3);
vc is the maximum design shear resistance of the concrete (the value obtainable from 4.3.4);
V and M are the design shear force and bending moment, respectively, at the section under consideration, and due to the particular ultimate load condition; and
b is the width or effective width of rectangular section or the width of the rib.
The value of Vcr should be taken as at least 0,1bd fcu .
5.3.4.3.2 The value of Vcr at a particular section, calculated using equation (18), may be assumed to be constant for a distance equal to d/2, measured in the direction of increasing moment, from that particular section.
5.3.4.3.3 For a section cracked in flexure and with inclined tendons or compression cords, the design shear forces produced should be combined with the external design load effects where these effects are increased.
5.3.4.4 Shear reinforcement
5.3.4.4.1 When V, the shear force due to the design ultimate loads, is less that Vc, which is the shear force that can be carried by the concrete, shear reinforcement need not be provided in the following cases:
a) where V is less than 0,5 Vc;
b) in elements of minor importance; and
c) where tests carried out in accordance with 3.4.5 have shown that shear reinforcement is not required.
5.3.4.4.2 In all cases except those in 5.3.4.4.1, minimum shear reinforcement in the form of links should be provided such that
Asv
sv 0,4b 0,87 fyv where
fyv is the characteristic strength of the reinforcement (but not more than 450 MPa);
b is as in equation (18);
Asv is the cross-sectional area of the two legs of a link; and sv is the link spacing along length of beam.
5.3.4.4.3 When V, the shear force due to the design ultimate loads, exceeds Vc, ensure that the shear reinforcement provided in addition is such that
Asv
sv V Vc 0,87 fyv dt
where dt is taken as the depth from the extreme compression fibre, to the greater of either the longitudinal bars (tendons, group of tendons) or the centroid of the tendons.
5.3.4.5 Arrangement of shear reinforcement
5.3.4.5.1 In rectangular beams, at both corners in the tensile zone, a link should pass round a longitudinal bar, a tendon or a group of tendons having a diameter not less than the link diameter. A link should extend as close to the tension or compression faces as possible, with due regard to cover.
Ensure that the links provided at a cross-section enclose all the tendons and unstressed reinforcement provided at the cross-section and that they are adequately anchored (see 4.11.6.4).
5.3.4.5.2 Ensure that the spacing of links along a beam does not exceed 0,75dt or four times the web thickness for flanged beams. When V exceeds 1,8Vc, reduce the maximum spacing to 0,5dt. Ensure that the lateral spacing of the individual legs of the links provided at a cross-section does not exceed 0,75dt.
5.3.5 Torsional resistance of beams
In general, when it is considered that torsional resistance or stiffness of beams need not be taken into account in the analysis of the structure, no specific calculations for torsion will be necessary, adequate control of any torsional cracking being provided by the required nominal shear reinforcement.
Calculations are required when torsional resistance is necessary for equilibrium or when significant torsional stresses may occur. The method for reinforced concrete beams given in 4.3.5 may generally be used.
5.3.6 Deflection of beams
NOTE - See also annex A.
5.3.6.1 Class 1 and class 2 elements (see 3.2.3.3.1.2)
5.3.6.1.1 The instantaneous deflection due to service loads may be calculated with the use of elastic analysis based on the concrete section properties and on the value for the modulus of elasticity given in 3.4.2.1.
5.3.6.1.2 The total long-term deflection due to the prestressing force, self-weight load and any sustained imposed load may be calculated with the use of elastic analysis based on the concrete section properties and on an effective modulus of elasticity based on the creep of the concrete per unit length for unit applied stress after the period under consideration (specific creep). The values for specific creep given in 5.8.2.5 may in general be used unless a more accurate assessment is required.
Make due allowance for the loss of prestress after the period under consideration. Ensure that the deflections comply with the limits given in 3.2.3.2.
5.3.6.2 Class 3 elements
Where the permanent load is less than or equal to 25 % of the imposed load, the deflection of class 3 elements may be calculated in accordance with 5.3.6.1. Where the permanent load exceeds 25 % of the imposed load, the basic span/effective depth ratios given in 4.3.6 and table 10 should be complied with unless more rigorous calculations based on the moment curvature relationship are made.