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In the derivation of Table 4.3, it has been assumed that fcm = fck + 181 (N/mm2) (4.3)
(7) For simplification, a constant value Bc = fc may be adopted in the range £cl > (c > (cu (see Figure 4.1)
(8) Other idealized stress-strain diagrams may be used (e.g. bi-linear), provided they are effectively equivalent to the one described in (3) and (4).
b) Stress distribution for cross-section design
(9) The preferred idealization for cross-section design is the parabolic rectangular one, given in Figure 4.2.
In this diagram (cu max is taken as 3.5°/oo, and compressive stress and strain are taken as negative.
(10) Other idealized stress-diagrams may be used, provided they are effectively equivalent to the parabolic-rectangular diagram, with regard to the shape of the compression zone in the cross-section (e.g., the bi-linear diagram in Figure 4.3).
Figure 4.2 — Parabolic — rectangular stress-strain diagram for concrete in compression o -0.001 --------------- ------C
Figure 4.2 — Parabolic — rectangular stress-strain diagram for concrete in compression
(11) The design concrete strength is defined by f _ Ack fcd"*:
The design diagram is derived from the chosen idealized diagram by means of a reduction of the stress ordinate of the idealized diagram by a factor !/Yc, in which
Yc a is the partial coefficient for concrete (see 2.3.3.2)
is a coefficient taking account of long term effects on the compressive strength and of unfavourable effects resulting from the way the load is applied.
The additional reduction factor a for sustained compression may generally be assumed to be | 0.85 |.
(12) A rectangular stress distribution (as given in Figure 4.4) may be assumed. The a-factor as given for the idealized diagram is valid, except that it should be reduced to | 0.80 | when the compression zone decreases in width in the direction of the extreme compression fibre.
4.2.1.4 Time dependent behaviour
(1) The data given in Table 3.3 are mean values, valid in the temperature ranges of 3.1.2.5.5(3), and may be assumed to represent the final coefficients of creep and shrinkage for concrete, where compressive stresses do not exceed Bc = | 0.45 fck |.
(2) For detailed calculations involving loss of prestress, the general information in sections 3.1, 2.5.4 and 2.5.5 may be used where appropriate, in following the procedures set out in 4.2.3.5.5.
4.2.2 Reinforced concrete
4.2.2.0 Notation
For notation used in this section, see 1.6 and 1.7.
4.2.2.1 Reinforcing steel: general
P(1) Data on material properties given in this section are either representative values, corresponding to the relevant steel grade specified in appropriate Standards, or are idealisations suitable for design purposes.
(2) In general, the properties specified are those given in 3.2.1(5) and established in EN 10080 or other appropriate Standards.
P(3) Unless stated otherwise, design shall be based on a specified grade, represented by its characteristic yield stress (fyk).
(4) All types of reinforcing steel specified in 3.2, which satisfy the mechanical, physical and technological requirements of EN 10080 or other relevant Standards, may generally be used in design, in accordance with 4.2.2.2-4.2.2.4, unless greater accuracy is required.
4.2.2.2 Physical properties of reinforcing steel
(1) The values given in 3.2.3 may be used as design data. They may be assumed to be valid in the range from - 20 °C to 200 °C.
4.2.2.3 Mechanical properties of reinforcing steel
4.2.2.3.1 Strength
P(1) For all types of steel, the values for euk, fyk, (ft/fy)k and ftk shall be defined.
(2) Values for these properties and for defined types and grades of steel may be taken from EN 10080. For other types of steel, these properties should be established by testing.
(3) If not specified otherwise, the yield stress and the ultimate strength may be assumed to be equal in tension and in compression, for design purposes.
(4) Design calculations may be based on the nominal size and the nominal cross-sectional area of the reinforcing steel.
4.2.2.3.2 Stress-strain diagram
P(1) The general ductility requirements shall be in accordance with 3.2.4.2, and as specified in relevant Standards.
(2) For overall analysis, the idealised bi-linear diagram in Figure 4.5 may generally be used. This diagram is valid for temperatures from - 20° to 200 °C.
(3) Figure 4.5 may be modified, e.g. with a flatter or horizontal top branch, for local verifications and section design.
(4) Design values are derived from the idealised characteristic diagram, by dividing by Ys, the partial factor for reinforcing steel (see 2.3.3).
(5) For section design, either of the following assumptions may be made:
— a horizontal top branch to the design curve in Figure 4.5, i.e. the stress in the reinforcement is limited to fyk/Ys, with no limit to the steel strain, although in some cases it may be convenient to assume a limit.
— an inclined top branch, with the steel strain limited to | 0.011. 4.2.2.3.3 Fatigue
(1) For fatigue requirements for reinforcing steel, see Part 1E. 4.2.2.4 Technological properties of reinforcing steel
P(1) Bond and anchorage properties shall be derived from the shape of the surface characteristics of the bars and/or from the strength of welded joints of welded fabric.
(2) Bond requirements should comply with 5.2.2.
(3) Anchorage requirements should comply with 5.2.3-5.2.5.
(1) Reinforcing steel which complies with EN 10080 may be assumed to be weldable. 4.2.3.0 Notation (see also 1.6, 1.7 and 2.5.4.0)
Fpx Ultimate resisting force provided by the prestressing tendons in a cracked anchorage zone
k Unintentional angular displacement (per unit length) related to the profile of the tendons lba Anchorage length over which the ultimate tendon force Fpu in pretensioned members is fully transmitted to the concrete lbp Transmission length, over which the prestressing force from a pretensioned tendon is fully transmitted to the concrete lbpd Design value for transmission length lbpo Length of a neutralised zone at the ends of pretensioned members, in the case of sudden release lp.eff Dispersion length, over which the concrete stresses gradually disperse to a linear distribution across the section (effective transfer)
n1 Total number of wires or strands in a tendon n2 Number of wires or strands transferring the radial force of all wires or strands in the tendon to the deviator (Figure 4.7)
zcp Distance between the centre of gravity of the concrete section and the tendons
0 Sum of angular displacements over a distance, x (irrespective of direction or sign)
"b Coefficient relating transmission length of prestressing tendons to concrete strength
Es(t,to) Estimated shrinkage strain ao,max Maximum stress applied to a tendon apmo Stress in the tendon immediately after stressing or transfer
Gpgo Initial stress in the tendons due to prestress and permanent actions ocg Stress in the concrete adjacent to the tendons, due to self-weight and any other permanent actions acpo Initial stress in the concrete adjacent to the tendons, due to prestress
%Bp,c + s + r Variation of stress in the tendons due to creep, shrinkage and relaxation at location x, at time t
Aapr Variation of stress in the tendons at section x due to relaxation
4.2.3.1 Prestressing steel: general
P(1) Data on material properties given in this section are either representative values, corresponding to the relevant steel grade specified in appropriate Standards, or are idealisations suitable for design purposes.
(2) In general, the properties specified are those given in 3.3.1(5) and established in EN 10138 or other appropriate Standards.
P(3) Unless stated otherwise, design shall be based on a specified grade, represented by its characteristic 0.1 % proof stress (fp0.1k).
(4) All types of prestressing steel specified in 3.3, which satisfy the mechanical, physical and technological requirements of EN 10138 or other relevant Standards may generally be used in design, in accordance with the data given below, unless greater accuracy is required.
4.2.3.2 Physical properties of prestressing steel
(1) The values given in 3.3.3 may be used as design data. They may be assumed to be valid in the range from - 20 °C to 200 °C.
4.2.3.3 Mechanical properties of prestressing steel 4.2.3.3.1 Strength
P(1) For all types of prestressing steel the values for fp0.1k, (uk and fpk shall be defined.
(2) Relevant properties for defined types and grades of steel may be taken from EN 10138. For other types of steel, the properties are to be confirmed by technical approval documents.
(3) Design calculations may be based on the nominal size or the nominal cross-sectional area of the prestressing steel.
4.2.3.3.2 Modulus of elasticity
4.2.3.3.3 Stress-strain diagram
P(1) The general ductility requirements shall be in accordance with 3.3.4.3, and as specified in relevant Standards.
(2) An idealized bi-linear diagram is given in Figure 4.6. This diagram is valid for temperatures from - 20 °C to 200 °C.
(3) Figure 4.6 may generally be used for overall analysis, local verifications and the checking of section capacity.
(4) Figure 4.6 may be modified, e.g. with a flatter or horizontal top branch, for local verifications or section design.
(5) Design values for the steel stress are derived from the idealized characteristic diagram by dividing by Ys, the partial factor for prestressing steel (see 2.3.3).
(6) For section design, either of the following assumptions may be made:
— a horizontal top branch to the design curve in Figure 4.6, i.e. the stress in the prestressing steel is limited to 0.9 fpk/Ys with no limit to the steel strain, although in some cases it may be convenient to assume a limit.
— an inclined top branch, with the increasing steel strain limited to | 0.011.
(1) For structural analysis, if not stated otherwise, post-tensioned tendons may be assumed as having high ductility: pre-tensioned tendons are assumed as having normal ductility.
4.2.3.3.5 Fatigue
(1) For fatigue requirements for prestressing steel, see Part 1E.
(1) If not stated otherwise in technical approval documents, tendons assembled from prestressing steel satisfying the requirements of 3.3.4.6 may be considered to withstand the full specified tensile strength, if the bending radius of the saddle, which is supporting the tendon at its point of deviation, satisfies the requirements of Table 4.4.
(2) The values in Table 4.4 do not relate to the coefficients of friction in 4.2.3.5.5(8).
Type of tendon |
Ratio Minimum bending radius Nominal diameter |
Single wire or strand, deflected after tensioning |
|15| |
Single wire or strand, tensioned in smooth duct |
|20| |
Single wire or strand, tensioned in ribbed duct |
|40| |
Multi wire or strand tendon |
Preceding values multiplied by n1/n2 |
in which: n1 = total number of wires or strands in the tendon n2 = number of wires or strands transferring the radial force of all wires or strands in the tendon to the deviator. (See Figure 4.7 below). |
4.2.3.3.7 Anchorage or coupler assemblies of tendons
(1) Tendon anchorage assemblies and tendon coupler assemblies satisfying the performance requirements of 3.4.1.2 may be considered to withstand the full characteristic strength of the tendon.
4.2.3.4 Technological properties of prestressing steel
P(1) Certificates accompanying the consignments shall indicate the class and relevant relaxation data of the prestressing steel (see 3.3.5, and relevant Standards).
(2) For design calculations, the values which may be taken into account for losses at 1 000 h are either those given in the certificate or those assumed in Figure 4.8 for the three classes of steel shown. The long term values of the relaxation losses may be assumed to be | three | times the relaxation losses after 1 000 h.
(3) An indication of how relaxation losses increase between 0-1 000 hours is given in Table 4.5.
Table 4.5 — Indication of relationship between relaxation losses and time up to 1 000 hours
Table 4.5 — Indication of relationship between relaxation losses and time up to 1 000 hours
Time in hours |
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