Ultimate limit states

4.3.1 Ultimate limit states for bending and longitudinal force 4.3.1.0 Notation (See also 1.6 and 1.7)

Ag1 Area of tension reinforcement effective at a section

Ag2 Area of reinforcement in the compression zone at the ultimate limit state

£s1 Strain in tension reinforcement, for section analysis

£s2 Strain in compression reinforcement, for section analysis

£pm Steel strain corresponding to Pm,t (see 2.5.4.0)

%£p Variation of steel strain corresponding to %Pc (see 2.5.4.0)

4.3.1.1 General

P(1) This section applies to concrete beams or slabs which are either reinforced or prestressed with bonded internal tendons.

P(2) The applied design forces and moments shall be determined in accordance with the principles stated in Chapter 2.

P(3) Members shall be analysed at a sufficient number of cross-sections to ensure that the requirements of the Code are satisfied at all cross-sections along the member.

P(4) The cross-section shall be chosen and the reinforcement detailed so that its design resistance, determined in accordance with the assumptions set out in this section, is never less than that required to resist all combinations of the design values of the effects of actions determined in accordance with the provisions of Chapter 2.

(5) The effective width of T or L-beams should be assessed in accordance with 2.5.2.2.1. Where the flange of such beams is in tension (such as a T-beam over an intermediate support), the tensile reinforcement, required to provide the design flexural capacity, should be located in accordance with 5.4.2.1.2. P(6) The contribution of prestressing tendons to the design resistance shall be assessed taking into account the partial safety factors given in 2.3.3.1 Table 2.2 for the acting and resisting effects of prestress. P(7) If the direction of the principal stress deviates significantly from that of the main reinforcement, this shall be taken into account (See Appendix 2).

(8) For slabs, deviations between the direction of the principal stress and the main reinforcement of less than 15° may be ignored.

For greater deviations, the moments should be transformed to give equivalent moments in the main reinforcement directions.

4.3.1.2 Design resistance to bending and longitudinal force

P(1) In analysing a cross-section to determine its ultimate resistance, the assumptions given below shall be used:

i) Plane sections remain plane.

ii) The strain in bonded reinforcement, whether in tension or compression, is the same as that in the surrounding concrete.

iii) The tensile strength of the concrete is ignored.

iv) The stresses in the concrete in compression are derived from the design stress-strain curve in either Figure 4.2 or Figure 4.3.

v) The stresses in the reinforcement or prestressing steel are derived from the design curves in Figure 4.5 or Figure 4.6 respectively.

vi) The initial pre-strain in prestressing tendons is taken into account when assessing the stresses in the tendons at the ultimate limit state (see 2.5.4.4.3).

vii) For cross-sections subject to pure longitudinal compression, the compressive strain in the concrete is limited to - 0.002 (see Figure 4.2).

viii) For cross-sections not fully in compression, the limiting compressive strain is taken as — 0.0035. In intermediate situations, the strain diagram is defined by assuming that the strain is - 0.002 at a level 3/7 of the height of the section from the most compressed face.

(2) The adoption of the assumptions in P(1) above leads to the range of possible strain diagrams shown in Figure 4.11.

(3) In some cases where the interaction of local strength and deformation are significant, it may be convenient to assume a limiting tensile strain in the reinforcement and prestressing steel (see 4.2.2.3.2 and 4.2.3.3.3).

(4) As an alternative to the approach in P(1) above, the approach in 4.2.1.3.3(12) may be adopted.

(5) For prestressed members with permanently unbonded tendons, it is generally necessary to take the deformation of the whole member into account (see Part 1D). However, for buildings where unbonded tendons exist only during the construction phase, this is generally unnecessary.

(6) In the analysis of a cross-section which has to resist bending and only a small longitudinal force, the. effect of the design ultimate longitudinal compressive force may be ignored if it does not exceed 0.08 fck times the cross-sectional area.

(7) If changes in the position of the reinforcement, such as at a lap, can lead to a localised reduction in the effective depth, the most unfavourable value should be used in the cross-section analysis.

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