## Appendix Supplementary information on the ultimate limit states induced by structural deformations

Fv Sum of all vertical loads under service conditions fctk, 0.05 Lower characteristic value of tensile strength of concrete htot Total height of structure from top surface of foundation or non-deformable sub-stratum (in metres)

Msd1 First order design moment n Number of storeys

Nsd m Mean axial design force in columns in one storey

2m Mean slenderness ratio of columns within storey considered vu Longitudinal force coefficient for a member

### A3.1 Design procedures

P(1) The combinations of actions and the safety factors given in 2.3 shall normally be used. However, in multi-storey buildings,11-1 lower safety factors Yf than those given in 2.3 may be used for the calculation of structural deformations which lead to second order effects. This applies in particular for the calculation of creep deformations.

P(2) For some applications, the design format defined in 2.2.2.5 may be used.

P(3) In multi-storey buildings,1-* the deformations of the structure may be calculated by using a reduced safety coefficient Yc for concrete.

11) In the absence of national regulations, buildings may be considered multistorey, if their total height above ground exceeds |22| m.

(4) The partial safety factors Yf given in 2.3 may be reduced by about | 10 % | for multi-storey buildings.

(5) In multi-storey buildings, where a refined analysis of creep deformation is necessary [see A3.4 P(3) and (8)], the following safety coefficients Yf for quasi-permanent loading are recommended:

|Yf = I ll : for statically indeterminate structures

| Yf = 1.2l : for statically determinate structures and structural elements.

(6) A reduced safety coefficient Yc = | 1.35 | may be used in the analysis of frame structures by the general method (see 4.3.5.2).

(7) When applying 4.3.5 a refined general method or an appropriate proven simplified method may be used.

These methods may be classified as follows:

— General methods: Non-linear analyses using appropriate design models of the structure.

— Simplified methods: These may be either:

a) approximate non-linear second order analyses, simplified by assuming a distribution of the internal forces and moments and/or a deformed shape of the structure; or, b) first order analyses of cross-sections at the ultimate limit state for bending and longitudinal force modified by multiplying the first order internal forces and moments Nsd and/or Msdi by coefficients to cover the increase of Msdi due to deformations.

Simplified methods may be based on the real structure (e.g. height of columns between centres of restraints) or on fictitious design models (e.g. Model column, see 4.3.5.6.3). Appropriate design aids may be used.

(8) It is generally necessary to check the critical cross section about each of the two principle axes. For these two directions, different restraint conditions may be present at the ends of the member. These conditions should be represented in an adequate way.

P(9) The influence of the soil behaviour on the stability of a structure shall be considered and, if significant, taken into account in the design calculations.

(10) The design procedures envisaged in 4.3.5 are illustrated in the flow charts in Figure A3.1, Figure A3.2 and Figure A3.3 below.

First order analysis of the structure based on material data in 3. (2.5 & 3)

First order analysis of the structure based on material data in 3. (2.5 & 3)

Outline of design procedures

Outline of design procedures

Classify structure end structural elements (4.3.5 3)

Figure A3.1 — Flow Chart 1: General guide to application of section 4.3.5

Figure A3.2 — Flow Chart 2: Application of provisions of 4.3.5 and A3 to ultimate limit states due to deformation of the structure as a whole

A3.2 Non-sway structures

(1) Provided that the bracing elements are reasonably symmetrically distributed within the building, braced frames can be classified as non-sway if the flexural stiffness of the bracing elements satisfies the criterion below:Provided that the bracing elements are reasonably symmetrically distributed within the building, braced frames can be classified as non-sway if the flexural stiffness of the bracing elements satisfies the criterion below:

where:

n is the number of storeys htot is the total height of the structure in metres measured from the top surface of the foundation or from a non-deformable sub-stratum;

EcmIc is the sum of the nominal flexural stiffnesses of all the vertical bracing elements as defined in 4.3.5.3.2(1) acting in the direction under consideration. In the bracing elements, the concrete tensile stress under the relevant load combination in service conditions should not exceed the value fctk,0.05 defined in 3.1.2.3. If the stiffness of the bracing elements varies over their height, an equivalent stiffness should be used;

Fv is the sum of all vertical loads (i.e. acting both on the bracing elements and the braced sub-assembly) under service conditions, (i.e. Yf = 1).

It should be noted chat there are cases where the above equations will be conservative.

(2) If Equation (A3.1) or (A3.2) is not satisfied, the structure is classified as sway and should be designed accordingly.

(3) Frames without bracing elements may be considered non-sway structures if each vertical element of the frame which resists more than | 70 % | of the mean axial force NSd m = Yf Fv/n (n denotes the number of vertical elements in one storey) has a slenderness ratio 2 less than or equal to the greater of 25

A3.3 Bracing elements in braced structures

(1) In addition to 4.3.5.3.2(1) and in order to avoid horizontal forces in the braced sub-assembly

(e.g. columns), bracing elements should be designed to resist all horizontal loads acting on the structure

A3.4 Specific data

P(1) A stress-strain diagram for concrete shall be used which adequately represents the real behaviour. P(2) The contribution of the tensile strength of the concrete between cracks (tension stiffening) shall, if not stated otherwise, be taken into account.

P(3) Creep effects shall be considered if they are likely to reduce the structural stability significantly. P(4) The same stress-strain diagram for steel as used for cross-section design shall be adopted (see 4.2.2.3.2).

P(5) Plane sections shall be assumed to remain plane as stated in 4.3.1.2.

(6) For the concrete, the stress-strain diagram given in section 4.2.1.3.3 a) should be used with fc and Ec taken as:

For the safety factor Yc, A3.1, P(3) and (6), apply.

(7) It will always be conservative to ignore tension stiffening effects. However, when using the model column method (see 4.3.5.6.3), the contribution of tension stiffening should not be taken into account.

Figure A3.4 — Slenderness limits in frames (A3.2)

(8) For simplification, creep effects may be ignored if the increase in the first order bending moments due to creep deformations and longitudinal force do not exceed 110 % |. Where necessary, creep effects may be assessed by means of approximate methods based on 2.5.5 or, alternatively, by a modification of the stress-strain relationship for the concrete or by a correction of the additional eccentricity or unintentional inclination defined in 2.5.1.

The safety coefficients given in A3.1 should be used for the calculation of creep deformations, unless stated otherwise.

(9) In non-away buildings, creep deformations of slender compression members connected monolithically to slabs or beams at their two ends may normally be disregarded because their effects are generally compensated by other influences which are neglected in the design. In interior columns, the restraints at the column ends reduce the creep deformations significantly so that they can be neglected. In edge columns with different eccentricities at each end, creep increases the deformations but it does not decrease the bearing capacity because these deformations are not additional to the critical column deflections in the relevant failure state.

### A3.5 Sway frames

P(1) Sway frames shall be designed using the design data given in 4.3.5.4 and A3.4. Equivalent geometrical imperfections and, if necessary for reasons of structural stability, creep deformations shall be taken into account.

(2) The simplified methods defined in 4.3.5 may be used instead of a refined analysis, provided that the safety level required is ensured.

(3) For regular frames, simplified methods may be used which introduce, for example, increased horizontal design loads or bending moments which take account of second order effects in addition to the effects of geometrical imperfections. Regular frames are, for example, frames formed by beams and columns which have approximately equal nominal stiffnesses and a mean slenderness ratio 2m of each storey which is less than or equal to:

whichever is greater. where:

2m is the mean slenderness ratio of all columns within the storey considered (see 4.3.5.3.5).

(4) If the mean slenderness ratio, 2m, is greater than the value given by Equation (A3.4), reference should be made to appropriate literature to conform with P(1) and (2) above.

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