P Eurocode 2

Compression Member Restraint and Effective Length

El is the clear height of compression member between end restraints∞k = 0 is the theoretical limit for rigid rotational restraint, and k = represents the limit for no restraint atNote:all. Since fully rigid restraint is rare in practise, a minimum value of 0,1 is recommended for k and k .1 2(4) If an adjacent compression member (column) in a node is likely to contribute to the rotationΙ Ι Ιat buckling, then (E /l) in the definition of k should be replaced by [(E / l) +(E / l) ], a and ba brepresenting the compression member (column) above and below the node.(5) In the definition of effective lengths, the stiffness of restraining members should include theeffect of cracking, unless they can be shown to be uncracked in ULS.(6) For other cases than those in (2) and (3), e.g. members with varying normal force and/orcross section, the criterion in 5.8.3.1 should be checked with an effective length based on thebuckling load (calculated e.g. by a numerical

1. Equation (5.17):

Υ = πl ε / N (5.17)

2. Where:

E is a representative bending stiffness

IN is buckling load expressed in terms of this E IB (in Expression (5.14), i should also correspond to this)

EI (7)

3. The restraining effect of transverse walls:

The restraining effect of transverse walls may be allowed for in the calculation of the β effective length of walls by the factor given in 12.6.5.1. In Expression (12.9) and Table 12.1, lw is then substituted by l determined according to 5.8.3.2.

05.8.3.3 Global second order effects in buildings

4. As an alternative to 5.8.2 (6), global second order effects in buildings may be ignored if:

∑ ΥEn cd c ≤ σ σsF k (5.18)

V,Ed 1 + 2n L1,6s

5. Where:

F is the total vertical load (on braced and bracing members)

V,Ed is the number of storeys

L is the total height of building above level of moment restraint

E is the design value of the modulus of elasticity of concrete, see 5.8.6 (3)

cd is the second moment of area (uncracked concrete section) of bracing member(s)

I c The value of k

for use in a Country may be found in its National Annex. The recommended value isNote: 10,31.

Expression (5.18) is valid only if all the following conditions are met:

• torsional instability is not governing, i.e. structure is reasonably symmetrical
• global shear deformations are negligible (as in a bracing system mainly consisting of shear- walls without large openings)
• bracing members are rigidly fixed at the base, i.e. rotations are negligible
• EN 1992-1-1:2004 (E)
• the stiffness of bracing members is reasonably constant along the height
• the total vertical load increases by approximately the same amount per storey

(2) k in Expression (5.18) may be replaced by k if it can be verified that bracing members 1 2 are uncracked in ultimate limit state.

The value of k for use in a Country may be found in its National Annex. The recommended value isNote 1: 20,62. For cases where the bracing system has significant global shear deformations and/or end rotations,

Note 2:see Annex H (which also gives

The effect of creep shall be taken into account in second order analysis, with due consideration of both the general conditions for creep (see 3.1.4) and the duration of different loads in the load combination considered.

The duration of loads may be taken into account in a simplified way by means of an effective creep ratio, ϕ, which, used together with the design load, gives a creep deformation (curvature) corresponding to the quasi-permanent load:

ϕ ϕ ⋅M= / M (5.19)

where:

ϕ is the final creep coefficient according to 3.1.4

M is the first order bending moment in quasi-permanent load combination (SLS)

M is the first order bending moment in design load combination (ULS)

It is also possible to base on total bending moments M and M, but this requires iteration and a verification of stability under quasi-permanent load with ϕ.

If M / M varies in a

member or structure, the ratio may be calculated for the section _{0Eqp 0Ed} with maximum moment, or a representative mean value may be used. ϕ(4) The effect of creep may be ignored, i.e. = 0 may be assumed, if the following three conditions are met: ϕ ≤ 2- (∞,t0) λ ≤ 75- ≥M /N h- 0Ed Ed Here M is the first order moment and h is the cross section depth in the corresponding direction. If the conditions for neglecting second order effects according to 5.8.2 (6) or 5.8.3.3 are only just achieved, it may be too unconservative to neglect both second order effects and creep, unless the mechanical ω reinforcement ratio ( , see 5.8.3.1 (1)) is at least 0,25. 5.8.5 Methods of analysis (1) The methods of analysis include a general method, based on non-linear second order analysis, see 5.8.6 and the following two simplified methods: (a) Method based on nominal stiffness, see 5.8.7 (b) Method based on nominal curvature, see 5.8.8

Simplified Method (a) and (b) to be used in a Country may be found in its National^{Note 1}:Annex. Nominal second order moments provided by the simplified methods (a) and (b) are sometimes^{Note 2}:greater than those corresponding to instability. This is to ensure that the total moment is compatible with thecross section resistance.

(2) Method (a) may be used for both isolated members and whole structures, if nominalstiffness values are estimated appropriately; see 5.8.7.

(3) Method (b) is mainly suitable for isolated members; see 5.8.8. However, with realisticassumptions concerning the distribution of curvature, the method in 5.8.8 can also be used forstructures.

5.8.6 General method

(1)P The general method is based on non-linear analysis, including geometric non-linearity i.e.second order effects. The general rules for non-linear analysis given in 5.7 apply.

(2)P Stress-strain curves for concrete and steel suitable for overall analysis shall be used. Theeffect of creep shall be taken into

account.(3) Stress-strain relationships for concrete and steel given in 3.1.5, Expression (3.14) and 3.2.3(Figure 3.8) may be used. With stress-strain diagrams based on design values, a design valueof the ultimate load is obtained directly from the analysis. In Expression (3.14), and in the k-value, f is then substituted by the design compressive strength f and E is substituted bycm cd cmγE = E / (5.20)cd cm cE γThe value of for use in a Country may be found in its National Annex. The recommended value isNote: cE1,2.(4) In the absence of more refined models, creep may be taken into account by multiplying allϕstrain values in the concrete stress-strain diagram according to 5.8.6 (3) with a factor (1 + ),efϕwhere is the effective creep ratio according to 5.8.4.ef(5) The favourable effect of tension stiffening may be taken into account.This effect is favourable, and may always be ignored, for simplicity.Note:(6) Normally, conditions of equilibrium and strain compatibility

are satisfied in a number of cross sections. A simplified alternative is to consider only the critical cross section(s), and to assume a relevant variation of the curvature in between, e.g. similar to the first order moment or simplified in another appropriate way.

5.8.7 Method based on nominal stiffness

5.8.7.1 General

(1) In a second order analysis based on stiffness, nominal values of the flexural stiffness should be used, taking into account the effects of cracking, material non-linearity and creep on the overall behaviour. This also applies to adjacent members involved in the analysis, e.g. beams, slabs or foundations. Where relevant, soil-structure interaction should be taken into account.

(2) The resulting design moment is used for the design of cross sections with respect to bending moment and axial force according to 6.1, as compared with 5.8.6 (2).

5.8.7.2 Nominal stiffness

(1) The following model may be used to estimate the nominal stiffness of slender

compressionmembers with arbitrary cross section: K E + K E (5.21)EI I Ic cd c s s s

where:
E is the design value of the modulus of elasticity of concrete, see 5.8.6 (3)
cd is the moment of inertia of concrete cross section
I cE is the design value of the modulus of elasticity of reinforcement, 5.8.6 (3)
s is the second moment of area of reinforcement, about the centre of area of theI s concrete
K is a factor for effects of cracking, creep etc, see 5.8.7.2 (2) or (3)
cK is a factor for contribution of reinforcement, see 5.8.7.2 (2) or (3)
s ρ ≥(2)

The following factors may be used in Expression (5.21), provided 0,002:
K = 1s (5.22)
ϕK = k k / (1 + )c 1 2 ef

where:
ρ is the geometric reinforcement ratio, A /As c
A is the total area of reinforcements
A is the area of concrete section
cϕ is the effective creep ratio, see 5.8.4ef
k is a factor which depends on concrete strength class, Expression (5.23)1
k is a factor which depends on axial force and slenderness, Expression (5.24)2
k = f / 20

(MPa) (5.23)ck1 λ⋅ ≤nk = 0,20 (5.24)2 170

where:
n is the relative axial force, N / (A f )
Ed c cdλ is the slenderness ratio, see 5.8.3λ
If the slenderness ratio is not defined, k may be taken as2≤k = n⋅0,30 0,20 (5.25)2 ρ ≥

(3) As a simplified alternative, provided 0,01, the following factors may be used in
Expression (5.21):
K = 0s (5.26)
ϕK = 0,3 / (1 + 0,5 )c ef. The simplified alternative may be suitable as a preliminary step, followed by a more accurate calculation

Note
according to (2).
70 EN 1992-1-1:2004 (E)

(4) In statically indeterminate structures, unfavourable effects of cracking in adjacent members
should be taken into account. Expressions (5.21-5.26) are not generally applicable to such
members. Partial cracking and tension stiffening may be taken into account e.g. according to
7.4.3. However, as a simplification, fully cracked sections may be assumed. The stiffness
should be based on an effective concrete modulus:
ϕE = E /(1+ ) (5.27)
cd,eff

cd efwhere: E is the design value of the modulus of elasticity according to 5.8.6 (3)cd

ϕ is the effective creep ratio; same value as for columns may be usedef

5.8.7.3 Moment magnification factor

(1) The total design moment, including second order moment, may be expre