7 Structural analysis
7.3 Imperfections General General
(1) Appropriate allowances shall be considered to cover the effects of imperfections, including residual stresses and geometrical imperfections such as lack of verticality, lack of straightness, lack of flatness, lack of fit and any unspecified eccentricities present in joints of the unloaded structure.
NOTE Geometrical imperfections in accordance with the essential geometrical tolerances given in EN 1090-3 are considered in the resistance formulae, the buckling curves and the M-values in EN 1999.
(2) Equivalent geometric imperfections (see 7.3.2 and 7.3.3) should be used with values which reflect the possible effects of all type of imperfections. In the equivalent column method according to 7.3.4 the effects are included in the resistance formulae for member design.
(3) The following imperfections should be considered:
a) global imperfections for frames and bracing systems;
b) local imperfections for individual members.
Imperfections for global analysis of frames
(1) The assumed shape of global imperfections and local imperfections may be derived from the elastic buckling mode of a structure in the plane of buckling considered.
(2) Both in and out of plane buckling including torsional buckling with symmetric and asymmetric buckling shapes should be considered in the most unfavourable direction and form.
(3) For frames sensitive to buckling in a sway mode the effect of imperfections should be considered in the frame analysis by means of an equivalent imperfection in the form of an initial sway imperfection and individual bow imperfections of members. The imperfections may be determined from Formula (7.2) and Formula (7.3) as follows:
a) global initial sway imperfections, see Figure 7.1(d):
φ = φ0hm (7.2)
where
h is the reduction factor for height, h, applicable to columns: 2 2
h but h 1 0
3 ,
= h
where
h is the height of the structure in metres;
m is the reduction factor for the number of columns in a row: m 0 5 1, 1
(b) equivalent column length;
(c) first order moment;
Equivalent sway method:
(d) system load and sway imperfection;
(e) buckling length;
(f) second order moment
(g) initial local bow and buckling length for lateral-torsional buckling
If in the equivalent sway method initial bow imperfections are included in (d), the second order moments give the final result and no investigation according to (e) is required. In both methods the out-of-plane stability check (g) should be performed according to 7.2.2(5)b). If neither flexural torsional buckling nor lateral torsional buckling is involved it is sufficient to use the first order moments in this resistance verification.
Figure 7.1 — Equivalent buckling length and equivalent sway imperfections
b) When performing second order analysis including member imperfections the initial bow imperfections related to flexural buckling and torsional flexural buckling, e0, may be determined from Formula (7.3).
L is the member length. For in-plane and out-of-plane buckling of arches, L, may be taken as shown in Table 7.1.
χ is the reduction factor according to 8.3.1.3 for actual material buckling class BC, longitudinal weld, section type and slenderness 𝜆̄, where 𝜆̄ = 𝐿
𝑖𝜋√𝑓𝑜
𝐸 with i= √𝐼
𝐴; Mo,Rd is the design bending moment resistance according to Formula (8.28);
No,Rd is the design axial force resistance according to Formula (8.26);
β = η = 1 if a linear interaction formula is used;
for hollow sections and solid sections if Formula (8.73) is used and Mz,Ed = 0;
χy = 1 and χz = 1
in Formula (8.65), Formula (8.66) and Formula (8.73).
Table 7.1 — Shape and buckling length of global imperfection for in-plane and out-of-plane buckling of arches. The shape of imperfection is sinus or parabola
In-plane
Out-of-plane buckling
Ls
Ls
𝐿𝑠≤ 12𝑚 𝐿 = 𝐿𝑠
𝐿𝑠> 12𝑚 𝐿 = √12𝐿𝑠[𝑚]
(4) For building frames the sway imperfections may be disregarded where Formula (7.4) is satisfied:
HEd ≥ 0,15VEd (7.4)
where
HEd is the design value of the horizontal force;
VEd is design value of the vertical force.
(5) For the determination of horizontal forces to floor diaphragms the configuration of imperfections as given in Figure 7.2 should be applied, where is a sway imperfection obtained from Formula (7.2) assuming a single storey with height, h, see (3) a).
/ 2 NEd
/ 2
NEd
Hi = NEd
hh
NEd
Hi =NEd
h
NEd
a) Two or more storeys b) Single storey
Figure 7.2 — Configuration of sway imperfections, , for horizontal forces on floor diaphragms (6) When performing the global analysis for determining end forces and end moments to be used in member checks according to 8.3 local bow imperfections may be neglected. However, for frames sensitive to second order effects local bow imperfections of members additionally to global sway imperfections (see 7.2.1(3)) should be introduced in the structural analysis of the frame for each compressed member where the following conditions are met:
— presence of at least one moment resistant joint at the end of one member;
— o
Ed
0 5, Af
= N (7.5)
where
NEd is the design value of the compression force;
is the in-plane relative slenderness calculated for the member considered as hinged at its ends.
NOTE Local bow imperfections are taken into account in member checks, see 7.2.2 (3) and 7.3.4.
(7) The effects of initial sway imperfection and bow imperfections may be replaced by systems of equivalent horizontal forces, introduced for each column, see Figure 7.2 and Figure 7.3.
NEd
NEd
NEd
NEd
NEd
NEd
NEd
NEd
NEd
NEd
e0,d
4NEde0d
L
4NEde0d
L 8NEde0d
L2
L
a) Initial sway imperfections b) Initial bow imperfections Figure 7.3 — Replacement of initial imperfections by equivalent horizontal forces
(8) These initial sway imperfections should apply in all relevant horizontal directions, but need only be considered in one direction at a time.
(9) Where, in multi-storey beam-and-column building frames, equivalent forces are used, they should be applied at each floor and roof level.
(10) The possible torsional effects on a structure caused by anti-symmetric sways at the two opposite faces, should also be considered, see Figure 7.4.
A B
direction b) Faces A-A and B-B sway in opposite
direction Key
1 translational sway 2 rotational sway
Figure 7.4 — Translational and torsional effects (plan view)
(11) As an alternative to (3) and to (6) the shape of the elastic critical buckling mode, ηcr, of the frame structure or of the verified member may be applied as the unique global and local initial (UGLI) imperfection. The equivalent geometrical imperfection may be expressed in the form of Formula (7.6) and Formula (7.7):
m denotes the critical cross-section of the frame structure or of the verified member (see Note 5). Index m indicates belonging to the critical cross-section;
αm is the imperfection factor for the relevant buckling curve, see Table 8.4;
𝜆̄𝑚
= √𝑁𝑁Rk,m
cr,m is the relative slenderness of the frame structure or of the verified member and of the equivalent member (see Note 2);
𝜆0 is the limit of the horizontal plateau for the relevant buckling curve given in Table 8.4;
χm is the reduction factor for the relevant buckling curve, see 8.3.1.3 and the slenderness 𝜆̄𝑚;
Ncr,m = αcrNEd,m is the value of axial force in the critical cross-section, m, when the elastic critical buckling was reached, and it is also the critical axial force for the equivalent member;
αcr is the minimum force amplifier for the axial force configuration NEd in members to reach the elastic critical buckling of the structure;
MRk,m is the characteristic moment resistance of the cross-section, m, according to 8.2.5.1(2);
NRk,m is the characteristic normal force resistance of the cross-section, m, according to 8.2.4(2);
𝐸𝐼𝑚|𝜂cr″|𝑚 is the value in the critical cross-section, m, of the bending moments which would be necessary to bend the structure (in the state without axial forces) into the form of the buckling mode;
NOTE 1 Formula (7.6) is based on the requirement that the imperfection, ηinit,m(x), in the shape of the elastic buckling mode, ηcr(x), should have the same maximum curvature as supposed for the equivalent member in 8.3.1.
Therefore, the buckling resistance of the uniform members loaded in axial compression and calculated with the imperfection according to (7.6) and for second order effects is identical with the value, Nb,Rd, according to 8.3.1.2(2).
NOTE 2 The imperfection, ηinit,m(x), in the shape of the elastic critical buckling mode is applicable generally to all members in compression and to frames buckling in their plane. It is especially suitable for members with cross-section characteristics and/or axial force not constant along their length and for frames containing such members.
NOTE 3 For elastic second order analysis, Formula (7.6) gives exactly the same result as Formula (7.3) with all exponents equal to 1,0.
NOTE 4 The equivalent member has pinned ends, and its cross-section and axial force are the same as in the critical cross-section, m, of the frame. Its length is such that the critical force is equal with the axial force in the critical cross-section, m, at the critical loading of the structure.
(12) For calculating the amplifier, αcr, in Formula (7.6) the members of the structure may be considered to be loaded by axial forces, NEd, from the first order elastic analysis of the structure.
(13) The Formula 𝐸𝐼𝑚|𝜂cr″|𝑚in (7.6) may be replaced by |𝑀η.cr,mII |(𝛼cr− 1), where 𝑀η.cr,mII is the bending moment in cross-section, m, calculated by the second order analysis of the structure with the imperfection in the shape of the elastic critical buckling mode 𝜂cr and with arbitrary value of the maximum amplitude, ηcr,max.
(14) Generally, the position of the critical cross-section m in Formula (7.6) should be determined using an iterative procedure.
NOTE The selected position for the critical cross-section, m, is correct, if the utilization in the selected position, xm, is greater than the utilizations in all other points, x, of the structure.
Imperfection for analysis of bracing systems
(1) In the analysis of bracing systems which are required to provide lateral stability within the length of beams or compression members the effects of imperfections should be included by means of an equivalent geometric imperfection of the members to be restrained, in the form of an initial bow imperfection given by Formula (7.9) and (Formula (7.10):
α0 = αmL/500 (7.9)
where
L is the span of the bracing system and
m
0 5 1, 1
= + m
(7.10)
where
m is the number of members to be restrained.
(2) For convenience, the effects of the initial bow imperfections of the members to be restrained by a bracing system may be replaced by the equivalent stabilising force as given by Formula (7.11) and shown in Figure 7.5:
(
0 q)
0,d Ed 2
q N 8 e
L
= +
(7.11) where
δq is the in-plane deflection of the bracing system due to q0,d plus any external loads calculated from first order analysis, which may be taken equal to 0 when second order analysis is carried out.
NOTE As δq in Formula (7.11) depends on q0,d, it results in an iterative procedure.
(3) Where the bracing system is required to stabilize the compression flange of a beam of constant height, the force NEd in Figure 7.5 may be obtained from Formula (7.12):
NEd = MEd/h (7.12)
where
MEd is the maximum bending moment in the beam;
h is the overall depth of the beam.
Where a beam is subjected to an additional compression force, this should be taken into account.
The force NEd of Figure 7.5 should be assumed to be uniform within the span, L, of the bracing system.
In case of non-uniform forces, this assumption is slightly conservative.
(4) At points where beams or compression members are spliced, it should also be verified that the bracing system is able to resist a local force equal to mNEd/100 applied to it by each beam or compression member which is spliced at that point, and to transmit this force to the adjacent points at which that beam or compression member is restrained, see Figure 7.6.
(5) In verifying the local force according to (4), any external loads acting on bracing systems should also be included, but the forces arising from the imperfection given in (1) may be omitted.
e0
NEd NEd
q0,d
L
1
Key
e0 imperfection
q0,d equivalent force per unit length 1 bracing system
Figure 7.5 — Equivalent stabilising force
NEd
NEd
NEd
1
2
NEd
2NEd
Key 1 splice
2 bracing system φ = αmφ0: φ0 = 1/200 2φNEd = αmNEd/100
Figure 7.6 — Bracing forces at splices in compression members Member imperfections
(1) The effects of imperfections of members described in 7.3.1(1) are incorporated in the formulae given for buckling resistance for members, see 8.3.1.
(2) Where the stability of members is accounted for by second order analysis according to 7.2.2(5)a) for compression members imperfections, e0, according to 7.3.2(3)b) or 7.3.2(5) or (6) should be considered.
(3) For a second order analysis taking account of lateral torsional buckling of a member in bending the imperfections may be adopted as 0,5e0, where, e0, is the equivalent initial bow imperfection about the weak axis of the profile considered. In general, additional torsional imperfections need not to be considered.