Imperfections According to EN 1993-1-1 Clause 5.3.2: Bow Imperfection

Technical Article

According to EN 1993‑1‑1 [1], it is necessary to use the equivalent geometric imperfections with values that reflect the possible effects of all types of imperfections. EN 1993‑1‑1, Clause 5.3, specifies basic imperfections for the global analysis of frames as well as member imperfections.

Global Analysis of Frames

Even though the influences of bow imperfections in the equations for stability designs of members according to Clause 6.3 are already included, it is nevertheless important to consider a bow imperfection in the global analysis of frames if a structure or component is present which is sensitive to bow imperfections. This is defined in the German DIN standard DIN EN 1993‑1‑1:2010‑12 Clause 5.3.2 (6).

Figure 01 - Example

Equation (5.8) is used to decide if a member is sensitive to bow imperfections:
$\overline{\mathrm\lambda}\;>\;0.5\;\cdot\;\sqrt{\frac{\mathrm A\;\cdot\;{\mathrm f}_{\mathrm y}}{{\mathrm N}_{\mathrm{Ed}}}}$
where
$\overline{\mathrm\lambda}\;=\;\sqrt{\frac{{\mathrm N}_{\mathrm{pl}}}{{\mathrm N}_{\mathrm{cr}}}}\;=$ degree of slenderness of structural component in considered plane which is determined by assuming hinged support on both sides
and
${\mathrm N}_{\mathrm{cr}}\;=\;\frac{\mathrm\pi²\;\cdot\;\mathrm E\;\cdot\;\mathrm I}{{\mathrm s}_{\mathrm k}²}$
results in the known differentiation by using the member coefficient from DIN 18800:
${\mathrm s}_{\mathrm k}\;\cdot\;\sqrt{\frac{{\mathrm N}_{\mathrm{Ed}}}{\mathrm E\;\cdot\;\mathrm I}}\;=\;\mathrm\varepsilon\;>\;0.5\;\cdot\;\mathrm\pi\;\approx\;1.6$

Member Imperfections

Flexural buckling
The recommended bow imperfection's values result from DIN EN 1993‑1‑1:2010‑12 Clause 5.3.2 (3) Table 5.1 as follows:

Buckling curve according to
DIN EN 1993‑1‑1:2010‑12,
Table 6.2
Cross-section design
Elastic e0,d/LPlastic e0,d/L
a01/3501/300
a1/3001/250
b1/2501/200
c1/2001/150
d1/1501/100

If the determination of internal forces of the entire structure is based on the elastic analysis and if a cross-section design according to DIN EN 1993‑1‑1:2010‑12 Clause 6.2.1 (7) Equation (6.2) is performed, Table NA.2 of DIN EN 1993‑1‑1/NA:2015‑08 [2] can be used:

Buckling curve according to
DIN EN 1993‑1‑1:2010‑12,
Table 6.2
Cross-section design
Elastic e0,d/LPlastic e0,d/L
a01/600As for
elastic,
but
Mpl/Mel-fold
a1/550
b1/350
c1/250
d1/150

Lateral-torsional buckling
According to DIN EN 1993‑1‑1:2010‑12 Clause 5.3.4 (3), for the torsional buckling analysis of elements subjected to bending according to the second-order theory, the imperfection as bow imperfection about the weak axis with the value k ∙ e0,d (e0,d see flexural buckling) has to be assumed. The value of k = 0.5 is recommended. It is not necessary to apply any further torsional imperfection.

An economical approach is possible according to DIN EN 1993‑1‑1/NA:2015‑08 for I-sections. According to NDP of 5.3.4(3) the bow imperfection can be assumed with the values of Table NA.3 instead of k ∙ e0,d.

Cross-sectionDimensionsElastic
cross-section
ratio
e0,d/L
Plastic
cross-section
ratio
e0,d/L
Rolled
I-sections
h/b ≤ 2.01/5001/400
h/b > 2.01/4001/300
Welded
I-sections
h/b ≤ 2.01/4001/300
h/b > 2.01/3001/200

The specified values have to be doubled in the range of 0.7 ≤ λLT ≤ 1.3. This requirement goes back to a dissertation of 2008 at the German Ruhr University in Bochum and has been added to the updated DIN 18800:2008‑11.

Literature

[1] Eurocode 3: Design of steel structures - Part 1‑1: General rules and rules for buildings; EN 1993‑1‑1:2010‑12
[2] National Annex - Nationally determined parameters - Eurocode 3: Design of steel structures - Part 1‑1: General rules and rules for buildings; DIN EN 1993‑1‑1/NA:2015‑08
[3] Training Manual EC3. Leipzig: Dlubal Software, September 2017

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