Geometric imperfections
Equivalent imperfections for steel frames to EN 1993-1-1 clause 5.3 — global sway, member bow and bracing load.
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Tool information
What this calculator computes
The tool determines the equivalent geometric imperfections for steel frames to EN 1993-1-1 clause 5.3: the global sway, the member bow imperfection and the bracing system load.
Why it exists
Real structures are not straight. Columns have erection tolerances, rolling leaves residual stresses, and joints carry small eccentricities. The code does not ask you to model each deviation; it replaces them with a single equivalent imperfection that covers them together.
The practical consequence matters more than it looks: a perfectly straight frame under purely vertical load has no second-order moment at all in the model. The analysis says it is stable up to the critical load. With imperfections, an equivalent horizontal force of roughly 0.3–0.5% of the vertical load appears — small against the weight, but it produces the real moments at the column bases and in the foundations.
Three imperfections, not to be confused
Global sway (clause 5.3.2)
\(\varphi = \varphi_0 \cdot \alpha_h \cdot \alpha_m\)
\(\varphi_0 = 1/200\) is the basic value. The other two factors reduce it:
\(\alpha_h = \frac{2}{\sqrt{h}}, \quad \frac{2}{3} \le \alpha_h \le 1.0 \qquad \alpha_m = \sqrt{0.5\left(1 + \frac{1}{m}\right)}\)
\(\alpha_h\) is capped at both ends, and the lower limit is the one that matters. Without it the imperfection would tend to zero as the building grows — which would be absurd, since erection tolerances do not vanish with height. For any \(h \ge 9\) m, \(\alpha_h = 2/3\).
\(\alpha_m\) reflects that with more columns it becomes less and less likely that all lean the same way. The limit at infinity is \(\sqrt{0.5} = 0.707\) and is never reached. When counting \(m\), include only the columns carrying at least 50% of the average \(N_{Ed}\) in the row.
The result is applied as equivalent horizontal forces, \(H_{Ed} = \varphi \cdot N_{Ed}\) at each storey, where \(N_{Ed}\) is the vertical force passing through the storey, not the one it adds.
It is applied in one direction at a time — the most unfavourable — not simultaneously in both orthogonal directions.
Member bow (clause 5.3.4)
The member has an initial bow \(e_0\), taken from Table 5.1 according to its buckling curve and the type of analysis:
| Curve | Elastic analysis | Plastic analysis |
|---|---|---|
| a₀ | L/350 | L/300 |
| a | L/300 | L/250 |
| b | L/250 | L/200 |
| c | L/200 | L/150 |
| d | L/150 | L/100 |
Plastic analysis requires larger imperfections on every curve: plastic hinges amplify the effect of the initial deviation.
The factor \(k = 0.5\) of clause 5.3.4(3) halves it, but is permitted only when the global imperfection is accounted for separately.
Bracing system (clause 5.3.3)
The bracing must be able to straighten the members it stabilises, so it carries a load of its own:
\(e_0 = \alpha_m \frac{L}{500} \qquad q_d = \sum N_{Ed} \frac{8(e_0 + \delta_q)}{L^2}\)
The calculation is iterative. \(q_d\) produces a deflection \(\delta_q\), which adds to \(e_0\) and in turn increases \(q_d\). The first pass with \(\delta_q = 0\) is unconservative — the page flags it.
When second-order effects may be neglected
\(\alpha_{cr} = \frac{F_{cr}}{F_{Ed}} \ge 10 \text{ (elastic)}, \qquad \ge 15 \text{ (plastic)}\)
The threshold depends on the type of analysis, and the band between 10 and 15 is where that choice changes the answer: \(\alpha_{cr} = 12\) passes in elastic analysis and fails in plastic.
National annex
\(\varphi_0 = 1/200\), Table 5.1, \(k = 0.5\) and the \(\alpha_{cr}\) thresholds are nationally determined parameters. The defaults are those of SR EN 1993-1-1:2006/NA:2008, which adopts the values recommended in the code itself.
Assumptions and limitations
Covered: global sway (clause 5.3.2), the member bow imperfection of Table 5.1 (clause 5.3.4) and the bracing system load (clause 5.3.3).
Not covered: joint imperfections, the buckling-mode shape of clause 5.3.2(11), and the determination of \(\alpha_{cr}\) — that is entered, not computed. You get it from an eigenvalue analysis or, approximately, from expression 5.2.
Related calculators
- Flexural buckling — the buckling curve that \(e_0/L\) depends on
- Bending and compression — where the second-order moments end up
- Portal frame — the model the equivalent forces go into