Contents
What second-order effects are
A first-order analysis calculates the internal forces on the undeformed structure. In reality the structure deflects, and the vertical loads then act on a displaced geometry: a column that leans over by δ carries its axial force N with an extra moment N · δ. That moment increases the deflection, which increases the moment again. In a stiff frame this converges quickly and the effect is small. In a flexible frame it does not, and at the elastic critical load the frame buckles.
There are two levels. Global effects (P-Δ) come from the sway of the frame as a whole; member effects (P-δ) come from the bending of each member between its ends. Both come from axial compression acting on a deformed shape, and both start from an imperfect one.
When a first-order analysis is enough: αcr
Clause 5.2.1(3) allows a first-order analysis when the deformations do not increase the internal forces significantly. That may be assumed when:
For portal frames with shallow roofs and for beam-and-column plane frames in buildings, §5.2.1(4)B gives an approximation per storey:
The approximation assumes a roof no steeper than 1:2 (26°) and no significant axial compression in the beams or rafters. The compression counts as significant when λ̄ ≥ 0.3 √(A fy / NEd) (5.3), with λ̄ the in-plane slenderness of the beam as if hinged at its ends. Outside that scope, αcr comes from an elastic buckling analysis of the frame.
Imperfections: sway and bow
Real frames are never perfectly plumb and real members never perfectly straight. §5.3 covers this with equivalent geometric imperfections, which also stand in for residual stresses and small eccentricities. Two kinds matter for frames (FIG. 1).
Global sway imperfection
The sway may be modelled as an inclined geometry, or replaced by horizontal forces φ · NEd at the top of each column (§5.3.2(7)). It applies in every relevant horizontal direction, one direction at a time (§5.3.2(8)); in multi-storey frames the equivalent forces act at every floor and roof level (§5.3.2(9)). For building frames it may be ignored when HEd ≥ 0.15 VEd (5.7), that is, when the horizontal loads are already large compared with the vertical ones.
Local bow imperfection
The initial bow of a member, e0/L, depends on its buckling curve. The national annex may set its own values; Table 5.1 gives the recommended ones.
| Buckling curve | a0 | a | b | c | d |
|---|---|---|---|---|---|
| Elastic analysis | 1/350 | 1/300 | 1/250 | 1/200 | 1/150 |
| Plastic analysis | 1/300 | 1/250 | 1/200 | 1/150 | 1/100 |
In the global analysis of a frame that is sensitive to second-order effects, the bow has to be modelled only for compressed members with at least one moment-resisting joint and λ̄ > 0.5 √(A fy / NEd) (5.8). In all other cases the member checks of §6.3 cover it.
Three ways to deal with second-order effects
Clause 5.2.2(3) allows three routes, depending on the frame and the analysis.
- Everything in the global analysis. Second-order effects, the sway and the member bow imperfections are all in the model. No member buckling check to §6.3 is needed (§5.2.2(7)a); the cross-sections are checked with the forces from the analysis.
- Partly in the global analysis, partly in member checks. The analysis includes the global second-order effects and the sway imperfection. The members are then checked to §6.3 for what the analysis left out, using its end forces and moments, and the buckling length may be taken equal to the system length (§5.2.2(7)b).
- Member checks only. For basic cases: a first-order analysis without imperfections, with members checked to §6.3 using buckling lengths from the global buckling mode of the frame (§5.2.2(8)).
Route 3 is the classic hand method. Its weak point is the buckling length: in a sway frame it depends on the stiffness of the beams and joints and on the distribution of the axial forces, so it changes from one load combination to the next. Routes 1 and 2 avoid that question.
The amplification factor
For single-storey frames designed by elastic analysis there is a shortcut: §5.2.2(5)B allows the sway effects of a first-order analysis to be amplified instead of running a second-order analysis.
For αcr < 3.0 a more accurate second-order analysis applies. §5.2.2(6)B extends the method to multi-storey frames when all storeys have a similar distribution of vertical loads, horizontal loads and frame stiffness. FIG. 2 shows the factor.
Worked example: a portal frame
A single-storey frame with two columns, h = 6.0 m. Each column carries NEd = 120 kN, so VEd = 240 kN. The design wind load gives HEd = 12.0 kN at the top.
A first-order analysis under the wind load plus the equivalent forces, 12.85 kN in total, gives a horizontal displacement of 20 mm at the top. The roof is shallow and the compression in the rafter small, so (5.2) applies.
With more flexible columns, the same loads give a displacement of 40 mm.
In both cases the members are then checked to §6.3 for flexural and lateral-torsional buckling, with the end moments from the analysis. Because the analysis contains the global second-order effects and the sway imperfection, the buckling lengths may be taken equal to the system lengths.
How XFEM4U and XFrame2D handle it
In XFEM4U and XFrame2D, a Eurocode check for steel or timber always runs a geometrically non-linear (second-order) analysis, with the global sway imperfection of §5.3.2(3)a included automatically. XFEM4U applies it by shifting the node coordinates, once in the positive and once in the negative direction, so every ULS combination is calculated twice; XFrame2D applies the tilt in both directions of its plane. In both programs you enter m, the number of columns in a row, in the calculation settings.
The member bow imperfections of §5.3.2(3)b are not in the analysis. That is route 2: the members are checked to §6.3, and because the analysis contains the global second-order effects and the sway imperfection, the buckling length may be taken equal to the member length (§5.2.2(7)b). You do not have to derive buckling lengths from a buckling mode, and you do not have to classify the frame as sway or non-sway: if it is sensitive to second-order effects, that follows from the analysis.
Two limits. The second-order analysis applies to the beams; plates in XFEM4U are calculated geometrically linear. And the member checks still matter: a geometrically non-linear analysis without bow imperfections does not replace the buckling checks of §6.3.
Common mistakes
- One direction only. The sway imperfection applies in each relevant horizontal direction, one at a time (§5.3.2(8)). A frame that is safe leaning one way may not be safe leaning the other.
- No equivalent forces in αcr. HEd in (5.2) includes the equivalent forces of the imperfections, and δH,Ed has to come from the same loads.
- (5.2) outside its scope. It is an approximation for shallow roofs and beams without significant compression. For other frames, αcr comes from an elastic buckling analysis.
- System length after a first-order analysis. The shortcut of §5.2.2(7)b only holds when the analysis includes the global second-order effects and the sway imperfection.
- Amplifying below αcr = 3. Then (5.4) no longer applies and a more accurate second-order analysis is required.
- EN 1993-1-1:2005, Eurocode 3: Design of steel structures, Part 1-1 (in the Netherlands NEN-EN 1993-1-1+C2:2011 with its national annex): §5.2.1, §5.2.2, §5.3.2, §5.3.4, Table 5.1, §6.3.
- The XFEM4U manual and the XFrame2D manual: Calculation settings, and References, Background, Buckling.