--------|-----------|-------| | Pinned base + simple top connection | 1.0 | Conservative default | | Pinned base + partial fixity top (eta_1 = 0.3) | 0.85-0.90 | Flush end plates | | Rigid base + simple top connection | 0.75-0.85 | Ground floor column | | Rigid base + moment connection top | 0.65-0.80 | Portal frame column (braced in-plane) |

Unbraced Frames (Sway):

In the majority of UK multi-storey buildings, lateral stability is provided by a braced core or bracing system. Columns are therefore classified as braced (non-sway), and k = 1.0 is the standard assumption. Only where the column participates in the lateral load-resisting system (moment frames, portal frames) does the unbraced classification apply.

Worked Example -- Braced Frame Column

A 254 x 254 x 89 UC column in a braced multi-storey frame. The column passes continuously through the floor, with 533 x 210 x 92 UB beams framing into the column at each floor level from both sides.

Given:

• Column: I_c = 14,310 cm^4, L_c = 4.0 m, K_c = 14,310/400 = 35.8 cm^3
• Beams (two each side, total four at each joint): I_b = 65,750 cm^4 each, L_b = 6.0 m each
• Beam far-end condition: continuous beam with moment connection, kappa = 1.0

At the top of the column (eta_1): Effective beam stiffness at top joint: K_b_total = 2 x 1.0 x 65,750/600 + 2 x 1.0 x 65,750/600 = 4 x 109.6 = 438.4 cm^3

eta_1 = K_c / (K_c + K_b_total) = 35.8 / (35.8 + 438.4) = 35.8 / 474.2 = 0.075

This very low eta_1 indicates that the beams dominate the joint stiffness -- the column top is nearly fully restrained.

At the bottom of the column (eta_2): Same arrangement, assuming column continues to storey below and beam arrangement is identical. eta_2 = 0.075

Effective length factor: k = [1 - 0.2(0.075 + 0.075) - 0.12 x 0.075^2] / [1 - 0.8(0.075 + 0.075) + 0.6 x 0.075^2] = [1 - 0.03 - 0.0007] / [1 - 0.12 + 0.0034] = 0.969 / 0.883 = 1.098

k = 1.10 -- slightly above 1.0 because the column stiffness is not negligible relative to the beam stiffness. Lcr = 1.10 x 4.0 = 4.4 m.

For design, using k = 1.0 would give Lcr = 4.0 m, a 10% underestimate. The Annex E calculation reveals the need for a slightly longer effective length.

Worked Example -- Sway Frame Column

The same 254UC column but in an unbraced frame where the column participates in the lateral load-resisting system.

eta_1 = eta_2 = 0.075 (same beam-to-column stiffness ratio)

Effective length factor (sway): k = sqrt[0.969 / 0.883] = sqrt[1.098] = 1.048

In the sway case, k is only slightly above 1.0 because the beam stiffness is very high relative to the column. If the beam stiffness were lower (e.g., 305UB instead of 533UB), eta would increase and k would rise significantly.

For a sway frame column with K_b reduced to one quarter (use 305 x 165 x 40 UB, I_b = 8,500 cm^4, K_b = 8,500/600 = 14.2 cm^3): K_b_total = 4 x 14.2 = 56.8 cm^3 eta = 35.8 / (35.8 + 56.8) = 0.387 k = sqrt[(1 - 0.155 - 0.018) / (1 - 0.620 + 0.090)] = sqrt[0.827/0.470] = 1.33

With the lighter beams, k increases to 1.33 -- a 33% increase in effective length, highlighting the importance of beam sizing for sway frame stability.

UK National Annex Guidance

The UK NA to BS EN 1993-1-1 confirms Annex E without modification and adds the following guidance:

1. Braced frame classification: A frame may be classified as braced (non-sway) only if the bracing system reduces the horizontal displacement by at least 80% compared with the unbraced frame. If this criterion is not satisfied, the frame must be classified as unbraced.

2. Simple connections: The UK NA confirms that nominally pinned connections (fin plates, partial-depth end plates, web cleats) provide negligible rotational restraint. For columns with simple connections at both ends, k = 1.0 must be used.

3. Column bases: Nominally pinned base plates (typical UK detail with two or four holding-down bolts inside the column footprint) do not provide significant rotational restraint. For braced frames with pinned bases, eta_2 = 1.0 at the base.

4. Continuous columns: For columns continuous through a floor level, the column stiffness K_c should be calculated using the full column length between lateral restraints, not the individual storey height, because the column is continuous at the floor.

Design Resources

• UK Column Buckling Reference -- Buckling curve methodology
• UK Steel Properties -- fy, fu tables
• UK UC and UB Section Properties -- I, i values for stiffness
• UK Combined Loading Design -- Beam-column interaction
• All UK Steel Design References -- complete library

Frequently Asked Questions

What effective length factor should I use for a braced frame column with simple connections?

Use k = 1.0 (Lcr = storey height) for braced frame columns with nominally pinned connections at both ends. This is the standard conservative assumption in UK practice and is confirmed by the UK NA. The alignment chart with typical K_c/K_b ratios (K_c << K_b for simple connections with stiff beams) gives k approaching 1.0 from below, so k = 1.0 provides a small but safe margin.

How does a pinned base affect the effective length of a braced frame column?

A pinned base (eta = 1.0 at the base) increases k by approximately 10-15% compared with a rigid base. For a typical braced frame column with eta_1 = 0.2 (top) and eta_2 = 1.0 (pinned base): k_braced approximately (1 - 0.24 - 0.024)/(1 - 0.96 + 0.12) = 0.736/0.16 = 4.6 -- no, this is wrong. Let me recalculate: with eta_1 = 0.2, eta_2 = 1.0: k = [1 - 0.2(1.2) - 0.12(0.2)]/[1 - 0.8(1.2) + 0.6(0.2)] = [1 - 0.24 - 0.024]/[1 - 0.96 + 0.12] = 0.736/0.16 = 4.6. This shows that a pinned base with a stiff top gives a very high k in a sway frame, but in a braced frame, the k is bounded at 1.0 because the lateral restraint prevents the amplification. For braced frames, k ranges from 0.5 to 1.0 regardless of the individual eta values.

When does a frame qualify as 'braced' under the UK NA?

A frame qualifies as braced if the bracing system reduces the horizontal displacement by at least 80%, per the UK NA to BS EN 1993-1-1 Clause 5.2.1. In practice, this means the lateral stiffness of the bracing system (concrete core, steel braced bay, or shear wall) must be at least 5 times the lateral stiffness of the frame acting alone. If the frame provides more than 20% of the total lateral stiffness, it must be classified as unbraced and Annex E sway k factors apply.

Educational reference only. All design values are per BS EN 1993-1-1:2005 + UK National Annex and BS EN 10025-2:2019. Verify all values against the current editions of the standards and the applicable National Annex for your project jurisdiction. Designs must be independently verified by a Chartered Structural Engineer registered with the Institution of Structural Engineers (IStructE) or the Institution of Civil Engineers (ICE). Results are PRELIMINARY -- NOT FOR CONSTRUCTION without independent professional verification.