Columns

How a reinforced concrete column is designed to EN 1992-1-1 on the Eurocode 2 route of the STATIX RC column calculator: braced and unbraced, slenderness and its limit λ_lim, imperfections, second-order moments by the nominal curvature method, unequal end moments and biaxial bending.

1. Learning outcomes

After this lesson you can:

find the slenderness λ and the limit λ_lim with A, B and C;
add the imperfection e_i to the first-order moment;
work out the second-order moment by nominal curvature, with creep;
apply M_0e for unequal end moments;
check biaxial bending with eq 5.38 and 5.39, and read the calculator’s column results.

2. Before you start

Lesson 1: design strengths and the stress block.
The RC column calculator, set to Eurocode 2 (EN 1992-1-1).
A pocket calculator and about 60 minutes.
The same lesson on the SA code: RC Design 1, lesson 5 (SANS 10100-1).

What the calculator’s Eurocode 2 route uses for columns: the nominal curvature method of 5.8.8 (not the nominal stiffness method of 5.8.7), with K_r = 1 and K_φ from your effective creep ratio (2.0 by default); λ_lim of 5.8.3.1 with A = 1/(1 + 0.2φ_ef) but not above 0.7, B = 1.1 and C = 1.7 − r_m (0.7 unbraced); imperfections e_i = l₀/400 (θ₀ = 1/200, α_h = α_m = 1); e₀ = max(h/30, 20 mm); M_0e of 5.8.8.2(2); biaxial 5.8.9; α_cc 0.85; the recommended values elsewhere. The axial load and moments you enter are design (ULS) values.

3. Braced, unbraced and slender

Braced or unbraced. A column is braced when walls, cores or bracing carry the sideways load and stop the frame from swaying. An unbraced column must resist the sway itself, so its effective length is longer and its second-order moments larger.

The effective length l₀. It is the length of the pinned column that buckles at the same load. Stiff beams at the ends shorten it; sway lengthens it. EN 1992-1-1 5.8.3.2 gives formulas from the end stiffnesses; the calculator takes l₀ as you enter it, in each direction.

Slenderness. The Eurocode measures slenderness with the radius of gyration, so λ is about 3.5 times the l₀/h ratio of other codes. A column may be short in one direction and slender in the other.

The slenderness limit (5.8.3.1). A heavily loaded column (large n) may be less slender before the second-order effects count. The calculator works A from the creep ratio you give but never above 0.7, takes B = 1.1, and C from the end moment ratio.

End moments. A column bent into a single curve (both end moments the same way) is the worst case: r_m = 1 gives C = 0.7. In double curvature r_m is negative, C up to 2.7, and the column may count as short.

4. The design moment

Imperfections (5.2). No column is perfectly straight or plumb. The Eurocode adds an eccentricity e_i = l₀/400 to the first-order moment. In biaxial bending it is added in the more unfavourable direction only (5.8.9(2)).

Nominal curvature (5.8.8). The calculator’s method: assume the column bends to the curvature at which the steel just yields, find the deflection e₂ from it, and add M₂ = N e₂. K_r could reduce the curvature for high axial load; the page keeps it at 1.

Creep. A column under sustained load creeps and bows further. K_φ raises the curvature for it. The calculator asks for the effective creep ratio, with 2.0 as a cautious default.

Unequal end moments. M₂ is largest at mid-height, while the larger end moment is at the top or bottom. The equivalent moment M_0e is the first-order moment near mid-height; the column is designed for the larger of the end moment and M_0e + M₂.

5. The section and biaxial bending

The section. The bars are checked on the N–M interaction curve of the section, worked out by strain compatibility over the bars as placed; the design point must lie inside. Every column carries at least N e₀ (6.1(4)), and A_s ≥ max(0.10 N_Ed/f_yd, 0.002 A_c) (9.5.2).

Biaxial bending. When there are moments about both axes (including the second-order moments of a column slender both ways), the Eurocode either finds one moment dominant (eq 5.38) or checks the interaction 5.39. The calculator does both, and also sizes the bars on a single enhanced moment (a cautious practice), so its bars may be a little more than the minimum.