Columns: short and slender

Columns carry the building down to its foundations. This lesson shows how a column carries axial load and bending together, when its height starts to matter, and how to design one to SANS 10100-1, by hand and in STATIX.

1. Learning outcomes

After this lesson you can:

explain how a column carries N and M, and what braced and unbraced mean;
decide whether a column is short or slender from l_e/h;
work out the design moment, with the minimum eccentricity and, for slender columns, the additional moment;
check a section with strain compatibility and read an N-M interaction diagram;
choose the bars and the links.

2. Before you start

Lesson 1: the stress block and strain compatibility.
The free STATIX RC column calculator, open in a second tab.
A pocket calculator and about 60 minutes.

3. What a column does

A column collects the floor loads above it and carries them down as an axial force N. The beams framing into it also bend it, so a column always carries N with a moment M.

Even a column with no beam moment is designed for some bending, because no column is perfectly straight and no load is perfectly central. SANS 10100-1 sets a minimum eccentricity (4.7.2.3): the moment is at least N × e_min, with e_min = 0.05h but not more than 20 mm.

4. Short or slender?

What happens as a column gets taller for its size. A short column fails when the concrete and steel crush. A slender column bows sideways; the axial load then acts at a lever arm a and adds a moment, so it fails sooner.

Whether a column bows depends on how tall it is compared to its depth, and on how its ends are held. Two ideas help: bracing and effective height.

Braced or unbraced (SANS 10100-1, 4.7.1). In a braced frame, walls or bracing stop the floors moving sideways, so the columns only carry vertical load and their end moments. In an unbraced frame the columns themselves resist sway, which makes slenderness more serious.

Effective height l_e: the length of the buckled shape between its points of no bending. Ends held stiffly shorten it; a free top makes it much longer than the storey. In STATIX you type l_e directly.

Short or slender (SANS 10100-1, 4.7.1.4)? Work it out about each axis. For a braced column bent in single curvature with equal end moments (M1/M2 = 1) the limit is 17 − 7 = 10; double curvature raises it.

5. The additional moment of a slender column

The additional moment (SANS 10100-1, 4.7.3). The code estimates the bow a_u from the slenderness and adds N × a_u to the design moment. STATIX takes the reduction factor K as 1, which is on the safe side.

End moments. When both ends bend the column the same way it bows into one big curve (single curvature, M1/M2 positive). When they bend it opposite ways it forms an S (double curvature, M1/M2 negative) and bows much less. For braced slender columns SANS uses M_i = 0.4 M1 + 0.6 M2 in the middle (4.7.3.2).

a_u = (1/2000) × (l_e/h)² × h (K = 1) M_add = N × a_u braced: M_i = 0.4 M1 + 0.6 M2; M = the larger of M2 and M_i + M_add

M2 is the larger end moment, first raised to at least N e_min. This is how STATIX applies 4.7.3; the code also allows a factor K below 1 when the axial load is high, which STATIX does not take (it is on the safe side).

6. Checking the section

How a column section is checked: strain compatibility, the same three steps as the beam in lesson 1. Pick a neutral axis depth x; draw the strains; turn them into stresses (concrete block, steel up to its limit: 0.87 f_y in tension and f_yc = f_y/(1.15 + f_y/2000) in compression, as STATIX uses); add the forces to get N and take moments to get M. Move x until N matches the load.

The N-M interaction diagram of one section: every point on the curve is a combination the column can just carry. High N with little M fails by crushing; low N with large M fails like a beam; the nose of the curve is the balance point. A design load (N, M) inside the curve passes. Shape only, not a real column.

For a column with no moment at all, the squash load is simple (STATIX):

N_uz = 0.45 f_cu (A_c − A_sc) + 0.75 f_y A_sc

With moment, you need the whole curve. By hand you check one bar layout at the design N: find x so the forces add up to N, then compare the section's moment with the design moment. That is what the worked examples below do.

Biaxial bending. When a column is bent about both axes, SANS 10100-1 lets you design for one enhanced moment about the more critical axis (shown here for Mx/h ≥ My/b). STATIX does this for you, per load case.

7. Bars and links

Links in columns (SANS 10100-1, 4.11). Compression bars buckle outwards unless a link holds them. Every corner bar and every alternate bar needs a link corner or a tie; a bar may be unheld only if it is within 150 mm clear of a held bar. Links are at least a quarter of the largest bar (and 6 mm), spaced at most 12 times the smallest bar.

Longitudinal steel between 0.4% and 6% of the section (STATIX; laps may need less than 6% at the lap).
At least four bars in a rectangular column, 12 mm or larger.
Links at least ¼ of the largest bar and 6 mm; spacing at most 12 times the smallest bar.