Buckling behaviour of columns

Buckling of columns

A column under axial compression stays straight until a critical load is reached; beyond this it suddenly bends sideways (buckles) with a large lateral deflection, even though the stress may be well below the yield stress. Buckling is a loss of stability, not a material failure.

Euler's behaviour of an ideal (perfect) column

Euler's critical load for a perfectly straight, elastic, pin-ended column:

$$P_{cr}=\frac{\pi^2EI}{(KL)^2},\qquad f_{cc}=\frac{\pi^2E}{\lambda^2},\ \ \lambda=\frac{KL}{r}$$

• The load is directly proportional to $EI$ and inversely proportional to the square of the effective length $KL$.
• A slender column (large $\lambda$) buckles elastically at $f_{cc}<f_y$; a stocky column (small $\lambda$) yields before it buckles.
• Buckling occurs about the axis of least resistance (largest slenderness).

 stress          Euler hyperbola
  fy |--------.
     |         \.
     |   real    \.   Euler curve
     |   curve ..  `---.____  fcc = pi^2 E/lambda^2
     +----------------------------> lambda

Behaviour of real columns

Real columns differ from Euler's ideal column because of:

1. Initial crookedness (out-of-straightness), usually about $L/1000$.
2. Eccentricity of load and non-uniform end restraint.
3. Residual stresses from rolling/welding, which cause early yielding of parts of the section.
4. Material non-linearity: yielding in the inelastic range at intermediate slenderness.
5. Local buckling of thin flanges or webs, and torsional / flexural-torsional buckling of open sections (angles, channels, tees).

For intermediate $\lambda$ the strength falls below both the yield load and the Euler load, so a smooth curve is used (Perry-Robertson):

$$f_{cd}=\frac{f_y/\gamma_{m0}}{\phi+\sqrt{\phi^2-\lambda_n^2}},\quad \phi=0.5[1+\alpha(\lambda_n-0.2)+\lambda_n^2],\ \ \lambda_n=\sqrt{f_y/f_{cc}}$$

The imperfection factor $\alpha$ depends on the buckling class: a (0.21), b (0.34), c (0.49), d (0.76), chosen from IS 800:2007 Table 10 according to the section type, axis and flange thickness.

Effective length

Support conditions change the buckled shape (IS 800:2007 Table 11, theoretical value with recommended design value in brackets): pinned-pinned $K=1.0$ (1.0); fixed-fixed $0.5$ (0.65); fixed-pinned $0.7$ (0.8); fixed-free $2.0$ (2.0). A column that is free to sway has a larger $KL$ (e.g. both ends fixed in rotation but free to sway: 1.0, design 1.2).

Remedies

Reduce $KL$ (bracing, intermediate restraints), increase $r$ (use a deeper, hollow or built-up section), use stocky compact sections, and provide stiff end connections.