Columns & Struts Notes

Last reviewed 16 Sept 2026 · 7 min read

Columns, struts and modes of failure

A column is a vertical compression member; a strut is a compression member in any direction (e.g. a member of a roof truss). A post or pillar is a column in a building.

How a compression member fails depends on its slenderness:

Type	Behaviour	Failure load
Short column (stocky)	Fails by crushing of the material	$P_{c} = f_{c} A$
Long column (slender)	Fails by buckling (sudden lateral bending) at a stress well below the crushing stress	Euler's load $P_{e}$
Intermediate column	Fails by a combination of crushing and buckling	Rankine–Gordon or empirical formulas

Slenderness ratio and radius of gyration

The least radius of gyration of the section is

k = r = \frac{I _{min}}{A}

The slenderness ratio is

λ = \frac{L _{e}}{k}

where $L_{e}$ is the effective length. A column always buckles about the axis with the least moment of inertia (largest slenderness), unless its restraints differ about the two axes.

Section	Least radius of gyration $k$
Solid circle, diameter $d$	$d /4$
Hollow circle, $D$ and $d$	$D^{2} + d^{2} /4$
Rectangle $b \times d$ ( $b < d$ )	$b / 12 = 0.289 b$
Square of side $a$	$a / 12$

Euler's theory of long columns

Assumptions

The column is initially perfectly straight and the load is exactly axial.
The material is homogeneous, isotropic and obeys Hooke's law (elastic buckling).
The cross-section is uniform along the length.
Self-weight is neglected; failure is by buckling alone.
The column's length is very large compared with its lateral dimensions.
Ends are frictionless (for pinned ends).

Derivation for both ends pinned

At a section distance $x$ from one end with lateral deflection $y$ , the moment is $M = - P y$ :

E I \frac{d ^{2} y}{d x ^{2}} + P y = 0 \Rightarrow y = C_{1} cos (\frac{P}{E I} x) + C_{2} sin (\frac{P}{E I} x)

$y (0) = 0$ gives $C_{1} = 0$ . $y (L) = 0$ with $C_{2} \neq = 0$ requires $P / E I L = nπ$ . The smallest non-zero load ( $n = 1$ ) is the critical load:

FormulaEuler's buckling load

P_{e} = \frac{π ^{2} E I _{min}}{L _{e}^{2}}, σ_{e} = \frac{P _{e}}{A} = \frac{π ^{2} E}{( L _{e} / k ) ^{2}}

The Euler stress depends only on $E$ and the slenderness ratio — not on the strength of the material. A high-strength steel buckles at the same load as mild steel of the same $E$ and geometry.

Effective length for end conditions

The effective length is the length of an equivalent pin-ended column with the same buckling load — the distance between points of zero moment (inflexion points) in the buckled shape.

End conditions	Theoretical $L_{e}$	Euler load	Relative strength
Both ends hinged (pinned)	$L$	$π^{2} E I / L^{2}$	1
One end fixed, other free	$2 L$	$π^{2} E I /4 L^{2}$	1/4
One end fixed, other hinged	$L / 2 \approx 0.707 L$	$2 π^{2} E I / L^{2}$	2
Both ends fixed	$L /2$	$4 π^{2} E I / L^{2}$	4

Code ProvisionIS 456 — recommended effective lengths

IS 456 gives design values of effective length that are larger than the theoretical ones for fixed conditions, because real fixity is never perfect. For example: both ends fixed — theoretical $0.50 L$ , recommended $0.65 L$ ; one end fixed and the other hinged — theoretical $0.70 L$ , recommended $0.80 L$ ; both ends hinged — $1.00 L$ ; one end fixed, other free — theoretical $2.0 L$ , recommended $2.0 L$ . IS 800 gives similar recommended values for steel members.

Exam TipRatio questions

For the same column, Euler loads for (fixed-free) : (pinned-pinned) : (fixed-pinned) : (fixed-fixed) $= \frac{1}{4} : 1 : 2 : 4$ .

Limitation of Euler's formula

Euler's formula is valid only when buckling occurs within the elastic range, i.e. $σ_{e} \leq$ proportional limit (approximately the yield or crushing stress $f_{c}$ ). The limiting slenderness ratio is

(\frac{L _{e}}{k})_{l im} = π \frac{E}{f _{c}}

For mild steel with $E$ = 200 GPa and $f_{c}$ = 320 N/mm² (a value often used in textbooks), $λ_{l im} = π 200 000/320 \approx 80$ . Hence the textbook statement: Euler's formula applies to mild-steel pin-ended columns with slenderness ratio greater than about 80. For short columns it predicts loads larger than the crushing load, which is meaningless.

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Columns & Struts

Columns, struts and modes of failure

Slenderness ratio and radius of gyration

Euler's theory of long columns

Assumptions

Derivation for both ends pinned

Effective length for end conditions

Limitation of Euler's formula

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