Column buckling and slenderness calculator

What it computes

The inputs are the width and depth of the section, the length, the end conditions (pinned at both ends, fixed and pinned, fixed at both ends, or fixed and free), the material, the axial load and a factor of safety. The sheet reports the effective length, the smallest radius of gyration, the slenderness ratio and the transition slenderness, a sentence saying whether the column is short, intermediate or slender, the Euler stress and load, the critical stress and load, the allowed load and the utilisation.

The formulas

• Effective length L_e = k L, with k = 1, 0.7, 0.5 or 2 for the four end conditions.
• Smallest radius of gyration r = b_min / sqrt(12) for a solid rectangle, where b_min is the smaller side.
• Slenderness lambda = L_e / r. Transition slenderness lambda_c = pi sqrt(2 E / f_y).
• Above lambda_c the column buckles elastically (Euler): sigma_cr = pi^2 E / lambda^2, and the Euler load is P_E = pi^2 E I_min / L_e^2.
• At or below lambda_c the parabolic (Johnson) formula applies: sigma_cr = f_y (1 - f_y lambda^2 / (4 pi^2 E)).
• Critical load P_cr = sigma_cr A, allowed load P_allow = P_cr / FS, utilisation = N / P_allow.

The two curves meet at lambda_c, where both give half the yield stress, so the critical stress never jumps. A slenderness of 50 or less is reported as short, where crushing is likely to govern.

Worked example

Defaults: a 60 mm by 100 mm steel bar (E = 200 GPa, yield stress 250 MPa), 2.4 m long and pinned at both ends, carrying 100 kN with a factor of safety of 2. Results from the engine:

The slenderness of 138.56 is above the transition value of 125.66, so the column is slender and the Euler formula governs. The allowed load of 308.43 kN is well above the 100 kN applied, and the sheet prints OK.

Change the end conditions

Fixing both ends halves the effective length to 1.2 m. The slenderness drops to 69.28, below the transition, so the sheet switches to the parabolic formula: the critical stress is 212 MPa instead of the Euler value of 411.23 MPa, which would be above the yield stress. The allowed load rises to 636.01 kN.

A column fixed at the base and free at the top, like a post or a flagpole, doubles the effective length to 4.8 m instead. The slenderness becomes 277.13, the allowed load falls to 77.11 kN, and the utilisation of 1.3 fails, so the sheet suggests a bigger section, a shorter column or fixed ends.

What this is, and is not

The sheet source

This is the template exactly as it ships. The lines with a question mark are the inputs.

# Rectangular column: slenderness and buckling

A template with inputs. Choose **Input** in the results bar to fill in the form. A solid
rectangular column of length $L$ with an effective length factor $k$ has a slenderness
$\lambda = k L / r$, where $r = b_{min}/\sqrt{12}$ is the smallest radius of gyration. Above the
transition slenderness $\lambda_c = \pi \sqrt{2E/f_y}$ the column buckles elastically (Euler),
$\sigma_{cr} = \pi^2 E / \lambda^2$. Below it the parabolic (Johnson) formula applies,
$\sigma_{cr} = f_y \left(1 - f_y \lambda^2 / (4 \pi^2 E)\right)$. The load that is allowed is
$\sigma_{cr} A$ divided by a factor of safety.

## Inputs
b:    ? 60 mm | section width
d:    ? 100 mm | section depth
L:    ? 2.4 m | column length
k:    ? [Pinned at both ends = 1, Fixed and pinned = 0.7, Fixed at both ends = 0.5, Fixed and free = 2] | end conditions
mat:  ? [Structural steel = 1, Aluminium = 2, Timber = 3] | material
N:    ? 100 kN | axial load
FS:   ? 2 | factor of safety

## Material properties
E_tab:  [200 GPa, 70 GPa, 11 GPa] | modulus of elasticity
fy_tab: [250 MPa, 160 MPa, 24 MPa] | yield strength, or crushing strength for timber
E:  E_tab.mat -> GPa
f_y: fy_tab.mat -> MPa

if mat == 1: "The material is structural steel."
if mat == 2: "The material is aluminium."
if mat == 3: "The material is timber."

## Section
A:   b * d -> cm^2 | area
b_min: min(b, d) | smaller side
I_min: max(b, d) * b_min^3 / 12 -> cm^4 | smaller second moment of area
r:   b_min / sqrt(12) -> mm | smallest radius of gyration

## Slenderness
L_e:      k * L -> m | effective length
lambda:   L_e / r | slenderness ratio
lambda_c: π * sqrt(2 * E / f_y) | transition slenderness

if lambda <= 50:
    The column is short: crushing is likely to govern rather than buckling.
else if lambda <= lambda_c:
    The column is intermediate: it buckles inelastically and the parabolic formula applies.
else:
    The column is slender: it buckles elastically and the Euler formula applies.

## Buckling resistance
sigma_E: π^2 * E / lambda^2 -> MPa | elastic (Euler) buckling stress
P_E:     π^2 * E * I_min / L_e^2 -> kN | elastic (Euler) buckling load

if lambda <= lambda_c:
    sigma_cr: f_y * (1 - f_y * lambda^2 / (4 * π^2 * E)) -> MPa | critical stress, parabolic formula
else:
    sigma_cr: sigma_E | critical stress, Euler formula

P_cr:    sigma_cr * A -> kN | critical load
P_allow: P_cr / FS -> kN | allowed load
util:    N / P_allow | utilisation

if util <= 1: "OK: the applied load is within the allowed load." else: "Fails: the applied load exceeds the allowed load. Enlarge the section, shorten the column or fix its ends."

How to use it

1. Open the template. Choose Open this template on this page. The sheet opens in a new tab of CivilKit Calcs, with the inputs set to the values used on this page.
2. Switch to Input mode. In the results bar choose Input. The input lines become boxes, lists, buttons and ticks.
3. Enter your values. Change the inputs to your own. Size, length, end conditions, material, axial load and factor of safety are the inputs. Every change recalculates the whole page.
4. Read and export the result. Read the results and the OK or not OK lines under them. Use Report, then Print PDF, Word or Excel, to keep the calculation with its workings.