This calculator is used to calculate the area moment of inertia, or second moment of area, for a range of different beam cross-sections and bending axes. The area moment of inertia is a parameter that defines how much resistance the cross-section of a beam has to bending because of its geometry.

Circular Cross-Section | Bending axes pass through centroid

$I_{xx} = I_{yy} = \frac{\pi r^4}{4}$

Radius r:

Input units: mmcmmin

Calculate

Second moment of area I_xx: Second moment of area I_yy:

Output units: mm⁴cm⁴m⁴in⁴

Hollow Circular Cross-Section | Bending axes pass through centroid

$I_{xx} = I_{yy} = \frac{\pi}{4} ({r}^4 - (r - t)^4)$

Radius r: Thickness t:

Input units: mmcmmin

Calculate

Second moment of area I_xx: Second moment of area I_yy:

Output units: mm⁴cm⁴m⁴in⁴

Rectangular Cross-Section | Bending axes pass through centroid

$I_{xx} = \frac{bh^3}{12}$ $I_{yy} = \frac{b^3h}{12}$

Height h: Width b:

Input units: mmcmmin

Calculate

Second moment of area I_xx: Second moment of area I_yy:

Output units: mm⁴cm⁴m⁴in⁴

I-Beam Cross-Section | Bending axes pass through centroid

$I_{xx} = \frac{b(d+2t)^3}{12}-\frac{(b-t_w)d^3}{12}$ $I_{yy} = \frac{b^3t}{6}+\frac{{t_w}^3d}{12}$

Width b: Inner Height d: Flange Thickness t: Web Thickness t_w:

Input units: mmcmmin

Calculate

Second moment of area I_xx: Second moment of area I_yy:

Output units: mm⁴cm⁴m⁴in⁴

Remember that the parallel axis theorem can be used to calculate $I$ for different bending axes. It states that the area moment of inertia about any axis $I_x$ can be calculated from the area moment of inertia about a parallel axis $I_{xc}$ that passes through the centroid of the cross-section, the area of the cross-section $A$ and the distance between the two axes $d$. Have a read of our page on the area moment of inertia to learn more about how the parallel axis theorem works.

Related Topics

Area Moment of Inertia

The area moment of inertia describes how the material of a cross-section is distributed relative to a specific bending axis.

Learn more

Buckling

Buckling is a sudden deflection that occurs in slender columns or members that are loaded in compression.

Learn more

This calculator is used to calculate the critical buckling force, the force at which a column will begin to buckle. You can play around with it to see how the different input parameters like Young’s modulus, second moment of area, column length and the column end conditions all affect how much force a column can carry.

Critical Buckling Force Calculator

Enter your cross-section properties, material, column length and select your end conditions to calculate the critical buckling force for your column.

Cross-section Hollow CircleCircleRectangleSquareCustom (Enter I)

Radius r (in mm):

Thickness t (in mm)

Area Moment of Inertia I (in mm⁴):

Material SteelAluminumWoodCustom (Enter E)

Modulus of Elasticity E (in GPa):

Column Length L (in m)

End Conditions

Fixed-Fixed

Calculate

Critical Buckling Force

The critical buckling force is calculated using Euler’s buckling formula, shown below. This equation and the calculator above do not include any safety factors. Have a read of our page on buckling to learn more about this equation and the buckling of columns.

Related Topics

Buckling

Buckling is a sudden deflection that occurs in slender columns or members that are loaded in compression.

Learn more

Torsion

Torsion is the twisting of an object caused by a moment applied about a longitudinal axis (torque).

Learn more

Buckling is an important topic that any engineer designing structures that carry compressive loads must understand.

This page will guide you through the basics of buckling, from Euler’s formula for predicting the onset of buckling to more complex topics like slenderness ratios and inelastic buckling. So keep reading, or watch the animated video below, to get up to speed.

What is Buckling?

Buckling is the sudden deformation of a structural member that is loaded in compression, that occurs when the compressive load in the member reaches a critical value.

Buckling often occurs suddenly, and can produce large displacements. This doesn’t always result in yielding or fracture of the material, but buckling is still considered to be a failure mode since the buckled structure can no longer support a load in the way it was originally intended to.

Any long structural member that is loaded in compression is at risk of failing due to buckling. Columns are common examples of structures that may fail by buckling. Individual members in trusses are frequently loaded in compression, so trusses are another example of a structure at risk of failure due to buckling.

Euler’s Buckling Formula

Euler’s buckling formula is a simple equation that is used to calculate the axial load $P_{cr}$ at which a column or beam will buckle. At the critical buckling load any small perturbation, whether it’s a lateral force or a small imperfection in the column geometry, will cause the column to buckle.

The critical load depends on only three parameters, the Young’s modulus $E$ of the column material, the second moment of area $I$ of the column cross-section, and the effective length $L_e$ of the column.

You can use our critical buckling force calculator to see how the different input parameters affect how much force a column can carry.

The figure below illustrates how a column with a critical buckling load of 1000 kN will buckle.

Effective Length and End Conditions

The way the ends of a column or beam are restrained will affect the critical buckling load. A column that is fixed at one end and free at the other will clearly only be able to support a much smaller load before buckling compared to one that is pinned at both ends. The buckled shapes are also quite different. 

Euler’s formula accounts for different end conditions using the effective length parameter $L_e$.

A column that is pinned at both ends has an effective length that is equal to the length of the column $L$. Other end conditions result in an effective length that is equal to the length $L$ multiplied by a specific factor. These factors are shown in the table below for a few different end conditions.

End Condition
Effective Length	$L_e = 0.5L$	$L_e = L$	$L_e = 2L$	$L_e = 0.7L$

The Efficient Engineer Summary Sheets

The Efficient Engineer summary sheets are designed to present all of the key information you need to know about a particular topic on a single page. It doesn’t get more efficient than that!

Get The Summary Sheets!

Slenderness Ratio

Even without knowing anything about Euler’s formula it’s intuitively quite obvious that slender columns are at much greater risk of buckling than stocky ones. This is why members of a truss that are in compression are sometimes designed to be thicker than those in tension, and why bracing members are used to prevent buckling of long compressive members.

Euler’s formula confirms this intuition – the length term in the equation is squared, and as a result doubling the length of a column means that it can only support a quarter of the load before buckling.

To better understand the effect of slenderness it’s useful to introduce a non-dimensional parameter called the slenderness ratio, that defines how slender the column is (i.e. how thin it is compared to its length).

The $r$ term is the radius of gyration of the column and is defined as the square root of the second moment of area $I$ divided by the cross-sectional area $A$.

$$r = \sqrt{\frac{I}{A}}$$

If we divide the Euler critical buckling force by the column cross-sectional area we obtain the Euler critical buckling stress $\sigma_{cr}$, which is the normal stress in the column at which buckling will occur.

$$\sigma_{cr} = \frac{P_{cr}}{A}$$

$$\sigma_{cr} = \frac{\pi ^{2}EI}{A{L_e}^2}$$

By introducing the slenderness ratio we obtain this form of the equation for $\sigma_{cr}$:

$$\sigma_{cr} = \frac{\pi ^{2}E}{({L_{e}}/r)^{2}}$$

As mentioned earlier it’s intuitive that the slenderness of a column will significantly affect how much load it can support before buckling, so it’s interesting to take a look at how the Euler critical buckling stress varies with the slenderness ratio – this is shown in the graph below.

Very slender columns have a large slenderness ratio and a very low critical buckling stress. For stocky columns with low slenderness ratios the critical buckling stress will be very large.

The dashed horizontal line on the graph above shows the compressive yield strength of the column material, $\sigma_{y}$. For very low slenderness ratios the strength of the material will be exceeded before the buckling limit is reached. This means we can define two distinct regions, one where columns fail by crushing because the stress in the column exceeds the material yield strength (i.e. for stocky columns), and one where they fail due to buckling (i.e. for slender columns).

This curve only represents the theoretical behaviour of columns. If we plot buckling stresses determined experimentally for real columns we can see it doesn’t exactly match the theoretical behaviour – this is shown in the graph below. In particular the transition between plastic failure (crushing) and elastic failure (buckling) is much more gradual. This is because for columns in this transition range, buckling is actually a complex combination of these two failure modes. Buckling in this transition region is called inelastic buckling, and it should be modelled using methods like Engesser’s theory or Shanley’s theory, rather than Euler’s formula.

Many industry design codes include curves similar to the ones shown above that can be used for the design of members loaded in compression.

Related Topics

Torsion

Torsion is the twisting of an object caused by a moment applied about a longitudinal axis (torque).

Learn more

The Finite Element Method

The finite element method is a numerical method used to approximate the solution to complex problems governed by differential equations.

Learn more