Short Answer:

Euler’s formula is used to determine the critical buckling load at which a long, slender column will buckle under an axial compressive load. It was derived by the scientist Leonhard Euler in 1757. The formula helps predict the maximum load that a column can safely carry before it becomes unstable.

The formula is expressed as:

where is the critical buckling load, is the modulus of elasticity, is the least moment of inertia, and is the effective length of the column depending on end conditions.

Detailed Explanation :

Euler’s Formula

Euler’s formula is a fundamental concept in mechanical and structural engineering that helps determine the buckling strength of slender columns. It provides a mathematical relationship between the load applied on a column and its geometric and material properties. Buckling is a form of failure that occurs in long, thin columns subjected to compressive forces. Unlike crushing, which is caused by exceeding the material’s strength, buckling happens due to instability — the column suddenly bends or deflects sideways even though the material may not have reached its yield point.

The formula developed by Leonhard Euler predicts the critical load, also known as the Euler’s buckling load, at which the column starts to buckle. The formula is given as:

Where,

= Critical buckling load (N)
= Modulus of elasticity of the material (N/mm²)
= Least moment of inertia of the column’s cross-section (mm⁴)
= Effective length of the column (mm)

The formula assumes that the column is perfectly straight, made of a homogeneous material, and loaded axially without any eccentricity.

Concept of Effective Length

The effective length of a column depends on its end conditions. Euler showed that the ability of a column to resist buckling changes with how it is supported at both ends. The effective length is a fraction or multiple of the actual length depending on these conditions:

Both ends hinged:
Both ends fixed:
One end fixed and other free:
One end fixed and other hinged:

Columns with fixed ends have a smaller effective length and hence can carry a larger critical load before buckling, while columns with free ends are the weakest and buckle easily.

Assumptions in Euler’s Formula

Euler’s formula is based on several ideal assumptions:

The column is perfectly straight, homogeneous, and elastic.
The load is applied axially and centrally.
The material obeys Hooke’s law (stress is proportional to strain).
The column is long and slender, meaning it fails by buckling, not by crushing.
The cross-section of the column remains uniform throughout its length.

These assumptions make Euler’s theory accurate for long and slender columns but not suitable for short or intermediate columns.

Derivation Outline

Euler’s formula is derived by analyzing the deflection curve of a column under compressive load using bending theory.

Consider a column under an axial load .
As the load increases, the column deflects slightly, and bending moment at a point is .
According to bending theory, .
Combining the two gives the differential equation:

Solving this equation with boundary conditions (for example, both ends hinged) gives the critical load as:

This shows that the critical load depends on the stiffness of the material (E), the moment of inertia (I), and the square of the effective length (L_eff²).

Effect of Slenderness Ratio on Euler Load

The slenderness ratio (λ) plays a major role in determining the buckling load. It is defined as:

where is the radius of gyration.
By substituting and into Euler’s equation, we get:

From this, it is clear that as the slenderness ratio increases, the buckling load decreases. Hence, longer and thinner columns buckle more easily than short, thick ones.

Limitations of Euler’s Formula

Euler’s formula is not applicable for all types of columns. Its limitations include:

It applies only to long and slender columns where buckling dominates.
It does not consider imperfections, such as initial curvature or eccentric loading.
It assumes linear elasticity and neglects plastic deformation.
It gives inaccurate results for short columns, where crushing failure occurs before buckling.

For intermediate columns, neither crushing nor buckling alone governs the failure; hence, Rankine’s formula is used, which combines both effects.

Practical Applications

Euler’s formula is widely used in the design of columns and compression members in structures and machines. Some examples include:

Design of steel and concrete columns in buildings and bridges.
Analysis of machine frames, struts, and supports.
Design of aircraft and vehicle components where lightweight compression members are used.
Determining safe loading conditions for tall structures and towers.

By using Euler’s formula, engineers ensure that columns are designed to resist buckling safely without material failure.

Conclusion:

Euler’s formula provides a clear and mathematical way to calculate the critical buckling load of slender columns. It shows that the buckling strength depends on material stiffness, column geometry, and end conditions. Although it has some limitations for short columns, it remains one of the most important theories for understanding column stability. Proper application of Euler’s formula ensures safety, strength, and efficiency in the design of structures and mechanical components.