Short Answer:

Euler’s formula for buckling load gives the critical load at which a long, slender column just begins to buckle under axial compression. It was derived by the mathematician Leonhard Euler and applies to perfectly straight, elastic columns with pinned or fixed end conditions.

Mathematically, Euler’s buckling load () is expressed as:

where  is the modulus of elasticity,  is the least moment of inertia of the column section, and  is the effective length of the column, depending on its end conditions.

Detailed Explanation :

Euler’s Formula for Buckling Load

When a long column is subjected to an axial compressive load, it remains straight up to a certain limit. Beyond this limit, even a small increase in load causes the column to bend or deflect sideways. This sudden bending is known as buckling, and the load at which it occurs is called the critical buckling load.

Euler’s formula defines this critical load mathematically and is used to predict the strength and stability of long, slender columns. It assumes that the material remains elastic, the load is applied perfectly axially, and the column has uniform cross-section and homogeneous material.

Derivation of Euler’s Formula

Consider a long, slender column of length , with both ends hinged, and subjected to an axial compressive load .
When the column bends, the deflection at a distance  from one end is denoted by .

From the bending theory,

where  is the bending moment at a section.

For a column under an axial compressive load :

Substituting this into the bending equation:

This is a second-order differential equation representing the deflection curve of the column.

Solving the Equation:

Dividing both sides by :

Let,

Then,

The general solution of this equation is:

where  and  are constants determined from boundary conditions.

Applying Boundary Conditions

For a column with both ends hinged:

• At ,  →
• At ,  →

For a non-trivial solution ():

 

where

For the first buckling mode, :

Substituting :

Thus, the Euler’s critical buckling load is:

This formula gives the minimum load at which the column becomes unstable and buckles.

Effective Length for Different End Conditions

The value of  (effective length) varies with support conditions because different end constraints change the column’s buckling tendency.

End Condition	Effective Length ()	Critical Load ()
Both ends hinged
Both ends fixed
One end fixed, other free
One end fixed, other hinged

From this table, it is clear that columns with fixed supports can carry much higher loads than those with free or hinged ends.

Important Observations

1. Dependence on Slenderness Ratio:
   The slenderness ratio () determines whether the column will fail by buckling or crushing.
   Substituting  in Euler’s formula gives:

Hence,  decreases rapidly as the slenderness ratio increases.

1. Applicable for Long Columns Only:
   Euler’s formula is valid for long and slender columns, where failure occurs by buckling. It is not applicable for short columns that fail by crushing.
2. Effect of Material Property:
   A higher modulus of elasticity (E) increases the critical load, meaning stiffer materials can resist buckling better.
3. Effect of Cross-Section Shape:
   Columns with a large moment of inertia (I) about the buckling axis are more stable and can carry higher loads.

Limitations of Euler’s Formula

1. Applicable only to long, slender columns (high slenderness ratio).
2. Assumes perfect straightness and central loading (no eccentricity).
3. Material must remain within elastic limits.
4. Does not consider imperfections or residual stresses.
5. Not suitable for intermediate columns, where empirical formulas like Rankine’s are preferred.

Practical Example

A steel column has , , and .
For both ends hinged:

 

Hence, the critical load is 17.55 MN, meaning the column will start to buckle if the load exceeds this value.

Importance of Euler’s Formula

• Helps in predicting buckling load and ensuring safety in design.
• Provides a mathematical relationship between stiffness, geometry, and stability.
• Serves as the foundation for modern column design theories.
• Ensures that compression members are designed with adequate stability margins to prevent sudden failure.

Conclusion

Euler’s formula for buckling load provides the critical axial load at which a column becomes unstable and buckles. It is given by

This formula highlights the influence of material stiffness, cross-sectional geometry, and support conditions on a column’s stability. Although it applies mainly to long, slender columns, it remains a fundamental concept in strength of materials, ensuring safe and efficient design of compression members in mechanical and structural systems.