Short Answer:

Rankine’s formula is an empirical formula used to find the safe load or critical load of a column by combining both crushing and buckling effects. It is used for columns of all lengths—short, medium, and long—because it gives more accurate results than Euler’s formula for real practical conditions.

This formula was developed by W.J.M. Rankine and is a combination of Euler’s theory (for long columns) and crushing strength (for short columns). It gives a better estimate of column strength by considering both material failure and instability due to buckling.

Detailed Explanation :

Rankine’s Formula

Rankine’s formula is a semi-empirical relation used to calculate the ultimate load or safe load that a column can carry without failure. It was proposed to overcome the limitations of Euler’s formula, which is valid only for long, slender columns. In real-life applications, columns are not perfectly straight, and the material may not behave ideally elastic. Hence, Rankine’s formula provides a more realistic approach by combining the effects of both crushing and buckling.

Expression of Rankine’s Formula

The general form of Rankine’s formula is:

Where,

= Crushing load or actual safe load on the column
= Crushing stress (compressive strength of material)
= Cross-sectional area of the column
= Euler’s critical buckling load

From Euler’s theory,

Substituting this value into Rankine’s formula gives:

Where,

= Rankine’s constant (depends on material)
= Effective length of the column
= Radius of gyration

This is the Rankine–Gordon formula commonly used in engineering design.

Meaning of the Formula

The Rankine formula shows that the load-carrying capacity of a column depends on both its crushing strength (material property) and slenderness ratio (geometric property).

For short columns, the term is small, so the denominator becomes nearly 1, and the load . Hence, the failure occurs mainly due to crushing.
For long columns, the term is large, so the denominator increases, reducing the value of . In this case, failure occurs due to buckling.

Thus, Rankine’s formula provides a smooth transition between short and long column behavior.

Values of Rankine’s Constant (a)

The value of Rankine’s constant ‘a’ varies for different materials and is determined experimentally. Common values are:

Mild Steel:
Wrought Iron:
Cast Iron:
Timber:

These constants help engineers to calculate safe loads accurately for specific materials.

Advantages of Rankine’s Formula

Applicable to all types of columns:
Rankine’s formula works for short, medium, and long columns, unlike Euler’s formula which is valid only for long columns.
Considers both failure modes:
It combines the crushing strength (material property) and buckling load (geometric property), giving more practical results.
Accurate for real structures:
Since it accounts for imperfections, eccentricity, and material irregularities, it is used in real-world design applications.
Easy to apply:
The formula is simple and can be used for different materials by adjusting the constant ‘a’.

Limitations of Rankine’s Formula

It assumes a perfect column shape and uniform material, which may not always be true in real life.
The constant ‘a’ must be determined experimentally for each material.
It does not consider the effects of eccentric loading or lateral forces.
It gives approximate results and may not be accurate for columns with complex end conditions or variable cross-sections.

Comparison with Euler’s Formula

Feature	Euler’s Formula	Rankine’s Formula
Type of Column	Long and slender	All types (short, medium, long)
Basis	Theoretical (buckling only)	Empirical (buckling + crushing)
Accuracy	Less for real columns	More accurate for practical use
Failure Mode	Buckling	Buckling and crushing both
Use	Ideal conditions	Real engineering conditions

From this comparison, it is clear that Rankine’s formula is more versatile and practical than Euler’s theory for design purposes.

Example Application

Suppose a mild steel column has the following properties:

Then using Rankine’s formula:

This shows that the actual load capacity reduces as the slenderness ratio increases.

Conclusion

Rankine’s formula is a very useful empirical relation for determining the safe load of columns under compression. It effectively combines the results of Euler’s theory and crushing strength to predict the failure of columns of any length. By including both material and geometric effects, it provides more reliable and accurate results for real engineering structures like pillars, struts, and machine frames.