Short Answer:

Rankine’s formula is an empirical equation used to determine the crushing or buckling load of columns, whether they are short, medium, or long. It combines the effects of both Euler’s buckling theory (for long columns) and direct crushing stress (for short columns) to give a more accurate and practical value of load at which the column will fail.

In simple terms, Rankine’s formula provides a single mathematical relationship that works for all types of columns, unlike Euler’s formula, which applies only to long and slender ones. It helps engineers design safe columns by considering both crushing and buckling effects together.

Detailed Explanation :

Rankine’s Formula

The Rankine’s formula (also known as Rankine–Gordon formula) is used to calculate the ultimate load-carrying capacity of a column under compression. It bridges the gap between Euler’s theory for long columns and crushing theory for short columns.

Euler’s formula applies only when columns are long and slender, failing by buckling, while short columns fail by crushing due to direct compression. For medium columns, both effects occur simultaneously, and Rankine’s formula successfully combines both failure modes to provide accurate results.

Expression for Rankine’s Formula

The Rankine’s formula for the crushing (or buckling) load is given as:

Where,

•  = Crippling or failure load on the column (N)
•  = Crushing strength of the material (N/mm²)
•  = Cross-sectional area of the column (mm²)
•  = Effective length of the column (mm)
•  = Least radius of gyration (mm)
•  = Constant depending on the material and end conditions

The term  is known as the slenderness ratio, and it plays a key role in determining whether the column will fail by crushing or buckling.

Derivation (Conceptual Form)

Rankine’s formula assumes that the total failure load is affected by both crushing and buckling, and combines these two effects using the reciprocal relation:

Where,

•  = Crushing load =
•  = Euler’s critical buckling load =

Substituting , we get:

Now substituting  and  in the reciprocal equation:

Simplifying:

To make it simpler, we define:

Thus, the final Rankine’s formula becomes:

This formula can be used for any type of column, whether short, medium, or long.

Values of Constant ‘a’ for Common Materials

Material	Value of ‘a’
Mild steel	1/7500
Cast iron	1/1600
Wrought iron	1/9000
Timber	1/750

The value of ‘a’ depends on the modulus of elasticity (E) and crushing strength (σc) of the material. Higher the stiffness (E), lower will be the value of ‘a’, indicating higher load-carrying capacity.

Meaning of Each Term

1. Crushing Strength (σc):
   It is the maximum compressive stress that the material can withstand before failure by crushing.
2. Effective Length (Leff):
   The actual length of the column adjusted for its end conditions (fixed, hinged, or free).
3. Radius of Gyration (r):
   It represents the stiffness of the column cross-section and is given by:

1. Slenderness Ratio (L/r):
   This dimensionless ratio determines the behavior of the column—whether it will fail by crushing or buckling.

Special Cases of Rankine’s Formula

1. For Short Columns (L/r is small):
   When the column is short, the value of  is very small.
   So,

This means the column fails by crushing only.

1. For Long Columns (L/r is large):
   When the column is very long,  becomes large, so:

Substituting :

Which is Euler’s buckling formula.

Hence, Rankine’s formula smoothly transitions between the two limiting cases — crushing and buckling.

Advantages of Rankine’s Formula

1. Applicable to All Columns:
   It gives reliable results for short, medium, and long columns.
2. Considers Both Crushing and Buckling:
   Provides a combined effect, making it more practical than Euler’s theory.
3. Simple to Use:
   It requires only basic material properties and geometrical parameters.
4. Accurate for Real Structures:
   It gives results closer to experimental data for actual columns with imperfections.

Limitations of Rankine’s Formula

1. It is empirical, so it relies on experimental constants rather than pure theory.
2. Assumes uniform material and cross-section, which may not hold for all structures.
3. Not suitable for columns under eccentric loading or non-axial compression.
4. Does not consider residual stresses or imperfections explicitly.

Despite these limitations, Rankine’s formula is widely used in design and analysis due to its accuracy and simplicity.

Example (Conceptual)

Let a steel column have:
and

Then:

 

Hence, the crippling load is 1.6 MN.

Conclusion

Rankine’s formula is an empirical relationship that combines the effects of crushing and buckling to determine the safe or crippling load of a column. It provides accurate results for all types of columns — short, intermediate, or long — by using a single equation. The formula effectively bridges the gap between Euler’s theory (for long columns) and crushing theory (for short columns), making it an essential tool in mechanical and structural engineering design for ensuring stability and safety under compressive loads.