# Rankine Formula for Columns

By BYJU'S Exam Prep

Updated on: September 25th, 2023

Euler’s theory provides the buckling load for long columns, but the theory does not apply to short columns. It only holds till the slenderness ratio of the column is greater than or equal to the critical slenderness ratio. To counter the limitation of Euler’s theory, we use the Rankine formula. Let us check the Rankine formula for different types of columns.

The **Rankine Gordon formula** is an empirical equation which is applicable for all types of columns (i.e., long or short). The equation is commonly known as the Rankine formula. In this article, we will understand the Rankine formula for columns.

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## What is Rankine Formula for Columns?

The long columns are assumed to fail due to buckling, whereas the short columns are assumed to fail due to crushing. However, practically, any column fails due to the combined effect of buckling and crushing. The Rankine Gordon formula assumes a combined failure mode, commonly known as the Rankine formula for columns. Hence, the Rankine formula is good for all columns (i.e., long or short).

The critical load, according to the Rankine formula, is given as

1/P=1/P_{c}+1/P_{e}

where

- P = Rankine’s critical load
- P
_{c}= crushing load - P
_{e}= Euler’s buckling load

## Alternate Form of Rankine Formula

By putting the values of crushing load Pc and Euler’s buckling load Pe in the Rankine formula mentioned in the previous section, we can get an alternative form of the Rankine formula for columns. We know that the Rankine formula is given as

1/P=1/P_{c}+1/P_{e}

P=(P_{c}P_{e})/P_{c}+P_{e}…(i)

We know that crushing load can be given as

P_{c}=σ_{c}A

where

- σ
_{c}= ultimate crushing stress - A = cross-sectional area of column

According to Euler’s theory, the buckling load can be given as

P_{e}=π^{2}EI/L_{e}^{2}=π^{2}EA/λ^{2}

where

- E = young’s modulus of elasticity
- I = moment of inertia
- L
_{e}= effective length of column - λ = slenderness ratio of the column

Putting the value of P_{c} and P_{e} in equation (i), we get

P=σ_{c}A/[1+σ_{c}A/(π^{2}EA/λ^{2})]

P=σ_{c}A/[1+(σ_{c}λ^{2}/π^{2}EA)]

P=σ_{c}/[1+(αλ^{2})]

where, α = Rankine’s constant = σ_{c}/π^{2}E

## What is Rankine’s Constant?

We have defined the term Rankine constant in the previous section while deriving the alternate form of the Rankine formula. The Rankine constant denoted by 𝜶 was given as

α=σ_{c}/π^{2}E

This Rankine constant is different for various materials. The Rankine constant for some of the materials used in the construction of columns is given in the table below:

Materials |
α |

Wrought Iron | 1/9000 |

Cast Iron | 1/1600 |

Mild Steel | 1/7500 |

Strong Timber | 1/750 |

## Rankine Formula for Short Columns

This section will determine how the Rankine formula for the critical load will modify for short columns. A short column will have a smaller value of effective length. We know that Euler’s equation for buckling load is given as

P_{e}=π^{2}EI/L_{e}^{2}

From the equation, it can be determined that for a smaller value of effective length (L_{e}), the buckling load (P_{e}) will be very high. So, in the Rankine formula for columns, the value of (1/P_{e}) will be very small and can be neglected. So the Rankine formula for the critical load can be written as

1/P=1/P_{c}

P=P_{c}

Therefore, the Rankine critical load for short columns will equal the crushing load.

## Rankine Formula for Long Columns

A long column will have a larger value of effective length. Therefore, according to Euler’s formula for buckling load, the buckling load (P_{e}) will be minimal. So, in the Rankine formula for columns, the value of (1/P_{e}) will be very high as compared to the value of (1/P_{c}), and hence (1/P_{c}) can be ignored. So the Rankine formula for the critical load can be written as

1/P=1/P_{e}

P=P_{e}

Therefore, the Rankine critical load for long columns will equal Euler’s buckling load.