Rankine Formula for Columns

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_cP_e)/P_c+P_e…(i)

We know that crushing load can be given as

P_c=σ_cA

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=π²EI/L_e²=π²EA/λ²

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_cA/[1+σ_cA/(π²EA/λ²)]

P=σ_cA/[1+(σ_cλ²/π²EA)]

P=σ_c/[1+(αλ²)]

where, α = Rankine’s constant = σ_c/π²E

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/π²E

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:

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=π²EI/L_e²

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.