For calculating the allowable stress of long columns $${\sigma _0} = \frac{{{\sigma _y}}}{n}\left[ {1 – a{{\left( {\frac{1}{r}} \right)}^2}} \right]$$ is the empirical formula, known as A. Straight line formula B. Parabolic formula C. Perry’s formula D. Rankine’s formula

The correct answer is: B. Parabolic formula

The formula ${\sigma _0} = \frac{{{\sigma _y}}}{n}\left[ {1 – a{{\left( {\frac{1}{r}} \right)}^2}} \right]$ is the parabolic formula for calculating the allowable stress of long columns. It is also known as the Rankine formula.

The formula is based on the assumption that the stress in a column is proportional to the strain. The constant of proportionality is called the modulus of elasticity. The formula takes into account the effect of the length of the column on the allowable stress. The longer the column, the lower the allowable stress.

The formula is used in the design of columns to ensure that they will not fail under the loads they are expected to carry.

The other options are incorrect because they do not represent the formula correctly.

• Option A is incorrect because the formula is not a straight line.
• Option C is incorrect because the formula is not Perry’s formula. Perry’s formula is a different formula for calculating the allowable stress of columns.
• Option D is incorrect because the formula is not Rankine’s formula. Rankine’s formula is the same as the parabolic formula.