What is Johnson’s Formulae for Columns?

A machine part subjected to an axial compressive force is called a strut. A strut may be horizontal, inclined, or even vertical. But a vertical strut is known as a column, pillar, or stanchion. The machine members that must be investigated for column action are piston rods, valve push rods, connecting rods, screw jack, side links of toggle jack, etc. In the previous articles, we have discussed Euler’s Column Theory, and the Rankine’s Formula for Columns to understand the design of long and short columns. with Johnson’s Formulae for Columns, we are going to study the short columns made of ductile materials.

Johnson’s Formulae for Columns

Professor J.B. Johnson proposed two formulas for short columns. Those are the Straight line formula and the Parabolic formula. Let us discuss these two formulae in detail.

Straight Line Formula

According to the straight-line formula proposed by Johnson, the critical or crippling load is

Where
A = Cross-sectional area of the column,
σ_y = Yield point stress,
C₁ = A constant, whose value depends upon the type of material as well as the type of ends
L/k = Slenderness ratio.

If the safe stress(W_cr /A) is plotted against the slenderness ratio(L/k), it works out to be a straight line, so it is known as the straight line formula.

Slenderness Ratio

In Euler’s formula, the ratio L / k is known as the slenderness ratio. It may be defined as the ratio of the effective length of the column to the least radius of gyration of the section.

It may be noted that the formula for crippling load, in the previous article is based on the assumption that the slenderness ratio l / k is so large, that the failure of the column occurs only due to bending, the effect of direct stress (i.e. W / A) being negligible.

Parabolic Formula

Prof. Johnson after proposing the straight-line formula found that the results obtained by this formula are very approximate. He then proposed another formula, according to which the critical or crippling load,

If a curve of safe stress(W_cr /A) is plotted against (L/k), it works out to be parabolic, so it is known as a parabolic formula.

The following figure shows the relationship of safe stress (W_cr / A) and the slenderness ratio (L / k) as given by Johnson’s formula and Euler’s formula for a column made of mild steel with both ends hinged (i.e. C = 1), having a yield strength, σ_y = 210 MPa.

• We see from the figure that points A (the point of tangency between Johnson’s straight-line formula and Euler’s formula) describes the use of two formulae.
• In other words, Johnson’s straight-line formula may be used when L / k < 180, and Euler’s formula is used when L / k > 180.
• Similarly, point B (the point of tangency between Johnson’s parabolic formula and Euler’s formula) describes using two formulae.
• In other words, Johnson’s parabolic formula is used when L / k < 140, and Euler’s formula is used when L / k > 140.
• Johnson’s parabolic formula is used for short columns made of ductile materials.

This is all about the Johnson’s Formulae for Columns. Let us know what you think about this article in the comment section below.