Johnson's parabolic formula

Euler's formula is displayed as such: ${\displaystyle \sigma _{cr}={P_{cr} \over A}={\pi ^{2}EI \over AL_{e}^{2}}={\pi ^{2}E \over \left({\frac {l}{k}}\right)^{2}}}$ where

${\displaystyle \sigma _{cr}=}$ critical stress,

${\displaystyle P_{cr}=}$ critical force,

${\displaystyle A=}$ area of cross section,

${\displaystyle L_{e}=}$ Effective length of the rod,

${\displaystyle E=}$ modulus of elasticity,

${\displaystyle I=}$ area moment of inertia of the cross section of the rod,

${\displaystyle {l \over k}}$ = slenderness ratio.

Euler's equation is useful in situations such as an ideal pinned-pinned column, or in cases in which the effective length can be used to adjust the existing formula (ie. Fixed-Free).[3]

	Pinned-Pinned	Fixed-Fixed	Fixed-Pinned	Fixed-Free
Effective Length, ${\displaystyle L_{e}}$	1L	0.5L	0.7L	2L

(L is the original length of the specimen before the force was applied.)

However, certain geometries are not accurately represented by the Euler formula. One of the variables in the above equation that reflects the geometry of the specimen is the slenderness ratio, which is the column's length divided by the radius of gyration.[4]

The slenderness ratio of the member can be found with ${\displaystyle \left({\frac {l}{k}}\right)=L_{e}{\sqrt {A \over I}}}$ while the critical slenderness ratio is ${\displaystyle {\left({\frac {l}{k}}\right)}_{cr}={\sqrt {2\pi ^{2}E \over \sigma _{y}}}}$

In practical terms, the slenderness ratio is an indicator of the specimen's resistance to bending and buckling, due to its length and cross section. If the slenderness ratio is less than the critical slenderness ratio, the column is considered to be a short column. In these cases, the Johnson parabola is more applicable than the Euler formula.[5]

Graph of Johnson's parabola (plotted in red) against Euler's formula, with the transition point indicated. The Johnson parabola creates a new region of failure.

Johnson's formula rounds out the function given by Euler's formula. It creates a new failure border by fitting a parabola to the graph of failure for Euler buckling.

${\displaystyle \sigma _{cr}=\sigma _{y}-{1 \over E}{\left({\frac {\sigma _{y}}{2\pi }}\right)}^{2}{\left({\frac {l}{k}}\right)}^{2}}$

There is a transition point on the graph of the Euler curve, located at the critical slenderness ratio. At slenderness values lower than this point (occurring in specimens with a relatively short length compared to their cross section), the graph will follow the Johnson parabola; in contrast, larger slenderness values will align more closely with the Euler equation.

One common material in aerospace applications is Al 2024. Certain material properties of Al 2024 have been determined experimentally, such as the tensile yield strength (324 MPa) and the modulus of elasticity (73.1 GPa). [6] The Euler formula could be used to plot a failure curve, but it would not be accurate below a certain ${\displaystyle {\frac {l}{k}}}$ value, the critical slenderness ratio.

${\displaystyle {\left({\frac {l}{k}}\right)}_{cr}={\sqrt {2\pi ^{2}E \over \sigma _{y}}}={\sqrt {\frac {2\pi ^{2}\cdot 73.1\cdot 10^{9}}{324\cdot 10^{6}}}}=66.7}$

Therefore, the Euler equation is applicable for values of ${\displaystyle {\frac {l}{k}}}$ greater than 66.7.

Euler: ${\displaystyle \sigma _{cr}={\pi ^{2}E \over \left({\frac {l}{k}}\right)^{2}}={\pi ^{2}\cdot 73.1\cdot 10^{9} \over \left({\frac {l}{k}}\right)^{2}}}$ for ${\displaystyle {\frac {l}{k}}>66.7}$

(units in Pascals)

Johnson's parabola takes care of the smaller ${\displaystyle {\frac {l}{k}}}$ values.

Johnson: ${\displaystyle \sigma _{cr}=\sigma _{y}-{1 \over E}{\left({\frac {\sigma _{y}}{2\pi }}\right)}^{2}{\left({\frac {l}{k}}\right)}^{2}=324\cdot 10^{6}-{1 \over 73.1\cdot 10^{9}}{\left({\frac {324\cdot 10^{6}}{2\pi }}\right)}^{2}{\left({\frac {l}{k}}\right)}^{2}}$ for ${\displaystyle 0\leq {\frac {l}{k}}\leq 66.7}$

(units in Pascals)

Plot of both Euler and Johnson formulas for Al 2024. The transition point occurs at the critical slenderness ratio.