Enter the column diameter, length, modulus of elasticity, and yield strength (or proportional limit) into the calculator to determine the critical buckling load using Johnson’s (parabolic) column formula. For very slender columns, the calculator automatically switches to Euler buckling.
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Johnson’s Formula
Johnson’s formula estimates the critical buckling load of an intermediate-length column. It fills the gap between two other failure modes: short columns that fail mainly by crushing or yielding, and very slender columns that buckle elastically according to Euler’s formula. This calculator assumes a solid circular column, uses the selected end-condition factor to determine effective length, and automatically applies the Johnson or Euler equation based on the column’s slenderness.
P_{\mathrm{cr}}=
\begin{cases}
A\,\sigma_y\left[1-\dfrac{\sigma_y}{4\pi^2E}\left(\dfrac{KL}{k}\right)^2\right], & \dfrac{KL}{k}\le C_c \\[6pt]
\dfrac{\pi^2EI}{(KL)^2}, & \dfrac{KL}{k}>C_c
\end{cases}The transition between Johnson buckling and Euler buckling is controlled by the critical slenderness value below.
C_c=\sqrt{\dfrac{2\pi^2E}{\sigma_y}}Solid Circular Column Relations Used by the Calculator
Because this tool is built around a solid round column, it derives the area, moment of inertia, and radius of gyration directly from the entered diameter.
\begin{aligned}
A&=\dfrac{\pi d^2}{4}\\[4pt]
I&=\dfrac{\pi d^4}{64}\\[4pt]
k&=\sqrt{\dfrac{I}{A}}=\dfrac{d}{4}
\end{aligned}| Symbol | Meaning | Typical Units |
|---|---|---|
| Pcr | Critical buckling load | lbf, N |
| E | Modulus of elasticity | psi, Pa, MPa, GPa |
| σy | Yield strength or proportional limit | psi, Pa, MPa, GPa |
| L | Unsupported column length | in, ft, mm, m |
| K | Effective length factor based on end restraint | dimensionless |
| A | Cross-sectional area | in², mm², m² |
| I | Area moment of inertia | in⁴, mm⁴, m⁴ |
| k | Radius of gyration | in, mm, m |
| Cc | Transition slenderness ratio | dimensionless |
How to Use the Johnson’s Formula Calculator
- Enter the column diameter.
- Enter the unsupported length of the member.
- Input the material’s modulus of elasticity.
- Input the yield strength or proportional limit.
- Select or enter the effective length factor K that matches the end conditions.
- Read the resulting critical load, which represents the estimated axial load at which buckling begins.
Common Effective Length Factors
| End Condition | Typical K Value | Buckling Effect |
|---|---|---|
| Pinned–Pinned | 1.0 | Baseline case for many textbook examples |
| Fixed–Free | 2.0 | Longest effective length and lowest buckling capacity |
| Fixed–Pinned | 0.7 | Shorter effective length and higher capacity than pinned–pinned |
| Fixed–Fixed | 0.5 | Shortest effective length and highest theoretical buckling resistance |
How to Interpret the Result
- A larger diameter greatly improves buckling resistance because round-section stiffness rises rapidly as diameter increases.
- A longer unsupported length lowers the critical load because slenderness increases.
- A larger K factor reduces capacity because the column behaves as if it were effectively longer.
- Higher elastic modulus generally improves buckling resistance, especially in the Euler region.
- Higher yield strength improves Johnson-region capacity and also increases the transition slenderness.
Example
Suppose a solid circular steel column has a diameter of 4 in, an unsupported length of 10 ft, a modulus of elasticity of 29,000,000 psi, a yield strength of 36,000 psi, and pinned–pinned end conditions so that K = 1.0.
\begin{aligned}
A&=\dfrac{\pi(4)^2}{4}=12.566\ \mathrm{in}^2\\[4pt]
I&=\dfrac{\pi(4)^4}{64}=12.566\ \mathrm{in}^4\\[4pt]
k&=\dfrac{4}{4}=1.0\ \mathrm{in}
\end{aligned}\dfrac{KL}{k}=\dfrac{1.0\times120}{1.0}=120C_c=\sqrt{\dfrac{2\pi^2(29{,}000{,}000)}{36{,}000}}=126.10Because the slenderness ratio is below the transition value, the column is in the Johnson range, not the Euler range.
P_{\mathrm{cr}}\approx 2.48\times10^5\ \mathrm{lbf}Practical Notes and Limitations
- This calculator is intended for straight, axially loaded, solid circular columns.
- It does not account for load eccentricity, initial crookedness, residual stress, local buckling, or connection flexibility beyond the selected K value.
- Use consistent units for stress and length inputs. If stress is entered in psi, section dimensions should be interpreted in inches for hand checks.
- The critical load is a buckling threshold, not automatically a safe design load. Real designs usually require an appropriate safety factor and compliance with the governing design code.
- If your column is not solid and circular, the governing equations are still based on area, inertia, and radius of gyration, but the section properties must come from the actual shape rather than the round-section relations above.
Why Johnson’s Formula Matters
Many real columns do not fall cleanly into the “short” or “very slender” category. Johnson’s formula is useful because it captures the transition where material strength and buckling behavior both influence failure. That makes it a practical method for estimating column capacity when Euler alone would be too optimistic and simple compressive strength alone would be too conservative or incomplete.
