Learning topic
Critical Buckling Stress, Euler Limits and Inelastic Column Formulas
Column buckling curves, Euler limiting slenderness, inelastic buckling, Johnson, Tetmajer, Rankine–Gordon and design-code approaches.
Critical buckling stress is the average compressive stress at which a column becomes unstable. Plotting it against slenderness produces a column-strength or buckling curve that distinguishes stocky, intermediate and slender members.
Euler's critical stress can be written in terms of the slenderness ratio:
$$\sigma_{cr,E}=\frac{N_{cr}}{A}=\frac{\pi^2E}{\lambda^2}.$$
Euler's formula is used in the elastic range when the critical stress does not exceed the proportional limit $\sigma_p$. The limiting slenderness follows from $\sigma_{cr,E}=\sigma_p$:
$$\lambda_{lim}=\pi\sqrt{\frac{E}{\sigma_p}}.$$
Thus, for $\lambda\ge\lambda_{lim}$, the Euler model is consistent with elastic buckling. For lower slenderness, another relation or a code-based buckling curve is required.
Elastic Euler range
For a slender elastic column,
\[\sigma_{cr,E}=\frac{P_{cr}}{A}=\frac{\pi^2E}{\lambda^2}.\]
Equating the Euler stress to the proportional limit \(\sigma_p\) gives the limiting slenderness
\[\lambda_{lim}=\pi\sqrt{\frac{E}{\sigma_p}}.\]
Euler's formula is applicable when the critical state remains within the elastic range.
Intermediate columns and inelastic buckling
Intermediate columns require empirical, semi-empirical, or code-based relations. The linear relation traditionally called the Yasinsky formula in Ukrainian and Eastern European mechanics courses is not commonly identified by that name in English-language engineering literature.
For columns of intermediate slenderness, an empirical Yasinsky relation may be used in educational engineering calculations:
$$\sigma_{cr}=a-b\lambda.$$
The corresponding critical load is:
$$N_{cr}=A\sigma_{cr}=A(a-b\lambda).$$
- $\sigma_{cr}$ — critical compressive stress;
- $N_{cr}$ — critical compressive load;
- $A$ — cross-sectional area;
- $\lambda$ — column slenderness ratio;
- $a$, $b$ — empirical coefficients depending on the material and the adopted reference or calculation method.
The coefficients $a$ and $b$ are not universal. The relation must be used only within the specified slenderness range for the particular material; outside that range, another model or design-code relation is required.
English terminology and alternative formulas
- inelastic column buckling is the usual general term for this range;
- Tetmajer or Euler–Tetmajer formula may denote historical empirical column curves used in Central European literature;
- Johnson parabolic formula is a common parabolic approximation for intermediate columns;
- Rankine–Gordon formula interpolates between crushing resistance and Euler buckling;
- tangent-modulus theory represents reduced material stiffness in the inelastic range;
- modern design standards use calibrated column buckling curves and reduction factors.
Short columns
For low slenderness, overall buckling may no longer govern. Resistance can instead be controlled by yielding, crushing, local buckling, or another material or sectional limit state.
Model-selection procedure
- Calculate slenderness and the Euler critical stress.
- Compare the result with the material proportional limit.
- Use Euler's formula for sufficiently slender elastic columns.
- Use the permitted inelastic or code-based relation for intermediate columns.
- Check strength and local stability for stocky members.
About this topic
Euler's formula is valid only when critical stresses remain within a material's proportional limit. For intermediate and short columns buckling in the inelastic domain, empirical equations like the Yasinsky formula apply. This section details applicability boundaries and stress reduction factors.