15.2 Euler's formula and its limitations

Euler's Formula for Slender Columns

Calculating Critical Load

Euler's formula gives you the critical load ($P_{cr}$), the maximum axial load a slender column can carry before it buckles elastically. The formula is:

$P_{cr} = \frac{\pi^2 E I}{L_e^2}$

Three quantities drive the result:

• Modulus of elasticity ($E$): the material's stiffness
• Moment of inertia ($I$): the cross-section's resistance to bending
• Effective length ($L_e$): the column's length adjusted for its end conditions

Example: A steel column has $E = 200 \text{ GPa}$, $I = 1.2 \times 10^{-4} \text{ m}^4$, and $L_e = 3 \text{ m}$. Plugging in:

$P_{cr} = \frac{\pi^2 (200 \times 10^9)(1.2 \times 10^{-4})}{3^2} = \frac{\pi^2 (2.4 \times 10^7)}{9} \approx 26.3 \text{ MN}$

Notice that $L_e$ is squared in the denominator. That means doubling the effective length cuts the critical load to one-quarter. Length has a huge influence on buckling capacity.

Moment of Inertia and Effective Length

The moment of inertia ($I$) depends on the cross-sectional shape and dimensions. A column always buckles about the axis with the smallest $I$, so you need to check both principal axes.

Common formulas:

• Solid circular section (radius $r$): $I = \frac{\pi r^4}{4}$
• Solid rectangular section (width $b$, height $h$): $I = \frac{b h^3}{12}$

The effective length ($L_e$) is the equivalent pinned-pinned length that would buckle at the same critical load as the actual column with its real end conditions. It accounts for how much the supports restrain rotation and translation. (More on how to calculate $L_e$ in the End Conditions section below.)

Elastic Buckling and Critical Stress

Elastic buckling is a sudden lateral deflection that occurs when the axial load reaches $P_{cr}$. The column doesn't get progressively more loaded; instead, it snaps sideways at a well-defined load. All of this happens while the material is still in its elastic range (stresses below the proportional limit).

You can convert the critical load to a critical stress by dividing by the cross-sectional area:

$\sigma_{cr} = \frac{P_{cr}}{A}$

Example: For a column with $P_{cr} = 200 \text{ kN}$ and $A = 0.005 \text{ m}^2$:

$\sigma_{cr} = \frac{200 \times 10^3}{0.005} = 40 \text{ MPa}$

This critical stress must stay below the material's proportional limit for Euler's formula to be valid. If $\sigma_{cr}$ exceeds the proportional limit, the column will yield before it can buckle elastically, and Euler's formula overpredicts the actual failure load.

Assumptions and Limitations of Euler's Formula

Ideal Column Assumptions

Euler's formula is derived for an ideal column. That means it assumes:

• The column is perfectly straight with no initial curvature or crookedness.
• The material is homogeneous and linearly elastic (obeys Hooke's law), with stresses remaining below the proportional limit.
• The axial load is applied concentrically, exactly through the centroid of the cross-section, with no eccentricity or applied moments.
• The column is free from residual stresses and manufacturing imperfections.

Real columns never satisfy all of these perfectly, which is why actual buckling loads are typically lower than Euler's prediction.

Slenderness Ratio Limitations

The slenderness ratio is defined as $L/r$, where $L$ is the unsupported length and $r$ is the radius of gyration. Euler's formula is only valid for columns with high slenderness ratios, typically $L/r > 100$ or so (the exact threshold depends on the material).

Why the cutoff? For stocky columns with low slenderness ratios, the calculated $\sigma_{cr}$ from Euler's formula exceeds the material's yield strength. The material yields before elastic buckling can occur. This is called inelastic buckling, and Euler's formula does not account for it.

Other Limitations

• Shear deformation: Euler's formula neglects shear effects, which can matter in deep, short members.
• Local buckling: Thin-walled or open cross-sections (like channels or angles) can buckle locally at a flange or web before the whole column buckles globally. Euler's formula doesn't capture this.
• Complex cross-sections: Non-uniform material properties or unusual shapes may require finite element analysis or other advanced methods.
• Initial imperfections and residual stresses: These reduce the actual buckling load below the Euler prediction. Design codes handle this by applying safety factors or using empirical column curves.

Applicability of Euler's Formula

Slenderness Ratio and Column Classification

The slenderness ratio ($L/r$) is the primary tool for deciding whether Euler's formula applies. The radius of gyration is:

$r = \sqrt{\frac{I}{A}}$

It represents how far the cross-section's area is spread from the bending axis. A larger $r$ means more resistance to buckling.

Columns are classified into three categories based on slenderness:

Material Properties and Cross-Sectional Shape

The ratio $E/\sigma_y$ (modulus of elasticity to yield strength) affects where the transition from elastic to inelastic buckling occurs. Materials with a high $E/\sigma_y$ ratio (like mild steel or aluminum) have a wider range of slenderness ratios where Euler's formula works well. Materials with a low ratio transition to inelastic buckling sooner.

Cross-sectional shape matters too:

• Solid or compact sections (solid rectangles, solid circles) tend to buckle globally, which is what Euler's formula predicts.
• Thin-walled or open sections (wide-flange I-beams, channels) are more susceptible to local buckling of individual flanges or webs, which can trigger failure before the Euler load is reached. For these shapes, you need to check local buckling limits in addition to the global Euler load.

End Conditions and Critical Load

Effective Length Factor

End conditions have a dramatic effect on a column's buckling capacity. The effective length factor ($K$) converts the actual unsupported length $L$ into the effective length used in Euler's formula:

$L_e = K \cdot L$

Since $L_e$ is squared in the denominator of Euler's formula, even small changes in $K$ significantly change $P_{cr}$.

Ideal End Conditions

The four standard cases and their $K$ values:

A cantilever column (fixed-free) is the weakest configuration. Its effective length is twice the actual length, so its critical load is only one-quarter that of a pinned-pinned column.

Partial Fixity and End Restraints

In practice, connections are rarely perfectly pinned or perfectly fixed. Most real supports provide partial rotational restraint, giving an effective length factor somewhere between the ideal cases.

• A column with partial fixity at both ends might have $K$ between 0.5 and 1.0.
• Example: A column bolted at both ends with some rotational stiffness might use $K = 0.8$, giving $L_e = 0.8L$.

When in doubt, design codes often recommend using conservative (higher) $K$ values. Overestimating $K$ gives a lower $P_{cr}$, which is the safe direction for design.