Learning topic
Column Slenderness and Weak Buckling Axis
Calculate radius of gyration, effective length and column slenderness ratio, and identify the weak axis that governs buckling.
The slenderness ratio measures a compression member's tendency to buckle by combining its effective length with the geometric distribution of its cross-sectional area.
The radius of gyration of a cross-sectional area about a selected centroidal axis is:
$$i=\sqrt{\frac{I}{A}}.$$
The column slenderness ratio is:
$$\lambda=\frac{l_{eff}}{i}=\frac{\mu L}{i}.$$
For buckling assessment, use the minimum radius of gyration:
$$i_{min}=\sqrt{\frac{I_{min}}{A}},\qquad \lambda_{max}=\frac{\mu L}{i_{min}}.$$
- $i$ — radius of gyration;
- $I$ — centroidal second moment of area;
- $A$ — cross-sectional area;
- $\lambda$ — slenderness ratio;
- $l_{eff}=\mu L$ — effective length;
- $i_{min}$, $I_{min}$ — minimum radius of gyration and minimum centroidal second moment of area.
The larger $\lambda$ is, the more slender the member and the more likely its load capacity is to be governed by buckling rather than material strength alone.
The calculator below evaluates radius of gyration, second moment of area, or cross-sectional area. Slenderness $\lambda$ is not included because the calculator registry has no separate dimensionless quantity for this parameter.
Radius of gyration
For a selected centroidal axis,
\[r=\sqrt{\frac{I}{A}},\]
where \(I\) is the second moment of area and \(A\) is the cross-sectional area.
Slenderness in both planes
Calculate
\[\lambda_x=\frac{L_{eff,x}}{r_x},\qquad \lambda_y=\frac{L_{eff,y}}{r_y}.\]
If the end restraints are identical, the smaller radius of gyration usually defines the weak buckling axis.
Governing axis
The governing direction is the one with the lower critical load. It is not always enough to select the smaller \(I\): different effective lengths or restraints in the two planes may change the result. Compare the complete slenderness ratios or critical loads.
Calculation sequence
- Determine \(A\), \(I_x\), and \(I_y\).
- Calculate \(r_x\) and \(r_y\).
- Determine effective lengths in both planes.
- Calculate \(\lambda_x\) and \(\lambda_y\).
- Use the governing case in the stability check.
About this topic
Column slenderness is the ratio of effective length to radius of gyration. Both principal axes must be checked because end restraints may differ by plane.