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Radius of Gyration is a measure that describes how the cross-sectional area of a column is distributed about its centroidal axis. It is given by the equation

The Slenderness Ratio of a column is a measure of its susceptibility to buckling, defined as the effective length of the column ($KL$) divided by the radius of gyration ($r$),

The Column Curve is a graphical representation showing the relation between the slenderness ratio of a column and the stress at which buckling occurs. It highlights the transition from material-dominated failure (yielding) to geometry-dominated failure (buckling).

Imperfections and Initial Deflections in a structure can significantly reduce its buckling load capacity. These imperfections act as stress concentrators and can initiate buckling at loads lower than the theoretical critical load predicted by idealized formulas.

Material Yielding occurs when a material exceeds its yield strength and undergoes plastic deformation. In contrast, Buckling occurs due to instability under loading conditions. While yielding depends on the material properties, buckling is also influenced by geometry and support conditions.

Euler's Critical Load is the axial force at which a slender column will buckle. It is given by the formula

Load Eccentricity refers to the condition where the line of action of the load does not pass through the centroid of the cross-section of a structural member. This eccentricity can introduce additional bending moments, thereby reducing the buckling strength of the member.

The Effective Length Factor, $K$, accounts for the boundary conditions affecting a column's buckling behavior. It varies based on the support conditions; for example, a column with both ends pinned has $K = 1.0$, while a fixed-free column has $K = 2.0$. Different $K$ values influence the critical buckling load.

Buckling Modes refer to the different shapes that a column can assume upon buckling. The first mode of buckling is typically the most critical and has the lowest load capacity, but higher modes can become relevant in certain scenarios.