Editing Column (section)

===Equilibrium, instability, and loads===
{{Main|Buckling#columns}}
{{Mechanical_failure_modes}}
[[File:ColumnEffectiveLength.png|thumb|220px|Table showing values of K for structural columns of various end conditions (adapted from Manual of Steel Construction, 8th&nbsp;edition, American Institute of Steel Construction, Table C1.8.1)]]
As the axial load on a perfectly straight slender column with elastic material properties is increased in magnitude, this ideal column passes through three states: stable equilibrium, neutral equilibrium, and instability. The straight column under load is in stable equilibrium if a lateral force, applied between the two ends of the column, produces a small lateral deflection which disappears and the column returns to its straight form when the lateral force is removed. If the column load is gradually increased, a condition is reached in which the straight form of equilibrium becomes so-called neutral equilibrium, and a small lateral force will produce a deflection that does not disappear and the column remains in this slightly bent form when the lateral force is removed. The load at which neutral equilibrium of a column is reached is called the critical or [[buckling]] load. The state of instability is reached when a slight increase of the column load causes uncontrollably growing lateral deflections leading to complete collapse.

For an axially loaded straight column with any end support conditions, the equation of static equilibrium, in the form of a differential equation, can be solved for the deflected shape and critical load of the column. With hinged, fixed or free end support conditions the deflected shape in neutral equilibrium of an initially straight column with uniform cross section throughout its length always follows a partial or composite sinusoidal curve shape, and the critical load is given by

<math>f_{cr}\equiv\frac{\pi^2\textit{E}I_{min}}{{L}^2}\qquad (1)</math>

where ''E'' = [[elastic modulus]] of the material, ''I<sub>min</sub>'' = the minimal moment of inertia of the cross section, and ''L'' = actual length of the column between its two end supports. A variant of (1) is given by

<math>f_{cr}\equiv\frac{\pi^{2}E_T}{(\frac{KL}{r})^{2}}\qquad (2)</math>

where ''r'' = [[radius]] of gyration of column cross-section which is equal to the square root of (I/A), ''K'' = ratio of the longest half [[sine]] wave to the actual column length, ''E''<sub>''t''</sub> = tangent modulus at the stress ''F''<sub>cr</sub>, and ''KL'' = effective length (length of an equivalent hinged-hinged column). From Equation&nbsp;(2) it can be noted that the buckling strength of a column is inversely proportional to the square of its length.

When the critical stress, ''F''<sub>cr</sub> (''F''<sub>cr</sub> =''P''<sub>cr</sub>/''A'', where ''A''&nbsp;= cross-sectional area of the column), is greater than the proportional limit of the material, the column is experiencing inelastic buckling. Since at this stress the slope of the material's stress-strain curve, ''E''<sub>''t''</sub> (called the [[tangent modulus]]), is smaller than that below the proportional limit, the critical load at inelastic buckling is reduced. More complex formulas and procedures apply for such cases, but in its simplest form the critical buckling load formula is given as Equation&nbsp;(3),

<math>f_{cr}\equiv{F_y}-\frac{F^{2}_{y}}{4\pi^{2}E}\left(\frac{KL}{r^2}\right)\qquad (3)</math>

A column with a cross section that lacks symmetry may suffer torsional buckling (sudden twisting) before, or in combination with, lateral buckling. The presence of the twisting deformations renders both theoretical analyses and practical designs rather complex.

Eccentricity of the load, or imperfections such as initial crookedness, decreases column strength. If the axial load on the column is not concentric, that is, its line of action is not precisely coincident with the centroidal axis of the column, the column is characterized as eccentrically loaded. The eccentricity of the load, or an initial curvature, subjects the column to immediate bending. The increased stresses due to the combined axial-plus-flexural stresses result in a reduced load-carrying ability.

Column elements are considered to be massive if their smallest side dimension is equal to or more than 400&nbsp;mm. Massive columns have the ability to increase in carrying strength over long time periods (even during periods of heavy load). Taking into account the fact, that possible structural loads may increase over time as well (and also the threat of progressive failure), massive columns have an advantage compared to non-massive ones.