## 312 Basic Concept

Mathematically, the effective length factor or the elastic K-factor is defined as jn2EI

p2EI

where Pe is the Euler load, the elastic buckling load of a pin-ended compression member or column, Pcr is the elastic buckling load of an end-restrained compression member or column, E is the modulus of elasticity of the material, I is the moment of inertia in the flexural buckling plane, and L is the unsupported length of the compression member or column.

Physically, the K-factor is a factor that when multiplied by the actual length of the end-restrained column (Figure 31.1a) gives the length of an equivalent pin-ended column (Figure 31.1b) whose buckling load is the same as that of the end-restrained column. It follows that effective length KL of an end-restrained column is the length between the adjacent inflection points of its pure flexural buckling shape.

Practically, the design specifications usually provide the resistance equations for pin-ended columns, while the resistance of framed columns can be estimated through the K-factor to the pin-ended column strength equation. Theoretical K-factor is determined from an elastic eigenvalue analysis of the entire structural system, while practical methods for the K-factor are based on an elastic eigenvalue analysis of selected subassemblages. The effective length concept is the only tool currently available for the design of compression members in engineering structures and it is an essential part of analysis and design procedures.

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