Formulas: effective length method

Formula 1

The effective length, $L_{c}=K L$, for column buckling based upon elastic or inelastic stability theory, or alternatively the equivalent elastic column buckling stress, $F_{e}=\pi^{2} E /\left(L_{c} / r\right)^{2}$, is used to calculate an axial compressive strength, $P_{c}$, through an empirical column curve that accounts for geom

Formula 2

The effective length, $L_{c}=K L$, for column buckling based upon elastic or inelastic stability theory, or alternatively the equivalent elastic column buckling stress, $F_{e}=\pi^{2} E /\left(L_{c} / r\right)^{2}$, is used to calculate an axial compressive strength, $P_{c}$, through an empirical column curve that accounts for geometric imperfections and distributed yielding including the effects of residual stresses.

Formula 3

If $K<1.0$ is used for the calculation of the nominal compressive strength, $P_{n}$, in braced frames, the additional demands on the stability bracing systems and the influence on the second-order moments in beams providing restraint to the columns must be considered.

Equation C-A-7-2

The alignment chart for sidesway uninhibited frames shown in Figure C-A-7.2 is based on the following equation:

Equation C-A-7-3

Equation C-A-7-4

Equation C-A-7-5

Equation C-A-7-6

where

$P_{m f}, P_{\text {story }}$, and $H$ are as defined in Appendix 8, Section 8.1.3.

Equation C-A-7-7

Equation C-A-7-8

where

$K_{n 2}$ is defined as the value of $K$ determined directly from the alignment chart in Figure C-A-7.2.

Equation C-A-7-9

Equation C-A-7-10

Equation C-A-7-11

In this equation, $M_{1}$ and $M_{2}$ are the smaller and larger end moments, respectively, in the column. These moments are determined from a first-order analysis of the frame under lateral load. Column inelasticity is considered in the derivation of this equation. The unconservative error in $P_{c}$, when it is based on $K_{2}$ determined from Equation C-A-7-10, is less than $3 \%$ if the following inequality is satisfied:

where

$\Sigma P_{y m f}$ is the sum of the axial yield strengths of all columns in the story that are part of moment frames, if any, in the direction of translation being considered.

Formula 14

For slender columns, the calculation of the effective length, $L_{\mathrm{c}}=K L$, and $P_{n K L}$ is critical to achieving an accurate solution when using the effective length method.

Found in

• EFFECTIVE LENGTH METHOD