Annex E (Clause 25.2) effective length of columns

E-1

In the absence of more exact analysis, the effective length of columns in framed structures may be obtained from the ratio of effective length to unsupported length $l_{\mathrm{e}} / l$ given in Fig. 26 when relative displacement of the ends of the column is prevented and in Fig. 26 when relative lateral displacement of the ends is not prevented. In the latter case, it is recommended that the effective length ratio $l_{\mathrm{e}} / l$ may not be taken to be less than 1.2 .

NOTE

• 1 Figures 26 and 27 are reproduced from 'The Structural Engineer' No. 7, Volume 52, July 1974 by the permission of the Council of the Institution of Structural Engineers U.K.

ΣK

• 2 In Figs. 26 and 27, $\beta_{1}$ and $\beta_{2}$ are equal to $\frac{2 K_{c}}{\Sigma K_{c}+\Sigma K}$

where the summation is to be done for the member framing into a joint at top and bottom respectively; and $K_{\mathrm{c}}$ and $K_{\mathrm{b}}$ being the flexural stiffness for column and beam respectively.

E-2

To determine whether a column is a no sway or a sway column, stability index Q may be computed as given below:

$Q=\frac{\Sigma P_{\mathrm{u}} \Delta}{H_{\mathrm{i}} h}$

where

• $\begin{aligned} \sum P_{u} & =\text { sum of axial loads on all column in the } \\ & \text { storey }\end{aligned}$
• Δe = elastically computed first order lateral deflection.
• $H_{u}=$ total lateral force acting within the storey
• $h=$ height of the storey

If $Q \leq 0.04$, then the column in the frame may be taken as no sway column, otherwise the column will be considered as sway column.

E-3

For normal usage assuming idealized conditions the effective length $l_{e}$ of in a given plane may be assessed on the basis of Table 28.

Figure description:

EFFECTIVE LENGTH RATIO l_ef/l

EFFECTIVE LENGTH RATIO l_ef/l

Annotations

• FIXED (text_label - position: beta_1 = 0, beta_2 = 0)
• HINGED (text_label - position: beta_1 = 1.0, beta_2 = 1.0)
• EFFECTIVE LENGTH RATIO l_ef/l (text_label - position: Center of plot area)

FIG. 26 EFFECTIVE LENGTH RATIOS FOR A COLUMN IN A FRAME WITH NO SWAY

Figure description:

Effective Length Factor Nomogram for Unbraced Columns

Relationship between β1, β2 and Effective Length Ratio l/L

Legend

• Isolines of Effective Length Ratio (l/L — color: black; symbol: solid curve

Annotations

• HINGED (text_label - position: β1 = 1.0)
• FIXED (text_label - position: β1 = 0)
• HINGED (text_label - position: β2 = 1.0)
• FIXED (text_label - position: β2 = 0)
• EFFECTIVE LENGTH RATIO l/L = (text_label - position: interior label)

FIG. 27 EFFECTIVE LENGTH RATIOS FOR A COLUMN IN A FRAME WITHOUT RESTRAINT AGAINST SWAY