# 1.8 Effective length and frames

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#### A different effective length for frames

Effective length can be found easily on isolated columns by using Table C-C2 in the Commentary of the SpecificationSection of the Manual . However, this table will not work very well with rigid frames. Columns ina frame are not independent, they are continuous. The buckling of one member will affect all the members aroundit. Therefore, the end conditions necessary for using Table C-C2 are not sufficient. It is important to account for thedegree of restraint by connecting members of a column in a frame.

A frame can be unbraced or braced, where unbraced means horizontal displacement in possible. A frame can also havesidesway.

"The rotational restraint provided by beams, or girders, at the end of a column is a function of rotational stiffnessesof the members intersecting at the joint." LRFD Steel Design Second Edition -- William T. Segui, 1999. The restraint is proportional to $\frac{EI}{L}$ .

$G=\frac{\sum \frac{{E}_{c}{I}_{c}}{{L}_{c}}}{\sum \frac{{E}_{g}{I}_{g}}{{L}_{g}}}=\frac{\sum \frac{{I}_{c}}{{L}_{c}}}{\sum \frac{{I}_{g}}{{L}_{g}}}$

$K$ , then depends on the ratio of column stiffness to the girder stiffness at each end.

#### G and k relationships

$K$ is relatively small when a slender column is connected to a girder of large crosssection. This is because the girder effectively prevents rotation and acts as a fixed end. The $G$ value for this case is rather small too. $K$ (or $G$ ) is relatively large when the ends of very stiff columns are connected to ratherflexible beams. This is because the ends of the column can more freely rotate and approach the pinned condition.

"The relationship between $G$ and $K$ has been quantified in the Jackson-Mooreland Alignment Charts (Johnston, 1796),which are reproduced in Figure C-C2.2 in the Commentary. To obtain a value of $K$ from one of the nomograms, first calculate the value of $G$ at each end of the column, letting one value be ${G}_{A}$ and the other be ${G}_{B}$ . Connect ${G}_{A}$ and ${G}_{B}$ with a straight line, and read the value of $K$ on the middle scale. The effective length factor obtained in this manner is withrespect to the axis of bending, which is the axis perpendicular to the plane of the frame. A separateanalysis must be made for buckling about the other axis. Normally the beam-to-column connections in this directionwill not transmit moment, sidesway is prevented by bracing, and $K$ can be taken as 1.0.

#### Value of g for pinned support

$G$ can be taken as 10 at a pinned support because at a pin connection, the situation isjust like a very stiff column attached to infinitely flexible girders. This means the girders have zerostiffness. Then, the ratio of column stiffness to girder stiffness would be infinite for a perfectly frictionlesshinge. This end condition can only be approximated in practice, so the discussion accompanying the alignment chartrecommends that $G$ be taken as 10.

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