In theory at least, the design of a column or of a beam-column starts with the evaluation of the elastic restraints at both ends of the column, from which the effective length factor k is then derived. To get a k-factor, the designer is much more likely to use the two charts provided in the Column Design section
of the AISC Manuals 1,2 rather than to solve the transcendental equations on which the charts are based.
However, having to read k-factors from an alignment chart in the middle of an electronic computation, in a spreadsheet for instance, prevents full automation and can be a source of errors. The fact that spreadsheets cannot accept so-called circular references makes their use awkward for the automatic
solution of transcendental equations. A side benefit of an excellent article by Barakat and Chen was the demonstration of how powerful an engineering tool the electronic spreadsheet can be: it automates many routine calculations, and it is well suited for tedious column and beam-column calculations.
Barakat and Chen did not elaborate on how they obtained the k-factors used in their examples; from the context, it seems that the factors were manually entered into the spreadsheet. Obviously, it would be convenient to have simple
equations take the place of the charts in the AISC Manuals. The American Concrete Institute does publish equations, but their lack of accuracy may be why they seem not to be used in steel design. Better equations have been available in the
French Design Rules for Steel Structures since 1966, and have been included in the European Recommendations of 1978, with only a change in notation. These equations are accurate, yet simple enough to be easily programmed within the confines of a spreadsheet cell. For this reason, they may be useful to North American engineers.
1. EXACT AND APPROXIMATE EQUATIONS
Consider a column AB elastically restrained at both ends. The rotational restraint at one end, A for instance, is represented by a restraint factor G4, expressing the relative stiffness of all the columns connected at A to that of all the beams framing into A:
1.1. Braced Frames
Braced frames are frames in which the sidesway is effectively prevented, and, therefore, the ^-factor is never greater than 1.0. The "sidesway inhibited" alignment chart is the graphic solution of the following mathematical equation:
This equation is mathematically exact, in that certain physical assumptions are exactly translated in mathematical terms. Whether these assumptions can be reasonably extended to a specific structure is a matter for the designer to
decide.
For the transcendental Eq. 2, which can only be solved by numerical methods, the French Rules propose the following approximate solution:
Slightly simpler equations apply to special cases. If the column is hinged at B,GB is infinitely large, and 1 / GB = 0
If, instead, the column is fully fixed at B, GB = 0:
Finally, in the not infrequent case where GA = GB= G:
1.2. Sway Frames
If a rigid frame depends solely on frame action to resist lateral forces, its sidesway is not prevented. In this case, the K-factor is never smaller than 1.0. The mathematical equation for the "sway uninhibited" case is:
Although simpler than Eq. 2, this equation cannot be solved in closed form either. The French Rules recommend the following approximate solution:
For a hinge at B, the formula simplifies to:
For complete fixity at B, the approximation is:
When GA = GB= G:
1.3. Accuracy of Equations
The accuracy that we can readily measure is of course the mathematical accuracy, that is, the comparison of the results given by an approximate formula to those obtained by solving the corresponding "exact" equation. The accuracy of the alignment charts depends essentially on the size of the charts,
and on the reader's sharpness of vision. For the small charts in the Column Section of the AISC Manuals, this accuracy may be about five percent. In view of the many simplifying assumptions needed to arrive at Eqs. 2 and 7, this accuracy is certainly sufficient.
The formula proposed by the ACI for braced frames gives K = 0.7 for a beam fully fixed at both ends, instead of 0.5. If GA = GB = 3.0, it yields K=l.O, instead of the expected 0.89.
The equations for unbraced frames are somewhat better: for GA = GB = 2.0 for instance, they yield K = 1.56, instead of 1.61. The French Rules indicate that Eq. 3 has an accuracy of -0.5 percent to +1.5 percent, while Eq. 8 is accurate within two percent. Tables 1 and 2 report the accuracies found at a few sample points. Again because of the nature of the surrounding assumptions, Eqs. 3 and 8 may be considered mathematically exact.
