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For unbraced frames where

Equation 31.28 shall be used to first calculate K. If the value of K calculated by Equation 31.28 is greater than 2, Equation 31.29 shall then be used.

31.4.3.3 French Equations

For braced frames

For unbraced frames f

Equations 31.32 and 31.33 first appeared in the French Rules for the Design of Steel Structure in 1966 (CM 1966) and were later incorporated into the European Recommendation for Steel Construction in 1978 (ECCS 1978). Neither French CM 66 Rules nor the European Recommendations give the origin of these two formulas. They provide a good approximation to the alignment charts (Dumonteil 1992). For braced frames, Equation 31.32 underestimates K by not more than 0.5% and overestimates it by 1.5%. For unbraced frames, Equation 31.33 approximates K within 2% (Dumonteil 1999).

31.4.3.4 Donnell Equation for Braced Frames

The Donnell equation, developed in 1949 as reported by Rondal (1988), has the following format:

This is the formula presented by Dumonteil and Valley (1995) in their discussion paper. Its accuracy ranges from 0.4 to 1.3%.

31.4.3.5 Newmark Equation for Braced Frames

Introducing the G-factor, the Newmark (1949) equation has the following form:

While the accuracy of Equation 31.35 is remarkable, it could still be slightly improved by replacing the term 4/p2 with 0.41, and the Newmark equation becomes (Dumonteil 1999)

Equation 31.36 underestimates K by no more than 0.1% and overestimates it by less than 1.5%.

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