C-7.37.3 first-order analysis method

This section provides a method for designing frames using a first-order elastic analysis with the effective length, $L_{c}$, taken as the laterally unbraced length with $K=1.0$, provided the limitations in Section 7.3.1 are satisfied. This method is derived from the direct analysis method by mathematical manipulation so that the second-order internal forces and moments are determined directly as part of the first-order analysis. AISC Design Guide 28, Stability Design of Steel Buildings (Griffis and White, 2013), presents details of the method. It is based upon a target maximum drift ratio, $\Delta / L$, and assumptions including the following:

• (a) The sidesway amplification, $\Delta_{2 n d \text {-order }} / \Delta_{1 s t \text {-order }}$ or $B_{2}$, is assumed equal to 1.5 .
• (b) The initial out-of-plumbness in the structure is assumed as $\Delta_{o} / L=1 / 500$, but the initial out-of-plumbness does not need to be considered in the calculation of $\Delta$.

The first-order analysis method can be useful for rapid approximate, typically conservative, assessments of basic structures. The various approximations involved in the development of this method can make its accuracy suspect for general moment frames involving nonrectangular configurations of the framing and/or cases such

as clear-span portal frames where the roof girders or rafters resist similar or larger levels of axial compression compared to the nominally vertical members of the framing. Frames in which the axial loads in these nominally horizontal members are larger than approximately 10% of the elastic critical buckling strength of the member in the plane of bending, $P_{e 1}$, given by Equation A-8-5 with $E I^{*}$ taken as $0.8 E I$ and with $L_{c 1}$ taken as the laterally unbraced length of the member with $K=1.0$, can have potential significant stability interactions between the nominally horizontal and nominally vertical members that are not captured within the first-order elastic analysis method procedures. Equation A-7-1 represents a simplification of these details.

The first-order analysis is performed using the nominal (unreduced) stiffness; stiffness reduction is accounted for solely within the calculation of the amplification factors. The nonsway amplification of beam-column moments is addressed within the procedure specified in this section by applying the $B_{1}$ amplifier of Appendix 8, Section 8.1.2, conservatively to the total member moments. In many cases involving beam-columns not subjected to transverse loading between supports in the plane of bending, $B_{1}=1.0$.

The target maximum drift ratio, corresponding to drifts under either the LRFD strength load combinations or 1.6 times the ASD strength load combinations, can be assumed at the start of design to determine the additional lateral load, $N_{i}$. If that drift ratio is not exceeded at any strength load level, the design will be conservative.

If this approach is employed, it can be shown that, for $B_{2} \leq 1.5$ and $\tau_{b}=1.0$, the required additional lateral load to be applied with other lateral loads in a first-order analysis of the structure, using the nominal (unreduced) stiffness, is

(C-A-7-12)

where these variables are as defined in Chapter C, Appendix 7, and Appendix 8. Note that if $B_{2}$ based on the unreduced stiffness is set equal to the 1.5 limit prescribed in Chapter C, then

(C-A-7-13)

This is the additional lateral load required in Section 7.3.2. The minimum value of $N_{i}$ of $0.0042 Y_{i}$ assumes a minimum first-order drift ratio, due to any effects, of $\Delta / L=1 / 500$.