7.3first-order analysis method

7.3.1 Limitations

When using the first-order analysis method, the following conditions shall be met:

• (a) The structure supports gravity loads primarily through nominally vertical columns, walls, or frames.

• (b) The required axial compressive strengths in nominally horizontal members in moment frames subjected to bending satisfy the following limitation:

(A-7-1)

where

• $\alpha=1.0$ (LRFD); $\alpha=1.6$ (ASD)
• $P_{e}=\pi^{2} E I / L^{2}$, kips (N)
• $E=\operatorname{modulus}$ of elasticity of steel, ksi (MPa)
• = 29,000 ksi (200 000 MPa)
• $I \quad=$ moment of inertia in the plane of bending, in. ${ }^{4}\left(\mathrm{~mm}^{4}\right)$
• $L$ = height of story, in. (mm)
• $P_{r}=$ required axial compressive strength using LRFD or ASD load combinations, kips (N)
• (c) The ratio of maximum second-order drift to maximum first-order drift (both determined for LRFD load combinations or 1.6 times ASD load combinations, with stiffness not adjusted as specified in Section C2.3) in all stories is equal to or less than 1.5.

User Note: The ratio of second-order drift to first-order drift in a story may be taken as the $B_{2}$ multiplier, calculated as specified in Appendix 8.

• (d) The required axial compressive strengths of all members whose flexural stiffnesses are considered to contribute to the lateral stability of the structure satisfy the following limitation:

(A-7-2)

where

$\begin{aligned} P_{n s} & =\text { cross-section compressive strength; for nonslender-element sections, } \\ P_{n s} & =F_{y} A_{g} \text {, and for slender-element sections, } P_{n s}=F_{y} A_{e} \text {, where } A_{e} \text { is as } \\ & \text { defined in Section E7 with } F_{n}=F_{y}, \text { kips (N) }\end{aligned}$

7.3.2 Required Strengths

The required strengths of components shall be determined from a first-order analysis, with additional requirements (a) and (b) given in the following. The analysis shall consider flexural, shear, and axial member deformations, and all other deformations that contribute to displacements of the structure.

(a) All load combinations shall include an additional lateral load, $N_{i}$, applied in combination with other loads at each level of the structure:

(A-7-3)

where

$\alpha=1.0$ (LRFD); $\alpha=1.6$ (ASD)

$Y_{i}=$ gravity load applied at level $i$ from the LRFD load combination or ASD load combination, as applicable, kips (N)

$\Delta / L=$ maximum ratio of $\Delta$ to $L$ for all stories in the structure

• $\Delta$ = first-order interstory drift due to the LRFD or ASD load combination, as applicable, in. (mm). Where $\Delta$ varies over the plan area of the structure, $\Delta$ shall be the average drift weighted in proportion to vertical load or, alternatively, the maximum drift.
• $L \quad$ = height of story, in. (mm)

The additional lateral load at any level, $N_{i}$, shall be distributed over that level in the same manner as the gravity load at the level. The additional lateral loads shall be applied in the direction that provides the greatest destabilizing effect.

User Note: For most building structures, the requirement regarding the direction of $N_{i}$ may be satisfied as follows: (a) For load combinations that do not include lateral loading, consider two alternative orthogonal directions for the additional lateral load in a positive and a negative sense in each of the two directions, same direction at all levels; (b) for load combinations that include lateral loading, apply all the additional lateral loads in the direction of the resultant of all lateral loads in the combination.

• (b) The nonsway amplification of beam-column moments shall be included by applying the $B_{1}$ amplifier of Appendix 8 to the total member moments.

User Note: Since there is no second-order analysis involved in the first-order analysis method for design by ASD, it is not necessary to amplify ASD load combinations by 1.6 before performing the analysis, as required in the direct analysis method and the effective length method.

7.3.3 Available Strengths

The available strengths of members and connections shall be calculated in accordance with the provisions of Chapters D through K, as applicable.

The effective length for flexural buckling of all members shall be taken as the unbraced length unless a smaller value is justified by rational analysis.

Bracing intended to define the unbraced lengths of members shall have sufficient stiffness and strength to limit member movement at the braced points.

User Note: Methods of satisfying this requirement are provided in Appendix 6. The requirements of Appendix 6 are not applicable to bracing that is included in the analysis of the overall structure as part of the overall force-resisting system.