8.1approximate second-order elastic analysis

Second-order effects in structures may be approximated by amplifying the required strengths determined by two first-order elastic analyses. The use of this procedure is limited to structures that support gravity loads primarily through nominally vertical columns, walls, or frames, except that it is permissible to use the procedure specified for determining $P-\delta$ effects for any individual compression member. This method is not permitted for design by advanced analysis using the provisions of Appendix 1.

User Note: The two first-order elastic analyses are (1) restrained against lateral translation (nt-no translation), and (2) lateral translation (lt), using the subscript notation in Equations A-8-1 and A-8-2.

8.1.1 Calculation Procedure

The required second-order flexural strength, $M_{r}$, and axial strength, $P_{r}$, of all members shall be determined as

where

• $B_{1}=$ multiplier to account for $P-\delta$ effects, determined for each member subjected to compression and flexure, and each direction of bending of the member in accordance with Appendix 8, Section 8.1.2. $B_{1}$ shall be taken as 1.0 for members not subjected to compression.

• $B_{2}=$ multiplier to account for $P-\Delta$ effects, determined for each story of the structure and each direction of lateral translation of the story in accordance with Appendix 8, Section 8.1.3.

• $M_{l t}=$ first-order moment using LRFD or ASD load combinations, due to lateral translation of the structure only, kip-in. (N-mm)

• $M_{n t}=$ first-order moment using LRFD or ASD load combinations, with the structure restrained against lateral translation, kip-in. (N-mm)

• $M_{r}=$ required second-order flexural strength using LRFD or ASD load combinations, kip-in. (N-mm)

• $P_{l t}=$ first-order axial force using LRFD or ASD load combinations, due to lateral translation of the structure only, kips (N)

• $P_{n t}=$ first-order axial force using LRFD or ASD load combinations, with the structure restrained against lateral translation, kips (N)

• $P_{r}=$ required second-order axial strength using LRFD or ASD load combinations, kips (N)

User Note: Equations A-8-1 and A-8-2 are applicable to all members in all structures. Note, however, that $B_{1}$ values other than unity apply only to moments in beam-columns; $B_{2}$ applies to moments and axial forces in components of the lateral force-resisting system (including columns, beams, bracing members, and shear walls). See the Commentary for more on the application of Equations A-8-1 and A-8-2.

8.1.2 Multiplier $B_{1}$ for $P-\delta$ Effects

The $B_{1}$ multiplier for each member subjected to compression and each direction of bending of the member is calculated as follows:

(A-8-3)

where

• $\alpha=1.0$ (LRFD); $\alpha=1.6$ (ASD)
• $C_{m}=$ equivalent uniform moment factor, assuming no relative translation of the member ends, determined as follows:
• (a) For beam-columns not subjected to transverse loading between supports in the plane of bending

(A-8-4)

where $M_{1}$ and $M_{2}$, calculated from a first-order analysis, are the smaller and larger moments, respectively, at the ends of that portion of the member unbraced in the plane of bending under consideration. $M_{1} / M_{2}$ is positive when the member is bent in reverse curvature and negative when bent in single curvature.

• (b) For beam-columns subjected to transverse loading between supports, the value of $C_{m}$ shall be determined either by analysis or conservatively taken as 1.0 for all cases.
• $\begin{aligned} P_{e 1} & =\text { elastic critical buckling strength of the member in the plane of bending, } \\ & \text { calculated based on the assumption of no lateral translation at the member } \\ & \text { ends, kips (N) }\end{aligned}$

$=\frac{\pi^{2}EI^{*}}{\left(L_{c1}\right)^{2}}$ (A-8-5)

• $\begin{aligned} E I^{*} & =\text { flexural rigidity required to be used in the analysis, kip-in. }^{2}(\mathrm{~N}-\mathrm{mm}^{2}) \\ & \left(=0.8 \tau_{b} E I \text { when used in the direct analysis method, where } \tau_{b} \text { is as defined }\right. \\ & \left.\text { in Chapter C; }=E I \text { for the effective length and first-order analysis methods }\right)\end{aligned}$

• $E \quad=$ modulus of elasticity of steel

• = 29,000 ksi (200 000 MPa)

• $I \quad=$ moment of inertia in the plane of bending, in. ${ }^{4}\left(\mathrm{~mm}^{4}\right)$

• $L_{c 1}=$ effective length in the plane of bending, calculated based on the assumption of no lateral translation at the member ends, set equal to the laterally unbraced length of the member unless analysis justifies a smaller value, in.

(mm)

It is permitted to use the first-order estimate of $P_{r}$ (i.e., $P_{r}=P_{n t}+P_{l t}$ ) in Equation A-8-3.

8.1.3 Multiplier $B_{2}$ for $P-\Delta$ Effects

The $B_{2}$ multiplier for each story and each direction of lateral translation is calculated as follows:

(A-8-6)

where

• $\alpha \quad=1.0$ (LRFD); $\alpha=1.6$ (ASD)
• $P_{\text {story }}=$ total vertical load supported by the story using LRFD or ASD load combinations, as applicable, including loads in columns that are not part of the lateral force-resisting system, kips (N)
• $P_{e \text { story }}=$ elastic critical buckling strength for the story in the direction of translation being considered, kips (N), determined by sidesway buckling analysis or as given in Equation A-8-7.

(A-8-7)

• $H$ = total story shear, in the direction of translation being considered, produced by the lateral forces used to compute $\Delta_{H}$, kips (N)
• $L \quad$ =height of story, in. (mm)

(A-8-8)

• $\begin{aligned} P_{m f} & =\text { total vertical load in columns in the story that are part of moment frames, } \\ & \text { if any, in the direction of translation being considered } (=0 \text { for braced- } \\ & \text { frame systems), kips (N) }\end{aligned}$
• $\Delta_{H}$ = first-order interstory drift, in the direction of translation being considered, due to lateral forces, in. (mm), computed using the stiffness required to be used in the analysis. (When the direct analysis method is used, stiffness is reduced according to Section C2.3.) Where $\Delta_{H}$ varies over the plan area of the structure, it shall be the average drift weighted in proportion to vertical load or, alternatively, the maximum drift.

User Note: The story gravity load ( $P_{\text {story }}$ and $P_{m f}$ ) includes loading from levels above and on nonframe columns and walls, and the weight of wall panels laterally supported by the lateral force-resisting system; it need not include the vertical component of the seismic force.

User Note: $R_{M}$ can be taken as 0.85 as a lower-bound value for stories that include moment frames, and $R_{M}=1$ if there are no moment frames in the story. $H$ and $\Delta_{H}$ in Equation A-8-7 may be based on any lateral loading that provides a representative value of story lateral stiffness, $H / \Delta_{H}$.