1.3design by inelastic analysis

User Note: Design by the provisions of this section is independent of the require- ments of Section 1.2.

1.3.1 General Requirements

The design strength of the structural system and its members and connections shall equal or exceed the required strength as determined by the inelastic analysis. The provisions of Section 1.3 do not apply to seismic design.

The inelastic analysis shall take into account (a) flexural, shear, axial, and torsional member deformations, and all other component and connection deformations that contribute to the displacements of the structure; (b) second-order effects (including $P-\Delta, P-\delta$, and twisting effects); (c) geometric imperfections; (d) stiffness reductions due to inelasticity, including partial yielding of the cross section that may be accentuated by the presence of residual stresses; and (e) uncertainty in system, member, and connection strength and stiffness.

Strength limit states detected by an inelastic analysis that incorporates all of the preceding requirements in this section are not subject to the corresponding provisions of this Specification when a comparable or higher level of reliability is provided by the analysis. Strength limit states not detected by the inelastic analysis shall be evaluated using the corresponding provisions of Chapters D through K.

Connections shall meet the requirements of Section B3.4.

Members and connections subject to inelastic deformations shall be shown to have ductility consistent with the intended behavior of the structural system. Force redis- tribution due to rupture of a member or connection is not permitted.

Any method that uses inelastic analysis to proportion members and connections to satisfy these general requirements is permitted. A design method based on inelastic analysis that meets the preceding strength requirements, the ductility requirements of Section 1.3.2, and the analysis requirements of Section 1.3.3 satisfies these general requirements.

1.3.2 Ductility Requirements

Members and connections with elements subject to yielding shall be proportioned such that all inelastic deformation demands are less than or equal to their inelastic deformation capacities. In lieu of explicitly ensuring that the inelastic deformation demands are less than or equal to their inelastic deformation capacities, the following requirements shall be satisfied for steel members subject to plastic hinging.

1.3.2a Material

The specified minimum yield stress, Fy, of members subject to plastic hinging shall not exceed 65 ksi (450 MPa).

1.3.2b Cross Section

The cross section of members at plastic hinge locations shall be doubly symmetric with width-to-thickness ratios of their compression elements not exceeding $\lambda_{p d}$, where $\lambda_{p d}$ is equal to $\lambda_{p}$ from Table B4.1b, except as modified in the following:

• (a) For the width-to-thickness ratio, $h / t_{w}$, of webs of I-shaped members, rectangular HSS, and box sections subjected to combined flexure and compression

(1) When $P_{u} / \phi_{c} P_{y} \leq 0.125$

(A-1-1)

• (2) When $P_{u} / \phi_{c} P_{y}>0.125$

(A-1-2)

where

• $\begin{aligned} P_{u} & =\text { required axial strength in compression, using LRFD load combinations, } \\ & \text { kips (N) }\end{aligned}$

• $\begin{aligned} P_{y} & =\text { axial yield strength, kips (N) } \\ & =F_{y} A_{g}\end{aligned}$

• $h \quad$ = as defined in Section B4.1, in. (mm)

• $t_{w}=$ thickness of web, in. (mm)

• = resistance factor for compression = 0.90

• (b) For the width-to-thickness ratio, $b / t$, of flanges of rectangular HSS and box sections, and for flange cover plates between lines of fasteners or welds

$\quad \sigma_{p d}=0.94 \sqrt{E / F_{y}} \quad \text { (A-1-3) }$

where

$b=$ as defined in Section B4.1, in. (mm)

$t=$ as defined in Section B4.1, in. (mm)

• (c) For the diameter-to-thickness ratio, $D / t$, of round HSS in flexure

where

D = outside diameter of round HSS, in. (mm)

1.3.2c Unbraced Length

In prismatic member segments that contain plastic hinges, the laterally unbraced length, $L_{b}$, shall not exceed $L_{p d}$, determined as follows. For members subjected to flexure only, or to flexure and axial tension, $L_{b}$ shall be taken as the length between points braced against lateral displacement of the compression flange, or between points braced to prevent twist of the cross section. For members subjected to flexure and axial compression, $L_{b}$ shall be taken as the length between points braced against both lateral displacement in the minor-axis direction and twist of the cross section.

(a) For I-shaped members bent about their major axis

(A-1-5)

where

$r_{y}=$ radius of gyration about minor axis, in. (mm)

• (1) When the magnitude of the bending moment at any location within the unbraced length exceeds $M_{2}$

$M_{1}^{\prime} / M_{2}=+1$ (A-1-6a)

Otherwise,

• (2) When $M_{\text {mid }} \leq\left(M_{1}+M_{2}\right) / 2$

(A-1-6b)

• (3) When $M_{\text {mid }}>\left(M_{1}+M_{2}\right) / 2$

(A-1-6c)

where

$M_{\text {mid }}=$ moment at middle of unbraced length, kip-in. (N-mm)

$\begin{aligned} M_{1} & =\text { smaller moment at end of unbraced length, kip-in. (N-mm) }\end{aligned}$

• $M_{2} \quad=$ larger moment at end of unbraced length, kip-in. (N-mm) (shall be taken as positive in all cases)
• $\begin{aligned} M_{1}^{\prime} & =\text { effective moment at end of unbraced length opposite from } M_{2} \\ & \text { kip-in. (N-mm) }\end{aligned}$

The moments M1 and Mmid are individually taken as positive when they cause compression in the same flange as the moment M2, and taken as negative otherwise.

• (b) For solid rectangular bars and for rectangular HSS and box sections bent about their major axis

ME E Lpd=0.17-0.10- -ry ≥ 0.10- ry (A-1-7) M2) Fy Fy

For all types of members subjected to axial compression and containing plastic hinges, the laterally unbraced lengths about the cross-section major and minor axes shall not exceed $4.71 r_{x} \sqrt{E / F_{y}}$ and $4.71 r_{y} \sqrt{E / F_{y}}$, respectively.

There is no $L_{p d}$ limit for member segments containing plastic hinges in the following cases:

• (a) Members with round or square cross sections subjected only to flexure or to combined flexure and tension
• (b) Members subjected only to flexure about their minor axis or combined tension and flexure about their minor axis
• (c) Members subjected only to tension

1.3.2d Axial Force

To ensure ductility in compression members with plastic hinges, the design strength in compression shall not exceed $0.75 F_{y} A_{g}$.

1.3.3 Analysis Requirements

The structural analysis shall satisfy the general requirements of Section 1.3.1. These requirements are permitted to be satisfied by a second-order inelastic analysis meeting the requirements of this section.

Exception: For continuous beams not subjected to axial compression, a first-order inelastic or plastic analysis is permitted and the requirements of Sections 1.3.3b and 1.3.3c are waived.

User Note: Refer to the Commentary for guidance in conducting a traditional plastic analysis and design in conformance with these provisions.

1.3a Material Properties and Yield Criteria

The specified minimum yield stress, $F_{y}$, and the stiffness of all steel members and connections shall be reduced by a factor of 0.9 for the analysis, except as stipulated in Section 1.3.3c.

The influence of axial force, major-axis bending moment, and minor-axis bending moment shall be included in the calculation of the inelastic response.

The plastic strength of the member cross section shall be represented in the analysis either by an elastic-perfectly-plastic yield criterion expressed in terms of the axial force, major-axis bending moment, and minor-axis bending moment, or by explicit modeling of the material stress-strain response as elastic-perfectly-plastic.

1.3b Geometric Imperfections

In all cases, the analysis shall directly model the effects of initial imperfections due to both points of intersection of members displaced from their nominal locations (system imperfections) and initial out-of-straightness or offsets of members along their length (member imperfections). The magnitude of the initial displacements shall be the maximum amount considered in the design; the pattern of initial dis- placements shall be such that it provides the greatest destabilizing effect.

1.3c Residual Stress and Partial Yielding Effects

The analysis shall include the influence of residual stresses and partial yielding. This shall be done by explicitly modeling these effects in the analysis or by reducing the stiffness of all structural components as specified in Section C2.3.

If the provisions of Section C2.3 are used, then

• (a) The 0.9 stiffness reduction factor specified in Section 1.3.3a shall be replaced by the reduction of the modulus of elasticity, E, by 0.80 as specified in Section C2.3, and
• (b) The elastic-perfectly-plastic yield criterion, expressed in terms of the axial force, major-axis bending moment, and minor-axis bending moment, shall satisfy the cross-section strength limit defined by Equations H1-1a and H1-1b using Pc = 0.9Py, Mcx = 0.9Mpx, and Mcy = 0.9Mpy.