If bracing elements resisting lateral movement of a story have a total stiffness of at least 12 times the gross lateral stiffness of the columns in the direction considered, it shall be permitted to consider columns within the story to be braced against sidesway.

6.2.5.1

The radius of gyration, r, shall be permitted to be calculated by (a), (b), or (c):

(a)

(6.2.5.1)

(b) 0.30 times the dimension in the direction stability is being considered for rectangular columns

6.2.5.2

For composite columns, the radius of gyration, r, shall not be taken greater than:

(6.2.5.2)

Longitudinal bars located within a concrete core encased by structural steel or within transverse reinforcement surrounding a structural steel core shall be permitted to be used in calculating A_sx and I_sx.

6.2.6

Unless slenderness effects are neglected as permitted by 6.2.5, the design of columns, restraining beams, and other supporting members shall be based on the factored forces and moments considering second-order effects in accordance with 6.6.4, 6.7, or 6.8. M_u including second-order effects shall not exceed 1.4M_u due to first-order effects.

6.3 Modeling Assumptions

6.3.1 General

6.3.1.1

Relative stiffnesses of members within structural systems shall be based on reasonable and consistent assumptions.

6.3.1.2

To calculate moments and shears caused by gravity loads in columns, beams, and slabs, it shall be permitted to use a model limited to the members in the level being considered and the columns above and below that level. It shall be permitted to assume far ends of columns built integrally with the structure to be fixed.

6.3.1.3

The analysis model shall consider the effects of variation of member cross-sectional properties, such as that due to haunches.

6.3.2 T-Beam Geometry

6.3.2.1

For nonprestressed T-beams supporting monolithic or composite slabs, the effective flange width b_f shall include the beam web width b_w plus an effective overhanging flange width in accordance with Table 6.3.2.1, where h is the slab thickness and s_w is the clear distance to the adjacent web.

Table 6.3.2.1—Dimensional limits for effective overhanging flange width for T-beams

Flange location	Effective overhanging flange width, beyond face of web
Each side of web	Least of:	8h
		s_w/2
		ℓ_n/8
One side of web	Least of:	6h
		s_w/2
		ℓ_n/12

6.3.2.2

Isolated nonprestressed T-beams in which the flange is used to provide additional compression area shall have a flange thickness greater than or equal to 0.5b_w and an effective flange width less than or equal to 4b_w.

6.3.2.3

For prestressed T-beams, it shall be permitted to use the geometry provided by 6.3.2.1 and 6.3.2.2.

6.4 Arrangement of Live Load

6.4.1

For the design of floors or roofs to resist gravity loads, it shall be permitted to assume that live load is applied only to the level under consideration.

6.4.2

For one-way slabs and beams, it shall be permitted to assume (a) and (b):

(a) Maximum positive M_u near midspan occurs with factored L on the span and on alternate spans

(b) Maximum negative M_u at a support occurs with factored L on adjacent spans only

6.4.3

For two-way slab systems, factored moments shall be calculated in accordance with 6.4.3.1, 6.4.3.2, or 6.4.3.3, and shall be at least the moments resulting from factored L applied simultaneously to all panels.

6.4.3.1

If the arrangement of L is known, the slab system shall be analyzed for that arrangement.

6.4.3.2

If L is variable and does not exceed 0.75D, or the nature of L is such that all panels will be loaded simultaneously, it shall be permitted to assume that maximum M_u at all sections occurs with factored L applied simultaneously to all panels.

6.4.3.3

For loading conditions other than those defined in 6.4.3.1 or 6.4.3.2, it shall be permitted to assume (a) and (b):

(a) Maximum positive M_u near midspan of panel occurs with 75 percent of factored L on the panel and alternate panels

(b) Maximum negative M_u at a support occurs with 75 percent of factored L on adjacent panels only

6.5 Simplified Method of Analysis for Nonprestressed Continuous Beams and One-Way Slabs

6.5.1

It shall be permitted to calculate M_u and V_u due to gravity loads in accordance with this section for continuous beams and one-way slabs satisfying (a) through (e):

(a) Members are prismatic

(b) Loads are uniformly distributed

(d) There are at least two spans

(e) The longer of two adjacent spans does not exceed the shorter by more than 20 percent

6.5.2

M_u due to gravity loads shall be calculated in accordance with Table 6.5.2.

Table 6.5.2—Approximate moments for nonprestressed continuous beams and one-way slabs

Moment	Location	Condition	M_u
Positive	End span	Discontinuous end integral with support	w_uℓ_n²/14
	End span	Discontinuous end unrestrained	w_uℓ_n²/11
	Interior spans	All	w_uℓ_n²/16
Negative^[1]	Interior face of exterior support	Member built integrally with supporting spandrel beam	w_uℓ_n²/24
	Interior face of exterior support	Member built integrally with supporting column	w_uℓ_n²/16
	Exterior face of first interior support	Two spans	w_uℓ_n²/9
	Exterior face of first interior support	More than two spans	w_uℓ_n²/10
	Face of other supports	All	w_uℓ_n²/11
	Face of all supports satisfying(a) or (b)	(a) slabs with spans not exceeding 10 ft (b) beams where ratio of sum of column stiffnesses to beam stiffness exceeds 8 at each end of span	w_uℓ_n²/12

^[1]To calculate negative moments, ℓ_n shall be the average of the adjacent clear span lengths.

6.5.3

Moments calculated in accordance with 6.5.2 shall not be redistributed.

6.5.4

V_u due to gravity loads shall be calculated in accordance with Table 6.5.4.

Table 6.5.4—Approximate shears for nonprestressed continuous beams and one-way slabs

Location	V_u
Exterior face of first interior support	1.15w_uℓ_n/2
Face of all other supports	w_uℓ_n/2

6.5.5

Floor or roof level moments shall be resisted by distributing the moment between columns immediately above and below the given floor in proportion to the relative column stiffnesses considering conditions of restraint.

6.6 First-Order Analysis

6.6.1 General

6.6.1.1

Slenderness effects shall be considered in accordance with 6.6.4, unless they are allowed to be neglected by 6.2.5.

6.6.1.2

Redistribution of moments calculated by an elastic first-order analysis shall be permitted in accordance with 6.6.5.

6.6.2 Modeling of Members and Structural Systems

6.6.2.1

6.6.2.2

For frames or continuous construction, consideration shall be given to the effect of floor and roof load patterns on transfer of moment to exterior and interior columns, and of eccentric loading due to other causes.

6.6.2.3

It shall be permitted to simplify the analysis model by the assumptions of (a), (b), or both:

(a) Solid slabs or one-way joist systems built integrally with supports, with clear spans not more than 10 ft, shall be permitted to be analyzed as continuous members on knife-edge supports with spans equal to the clear spans of the member and width of support beams otherwise neglected.

(b) For frames or continuous construction, it shall be permitted to assume the intersecting member regions are rigid.

6.6.3 Section Properties

6.6.3.1 Factored Load Analysis

6.6.3.1.1

Moment of inertia and cross-sectional area of members shall be calculated in accordance with Tables 6.6.3.1.1(a) or 6.6.3.1.1(b), unless a more rigorous analysis is used. If sustained lateral loads are present, I for columns and walls shall be divided by (1 + β_ds), where β_ds is the ratio of maximum factored sustained shear within a story to the maximum factored shear in that story associated with the same load combination.

Table 6.6.3.1.1(a)—Moment of inertia and crosssectional area permitted for elastic analysis at factored load level

Member and condition		Moment of Inertia	Cross-sectional area
Columns		0.70I_g	1.0A_g
Walls	Uncracked	0.70I_g
Walls	Cracked	0.35I_g
Beams		0.35I_g
Flat plates and flat slabs		0.25I_g

Table 6.6.3.1.1(b)—Alternative moments of inertia for elastic analysis at factored load

Member	Alternative value of I for elastic analysis
Member	Minimum	I	Maximum
Columns and walls	0.35I_g		0.875I_g
Beams, flat plates, and flat slabs	0.25I_g		0.5I_g

Notes: For continuous flexural members, I shall be permitted to be taken as the average of values obtained for the critical positive and negative moment sections. P_u and M_u shall be calculated from the load combination under consideration, or the combination of P_u and M_u that produces the least value of I.

6.6.3.1.2

For factored lateral load analysis, it shall be permitted to assume I = 0.5I_g for all members or to calculate I by a more detailed analysis, considering the reduced stiffness of all members under the loading conditions.

6.6.3.1.3

For factored lateral load analysis of two-way slab systems without beams, which are designated as part of the seismic-force-resisting system, I for slab members shall be defined by a model that is in substantial agreement with results of comprehensive tests and analysis and I of other frame members shall be in accordance with 6.6.3.1.1 and 6.6.3.1.2.

6.6.3.2 Service Load Analysis

6.6.3.2.1

Immediate and time-dependent deflections due to gravity loads shall be calculated in accordance with 24.2.

6.6.3.2.2

It shall be permitted to calculate immediate lateral deflections using a moment of inertia of 1.4 times I defined in 6.6.3.1, or using a more detailed analysis, but the value shall not exceed I_g.

6.6.4 Slenderness Effects, Moment Magnification Method

6.6.4.1

Unless 6.2.5 is satisfied, columns and stories in structures shall be designated as being nonsway or sway. Analysis of columns in nonsway frames or stories shall be in accordance with 6.6.4.5. Analysis of columns in sway frames or stories shall be in accordance with 6.6.4.6.

6.6.4.2

The cross-sectional dimensions of each member used in an analysis shall be within 10 percent of the specified member dimensions in construction documents or the analysis shall be repeated. If the stiffnesses of Table 6.6.3.1.1(b) are used in an analysis, the assumed member reinforcement ratio shall also be within 10 percent of the specified member reinforcement in construction documents.

6.6.4.3

It shall be permitted to analyze columns and stories in structures as nonsway frames if (a) or (b) is satisfied:

(a) The increase in column end moments due to second-order effects does not exceed 5 percent of the first-order end moments

(b) Q in accordance with 6.6.4.4.1 does not exceed 0.05

6.6.4.4 Stability Properties

6.6.4.4.1

The stability index for a story, Q, shall be calculated by:

(6.6.4.4.1)

where ∑P_u and V_us are the total factored vertical load and horizontal story shear, respectively, in the story being evaluated, and Δ_o is the first-order relative lateral deflection between the top and the bottom of that story due to V_us.

6.6.4.4.2

The critical buckling load P_c shall be calculated by:

(6.6.4.4.2)

6.6.4.4.3

The effective length factor k shall be calculated using E_c in accordance with 19.2.2 and I in accordance with 6.6.3.1.1. For nonsway members, k shall be permitted to be taken as 1.0, and for sway members, k shall be at least 1.0.

6.6.4.4.4

For noncomposite columns, (EI)_eff shall be calculated in accordance with (a), (b), or (c):

(a)	(6.6.4.4.4a)
(b)	(6.6.4.4.4b)
(c)	(6.6.4.4.4c)

where β_dns shall be the ratio of maximum factored sustained axial load to maximum factored axial load associated with the same load combination and I in Eq. (6.6.4.4.4c) is calculated according to Table 6.6.3.1.1(b) for columns and walls.

6.6.4.4.5

For composite columns, (EI)_eff shall be calculated by Eq. (6.6.4.4.4b), Eq. (6.6.4.4.5), or from a more detailed analysis.

(6.6.4.4.5)

6.6.4.5 Moment Magnification Method: Nonsway Frames

6.6.4.5.1

The factored moment used for design of columns and walls, M_c, shall be the first-order factored moment M₂ amplified for the effects of member curvature.

M_c = δM₂

(6.6.4.5.1)

6.6.4.5.2

Magnification factor δ shall be calculated by:

(6.6.4.5.2)

6.6.4.5.3

C_m shall be in accordance with (a) or (b):

(a) For columns without transverse loads applied between supports

(6.6.4.5.3a)

where M₁/M₂ is negative if the column is bent in single curvature, and positive if bent in double curvature. M₁ corresponds to the end moment with the lesser absolute value.

(b) For columns with transverse loads applied between supports.

C_m = 1.0

(6.6.4.5.3b)

6.6.4.5.4

M₂ in Eq. (6.6.4.5.1) shall be at least M_2,min calculated according to Eq. (6.6.4.5.4) about each axis separately.

M_2,min = P_u(0.6 + 0.03h)

(6.6.4.5.4)

If M_2,min exceeds M₂, C_m shall be taken equal to 1.0 or calculated based on the ratio of the calculated end moments M₁/M₂, using Eq. (6.6.4.5.3a).

6.6.4.6

Moment magnification method: Sway frames

6.6.4.6.1

Moments M₁ and M₂ at the ends of an individual column shall be calculated by (a) and (b).

(a) M₁ = M_1ns+ δ_sM_1s	(6.6.4.6.1a)
(b) M₂ = M_2ns+ δ_sM_2s	(6.6.4.6.1b)

6.6.4.6.2

The moment magnifier δ_s shall be calculated by (a), (b), or (c). If δ_s exceeds 1.5, only (b) or (c) shall be permitted:

(a)	(6.6.4.6.2a)
(b)	(6.6.4.6.2b)
(c) Second-order elastic analysis

where ∑P_u is the summation of all the factored vertical loads in a story and ∑P_c is the summation for all sway-resisting columns in a story. P_c is calculated using Eq. (6.6.4.4.2) with k determined for sway members from 6.6.4.4.3 and (EI)_eff from 6.6.4.4.4 or 6.6.4.4.5 as appropriate with β_ds substituted for β_dns.

6.6.4.6.3

Flexural members shall be designed for the total magnified end moments of the columns at the joint.

6.6.4.6.4

Second-order effects shall be considered along the length of columns in sway frames. It shall be permitted to account for these effects using 6.6.4.5, where C_m is calculated using M₁ and M₂ from 6.6.4.6.1.

6.6.5 Redistribution of Moments in Continuous Flexural Members

6.6.5.1

Except where approximate values for moments are used in accordance with 6.5, where moments have been calculated in accordance with 6.8, or where moments in two-way slabs are determined using pattern loading specified in 6.4.3.3, reduction of moments at sections of maximum negative or maximum positive moment calculated by elastic theory shall be permitted for any assumed loading arrangement if (a) and (b) are satisfied:

(a) Flexural members are continuous

(b) ε_t ≥ 0.0075 at the section at which moment is reduced

6.6.5.2

For prestressed members, moments include those due to factored loads and those due to reactions induced by prestressing.

6.6.5.3

At the section where the moment is reduced, redistribution shall not exceed the lesser of 1000ε_t percent and 20 percent.

6.6.5.4

The reduced moment shall be used to calculate redistributed moments at all other sections within the spans such that static equilibrium is maintained after redistribution of moments for each loading arrangement.

6.6.5.5

Shears and support reactions shall be calculated in accordance with static equilibrium considering the redistributed moments for each loading arrangement.

6.7 Elastic Second-Order Analysis

6.7.1 General

6.7.1.1

An elastic second-order analysis shall consider the influence of axial loads, presence of cracked regions along the length of the member, and effects of load duration. These considerations are satisfied using the cross-sectional properties defined in 6.7.2.