  Contents

# Chapter 6 Structural Analysis

### 6.1.1

This chapter shall apply to methods of analysis, modeling of members and structural systems, and calculation of load effects.

### 6.2.1

Members and structural systems shall be permitted to be modeled in accordance with 6.3.

### 6.2.2

All members and structural systems shall be analyzed for the maximum effects of loads including the arrangements of live load in accordance with 6.4.

### 6.2.3

Methods of analysis permitted by this chapter shall be (a) through (e):
(a) The simplified method for analysis of continuous beams and one-way slabs for gravity loads in 6.5
(b) First-order in 6.6
(c) Elastic second-order in 6.7
(d) Inelastic second-order in 6.8
(e) Finite element in 6.9

### 6.2.4

Additional analysis methods that are permitted include 6.2.4.1 through 6.2.4.4.

### 6.2.4.1

Two-way slabs shall be permitted to be analyzed for gravity loads in accordance with (a) or (b):
(a) Direct design method in 8.10
(b) Equivalent frame method in 8.11

### 6.2.4.2

Slender walls shall be permitted to be analyzed in accordance with 11.8 for out-of-plane effects.

### 6.2.4.3

Diaphragms shall be permitted to be analyzed in accordance with 12.4.2.

### 6.2.4.4

A member or region shall be permitted to be analyzed and designed using the strut-and-tie method in accordance with Chapter 23.

### 6.2.5

Slenderness effects shall be permitted to be neglected if (a) or (b) is satisfied:
(a) For columns not braced against sidesway (6.2.5a)
(b) For columns braced against sidesway (6.2.5b)
and (6.2.5c)
where M1/M2 is negative if the column is bent in single curvature, and positive for double curvature.
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
(c) 0.25 times the diameter of circular 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 Asx and Isx.

### 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. Mu including second-order effects shall not exceed 1.4Mu due to first-order effects.

### 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.

### 6.3.2.1

For nonprestressed T-beams supporting monolithic or composite slabs, the effective flange width bf shall include the beam web width bw plus an effective overhanging flange width in accordance with Table 6.3.2.1, where h is the slab thickness and sw 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
sw/2
n/8
One side of web Least of: 6h
sw/2
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 0.5bw and an effective flange width less than or equal to 4bw.

### 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.

### 6.4.1

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 Mu near midspan occurs with factored L on the span and on alternate spans
(b) Maximum negative Mu 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 Mu 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 Mu near midspan of panel occurs with 75 percent of factored L on the panel and alternate panels
(b) Maximum negative Mu at a support occurs with 75 percent of factored L on adjacent panels only

### 6.5.1

It shall be permitted to calculate Mu and Vu due to gravity loads in accordance with this section for continuous beams and one-way slabs satisfying (a) through (e):
(a) Members are prismatic
(c) L ≤ 3D
(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

Mu 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 Mu
Positive End span Discontinuous end integral with support wun2/14
Discontinuous end unrestrained wun2/11
Interior spans All wun2/16
Negative Interior face of exterior support Member built integrally with supporting spandrel beam wun2/24
Member built integrally with supporting column wun2/16
Exterior face of first interior support Two spans wun2/9
More than two spans wun2/10
Face of other supports All wun2/11
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 wun2/12
To calculate negative moments, 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

Vu 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 Vu
Exterior face of first interior support 1.15wun/2
Face of all other supports wun/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.

### 6.6.1.1

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.

### 6.6.2.1

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 and considering conditions of restraint.

### 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.

### 6.6.3.1.1

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 + βds), where βds 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 0.70Ig 1.0Ag
Walls Uncracked 0.70Ig
Cracked 0.35Ig
Beams 0.35Ig
Flat plates and flat slabs 0.25Ig
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
Minimum I Maximum
Columns and walls 0.35Ig 0.875Ig
Beams, flat plates, and flat slabs 0.25Ig 0.5Ig
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. Pu and Mu shall be calculated from the load combination under consideration, or the combination of Pu and Mu that produces the least value of I.

### 6.6.3.1.2

For factored lateral load analysis, it shall be permitted to assume I = 0.5Ig 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.

### 6.6.3.2.1

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 Ig.

### 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

### 6.6.4.4.1

The stability index for a story, Q, shall be calculated by: (6.6.4.4.1)
where Pu and Vus are the total factored vertical load and horizontal story shear, respectively, in the story being evaluated, and Δo is the first-order relative lateral deflection between the top and the bottom of that story due to Vus.

### 6.6.4.4.2

The critical buckling load Pc shall be calculated by: (6.6.4.4.2)

### 6.6.4.4.3

The effective length factor k shall be calculated using Ec 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 βdns 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)

### 6.6.4.5.1

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

Cm 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.
 Cm = 1.0 (6.6.4.5.3b)

### 6.6.4.5.4

M2 in Eq. (6.6.4.5.1) shall be at least M2,min calculated according to Eq. (6.6.4.5.4) about each axis separately.
 M2,min = Pu(0.6 + 0.03h) (6.6.4.5.4)
If M2,min exceeds M2, Cm 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 = M1ns+ δsM1s (6.6.4.6.1a) (b) M2 = M2ns+ δsM2s (6.6.4.6.1b)

### 6.6.4.6.2

The moment magnifier δs shall be calculated by (a), (b), or (c). If δs 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 Pu is the summation of all the factored vertical loads in a story and Pc is the summation for all sway-resisting columns in a story. Pc 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 βds substituted for βdns.

### 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 Cm is calculated using M1 and M2 from 6.6.4.6.1.

### 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) εt ≥ 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 1000εt 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.

### 6.7.1.1

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.

### 6.7.1.2

Slenderness effects along the length of a column shall be considered. It shall be permitted to calculate these effects using 6.6.4.5.

### 6.7.1.3

The cross-sectional dimensions of each member used in an analysis to calculate slenderness effects shall be within 10 percent of the specified member dimensions in construction documents or the analysis shall be repeated.

### 6.7.1.4

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

### 6.7.2.1.1

It shall be permitted to use section properties calculated in accordance with 6.6.3.1.

### 6.7.2.2.1

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

### 6.7.2.2.2

Alternatively, it shall be permitted to calculate immediate deflections using a moment of inertia of 1.4 times I given in 6.6.3.1, or calculated using a more detailed analysis, but the value shall not exceed Ig.

### 6.8.1.1

An inelastic second-order analysis shall consider material nonlinearity, member curvature and lateral drift, duration of loads, shrinkage and creep, and interaction with the supporting foundation.

### 6.8.1.2

An inelastic second-order analysis procedure shall have been shown to result in prediction of strength in substantial agreement with results of comprehensive tests of statically indeterminate reinforced concrete structures.

### 6.8.1.3

Slenderness effects along the length of a column shall be considered. It shall be permitted to calculate these effects using 6.6.4.5.

### 6.8.1.4

The cross-sectional dimensions of each member used in an analysis to calculate slenderness effects shall be within 10 percent of the specified member dimensions in construction documents or the analysis shall be repeated.

### 6.8.1.5

Redistribution of moments calculated by an inelastic second-order analysis shall not be permitted.

### 6.9.1

Finite element analysis to determine load effects shall be permitted.

### 6.9.2

The finite element model shall be appropriate for its intended purpose.

### 6.9.3

For inelastic analysis, a separate analysis shall be performed for each factored load combination.

### 6.9.4

The licensed design professional shall confirm that the results are appropriate for the purposes of the analysis.

### 6.9.5

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.

### 6.9.6

Redistribution of moments calculated by an inelastic analysis shall not be permitted.