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

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

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.

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

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

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

**/***M*_{1}**is negative if the column is bent in single curvature, and positive for double curvature.***M*_{2}For composite columns, the radius of gyration,

**, shall not be taken greater than:***r*(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

**and***A*_{sx}**.***I*_{sx}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.

**including second-order effects shall not exceed***M*_{u}**1.4**due to first-order effects.*M*_{u}Relative stiffnesses of members within structural systems shall be based on reasonable and consistent assumptions.

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.

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

For nonprestressed T-beams supporting monolithic or composite slabs, the effective flange width

**shall include the beam web width***b*_{f}**plus an effective overhanging flange width in accordance with Table 6.3.2.1, where***b*_{w}**is the slab thickness and***h***is the clear distance to the adjacent web.***s*_{w}**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/2_{w} | ||

ℓ/8_{n} | ||

One side of web | Least of: | 6h |

s/2_{w} | ||

ℓ/12_{n} |

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

**0.5**and an effective flange width less than or equal to*b*_{w}**4**.*b*_{w}For one-way slabs and beams, it shall be permitted to assume (a) and (b):

(a) Maximum positive

**near midspan occurs with factored***M*_{u}**on the span and on alternate spans***L*(b) Maximum negative

**at a support occurs with factored***M*_{u}**on adjacent spans only***L*If the arrangement of

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

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

**near midspan of panel occurs with 75 percent of factored***M*_{u}**on the panel and alternate panels***L*(b) Maximum negative

**at a support occurs with 75 percent of factored***M*_{u}**on adjacent panels only***L*It shall be permitted to calculate

**and***M*_{u}**due to gravity loads in accordance with this section for continuous beams and one-way slabs satisfying (a) through (e):***V*_{u}(a) Members are prismatic

(b) Loads are uniformly distributed

(c)

*L*≤ 3*D*(d) There are at least two spans

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

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

_{u}**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_{u}ℓ_{n}^{2}/14 |

Discontinuous end unrestrained | w_{u}ℓ_{n}^{2}/11 | ||

Interior spans | All | w_{u}ℓ_{n}^{2}/16 | |

Negative^{[1]} | Interior face of exterior support | Member built integrally with supporting spandrel beam | w_{u}ℓ_{n}^{2}/24 |

Member built integrally with supporting column | w_{u}ℓ_{n}^{2}/16 | ||

Exterior face of first interior support | Two spans | w_{u}ℓ_{n}^{2}/9 | |

More than two spans | w_{u}ℓ_{n}^{2}/10 | ||

Face of other supports | All | w_{u}ℓ_{n}^{2}/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_{u}ℓ_{n}^{2}/12 |

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.

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,

**for columns and walls shall be divided by (***I***1 + β**), where_{ds}**β**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._{ds}**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} | |

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 | ||
---|---|---|---|

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*and_{u}*M*shall be calculated from the load combination under consideration, or the combination of_{u}*P*and_{u}*M*that produces the least value of_{u}*I*.For factored lateral load analysis, it shall be permitted to assume

**for all members or to calculate***I*= 0.5*I*_{g}**by a more detailed analysis, considering the reduced stiffness of all members under the loading conditions.***I*For factored lateral load analysis of two-way slab systems without beams, which are designated as part of the seismic-force-resisting system,

**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.***I*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.

The stability index for a story,

**, shall be calculated by:***Q*(6.6.4.4.1) |

where

**∑**and*P*_{u}**are the total factored vertical load and horizontal story shear, respectively, in the story being evaluated, and***V*_{us}**Δ**is the first-order relative lateral deflection between the top and the bottom of that story due to_{o}**.***V*_{us}For noncomposite columns,

**(**shall be calculated in accordance with (a), (b), or (c):*EI*)_{eff}(a) | (6.6.4.4.4a) |

(b) | (6.6.4.4.4b) |

(c) | (6.6.4.4.4c) |

Magnification factor

**δ**shall be calculated by:(6.6.4.5.2) |

*C*shall be in accordance with (a) or (b):

_{m}(6.6.4.5.3a) |

where

**/***M*_{1}**is negative if the column is bent in single curvature, and positive if bent in double curvature.***M*_{2}**corresponds to the end moment with the lesser absolute value.***M*_{1}C = 1.0_{m} |
(6.6.4.5.3b) |

*M*

_{2}in Eq. (6.6.4.5.1) shall be at least

**calculated according to Eq. (6.6.4.5.4) about each axis separately.**

*M*_{2,min}M_{2,min} = P(0.6 + 0.03_{u}h) |
(6.6.4.5.4) |

If

**exceeds***M*_{2,min}**,***M*_{2}**shall be taken equal to 1.0 or calculated based on the ratio of the calculated end moments***C*_{m}**/***M*_{1}**, using Eq. (6.6.4.5.3a).***M*_{2}*Moment magnification method: Sway frames*

Moments

**and***M*_{1}**at the ends of an individual column shall be calculated by (a) and (b).***M*_{2}(a) M_{1} = M_{1ns}+ δ_{s}M_{1s} | (6.6.4.6.1a) |

(b) M_{2} = M_{2ns}+ δ_{s}M_{2s} | (6.6.4.6.1b) |

The moment magnifier

**δ**shall be calculated by (a), (b), or (c). If_{s}**δ**exceeds 1.5, only (b) or (c) shall be permitted:_{s}(a) | (6.6.4.6.2a) |

(b) | (6.6.4.6.2b) |

(c) Second-order elastic analysis |

where

**∑**is the summation of all the factored vertical loads in a story and*P*_{u}**∑**is the summation for all sway-resisting columns in a story.*P*_{c}**is calculated using Eq. (6.6.4.4.2) with***P*_{c}*k*determined for sway members from 6.6.4.4.3 and**(**from 6.6.4.4.4 or 6.6.4.4.5 as appropriate with*EI*)_{eff}**β**substituted for_{ds}**β**._{dns}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**at the section at which moment is reduced_{t}≥ 0.0075For prestressed members, moments include those due to factored loads and those due to reactions induced by prestressing.

At the section where the moment is reduced, redistribution shall not exceed the lesser of

**1000ε**percent and 20 percent._{t}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.

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

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.

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.

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.

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.

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

Finite element analysis to determine load effects shall be permitted.

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

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

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

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.

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