# 6.6 First-Order Analysis

(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.

**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 _{u}* and

*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 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 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***defined in 6.6.3.1, or using a more detailed analysis, but the value shall not exceed**

*I***.**

*I*_{g}**, shall be calculated by:**

*Q*(6.6.4.4.1) |

where **∑ P_{u}** and

**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}**(**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) |

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

**in Eq. (6.6.4.4.4c) is calculated according to Table 6.6.3.1.1(b) for columns and walls.**

*I***δ**shall be calculated by:

(6.6.4.5.2) |

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

_{m}(a) For columns without transverse loads applied between supports

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

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 ** M_{2,min}** exceeds

**,**

*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*

**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) |

**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 **∑ P_{u}** is the summation of all the factored vertical loads in a story and

**∑**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}(a) Flexural members are continuous

(b) **ε _{t} ≥ 0.0075** at the section at which moment is reduced

**1000ε**percent and 20 percent.

_{t}### Related Code Sections

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