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# A509.4 Displacement Coefficient Analysis Procedure

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This section presents a procedure for generalized nonlinear static analysis for verification of acceptable performance by comparing the available capacity to the earthquake demand.

Where inelastic torsional response is a dominant feature of overall response, the engineer shall use either a retrofit that reduces the torsional response or an alternative analysis procedure. Inelastic torsional response may be deemed to exist if there is torsional irregularity as defined in Section A508.2 present in any story.

The mathematical model of the building shall be determined in accordance with Section A509.1. The general procedure for execution of the displacement coefficient analysis shall be determined in accordance with Section A509.4.5.

Results of the displacement coefficient analysis procedure shall be checked using the applicable acceptance criteria specified in Section A509.1.2.

For three-dimensional analyses, the static lateral forces shall be imposed on the three-dimensional mathematical model corresponding to the mass distribution at each story level. Effects of accidental torsion shall be considered.

For two-dimensional analyses, the mathematical model describing the framing along each axis of the building shall be developed. The effects of horizontal torsion shall be considered by increasing the target displacement (see Section A509.4.2) by a displacement multiplier, η. The displacement multiplier is the ratio of the maximum displacement at any point on any floor diaphragm (including torsional effects for actual torsion and accidental torsion) to the average displacement on that diaphragm.

The behavior of foundation components and effects of soil-structure interaction shall be modeled or shown to be insignificant to building response.

Where inelastic torsional response is a dominant feature of overall response, the engineer shall use either a retrofit that reduces the torsional response or an alternative analysis procedure. Inelastic torsional response may be deemed to exist if there is torsional irregularity as defined in Section A508.2 present in any story.

The mathematical model of the building shall be determined in accordance with Section A509.1. The general procedure for execution of the displacement coefficient analysis shall be determined in accordance with Section A509.4.5.

Results of the displacement coefficient analysis procedure shall be checked using the applicable acceptance criteria specified in Section A509.1.2.

For three-dimensional analyses, the static lateral forces shall be imposed on the three-dimensional mathematical model corresponding to the mass distribution at each story level. Effects of accidental torsion shall be considered.

For two-dimensional analyses, the mathematical model describing the framing along each axis of the building shall be developed. The effects of horizontal torsion shall be considered by increasing the target displacement (see Section A509.4.2) by a displacement multiplier, η. The displacement multiplier is the ratio of the maximum displacement at any point on any floor diaphragm (including torsional effects for actual torsion and accidental torsion) to the average displacement on that diaphragm.

The behavior of foundation components and effects of soil-structure interaction shall be modeled or shown to be insignificant to building response.

The target displacement of the control node (typically the center of mass of the building's roof) shall be determined using the following equation:

where:

where:

C_{0} = | Modification factor to relate spectral displacement to expected building roof displacement. Value of C_{0 }can be estimated using any one of the following: | |

1. | The first modal participation factor at the level of the control node. | |

2. | The modal participation factor at the level of the control node computed using a shape vector corresponding to the deflected shape of the building at the target displacement. | |

3. | The appropriate value from Table A509.4.2. |

TABLE A509.4.2 VALUES OF MODIFICATION FACTOR,TABLE A509.4.2 VALUES OF MODIFICATION FACTOR,

*C*_{0} NUMBER OF STORIES | C_{0} |

1 | 1.0 |

2 | 1.2 |

3 | 1.3 |

5 | 1.4 |

10+ | 1.5 |

NOTE: Linear interpolation shall be used to calculate intermediate values. | |||||

C_{1} | = | Modification factor to relate expected maximum inelastic displacements to displacements for linear elastic response. C_{1} shall not be taken as less than 1.0. | |||

= | 1.0 for T_{e} ≥ T_{0} | ||||

= | [1.0 + (R - 1)T_{0}/T_{e}]/R for T_{e} < T_{0} | ||||

Where: | |||||

R | = | Strength ratio = | |||

V_{y} | = | Yield strength calculated using the results of static pushover analysis where the nonlinear base-shear roof-displacement curve of the building is characterized by a bilinear relation (see Section A509.4.5). | |||

T_{0} | = | Characteristic period of the response spectrum, defined as the period associated with the transition from the constant acceleration segment of the spectrum to the constant velocity segment of the spectrum. | |||

C_{2} | = | Modification factor to represent the effect of hysteresis shape on maximum displacement response. | |||

= | 1.3 where T > T_{0} | ||||

= | 1.1 where T ≥ T_{0} | ||||

Exception: Where the stiffness of the structural component in a lateral-force-resisting system, which resists no less than 30 percent of the story shear, does not deteriorate at the target displacement level, C_{2} may be assumed to be equal to 1.0. | |||||

S_{a} | = | Response spectral acceleration at the effective fundamental period and damping ratio of the building, g, in the direction under consideration. | |||

T_{e} | = | Effective fundamental period of the building in the direction under consideration, per Section A509.4.5. |

Two different vertical distributions of loads shall be used. The first load pattern, termed as the uniform pattern, shall be based on lateral forces proportional to the mass at each story level. The second pattern, called the modal pattern, shall be selected from one of the following:

- A lateral load pattern represented by
*C*, if more than 75 percent of mass participates in the fundamental mode in the direction under consideration._{vx}*C*is given by the following expression:_{vx}

where:*w*_{i}= Portion of the total building weight, *W*, located on or assigned to floor level*i*.*h*_{i}= Height in feet from base to floor level *i*.*w*_{x}= Portion of the total building weight, *W*, located on or assigned to floor level*x*.*h*_{x}= Height in feet from base to floor level *x*.*k*= 1.0 for *T*_{e}≤ 0.5 sec.= 2.0 for *T*_{e}≥ 2.5 sec.

Linear interpolation shall be used to estimate k for intermediate values of*T*._{e} - A lateral load pattern proportional to the story inertia forces consistent with the story shear distribution computed by combination of modal responses using response spectrum analysis of the building, including a sufficient number of modes to capture 90 percent of the total seismic mass and the appropriate ground motion spectrum.

The effective fundamental period,

where:

*T*, in the direction under consideration, shall be determined using the force-displacement relation of the nonlinear static pushover analysis. The nonlinear relation between the base shear and target displacement of the control node shall be replaced by a bilinear relation to estimate the effective lateral stiffness,_{e}*K*, and the yield strength,_{e}*V*, of the building. The effective lateral stiffness shall be taken as the secant stiffness calculated at a base shear force equal to 60 percent of the yield strength. The effective fundamental period,_{y}*T*, shall then be calculated as:_{e}where:

T_{i} | = | Elastic fundamental period in the direction under consideration calculated by elastic dynamic analysis. |

K_{i} | = | Elastic lateral stiffness of the building in the direction under consideration. |

K_{e} | = | Effective lateral stiffness of the building in the direction under consideration. |

The general procedure for the execution of the displacement coefficient analysis procedure shall be as follows:

- An elastic structural model shall be created that includes all components (existing and new) contributing significantly to the weight, strength, stiffness or stability of the structure, and whose behavior is important in satisfying the intended seismic performance.
- The structural model shall be loaded with gravity loads before application of the lateral loads.
- The mathematical model shall be subjected to incremental lateral loads using one of the lateral load patterns described in Section A509.4.3. At least two different load patterns shall be used in each principal direction.
- The intensity of the lateral load shall be monotonically increased until the weakest component reaches a deformation at which there is a significant change in its stiffness. The stiffness properties of this "yielded" component shall be modified to reflect the post-yield behavior, and the modified structure shall be subjected to an increase in lateral loads (for load control) or displacements (for displacement control) using the same lateral load pattern.
- The previous step shall be repeated as more components reach their yield strengths. At each stage, the internal forces and deformations (both elastic and plastic) of all components shall be computed.
- The forces and deformations from all previous loading stages shall be accumulated to obtain the total force and deformations of all components at all stages.
- The loading process shall be continued until unacceptable performance is detected or until a roof displacement is obtained that is larger than the maximum displacement expected in the design earthquake at the control node.
- A plot of the control node displacement versus base shear at various stages shall be created. This plot is indicative of the nonlinear response of the structure, and changes in the slope of this load-displacement curve are indicative of the yielding of various components.
- The load-displacement curve obtained in Item 8 shall be used to compute the effective period of the structure, which would then be used to estimate the target displacement (Section A509.4.2).
- Once the target displacement has been determined, the accumulated forces and deformations at this displacement shall be used to evaluate the performance of various components.
- If either the force-demands in the nonductile components or deformation-demands in the ductile components exceed the permissible values, then the component shall be deemed to violate the performance criterion, indicating that rehabilitation be performed for such elements.

The relation between base shear force and lateral displacement of the control node shall be established for control node displacements ranging between zero and 150 percent of the target displacement, δ._{t}

The inter-story drift between floors of the building and the corresponding strains in building components shall be checked at 150 percent of the target displacement, δ

*, to verify acceptability under the demand earthquake ground motion. Performance shall be considered acceptable if building response parameters do not exceed the limitations outlined in Section A509.1.2.*_{t}**Where the effective stiffness,**

Exception:Exception:

*K*, and the yield strength,_{e}*V*, of the building can be determined through rational analysis, the acceptance criteria may be determined based on 100 percent of the target displacement, δ_{y}*.*_{t}