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# A509.3 Capacity Spectrum Analysis Procedure

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This section presents an alternative procedure for a 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 torsional irregularity as defined in Section A508.2 is present in any story.

The behavior of foundation components and the 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 torsional irregularity as defined in Section A508.2 is present in any story.

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

Component initial stiffness shall be represented by a secant value defined by the effective yield point of the component. The effective initial stiffness shall be calculated using principles of mechanics, with due consideration of the effects of tensile cracking and compression strain.

**Component effective initial stiffness may be calculated using the approximate values shown in Table A508.1.**

Exception:Exception:

The strength of building components shall be calculated using the procedures outlined in the appropriate section of the building code.

**Component properties may be calculated using the principles of mechanics as verified by experimental results.**

Exception:Exception:

The deformability of building components shall be obtained from nonlinear load-deformation relationships that are appropriate for the component being considered. The nonlinear load-deformation relationship shall include information on the plastic deformation capacity at which lateral strength degrades, the plastic deformation capacity at which gravity-load resistance degrades, and the residual strength of the component after strength degradation.

The nonlinear load-deformation relationships of building components shall be determined from nonlinear analyses based on the principles of mechanics, experimental data or established values published in technical literature, as approved by the building official.

The nonlinear load-deformation relationships of building components shall be determined from nonlinear analyses based on the principles of mechanics, experimental data or established values published in technical literature, as approved by the building official.

The structure's capacity shall be represented by a capacity curve, which is a plot of the building's base shear versus roof displacement. The capacity curve shall be determined by performing a series of sequential analyses with increasing lateral load, using a mathematical model that accounts for reduced resistance of yielding components. The analysis should include the effect of gravity loads on the building's response to lateral loads.

Lateral forces shall be applied to the structure in proportion to the product of mass and fundamental mode shape.

Lateral forces shall be applied to the structure in proportion to the product of mass and fundamental mode shape.

Exceptions:

- For buildings with weak stories, the vertical distribution of lateral forces shall be modified to reflect the changed fundamental mode shape after yielding of the weak story.
- For buildings over 100 feet (30 480 mm) in height or buildings with irregularities that cause significant participation from modes of vibration other than the fundamental mode, the vertical distribution of lateral forces shall reflect the contribution of higher modes.

The capacity curve calculated in Section A509.3.3.1 shall be converted to the capacity spectrum, which is a representation of the capacity curve in the Acceleration-Displacement Response Spectra (ADRS) format. Each point on the capacity curve shall be converted using Equations A5-8 and A5-9.

| (Equation A5-8) |

| (Equation A5-9) |

| (Equation A5-10) |

| (Equation A5-11) |

A bilinear representation of the capacity spectrum curve obtained in Section A509.3.3.2 shall be used in estimating the appropriate reduction of spectral demand. The first segment of the bilinear representation of the capacity spectrum shall be a line from the origin at the initial stiffness of the building using the component initial stiffness specified in Table A508.1. The second segment of the bilinear representation of the capacity spectrum shall be a line back from the trial performance point,

*a*,_{pi}*d*, at a slope that results in the area under the bilinear representation being approximately equal to the area under the actual capacity spectrum curve. The intersection of the two segments of the bilinear representation of the capacity spectrum shall determine the yield point_{pi}*a*,_{y}*d*._{y}The demand spectrum is a plot of the spectral acceleration and spectral displacement of the demand earthquake ground motion in the Acceleration-Displacement Response Spectra (ADRS) format. The 5-percent damped acceleration response spectra in Section A506 shall be modified for use in the capacity spectrum analysis procedure as follows:

- In the constant acceleration region, the 5-percent damped acceleration spectra shall be multiplied by:
**(Equation A5-12)** - In the constant velocity region, the 5-percent damped acceleration spectra shall be multiplied by:
**(Equation A5-13)** - The spectral displacement ordinate,
*S*, for a corresponding spectral acceleration,_{d}*S*, shall be determined from:_{a}**(Equation A5-14)**

The performance point shall represent the maximum roof displacement expected for the demand earthquake ground motion. When the displacement of intersection of the capacity spectrum defined in Section A509.3.3.2 and the demand spectrum defined in Section A509.3.3.4 is within 5 percent of the displacement of the trial performance point,

*a*,_{pi}*d*, used in Section A509.3.3.3, the trial performance point shall be considered the performance point. If the intersection of the capacity spectrum and the demand spectrum is not within the acceptable tolerance of 5 percent, a new trial performance point shall be selected and the analysis shall be repeated._{pi}The inter-story drift between floors of the building and the corresponding strains in building components shall be checked at the performance point 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.

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