# 3104F.2.3.2.1 Coefficient Method

The Coefficient Method is based on the ASCE/SEI 41 [4.3] procedure.

The first step in the Coefficient Method requires idealization of the pushover curve to calculate the effective elastic lateral stiffness, k_{e}, and effective yield strength, F_{y}, of the structure as shown in Figure 31F-4-4.

The first line segment of the idealized pushover curve shall begin at the origin and have a slope equal to the effective elastic lateral stiffness, k_{e}. The effective elastic lateral stiffness, k_{e}, shall be taken as the secant stiffness calculated at the lateral force equal to 60 percent of the effective yield strength, F_{y}, of the structure. The effective yield strength, F_{y}, shall not be taken as greater than the maximum lateral force at any point along the pushover curve.

The second line segment shall represent the positive post-yield slope (α_{1}k_{e}) determined by a point (F_{d},Δ_{d}) and a point at the intersection with the first line segment such that the area above and below the actual curve area approximately balanced. (F_{d},Δ_{d}) shall be a point on the actual pushover curve at the calculated target displacement, or at the displacement corresponding to the maximum lateral force, whichever is smaller.

The third line segment shall represent the negative post-yield slope (α_{2}k_{e}), determined by the point at the end of the positive post-yield slope (F_{d}, Δ_{d}) and the point at which the lateral force degrades to 60 percent of the effective yield strength.

The target displacement shall be calculated from:

(4-1)

where:

S_{A} = spectral acceleration of the linear-elastic system at vibration period, which is computed from:

(4-2)

where:

m | = | seismic mass as defined in Section 3104F.2.3 |

k_{e} | = | effective elastic lateral stiffness from idealized pushover |

C_{1} | = | modification factor to relate maximum inelastic displacement to displacement calculated for linear elastic response. For period less than 0.2 s, C_{1} need not be taken greater than the value at T_{e} = 0.2 s. For period greater than 1.0 s, C_{1} = 1.0. For all other periods: |

(4-3)

where:

α | = | Site class factor |

= | 130 for Site Class A or B, | |

= | 90 for Site Class C, and | |

= | 60 for Site Class D, E, or F. | |

μ_{strength} | = | ratio of elastic strength demand to yield strength coefficient calculated in accordance with Equation (4-5). The Coefficient Method is not applicable where μ_{strength} exceeds μ_{max} computed from Equation (4-6). |

C_{2} | = | modification factor to represent the effects of pinched hysteresis shape, cyclic stiffness degradation, and strength deterioration on the maximum displacement response. For periods greater than 0.7s, C_{2} = 1.0. For all other periods: |

(4-4)

The strength ratio μ_{strength} shall be computed from:

(4-5)

where:

F_{y} = yield strength of the structure in the direction under consideration from the idealized pushover curve.

For structures with negative post-yield stiffness, the maximum strength ratio μ_{max} shall be computed from:

(4-6)

where:

Δ_{d} = larger of target displacement or displacement corresponding to the maximum pushover force,

Δ_{y} = displacement at effective yield strength

h = 1 + 0.15lnT_{e}, and

α_{e} = effective negative post-yield slope ratio which shall be computed from:

(4-7)

where:

α_{P-Δ}, and the maximum negative post-elastic stiffness ratio, α_{2}, are estimated from the idealized force-deformation curve, and λ is a near-field effect factor equal to 0.8 for sites with 1 second spectral value, S_{1} greater than or equal to 0.6g and equal to 0.2 for sites with 1 second spectral value, S_{1} less than 0.6g.

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