# 3104F.2.3.2 Nonlinear Static Demand Procedure

A nonlinear static procedure shall be used to determine the displacement demand for all concrete and steel structures, with the exception of irregular configurations with high or moderate spill classifications. A linear modal procedure is required for irregular structures with high or moderate spill classifications, and may be used for all other classifications in lieu of the nonlinear static procedure.

In the nonlinear static demand procedure, deformation demand in each element shall be computed at the target node displacement demand. The analysis shall be conducted in each of the two orthogonal directions and results combined as described in Section 3104F.4.2.

The target displacement demand of the structure, Δ_{d}, shall be calculated by multiplying the spectral response acceleration, S_{A}, corresponding to the effective elastic structural period, T_{e} (see Equation (4-2) or Equation (4-8)), by . If T_{e} < T_{0}, where T_{0} is the period corresponding to the peak of the acceleration response spectrum, a refined analysis (see Section 3104F.2.3.2.1 or 3104F.2.3.2.2) shall be used to calculate the displacement demand. In the refined analysis, the target node displacement demand may be computed from the Coefficient Method of ASCE/SEI 41 [4.3] that is based on the procedure presented in FEMA 440 [4.6], or the Substitute Structure Method presented in Priestley et al. [4.4]. Both of these methods utilize the pushover curve developed in Section 3104F.2.3.1.

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.

The Substitute Structure Method is based on the procedure presented in Priestley et al. [4.4] and is briefly summarized below.

- Idealize the pushover curve from nonlinear pushover analysis, as described in Section 3104F.2.3.2.1, and estimate the yield force, F
_{y}, and yield displacement, Δ_{y}. - Compute the effective elastic lateral stiffness, k
_{e}, as the yield force, F_{y}, divided by the yield displacement, Δ_{y}. Compute the structural period in the direction under consideration from:

(4-8)

where:

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

k

_{e}=effective elastic lateral stiffness in direction under considerationDetermine target displacement Δ

_{d}, from:(4-9)

S

_{A}= spectral displacement corresponding tostructural period, T_{e}*The ductility level,*μ_{Δ}, is found from Δ_{d}/Δ_{y}. Use the appropriate relationship between ductility and damping, for the component undergoing inelastic deformation, to estimate the effective structural damping, ξ_{eff}. In lieu of more detailed analysis, the relationship shown in Figure 31F-4-5 or Equation (4-10) may be used for concrete and steel piles connected to the deck through dowels embedded in the concrete.(4-10)

where:

r =ratio of second slope over elastic slope (see Figure 31F-4-7)

Equation (4-10) for effective damping was developed by Kowalsky et al. [4.5] for the Takeda hysteresis model of system's force-displacement relationship.

From the acceleration response spectra, create elastic displacement spectra, S

_{D}, using Equation (4-11) for various levels of damping.(4-11)

- Using the curve applicable to the effective structural damping, ξ
_{eff}, find the effective period, T_{d}(see Figure 31F-4-6). - In order to convert from a design displacement response spectra to another spectra for a different damping level, the adjustment factors in Section 3103F.4.2.9 shall be used.
The effective secant stiffness, k

_{eff}, can then be found from:(4-12)

where:

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

T

_{d}= effective structural periodThe required strength, F

_{u}, can now be estimated by:(4-13)

- F
_{u }*and*Δ_{d }can be plotted on the force-displacement curve established by the pushover analysis. Since this is an iterative process, the intersection of F_{u}and Δ_{d}most likely will not fall on the force-displacement curve and a second iteration will be required. An adjusted value of Δ_{d}, taken as the intersection between the force-displacement curve and a line between the origin and F_{u}and Δ_{d}, can be used to find μ_{Δ}. - Repeat the process until a satisfactory solution is obtained (see Figure 31F-4-7).
- From pushover data, calculate the displacement components of an element along the two axis of the system.