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, Fy, and yield displacement, Δy.
Compute the effective elastic lateral stiffness, ke, as the yield force, Fy, divided by the yield displacement, Δy.
Compute the structural period in the direction under consideration from:
ke =effective elastic lateral stiffness in direction under consideration
Determine target displacement Δd, from:
(4-9)
SA = spectral displacement corresponding tostructural period, Te
The ductility level,µΔ, is found from Δd /Δy. Use the appropriate relationship between ductility and damping, for the component undergoing inelastic deformation, to estimate the effective structural damping, ξeff. 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, SD, using Equation (4-11) for various levels of damping.
(4-11)
Using the curve applicable to the effective structural damping, ξeff, find the effective period, Td (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, keff, can then be found from:
Fu and Δd can be plotted on the force-displacement curve established by the pushover analysis. Since this is an iterative process, the intersection of Fu and Δd most likely will not fall on the force-displacement curve and a second iteration will be required. An adjusted value of Δd, taken as the intersection between the force-displacement curve and a line between the origin and Fu and Δd, can be used to find µΔ.
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.
FIGURE 31F-4-5
RELATION BETWEEN DUCTILITY, µΔ, AND EFFECTIVE DAMPING, ξeff [4.5]
, the target node
displacement demand may be computed from the Coefficient
Method (Section 3104F.2.3.2.1) or the SubstituteStructureMethod (Section 3104F ...
analysis, the target node
displacement demand may be computed from the Coefficient
Method (Section 3104F.2.3.2.1) or the SubstituteStructureMethod (Section ...
(4-8)where:m =seismic mass as defined in Section 3104F.2.3ke =effective elastic lateral stiffness in direction under consideration
(4-9)SA = spectral displacement corresponding tostructural period, Te
(4-10)where: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.
(4-11)
(4-12)where:m = seismic mass as defined in Section 3104F.2.3Td = effective structural period
(4-13)