# 3104F.2.3.2.2 Substitute Structure Method

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.