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.

1. 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.
2. Compute the effective elastic lateral stiffness, k_e, as the yield force, F_y, divided by the yield displacement, Δ_y.

3. 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 consideration

4. Determine target displacement Δ_d, from:

   (4-9)

   S_A = spectral displacement corresponding tostructural period, T_e

5. 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.

6. From the acceleration response spectra, create elastic displacement spectra, S_D, using Equation (4-11) for various levels of damping.

   (4-11)

7. Using the curve applicable to the effective structural damping, ξ_eff, find the effective period, T_d (see Figure 31F-4-6).
8. 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.

9. 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 period

10. The required strength, F_u, can now be estimated by:

    (4-13)

11. 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.
12. Repeat the process until a satisfactory solution is obtained (see Figure 31F-4-7).
13. From pushover data, calculate the displacement components of an element along the two axis of the system.