Displacement-Based Procedures

In displacement-based procedures, system behavior is represented by a force versus displacement relationship that is calculated through nonlinear static (i.e., pushover) analyses. A pushover analysis involves the application of static lateral loads distributed over the height of the structure, and calculation of the resulting

displacements in a model of the SSI system. A pushover analysis of a structure with a flexible base is schematically illustrated in Figure 4-3.

Figure 4-3 Schematic illustration of a pushover analysis and development of a pushover curve for a structure with a flexible base.

The cumulative lateral load resultant, H, is related to a reference displacement, , forming the nonlinear pushover curve. In some applications, the pushover curve is modified to an acceleration-displacement response spectrum (ADRS) by converting H to an equivalent spectral acceleration, and converting  to an equivalent spectral displacement (e.g, Chopra and Goel, 1999; Powell, 2006). At each point on the pushover curve, the deformations of all components in the structural system are related to the reference displacement.

Powell (2006) describes common ways by which the pushover curve is combined with a design response spectrum to estimate the seismic displacement in a structure.

Three such methods are known as the Capacity Spectrum Method (ATC, 1996), the Coefficient Method (FEMA, 1997; FEMA, 2000; and ASCE, 2007), and Equivalent Linearization (FEMA, 2005). These methods are illustrated in Figure 4-4.

Soil-structure interaction is considered in displacement-based analysis procedures through: (1) foundation springs used in the pushover model; (2) reduction of the free-field response spectrum for kinematic interaction effects; and (3) reduction of the response spectrum for flexible-base damping ratios, 0, that are greater than the fixed-base structural damping ratio, i. The manner by which these components are evaluated in displacement-based analysis procedures is described below. Slight modifications to the notation contained in reference engineering standards and guidelines have been made for consistency with the notation adopted in this report.

In general, soil-foundation springs used in pushover analyses are similar to those described in Chapter 2, Section 2.2, except that dynamic stiffness modifiers are neglected. The Pais and Kausel (1988) static stiffness equations listed in Table 2-2 are used. Distributed vertical springs are evaluated in a manner similar to that described in Chapter 2, Section 2.2.3. Horizontal springs are not distributed but are concentrated at the end of the foundation as shown in Figure 2-8 (in Chapter 2).

Kinematic interaction effects are represented in terms of ratios of response spectra (RRS) between the foundation and free-field motions. Equations for RRS as a function of period are given for the effects of base slab averaging and embedment are adapted from FEMA 440 as follows:

 

Figure 4-4 Schematic illustration of procedures used to combine pushover curves with design response spectra to estimate seismic displacements in a structure (Powell, 2006, with permission).

2 1

where V_sr is the strain-reduced shear wave velocity evaluated using the reduction factors in Table 2-1. In Equations 4-5, the equivalent foundation dimension B is _e^A expressed in units of meters. These equations are a curve-fit of the semi-empirical base-slab averaging transfer function described in Equation 3-3 (in Chapter 3). A shear wave velocity term does not appear in Equations 4-5 because the V_s terms cancel in the expression for b₀ in Equation 3-3. The resulting RRS curves for base-slab averaging are shown in Figure 4-5.

Figure 4-5 Ratios of response spectra (u_FIM/u_g) for base slab averaging using the semi-empirical formulation adopted in FEMA 440.

For embedment, the RRS in Equations 4-6 match Equation 3-4 (in Chapter 3) after re-writing in terms of period, T, instead of angular frequency, . In FEMA 440, the limiting period, fL, is taken as 5 Hz (0.2 sec). As of this writing, these equations are in the process of being revised in the next version of ASCE/SEI 41 (ASCE, 2013) to reflect the recommendations contained in Chapter 3 of this report.

In FEMA 440, the objective of the damping analysis is to estimate the foundation damping ratio, _f, which is then combined with the fixed-base structural damping ratio, i, to estimate 0 using Equation 4-4 (with n = 3). The principal challenge is to extract f from the results of the pushover analysis of the structure in both its fixed-base and flexible-fixed-base condition. As described in Chapter 2, Section 2.2.3,

foundation flexibility can significantly reduce radiation damping (_yy) from rotational vibration modes, which is considered in the FEMA 440 procedures.

First, the period lengthening ratio at small displacements is estimated using the initial stiffness of capacity diagrams for the fixed-base and flexible-base structures.

Assuming shaking in the x-direction, stiffness K_x is then evaluated using equations in Table 2-2 (the dynamic stiffness modifier, x, is assumed as unity). The effective rotational stiffness of the foundation system is then evaluated from a manipulation of Equation 2-7 as follows:

* 2

where K^*_fixed is the equivalent fixed-base stiffness of the structure evaluated from:

2

Note that dynamic stiffness modifier, yy, is also taken as unity. The value of Kyy

estimated from Equation 4-7 reflects the stiffness of the foundation structural elements as implemented in the pushover analysis, so no assumptions of foundation rigidity are required.

The next step is to reduce the period lengthening ratio from the small-displacement condition to the large-displacement (i.e., post-yield) condition (with elongated periods). Taking  as the expected ductility demand for the system (including structure and soil effects), the effective period lengthening in the post-yield state is computed as:

This effective period lengthening ratio can then be used with Figure 4-2 to estimate the foundation damping ratio, f.