Selection of Isolation System Parameters

Based on the previous formulation in Section 6.3.1, a designer can choose the following parameters to characterize a linear isolation system with supplemental nonlinear fluid viscous dampers: , cd, and η. Recall that defines the natural frequency of the isolated building assuming superstructure rigidity, and cd and η define the damping coefficient and velocity power law, respectively, of the generally nonlinear damper. Here it is assumed that all energy dissipation in the isolation system is due to the supplemental dampers, and therefore is the only parameter needed to characterize the isolation bearings.

The properties of the isolation system should be selected considering the seismic environment in which the structure is located, and the desired response. For any location in the U.S., a set of seismic hazard curves has been developed by the United States Geological Survey [USGS, 2007.] From these hazard curves, spectral ordinates (such as spectral displacement at damping ζ = 0.05) may be computed at any natural frequency , and for any mean annual frequency (MAF) ν. In current model building codes in the United States [ASCE, 2005] the isolation system and all interconnected components must remain stable under a Maximum Considered Event (MCE) ground motion. The level of hazard, where not controlled by a fault scenario-based deterministic event, is defined as a seismic event having a 2% probability of exceedance in a 50-year interval, or a 2475-year return period. This event corresponds to . Therefore, the seismic hazard curve for the site provides the parameter , which approximates the 5% damped MCE isolator displacement for a linear isolation system having natural frequency .

ω1 ω₁

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Typically, the isolation frequency is selected to provide substantial frequency separation from the superstructural natural frequency, and to minimize the base shear in the Design Basis Event (DBE). If we take as a result of the structural design and treat it as given, the bearing stiffness is simply , where m is the total mass of the isolated structure. This selection of leads to an estimate of the MCE isolator displacement, . However, this spectral displacement is typically computed at a damping of ζ = 0.05, which is generally considered appropriate for fixed-base structures with cladding, partitions, and other sources of nominal energy dissipation under small vibration. For a linear isolation system, composed of either natural rubber bearings or lubricated spherical sliding bearings, damping is very close to zero, so the estimate of MCE isolator displacement must be modified in the absence of supplemental energy dissipation. An approximation for the IBC-based damping reduction factor is given by Christopolous et al. [2006] as

(6.29)

where Sd is the spectral displacement and is the target damping ratio of the isolation system. If a displacement limit of the isolation system is defined as , then it is feasible to estimate the target viscous damping ratio for supplemental linear viscous dampers by imposing the constraint

(6.30)

If we assume a damping ratio in the linear isolation system without supplemental energy dissipation as being negligibly small, Equation (6.30) can be rearranged to solve for the target damping as

(6.31)

This estimate of the target damping ratio for a linear supplemental energy dissipation system can be used to compute the parameters cd and η for a nonlinear system by assuming the energy dissipation by the nonlinear supplemental dampers during a representative cycle equals the energy dissipated by the linear system with viscous damping ration .

ω1

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For a system comprised of nonlinear viscous damping elements having the same power for the velocity, recall that the damping force of the system is given by

(6.32)

where cd is the damping coefficient of all dampers in the system. Here we will estimate the peak velocity by considering a representative cycle of response having a frequency and peak displacement of the isolation system. Given the force-velocity relationship of Equation (6.32), and assuming one cycle of displacement demand is given by , the viscous damping force can be expressed as

(6.33)

We wish to compute the total energy dissipated in one cycle of displacement, and recognizing that, for an infinitesimal displacement du, the energy accumulation over that displacement is . Integrating over some displacement interval gives the expression for total energy dissipation in one cycle as

(6.34)

However, substituting , this integral becomes

(6.35)

Substituting the expressions for damper force and velocity yields

(6.36)

The cosine function is positive-valued for and , and negative-valued for , so Equation (6.37) may be expressed as

(6.37) This summation of integrals has a closed form solution, and is given by Soong and

Constantinou [1994] as

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(6.38)

where is defined by the Gamma function, and is given by

(6.39)

For the above displacement cycle, the peak damper force is for a peak displacement of . From Equation (6.38), Ed may be expressed as

(6.40)

The variable may be interpreted as a shape modification factor for the damper hysteresis. Taking two natural cases of the velocity exponent: (velocity independent) and (linear viscous), computing λ for each gives and . This same result is found by recognizing that for velocity-independent damping force, the hysteresis is rectangular, and for linear viscous damping force, the hysteresis is elliptical. In the former case, the peak damper force is in-phase with the peak displacement, whereas for the latter, it is exactly out of phase by π/2 radians. The implications of this for isolation structures will become clear in subsequent discussions regarding different mechanisms of energy dissipation at the isolation interface.

The final goal of this section is to compute a required set of damper parameters cd and η such that a nonlinear viscous damper dissipates the same energy as a linear one having a target damping ratio ζt. The approach is to equate the energy dissipation for linear viscous and nonlinear cases for a cycle of displacement having the same peak displacement and frequency. In light of the above derivations, this is a straightforward process. While this approach has been derived assuming rigid superstructure, it may be extended to the case of the non-rigid superstructure so long as there is sufficient period separation that earthquake-induced deformation is primarily concentrated at the isolation interface.

Consider the canonical 2-DOF isolated structure, having total mass m1 + m2 = m, isolated frequency , and negligible inherent damping. If a linear viscous damper is installed to act in parallel with the isolators, the viscous damping ratio is defined as . From Equation (6.38), taking η = 1, the peak displacement as and the

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frequency of response as , the energy dissipation per cycle is given as . Therefore, the damping ratio may be expressed as

(6.41)

While only valid for linear viscous damping, the expression above is often used to approximate the equivalent linear viscous damping coefficient for a nonlinear damping element if the proper energy dissipation per cycle, Ed, is substituted. This approach was first taken by Jacobsen and Ayre [1958] in their definition of equivalent viscous damping for inelastic systems subjected to harmonic vibrations. Substituting Ed from Equation (6.38) into Equation (6.41) yields

(6.42)

For earthquake excitation of isolated structures, the frequency of response is normally very close to the structure’s natural frequency during strong shaking. Therefore, it is reasonable for the level of approximation we are seeking here to assume that . Making this substitution and solving for the required normalized damping coefficient (as a fraction of the supported weight) to achieve a target effective damping ratio ζt gives

(6.43)

It is useful to define a new coefficient and examine its properties. The functional form of is

(6.44)

While this may be computed with relative ease using digital computers, a more tractable form amenable to preliminary design is preferred. A plot of for , a likely bound on values for structural engineering applications, is shown below in Figure 6.2.

Observation of the values of this function indicates a nearly linear relationship between and η over the given range. Also shown in Figure 6.2 is the linear least-squares

ω E_d =cπωu₀²

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estimate of over the same range. As a result, an appropriately accurate simplification of the required nonlinear damping exponent to achieve target damping ratio ζt is

(6.45)

Importantly, for the linear case of η = 1, the actual damping coefficient is computed as , the result from classic linear dynamics. For a velocity-independent device having η = 0, such as one based on metallic yielding or friction, the required normalized coefficient is . However, for such a device, the damping coefficient is equal to the yield (or slip) force of the device, and the term is close to the maximum restoring force of the isolation system, , for small amounts of added damping force.

Therefore, the necessary yield force of the device can be computed as . This can be taken a step further, since the restoring force of the isolator can be taken directly from the spectral acceleration at the natural frequency and target damping, or . Therefore, the necessary normalized yield force can be obtained from

(6.46)

This is a very useful form for the designer, since the yield force coefficient normalized yield force can be taken directly from the elastic response spectrum for the given site. By Equation (6.46), for an isolated structure with an elastic spectral acceleration of 0.2g at a target equivalent damping ratio of 0.15, the supplemental devices must be designed to yield at 0.047mg, or about 5% of the structure’s weight.

β η( )

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Figure 6.2: Exact equation for compared with linear approximation

Finally, we combine the discussion of target damping for a particular isolator displacement limit with equivalent damping for nonlinear viscous damping systems. Setting the expression for equivalent damping in Equation (6.42) equal to the target damping computed in Equation (6.31), we have the inequality

(6.47)

Since the target displacement is generally equal to the displacement limit of the isolation system in the MCE, it can be assumed that . The required normalized damping coefficient becomes

(6.48)

Careful inspection on the above relation reveals a possible further simplification. Noting that the target MCE spectral acceleration, in units of g, can be expressed as , Equation (6.48) becomes

(6.49)

where , and is the target MCE spectral acceleration. This form allows the designer to estimate the required damping coefficient of the nonlinear viscous damping

β (η )

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system as a function of the structure’s weight, which is conveniently analogous to the specification of the required base shear of a structure.