## Otq

where

Cd = the deflection amplification factor. Cd need not be greater than 2.0. ôxe = deflection determined by an elastic analysis, as denoted by the suffix e. I = occupancy importance factor ranging from 1.0 to 1.5.

For example, I = 1.5 if the building is placed in seismic use group III.

It should be noted that for structures in SDC C, D, E, or F having torsional or extreme torsional irregularities, the story drift A should be computed at the building corners, and not at the center of the mass. The calculation of interstory drift should include vertical deformation of the isolation systems and PA effects where required.

8.7.3.2. Dynamic Analysis

Both the response spectrum and the time-history analyses are permitted under dynamic analysis procedure.

8.7.3.2.1. Response Spectrum Analysis. This analysis is permitted subject to the following stipulations:

1. The structure is located on a class A, B, C, or D site

2. The isolation system meets the criteria of Item 7 of the equivalent lateral force procedure. (Section 8.7.3.1)

Response spectrum analysis should be performed using a modal damping value for the fundamental mode in the direction of interest not greater than the effective damping of the isolation system or 30% of critical, whichever is less. Damping values for higher modes should be selected consistent with those appropriate for response spectrum analysis of the structure above the isolation system on a fixed base.

Response spectrum analysis used to determine the total design displacement and the total maximum displacement should include simultaneous excitation of the model by 100% of the most critical direction of ground motion and 30% of the ground motion on the orthogonal axis. The maximum displacement of the isolation system must be calculated as the vectorial sum of the two orthogonal displacements.

The isolated building should be represented by a three-dimensional linear elastic structural model. The isolators should be represented by linear springs with stiffness Keff. The calculation of Keff may require multiple iterations, as Keff will be a function of the target displacement.

8.7.3.2.2. Time-History Analysis. Time-history analysis is permitted for the design of any seismically isolated structure. It is mandated for the design of all seismically isolated structures located in a class E or F site, and for isolation systems not meeting the criteria of Item 7 of Section 8.7.3.1.

Time-history analysis must be performed with at least three matched pairs of horizontal time-history components. Parameters of interest should be calculated for each time-history analysis pair. The parameters of interest should include member forces, connection forces, interstory drift, isolator displacements, and overturning forces. Each matched pair of horizontal ground motion records should be simultaneously applied to the mathematical model, considering the most disadvantageous location of mass eccentricity, to calculate the maximum displacements in the isolation system. Where orthogonal forces are applied simultaneously, such as in time-history analysis, the required 5% displacement of the center of mass should be applied for only one of the orthogonal forces at a time. An analysis where the 5% displacement is applied for both orthogonal directions concurrently would result in a double application of the torsional effect of accidental eccentricity and would not be consistent with the original intent for the use of accidental eccentricity.

Common design practice is to: 1) impose the orthogonal components along each of the principal axes of the building separately; and 2) repeat step 1 after changing the signs of the ground motion components (separately), for a total of eight analyses per matched pair per mass eccentricity.

To reduce the computational effort, preliminary analysis may be undertaken to identify: 1) the most advantageous location of the mass eccentricity; 2) the critical matched pair of ground motion records; and 3) the critical orientation of the matched pair identified in foregoing Item 2.

Much of the analysis and design work may be completed with this substantially reduced set of parameters. Once the analysis and design effort is near completion, the final design(s) may be analyzed using the unreduced set of parameters, if deemed necessary. Site-specific ground-motion spectra of the design earthquake and the maximum considered earthquake are required for design and analysis of all seismically isolated structures if any one of the following conditions apply:

1. The structure is located on a class F site.

2. The structure is located at a site with S1 greater than 0.60 g.

Site-specific spectra must be prepared for the design of long-period base-isolation systems, base-isolated buildings on a soft soils, and base-isolated buildings located either near an active fault or in seismic zones. This requirement stems from the uncertainties associated with the spectral shapes set forth in ASCE 7-02 design provisions.

Although the development of a site-specific spectrum is encouraged, the ordinates of the spectrum are not permitted to be less than 80% of the ordinates of the standard design spectrum to guard against the use of inappropriately generated site-specific spectra. This limit is imposed on the ordinates of both site-specific DBE and MCE.

8.7.3.2.3. Mathematical Model. Several modeling procedures have been developed for the analysis and design of seismic-isolated buildings. These procedures may not adequately capture the secondary forces that develop as a function of the horizontal displacement (often large) of the isolators. One key example is the moment, equal to the product of the load on the isolator and the isolator displacement, commonly known as the P-delta (PA) effect, that must be resisted by the isolator, the connections of the isolators to the structural framing above and/or below the isolator, and the structural framing above and/or below the isolator.

Three-dimensional elastic models of isolated buildings are commonly used for response spectrum analysis. The model of the superstructure should include all significant structural members in the building frame and accurately account for their stiffness and mass for both static and response spectrum analysis. The isolators are modeled as linear springs with stiffness equal to the effective stiffness, requiring an a priori estimate of the likely displacement in the isolators.

Another common procedure used for the design of isolated buildings using time-history analysis assumes nonlinear isolators and a linear elastic superstructure. The procedure is appropriate for buildings in which the superstructure is assumed to undergo none-to-minimal inelastic response. The recent development of analysis software packages that include three-dimensional elastic modeling of the superstructure and three-dimensional nonlinear modeling of the isolators has made the analysis of such structures simpler and more efficient.

Another procedure involves the development of a complete three-dimensional nonlinear model of the building. This level of effort is computationally intensive and likely rarely justified. This procedure should probably be used only for isolated buildings in which the superstructure is likely to experience substantial inelastic response, an assumption at odds with the stated performance goals for seismic-isolated buildings.

### 8.7.3.3. Design and Construction Review

Design review of both the analysis and the design of the isolation system, and the isolator testing program, is mandated by ASCE 7-02 for three key reasons:

• The consequences of isolator failure could be catastrophic.

• Isolator design, fabrication, testing methods, and technology are evolving rapidly, perhaps utilizing technologies unfamiliar to many design professionals.

• Isolation system analysis and design often involve use of complex procedures, e.g., nonlinear time-history analysis, which can be highly sensitive to assumptions and idealizations made during the analysis and design process.

Design review aims to minimize the possibility of inappropriate assumptions and procedures in the analysis and the design process. The review should be performed by: 1) a team independent of the design team and the project contractors; and 2) a review team composed of individuals with special expertise in one or more aspects of the design and implementation of seismic isolation systems. The review teams should be formed prior to the development of ground motion criteria and isolator design options. Further, the review team should be given complete access to all pertinent information such that the review team can work closely with all consultants and regulatory agencies involved in the project.

### 8.7.3.4. Required Tests for Isolation Systems

For each cycle of testing, the force-deflection behavior of the prototype test specimen must be recorded so that the data can be used to determine whether the isolation system complies with both these requirements and the specifications prepared by the engineer of record. The engineer of record and the independent review team should review all raw data from the prototype tests.

The total number of testing cycles of substantial response will likely be greater for soft sites and systems with small damping values. If the mechanical characteristics of the isolation system are dependent on the rate of loading, additional dynamic tests must be performed to characterize this dependence. Rate-dependence behavior will be exhibited by most sliding isolation systems (velocity-dependent) and selected elastomeric isolation systems (strain rate-dependent). Reduced-scale models of isolators can be used to capture rate effects on stiffness and damping values, provided that the reduced-scale isolators are fabricated using the same processes and quality control procedures as the full-size isolators. Dimensional relationships between full- and reduced-scale units should be established and verified prior to finalizing the testing program.

The implementation of a quality control program is key to the production of isolators of uniform quality with consistent mechanical properties. This quality control program should be implemented for both prototype and production isolators. If the production (quality control) testing results are to be based in any part on the results of the prototype tests, the production testing program should be completed on each of the prototype isolators prior to starting the prototype tests. Qualified and independent inspection/ monitoring of the testing and manufacture process is an important element of an adequate quality control.

The criteria used to judge whether the properties of an isolation system are dependent on the rate of loading are specified: Namely, the isolators are to be considered rate-dependent if the test data demonstrate that the effective stiffness of the isolator changes by more than plus or minus 10% when the cycling rate is varied from the effective frequency at the design displacement to any frequency within the range of 0.1 to 2.0 times the effective frequency. If the effective stiffness and damping of any of the isolators in the isolation system are dependent on the magnitude of the imposed orthogonal displacement, additional testing is required to quantify this dependence. Reduced-scale isolators may be used to substantiate this dependence, provided the reduced-scale isolators are fabricated using the same processes and quality control procedures as the full-size isolators. Dimensional relationships between full- and reduced-scale units should be established and verified prior to finalizing the testing program. The properties of the isolators may be considered to be independent of bilateral displacement if the effective stiffness at 100% bilateral displacement does not differ from the effective stiffness at 0% bilateral displacement by more than plus or minus 10%.

The static vertical load test is used to verify isolator stability at the total maximum displacement under maximum and minimum vertical loads. The maximum vertical load is calculated using 1.2 DL + 1.0 LL and the maximum downward seismic overturning load from the MCE. The minimum vertical load is calculated using 0.8 DL and the maximum upward seismic overturning load from the MCE. This is a static stability test; no cycling is required. Prototype tests are not required if the isolator unit is of similar dimensional characteristics, of the same type and material, and constructed using the same processes as a prototype isolator unit that has been previously tested using the specified sequence of tests. The independence engineering team should determine whether the results of previously tested units are suitable, sufficient, and acceptable.

8.7.3.5. Illustrative Example: Static Procedure

Up to this point we have discussed the basic principles of seismic isolation and the design provisions of ASCE 7-02. It should be clear from the discussions that a dynamic analysis is mandatory for almost all buildings because buildings that meet the requirements of regularity are indeed rare, even in high seismic zones. However, the design principles are best understood by working through a static example. We will do so here using design provisions given in ASCE 7-02. As mentioned previously, the design provisions also apply to IBC-03, since ASCE 7-02 has been adopted by IBC. Ample interpretation of ASCE 7-02 provisions is repeated to present the solution in a stand-alone format.

Given. A new four-story hospital building to be located in the outskirts of Los Angeles, CA. The owners of the facility have desired a building of superior earthquake performance and are willing to incur the special costs associated with the design, fabrication, and installation of seismic isolaters. A target building performance level of immediate occupancy or better is sought.

The structure is expected to outperform a comparable fixed-base building in moderate and large earthquakes. The intent is to limit damage to the structure and its contents by using seismic isolation that, in effect, permits an elastic response of the structure, while limiting the floor accelerations to low levels even in a large earthquake event.

Building Characteristics

• A single basement, four-story, regular configuration steel building. The building has no vertical or plan irregularities.

• Seismic bracing consists of steel eccentric-braced frame with nonmoment-resisting connections away from links.

• Response modification coefficient R = 7 (ASCE 7-02, Table 9.5.2.2).

• Building is located in the outskirts of Los Angeles, CA.

• From seismic hazard maps Ss = 1.5 g and S1 = 0.60 g for the building site.

• Importance factor I = 1.0. Observe that importance factor I for a seismic-isolated building is taken as 1.0, regardless of the occupancy category, since there is no design ductility demand on the structure.

• Building period calculated as a fixed-base building = 0.9 sec.

• Building plan dimensions are 120 x 120 ft.

• Calculated distance between the center of mass and the center of rigidity is 5 ft at each floor and at the roof.

• The project geotechnical engineer has established the building site as site class D.

• Building weight for seismic design = 7200 kips.

• The project structural engineer has established that, to achieve immediate occupancy performance goals, the isolation system should provide effective isolated periods of TD = 2.5 and TM = 3.0 sec, and a damping of 20% of the critical. A margin of ± 15% variation in stiffness of isolators from the mean values is considered acceptable.

Required. A preliminary design using the provisions of ASCE 7-02 for baseisolation of the building. For purposes of illustration, a friction pendulum system, FPS, is selected as the base-isolation system. It should be noted that, in practice, building ownership, particularly if it is a public entity, requires that the design accommodate alternative systems to secure competitive bids. However, for illustration purposes we will consider only the FPS, it being understood that other isolation systems such as high-damping rubber and lead-rubber isolators are equally viable alternatives.

As part of preliminary design determine

• Minimum design displacements Dd and DM under DBE and MCE. Also total displacements DTD and DTM which include effects of torsion.

• Base shear Vb for designing the structure below the isolation surface.

• Base shear Vs for designing the structure above the isolation surface.

• Maximum dimension of the isolators.

Solution. The restrictions placed on the use of the static lateral response procedures effectively require dynamic analysis for most isolated structures. Therefore one might ask, "Why perform, in this day and age of computers, a static analysis of a building with a sophisticated system such as base isolation?" The answer is quite simple: to establish a minimum level of design forces and displacement. Lower-bound limits on design displacements and design forces are specified in ASCE 7-02 as a percentage of the values prescribed by the static procedure. These lower-bound limits on key design parameters ensure consistency in the design of isolated structures and serve as a safety net against gross undersign.

As mentioned previously, seismic isolation, also referred to as base isolation, is a design concept based on the premise that a structure can be substantially "decoupled" from potentially damaging earthquake ground motions. By decoupling the structure from ground shaking, isolation reduces the level of response in the structure from a level that would otherwise occur in a conventional fixed-base building. Typically, decoupling is accomplished using an isolation system that makes the effective period of the isolated structure several times greater than the period of the structure above the isolation system.

In our case, the four-story example building with a fixed-base period of 0.9 sec and a standard damping of 5% would have experienced a first-mode acceleration of 0.48g (see Fig. 8.34a). By decoupling the building from the ground, the period of the building is expected to increase to 2.7 sec. Additionally, the base isolation is counted upon to increase the damping from a standard 5% to about 20% of the critical. Together, these two factors reduce the first mode acceleration to 0.12g, as shown in Fig. 8.34a.

The underlying philosophy behind isolated structures may be characterized as a combination of primary performance objective for fixed-base buildings, which is the provision of life safety in a major earthquake, and the additional performance objective of damage protection, an attribute provided by isolated structures. The design criteria are then a combination of life safety and damage protection goals summarized as follows:

• Two levels of earthquake, the design basis earthquake DBE and the maximum considered earthquake MCE, are typically considered in the design of isolated structures. The DBE is the same level of ground shaking as that recommended for design of fixed-base structures. The MCE is a higher level of earthquake ground motion defined as the maximum level of ground shaking that may be expected at the building site within the known geological framework.

• The isolators must be capable of sustaining loads and displacements corresponding to the MCE without failure.

• The structure above the isolation system must remain "essentially elastic" for the DBE.

From the criteria given above, it is seen that the performance objectives and design requirements for fixed-base and isolated buildings vary significantly. The performance objective for fixed-base construction is life safety in a DBE; the intent is to prevent substantial loss of life rather than control damage. For isolated buildings, the performance objectives are

1. Minimal to no damage in the design earthquake (thus providing life safety).

2. A stable isolation system in the maximum capable earthquake.

The performance of an isolated building in a design basis earthquake will likely be much better (less interstory drift, smaller floor accelerations) than its fixed-base counterpart. Further, isolated buildings can be designed to provide continued function following a design earthquake: a level of performance that is very difficult to achieve with conventional fixed-base construction.

Fixed-base buildings are generally designed using large response modification factors to reduce elastic spectral demands to a design level, a strategy predicated on significant inelastic deformation of the framing system and damage to nonstructural building element. Such buildings are checked for response in the design earthquake only; there is no design check for the MCE. In contrast, isolated buildings are designed using a dual level approach, namely, the framing system is designed to remain essentially elastic (no damage) in the design earthquake, and the isolators are designed (and tested) to remain stable in the MCE.

The subject building is a steel-braced frame building. Using the post-earthquake scenario given in FEMA 356 as a guide, our building is expected to have

• No permanent drift. Structure substantially retains original strength and stiffness.

• Negligible damage to nonstructural components.

• Minor hairline cracking in concrete frames. No crushing of concrete.

• Minor local yielding at a few places in steel frames. No fracture.

• Minor yielding or buckling of braces.

• Connections between deck units and framing intact. Minor distortions.

• Cladding connections may yield. No failure.

• Some cracked panes in glazing. None broken.

• Negligible damage in stairs and fire escapes.

• Elevators operate.

• Fire alarm systems and electrical equipment functional.

• Computer units undamaged and operable.

Before proceding with the illustrative example, certain design requirements touched upon briefly in the preceding sections will be explained in greater detail. The purpose is to delve into the design intent behind these provisions.

8.7.3.5.1. Effective Stiffness of Isolators. Typically, isolation systems are nonlinear, meaning that their effective stiffness is displacement- and/or velocity-dependent, as shown by an idealized force-deflection relationship in Fig. 834m.

The effective stiffness keff of a seismic isolator is calculated using the forces in the isolator at the maximum and minimum displacements as given in the following equation:

where F+ and F are the positive and negative forces at A+ and A-, respectively.

For isolators whose properties are independent of velocity, the forces in the isolator at the maximum and minimum displacements will generally be maximum and minimum forces, respectively. For isolators whose properties exhibit velocity-dependence, the forces in the isolator at the maximum and minimum displacements will generally be less than the maximum and minimum forces, respectively. However, it is usually assumed that maximum and minimum forces in an isolator are attained at maximum and minimum displacements, respectively. For most types of isolator, this assumption is reasonable.

The deformational characteristics of an isolation system determine: 1) the design displacements; and 2) the maximum forces transmitted to the isolated structure. Defor-mational characteristics are represented by the effective (secant) stiffness of the isolation system. Recognizing that force-displacement hysteresis of an isolation system may change over the course of an earthquake, the maximum effective stiffness is used to calculate the maximum force transmitted by the isolators, and the minimum effective stiffness is used to calculate the fundamental period of the isolated building. The reason for using minimum is to arrive at a conservative estimate of the design displacement. The limiting values are generally established in the design phase and are required to be confirmed by testing.

Effective stiffness of the isolation system is determined from the force-displacement (hysteresis) loops based on the results of cyclic testing of a selected sample of isolator. The values of maximum effective stiffness and minimum effective stiffness can be calculated, as shown in Fig. 8.34m, for both design and maximum displacement levels.

8.7.3.5.2. Effective Damping. The effective damping beff is used to quantify the energy dissipation furnished by the isolation system. The maximum effective stiffness of the isolation system is used to provide a lower-bound, i.e., conservative, estimate of the effective damping.

For the purpose of design, energy dissipation is characterized as an equivalent viscous damping. The following equation defines the equivalent viscous damping beff for

Figure 8.34m. Idealized force-displaement relationships for base-isolation systems: (1) hyster-etic system; (2) viscous system. In seismic isolation, hysteretic behavior is a term that describes intrinsic damping due to inelastic deformation of base isolators. Energy is dissipated through work done by the inelastic actions in the isolators. Viscoelastic or viscous behavior, on the other hand, typifies damping action of external devices that use viscous liquids to absorb energy. No inelastic deformations are involved: Energy is dissipated as heat. Effective stiffness keff of an isolator is calculated from test data by measuring forces F+ and F-, and corresponding deformations A+ and A-. The area enclosed by the force-displacement loop is used to calculate effective damping beff.

loop a single isolator:

b = 2_Ei eff p keff(IA+1 + IA-1)2 where beff = effective damping of the isolation system and isolator unit

Eloop = area enclosed by the force-displacement loop of a single isolator in a complete cycle of loading to maximum positive and maximum negative displacement, A+ and A-

A+ = maximum positive displacement of isolator during prototype testing A- = maximum negative displacement of isolator during prototype testing

8.7.3.5.3. Total Design Displacement. The design of isolated structures must consider additional displacements due to actual and accidental eccentricity, similar to those prescribed for fixed-base structures. Equations given below provide a simple means to combine translational and torsional displacement in terms of the gross plan dimensions of the building (i.e., dimensions b and d), the distance from the center of the building to the point of interest (i.e., dimension y), and the actual plus the accidental eccentricity, as follows:

where e = the sum of the actual and accidental eccentricities.

Notice that the design displacement Dd at the center of the building has been modified to account for additional displacement at the corners or edges of the building due to torsion. It is assumed that the stiffness of the isolation system is distributed in plan proportional to the distribution of the supported weight of the building.

Smaller values of DtD can be used for design if the isolation system is configured to resist torsion (e.g., if stiffer isolator units are positioned near the edges and corners of the building). However, the minimum value of DtD is set equal to 1.1 DD for all types of isolation systems. The total displacement Dtm is calculated in a manner similar to the calculation of DtD. The eccentricity e used for calculating torsional displacements is the actual eccentricity of the isolation system plus an allowance of 5% of the width of building to account for accidental torsion. The parameter y is the distance between the center of rigidity of the isolation system and farthest corners of the building.

It should be noted that the stiffness values KDmin and Kmmin are not known to the designer during the preliminary design stage, but are derived from the known or expected values of periods of the building. Since the expected periods may not turn out to be equal to the final values, the derived stiffness values are also preliminary. After completing a satisfactory preliminary design, typically prototype isolaters are tested to obtain values of

KDmin, KDmax, KMmin, and KMmai-

8.7.3.5.4. Minimum Design Lateral Forces.

1. Isolation System and Structural Elements at or Below Isolation Interface.

The design actions for elements at or below the isolation interface are based on the maximum forces delivered by the isolation system during the design basis earthquake. The building's foundation, the isolation system, and all structural elements at or below the isolation interface are required to be designed and constructed to withstand a minimum lateral force.

The maximum force Vb is the product of the maximum stiffness of the isolation system at the design displacement KDmax and the design displacement DD. The design force Vb represents strength level forces.

The previous equation for Vb is for use in regions of high seismicity, such as UBC zone 4, wherein the difference between the total design displacement and total maximum displacement is relatively small; that is, if a supporting element was designed for design basis earthquake forces at the strength level, it is probable that such a supporting element could resist the forces associated with the MCE without failure.

There are significant differences in values of Mm between regions of high and low seismicity: Values of MM may be less than 1.25 in regions of high seismicity, but may exceed 2.5 in regions of low seismicity. As such, in a region of low seismicity, a supporting element designed for DBE-induced forces may be unable to sustain forces associated with the MCE without significant distress or failure. Therefore, in these regions it may be prudent to consider MCE-level forces to check the design of the isolation system and the structural elements at or below the isolation interface.

Isolation interface is the boundary between the upper portion of the building, which is isolated, and the lower portion, which is rigidly attached to the foundation or ground. The isolation interface can be assumed to pass through the midheight of elastomeric bearings or the sliding surface of sliding bearings. Observe that the isolation interface need not be a horizontal plane, but could change elevation if the isolators are positioned at different elevations throughout the building.

The isolation system includes the isolator units, connections of isolator units to the structural system, and all structural elements required for isolator stability. Isolator units include bearings that support the building's weight and provide lateral flexibility. Typically, isolation system bearings provide damping and wind restraint as an integral part of the bearing. Isolator systems may also include supplemental damping devices. For example, an FPS of basic isolation may include viscous dampers.

Structural elements that are required for structural stability include all structural elements necessary to resist design forces at the connection of the structure to isolator units. For example, a column segment and a beam immediately above an isolator constitute elements of the isolation system because they are necessary to resist forces due to the lateral earthquake displacement of the isolators.

2. Structural Elements above Isolation System. The design of the framing above the isolation system is based on the maximum force delivered by the isolation system divided by a response reduction factor, RThe values assigned to R¡. reflect system overstrength only and no expected ductility demand. By using these values for R, a significant measure of damage control is afforded in the design earthquake, since the structure remains essentially elastic.

The minimum base shear for the design of the structure above the isolation is given by

Three limits are imposed for the calculation of Vs.

• Vs shall not be less than the base shear required for a fixed-base structure of the same weight w and a period equal to the isolated period.

• Vs shall not be less than the total shear corresponding to the design wind load. (In wind design, engineers seldom use the term base shear to define the total shear due to wind. However, base shear and total shear are one and the same.)

• Vs shall not be less than 150% of the lateral seismic force required to fully activate the system.

Thus there are three lower-bound limits set on the minimum seismic shear to be used for the design of the framing above the isolation system. The first limit requires design base shear to be at least that of a fixed-base building of comparable period. The second limit ensures that the elements above the isolation system remain elastic during a design windstorm. The third limit is designed to prevent the elements above the isolation system from deforming inelastically before the isolation system is activated.

3. Vertical Distribution of Vs. The vertical distribution of the seismic base shear is similar to that used for fixed-base buildings, namely, a distribution that approximates the first-mode shape of the fixed-base building. This distribution conservatively approximates the inertia force distributions measured from time-history analyses.

Continuation of Illustrative Problem. The effective periods TD and TM of the isolated building are

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