Accuracy of Analytical Models to Predict Primary and Secondary System Response in Seismically Isolated Buildings

Seismic isolation is generally considered an effective earthquake protection strategy. As application of seismic isolation increases, decisions on the use of one particular isolator versus another isolator increasingly depend on computed responses with complex analytical models. Accordingly, validation of analytical models to predict primary (structural) and secondary system (non-structural component) response in seismically isolated buildings becomes very important. This paper presents comparisons of experimental and analytical results on the primary and secondary system response of a building model in order to provide information on the accuracy of the predicted response. The tested model was congured as a 6-story building at quarter length scale in a moment-frame conguration, and with the following seismic isolation systems: a) Low damping elastomeric bearings with and without linear or nonlinear viscous dampers, b) Single Friction Pendulum (FP) bearings with and without linear or nonlinear viscous dampers, and c) Lead-rubber bearings. Response quantities compared include story drifts and isolator shear forces and displacements for the primary system, and peak oor total velocities and oor response spectra that relate to secondary system response. This paper presents samples results out of a total of 288 comparisons of experimental and analytical results presented in an MCEER report. It is shown that the primary and secondary system response is computed with sucient accuracy by the analytical models but some response quantities may be underestimated or overestimated by signicant amounts. bearings with and without linear or nonlinear viscous dampers, c) Lead-rubber bearings, and d) Low damping elastomeric bearings in combination with at sliding bearings. Response quantities compared included story drifts and shear forces, and isolator displacements for the primary system, and peak oor accelerations and velocities and oor response spectra that relate to secondary system response. A total of 227 experiments were conducted in 24 congurations (3 superstructures and 8 isolation systems) using 14 historic ground motions of far-fault and near-fault characteristics. This paper presents a sample of results from the Wolff and Constantinou (2004) study that concentrates on the structural system with the largest story drift and with contemporary congurations of isolation systems: the moment frame superstructure, elastomeric isolators with linear and nonlinear viscous dampers, friction pendulum isolators without and with linear and nonlinear viscous dampers and lead-rubber isolators. Also, results are presented for only two earthquake ground motions; the 1940 Imperial Valley, El Centro S00E record (peak acceleration of 0.34g, peak velocity of 0.34m/sec-scaled up by factor of 2 for the test and with time compressed by factor of 2), a far-eld motion, and the 1994 Northridge, Sylmar NR 90 0 record (peak acceleration of 0.60g, peak velocity of 0.77m/sec-used as recorded for the test and with time compressed by factor of 2), a near-fault


Introduction
Seismic isolation is a technology and construction strategy with a signi cant number of applications. Just in Japan, the number of applications exceeds 5000 large buildings and another 5000 one and twostory houses (Cilsalar et al. 2019). Worldwide applications of the technology include houses, apartment buildings, hospitals, museums, courthouses, emergency operation centers, LNG and other tanks, offshore oil and gas platforms, airport terminals, and high-rise buildings. The analysis and design of seismically isolated buildings is governed by codes and standards, and typically involves use of nonlinear response history analysis (e.g., Chap. 17 of standard ASCE 7-16; ASCE/SEI 2017). Results of the analysis used in design of seismically isolated buildings include the isolator resultant displacement, the base shear force, and story drift and shear forces. This paper compares analytical and experimental results on the isolator displacement, base shear force, and story drift that are used for the primary system design.
Codes and standards include speci cations that are used for the design of secondary systems.
Chapter 13 of ASCE 7-16 (ASCE/SEI 2017) presents equations for the design of non-structural components attached to oors. The equations include the peak oor acceleration that can be computed using prescriptive equations or response spectrum analysis procedures. For seismically isolated buildings, engineers often use these equations with the peak oor accelerations predicted by the nonlinear response history analysis procedures described in Chap. 17 of ASCE 7-16. Generally, the design of secondary systems is affected by many structural system response parameters, such as the structural drift (e.g., piping systems, cladding, etc.), the peak oor acceleration (e.g., rigidly attached equipment), the spectral oor acceleration (e.g., exibly attached equipment), and combination of response parameters (e.g., cabinets and equipment that are allowed to rock, and complex systems such as sprinkler systems, gas supply systems, etc.). Accordingly, this paper also considers and compares analytical and experimental results on oor accelerations, oor total velocities and oor response spectra which are used or may be used for secondary system design.
Many past analytical studies (e.g., Kelly 1982; Chen and Soong 1988;Fan and Ahmadi 1990) and experimental studies (e.g., Kelly andTsai 1985, 1988;Juhn et al. 1992;Soroushian et al. 2016;Ryan et al. 2016) considered the response of secondary systems attached to the primary system of seismically isolated buildings, where they established the potential bene ts of seismic isolation in reducing the response of secondary systems. However, there has been little work on assessing the validity and accuracy of methods of analysis of seismically isolated buildings with emphasis on the secondary system response. Apart from a limited in scope study by Juhn et al. (1992), Wolff and Constantinou (2004) presented an extensive study on the veri cation of accuracy of methods of dynamic analysis of secondary systems in seismically isolated structures by comparing to experimental data. More recently, studies by Fenz and Constantinou (2008)  In the study of Wolff and Constantinou (2004), on which this paper is based on, the tested model was 233kN in weight and con gured as a 6-story building at quarter length scale in three building con gurations (moment-frame, asymmetrically braced frame and symmetrically braced frame), and with the following seismic isolation systems: a) Low damping elastomeric bearings with and without linear or nonlinear viscous dampers, b) Single Friction Pendulum (FP) bearings with and without linear or nonlinear viscous dampers, c) Lead-rubber bearings, and d) Low damping elastomeric bearings in combination with at sliding bearings. Response quantities compared included story drifts and shear forces, and isolator displacements for the primary system, and peak oor accelerations and velocities and oor response spectra that relate to secondary system response. A total of 227 experiments were conducted in 24 con gurations (3 superstructures and 8 isolation systems) using 14 historic ground motions of far-fault and near-fault characteristics. This paper presents a sample of results from the Wolff and Constantinou (2004) study that concentrates on the structural system with the largest story drift and with contemporary con gurations of isolation systems: the moment frame superstructure, elastomeric isolators with linear and nonlinear viscous dampers, friction pendulum isolators without and with linear and nonlinear viscous dampers and lead-rubber isolators. Also, results are presented for only two earthquake ground motions; the 1940 Imperial Valley, El Centro S00E record (peak acceleration of 0.34g, peak velocity of 0.34m/sec-scaled up by factor of 2 for the test and with time compressed by factor of 2), a far-eld motion, and the 1994 Northridge, Sylmar NR 90 0 record (peak acceleration of 0.60g, peak velocity of 0.77m/sec-used as recorded for the test and with time compressed by factor of 2), a near-fault motion. These two earthquake ground motions resulted in the largest isolator displacements which were, in the length scale of the tests, about 65mm for the low damping elastomeric system and about 60mm for the single FP system. Only one directional seismic excitation was used, which simpli ed the complexity of the analytical prediction. All analytical models were developed in program SAP2000 (Computer and Structures 2016), which nowadays is the most commonly used program in the analysis of seismically isolated buildings. The interested reader is referred to an extensive presentation of 288 comparisons of analytical and experimental results which is available in an MCEER report (Wolff and Constantinou 2004). Another paper of the authors (Wolff et al. 2015) presented another sample of results from this study concentrating on comparisons of the experimental responses, and some analytical predictions, of the model structure with elastomeric and single friction pendulum isolators when dampers were added to the isolation system.

Description Of Test Structure And Experimental Program
The test structure is shown in Fig. 1. The model represents a section in the weak direction of a steel moment-resisting frame. It is attached to a rigid base comprised of two AISC W14x90 sections, 5.2 m long with six transversely connected beams. The model has six stories of 0.915m height each, giving a total height of 5.49m above the base. The model is three bays by one bay in plan, each bay being 1.22m wide, for total plan dimensions of 1.22m by 3.66m. Concrete blocks were used to add mass to satisfy similitude requirements, bringing the total weight, including the base, to 233kN. The structure was constructed with a length scale factor of 4 (prototype length to model length) and was tested with a time scale factor of 2. The modal properties of the tested frame are discussed later when the analytical model is presented. The seismic isolation system consisted of four isolators (either four elastomeric or four single friction pendulum or four lead-rubber isolators) and, when dampers were used, in combination with two linear or two nonlinear viscous dampers installed at angle of 40.4degrees with respect the direction of testing (varied between 38 and 43degrees during testing due to change of length of the dampers). The elastomeric bearings were 177mm in bonded diameter with a 25mm mandrel hole. They consisted of 18 rubber layers of 3.2mm thickness each for a total rubber thickness of 57mm. The rubber was natural rubber of grade 5, hardness 50 and shear modulus of approximately 0.7MPa. The lead-rubber bearings were identical in construction with the elastomeric bearings but with the mandrel hole tted with a lead core. Accordingly, they referred to as low damping elastomeric bearings with lead core. The friction pendulum bearings had an actual radius (also effective radius due to the construction of the slider) of 762mm and a displacement capacity of 200mm. The linear and the nonlinear dampers were designed so that they produce the same force at a velocity of approximately 350mm/sec. Table 1 lists the properties of each element of the isolation system.
The properties were determined in testing of individual isolators and dampers under dynamic conditions and over three cycles. The elastomeric and lead-rubber bearings were tested at 1Hz frequency, displacement amplitude corresponding to 100% rubber shear strain (57mm) and various loads including the gravity load during shake table testing (the frequency of 1Hz is related to the frequency of the model on the low damping elastomeric and lead-rubber isolation systems-1.1 to 1.25Hz based on the postelastic stiffness in Table 1). The viscous dampers were tested at various velocities up to 350mm/s and displacement amplitude of 50mm. The friction pendulum bearings were tested at the gravity load, amplitude of 80mm and peak velocities between 4.8 and 430mm/s (the range was needed to better calibrate the friction-velocity relationship). There was insigni cant variation of the characteristic strength of the low damping elastomeric bearings and of the damper force, and there was some minor variation of the characteristic strength of the lead-rubber bearings and the friction force of the friction pendulum bearings from cycle to cycle in the tests. This result for the lead-rubber and friction pendulum bearings is inconsistent with the behavior of full-size bearings tested in prototype scale (Constantinou et al. 2007; Constantinou 2008, 2009) but consistent with the scaling principles used in the shake table testing (as demonstrated in Kalpakidis and Constantinou 2009, thermodynamic similarity requires to increase the velocity of motion when testing at reduced length scale, which is currently impossible to accomplish in shake table testing except at full scale). Accordingly, modeling the behavior of the isolators in the shake table testing had fewer complexities than modeling the behavior of full-size isolators. The properties in Table 1 include the elastic and post-elastic stiffness and yield force for the elastomeric and lead-rubber bearings, damping constant C and damping exponent α for the dampers (Eq. 1), and effective radius of curvature, initial stiffness and friction coe cient model values for the friction pendulum isolators. The damper force F is related to the damper relative velocity V by: The coe cient of friction follows the following relation (Constantinou et al. 1988), where V is the velocity of sliding.
The properties of the isolation systems in terms of their characteristic strength over weight ratio (Q/W) and period based on the post-elastic stiffness, in the time scale of the shake table tests, were: 1. Low damping elastomeric bearings: Q/W = 0.024, period = 0.88sec.
3. Single friction pendulum: Q/W = 0.08, period = 1.75sec The damping ratio of the combined low damping elastomeric bearing-linear damper system (using the bearing's post-elastic stiffness of 0.3kN/mm) is 0.23.
The The tested structure was modelled in SAP2000 using a skeletal representation with lumped masses at each joint as shown in Fig. 2. The model included rigid offsets to better represent the behavior of beams, columns and splices. Also, the loads cells below the isolators were modeled as they had some limited exibility. Due to symmetry, only half of the tested structure was modelled having two instead of four isolators, and one damper instead of two.
The modal characteristics of the tested model when xed at it base were determined by processing of records acquired during shake table testing using banded (0 to 40Hz) white noise excitation of 0.05g peak acceleration. Details of the identi cation are presented in Wolff and Constantinou (2004). These modal characteristics are presented in Table 2 and compared to those obtained by the analytical model. There is very agreement between the analytically and experimentally obtained frequencies and mode shapes. Based on the SAP2000 model, only the rst three modes were necessary to achieve a 95% modal mass participation. The measured modal damping ratios were used in the global model of the isolated structure using a procedure that mirrored the one described in Sarlis et al. (2010).
Each isolator and each damper were explicitly modelled using the parameters in Table 1, which were determined experimentally. The same models were used in all analyses for all tests. There was no recalibration of the isolator and damper models based on the results obtained in the shake table testing. As an example, Fig. 3 shows a comparison of force-displacement loops for one of the lead-rubber bearings. histories of the isolation system displacement and second story drift, (b) isolation system forcedisplacement loops, (c) 5%-damped acceleration response spectra for the fth oor and the base, and (d) histories of the base total velocity.

Table 2 Experimental and Analytical Modal Characteristics of Fixed Base Model
The results in Figs. 4 to 15 (the interested reader is referred to Wolff and Constantinou, 2004 for many more comparisons) demonstrate good agreement between experimental and analytical results, although there are differences which are best depicted in the summary of results in Table 3. This table includes peak values of isolator displacement, second story drift ratio and base total velocity, and the fth oor and base spectral acceleration values (5%-damped spectra) at the frequency of 20Hz (which is larger than but close to the peak oor acceleration). It may be seen in Figs. 4 to 15 and in Table 3 that, in general drift and oor total velocities are predicted very well by the analytical model. Also, the isolator displacement is predicted well but in one case in Table 3 the analytical model overpredicts the displacement by 35%. This case of overprediction occurred for the low damping elastomeric system with nonlinear dampers. Wolff and Constantinou (2004) reported more similar cases of overprediction for the same system. Also, the analytical prediction often overpredicts or underpredicts the high frequency oor spectral acceleration values (which are close to the peak oor accelerations). The analytical model most often overpredicts accelerations (by as much as 50%) for the friction pendulum and the lead-rubber isolation systems, and most often underpredicts the accelerations (by as much as 65%) for the low damping elastomeric system with dampers. In most cases, the differences between analytical and experimental acceleration values are smaller, and typically not more than 25%. Such differences are small but they need be viewed on the basis of the fact that the analytical models were based on extensive data on the building and isolation hardware, that the tested model lacked the complexity of a real building and that the seismic excitation was the simplest possible. For real buildings, in which knowledge of properties is incomplete and the seismic excitation is more complex, errors in the analytical prediction of response will likely much exceed 25%. Accordingly, engineers should not make blind use of response history analysis results on peak values of accelerations and oor response spectra for assessing performance and for designing secondary systems.

Summary And Conclusions
This paper presented a comparison of a sample of experimental and analytical results on the response of seven different seismic isolation systems in two different strong ground motions. A signi cant larger collections of similar comparisons may be found in an MCEER report of the authors. The presented results are for a model of 233kN in weight, con gured as a 6-story moment frame building at quarter length scale, and with the following seismic isolation systems: a) Low damping elastomeric bearings with and without linear or nonlinear viscous dampers, b) Single Friction Pendulum (FP) bearings with and without linear or nonlinear viscous dampers, and c) Lead-rubber bearings. Response quantities compared in this paper include story drifts, and isolator shear forces and displacements for the primary system, and oor total velocities and oor acceleration response spectra that relate to secondary system response.
In general, the isolator displacements, the story drift and the oor total velocities are predicted very well by the analytical model. However, the analytical model overpredicted the isolator displacements in the low damping elastomeric system with nonlinear dampers (also, see Wolff and Constantinou, 2004 for more similar results for this system). Moreover, the analytical prediction often overpredicted or underpredicted the high frequency oor spectral acceleration values (which are close to the peak oor accelerations). The analytical model most often overpredicted accelerations for the friction pendulum and the lead-rubber isolation systems, and most often underpredicted the accelerations for the low damping elastomeric system with dampers. In most cases, the differences between analytical and experimental acceleration values were about 25%. Such differences are small but they need be viewed on the basis of the fact that the analytical models were based on extensive experimental data on the building and isolation hardware, that the tested model lacked the complexity of a real building and that the seismic excitation was the simplest possible. For real buildings, in which knowledge of properties is incomplete and the seismic excitation is more complex, errors in the analytical prediction of response will likely much exceed 25%. Accordingly, engineers should not make blind use of response history analysis results on peak values of accelerations and oor response spectra for assessing performance and for designing secondary systems.