Experimental and analytical studies on t

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS Earthquake Engng Struct Dyn 2009; 38:827–848 Published online 10 December 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/eqe.871 Experimental and analytical studies on the response of freestanding laboratory equipment to earthquake shaking Dimitrios Konstantinidis1, ‡ and Nicos Makris2,3, , , Đ, ả Department of Civil and Environmental Engineering, University of California, Berkeley, U.S.A of Civil Engineering, University of Patras, Patras GR-26500, Greece Earthquake Engineering Research Center, University of California, Berkeley, U.S.A Department SUMMARY This paper presents results of a comprehensive experimental program on the seismic response of fullscale freestanding laboratory equipment First, quasi-static experiments are conducted to examine the mechanical behavior of the contact interface between the laboratory equipment and floors Based on the experimental results, the response analysis that follows adopts two idealized contact friction models: the elastoplastic model and the classical Coulomb friction model Subsequently, the paper presents shake table test results of full-scale freestanding equipment subjected to ground and floor motions of hazard levels with corresponding displacements that can be accommodated by the shake table at the UC Berkeley Earthquake Engineering Research Center For the equipment tested, although some rocking is observed, sliding is the predominant mode of response, with sliding displacements reaching up to 60 cm Numerical simulations with the proposed models are performed Finally, the paper identifies a physically motivated intensity measure and the associated engineering demand parameter with the help of dimensional analysis and presents ready-to-use fragility curves Copyright q 2008 John Wiley & Sons, Ltd Received March 2007; Revised 10 September 2008; Accepted 15 September 2008 KEY WORDS: laboratory equipment; non-structural components; shake table experiments; rocking; sliding; PBEE; fragility ∗ Correspondence to: Nicos Makris, Department of Civil Engineering, University of Patras, Patras GR-26500, Greece E-mail: nmakris@upatras.gr Postdoctoral Scholar Đ Professor ả Senior Research Engineer † Contract/grant sponsor: Earthquake Engineering Research Centers Program of the National Science Foundation; contract/grant number: EEC-9701568 Copyright q 2008 John Wiley & Sons, Ltd 828 D KONSTANTINIDIS AND N MAKRIS INTRODUCTION During strong earthquake shaking, heavy equipment located at various floor levels of research laboratories, hospitals, and other critical facilities may slide appreciably, slide-rock, rock, or even overturn Rocking response is very sensitive to the geometry of the slender object and the kinematic characteristics of the ground Minor variations in the input can result in overturning—catastrophe [1–3] Even if overturning does not occur, the high acceleration spikes that develop during impact of the rocking equipment are a major concern, since they can result in serious damage or loss of the equipment contents Of the possible modes of response, sliding is the most favorable Nonetheless, excessive sliding displacements may block a path or doorway that services evacuation Large displacements of sensitive/heavy equipment that result in impact with walls or neighboring equipment should be avoided, since the resulting acceleration spikes endanger the contents or even the equipment itself In practice, excessive sliding is prevented by restraining the equipment—commonly by chaining it to the framing of the nearby wall Although this may succeed in reducing sliding displacements, it substantially amplifies accelerations The problem of equipment sliding has been studied in the past at various scales by Shao and Tung [4], Lopez Garcia and Soong [5, 6], and Hutchinson and Chaudhuri [7] The research presented in this paper is part of a wider study that was set out to apply the performance-based earthquake engineering (PBEE) methodology [8] as proposed by the Pacific Earthquake Engineering Research (PEER) Center on a specific testbed: an actual science laboratory building, herein referred to as UC Science Building The UC Science Building is located on the western part of the main UC Berkeley campus and is approximately km west of the Hayward fault It is a modern structure, completed in 1988 in order to provide high-tech research laboratories [9] The PEER PBEE methodology consists of four stages [8]: hazard analysis, structural analysis, damage analysis and loss analysis In this paper we present experimental and analytical studies that examine the seismic vulnerability of freestanding and restrained laboratory equipment located in the UC Science Building laboratories within several floor levels The equipment of interest includes low-temperature refrigerators, freezers, incubators and other heavy equipment The study investigates the response of equipment to moderately strong motions (50 and 10% probability of being exceeded in 50 years) which result to peak ground displacements or peak floor displacements that the Earthquake Engineering Research Center (EERC) shake table at UC Berkeley can accommodate Additional results are presented in the report by Konstantinidis and Makris [10] The results of tests conducted on the shake table were used to develop a dimensionless engineering demand parameter (EDP) (a parameter that quantifies the response of the equipment), as a function of the intensity measure (IM) (a parameter of the excitation that corresponds to a certain seismic hazard level) Since uncertainties are inherent in all stages of the PEER methodology [8], a probabilistic approach was taken An analysis was performed to generate fragility curves, which give the probability that the aforementioned EDP will exceed a specific limit c SEISMIC HAZARD, GROUND AND FLOOR MOTIONS The seismic hazard study for the UC Science testbed was performed by Somerville [11] The seismic hazard on the UC Berkeley campus is dominated by potential ground motions generated Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe LABORATORY EQUIPMENT TO EARTHQUAKE SHAKING 829 from the Hayward fault, which is located approximately km east of the site The Hayward fault is a strike-slip fault that has a potential to generate earthquakes having magnitudes as large as MW = 7.0 The ground motions for the site were selected and scaled to correspond to three hazard levels: (a) events with probability of exceedance (POE), equal to 50% in 50 years, (b) events with POE 10% in 50 years, and (c) events with POE 2% in 50 years For a hazard level equal to 50% in 50 years, the largest contributions come from earthquakes in the magnitude range of MW = 5.5 to MW = 6.0 For hazard level equal to 10% in 50 years and to 2% in 50 years, the largest contributions come from earthquakes in the magnitude range of MW = 6.5 to MW = 7.0 It is noteworthy that the higher 2% in 50 years not reflect larger magnitudes (as the Hayward fault can generate earthquakes only up to MW = 7.0) but rather stronger ground motions with the same magnitude (with larger standard deviation above the mean) [11] The motions listed in Table I have been selected to satisfy (to the extent possible) the magnitude and distance combination from a strike-slip earthquake on NEHRP-classified SC soil type for moderately strong motions (50 and 10% in 50 years) The seismic response of the UC Science laboratory building was analyzed by Lee and Mosalam [9, 12], who developed a sophisticated structural model of the building Their analyses resulted in simulated floor motions Floor motions are of unique interest in assessing the seismic response of building contents since they differ appreciably from ground motions Table I lists the recorded ground-acceleration motions and the simulated floor-acceleration motions that were used as input for the shake table experiments conducted in this study All input motions used for shake table tests in this study were one-directional Preliminary dynamic analyses of a sliding block resting on a base subjected to horizontal and vertical earthquake motion yielded peak sliding displacements that were only slightly amplified when the vertical component of the excitation was included The same observation is made in a study by Shao and Tung [4] This is because the high-frequency content of vertical acceleration signals in association with their lower amplitude renders them too feeble to impart a significant amplification of the peak sliding displacement There are studies [5, 6], however, that indicate that certain combinations of the horizontal and vertical components of the excitation may not be ignored GEOMETRIC AND PHYSICAL CHARACTERISTICS OF THE TEST EQUIPMENT 3.1 Geometric properties The equipment of interest included incubators, low-temperature freezers, refrigerators and other heavy laboratory equipment of the UC Science Building at the UC Berkeley campus In particular, three pieces of equipment were obtained from the building laboratories in order to examine their mechanical properties and to perform shake table tests Figure shows pictures of the equipment, while Table II lists their geometric and physical characteristics Figure is a schematic of a piece of equipment that shows the geometric quantities that are listed in Table II Each piece of equipment has two vertical faces, designated here by W for width and D for depth The stockiness angles W and D of a piece of equipment are defined by W Copyright q = tan−1 2008 John Wiley & Sons, Ltd W H and −1 D = tan D H (1) Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe Copyright q Aigion, Greece 15/6/1995, MW = 6.2, D = 5.0 km Coyote Lake, California 6/8/1979, MW = 5.7, D = 3.0 km Parkfield, California 27/6/1966, MW = 6.0, D = 8.0 km Parkfield, California 27/6/1966 Coyote Lake, California 6/8/1979, MW = 5.7, D = 3.0 km Coyote Lake, California 6/8/1979 Loma Prieta, California 17/10/1989, MW = 7.0, D = 9.5 km Loma Prieta, California 17/10/1989 Tottori, Japan 6/10/2000, MW = 6.6, D = 10.0 km Tottori, Japan 6/10/2000 Earthquake 2008 John Wiley & Sons, Ltd Kofu FN 6TH LEVEL Gavilan College FN 6TH LEVEL Kofu FN GROUND Gilroy Array #6 FN 6TH LEVEL Gavilan College FN GROUND 50 Cholome Array #8 FN 6TH LEVEL Gilroy Array #6 FN GROUND 0.54 0.69 10 10 0.75 0.66 10 10 0.78 0.47 50 50 0.56 50 Cholome Array #8 FN GROUND 0.71 0.47 Gilroy Array #6 FN PGA (g) 0.50 Hazard level (in 50 years) (%) OTE FP Record 9.7 2.0 7.9 34.3 9.4 6.6 17.5 Umax (cm) Leg failure 0.73 0.43 0.75 1.61 0.70 0.68 0.75 PTA (g) FORMA Face 0.81 0.64 0.76 0.76 0.74 1.66 0.75 0.69 0.76 PTA (g) 13.2 13.0 5.8 18.8 8.4 22.6 3.3 5.3 15.2 Umax (cm) KELVIN Face Equipment 1.08 1.07 0.67 0.77 0.77 0.76 1.7 0.77 0.73 0.86 PTA (g) 35.3 35.3 17.8 4.8 23.9 9.1 57.9 2.0 4.1 8.1 Umax (cm) ASP Face 0.85 1.07 0.67 0.79 0.80 0.75 1.68 0.70 0.70 0.73 PTA (g) 20.3 42.2 17.0 6.6 20.3 9.4 60.2 0.8 2.5 6.1 Umax (cm) Profile Table I Input motions for the shake table tests conducted on the three pieces of laboratory equipment Recorded peak table accelerations (PTA) and maximum sliding displacements (Umax ) of the freestanding equipment 830 D KONSTANTINIDIS AND N MAKRIS Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 831 LABORATORY EQUIPMENT TO EARTHQUAKE SHAKING FORMA Incubator Kelvinator Refrigerator ASP Refrigerator Figure The three pieces of freestanding heavy laboratory equipment that were obtained from the UC Science research facility for the purposes of this study Table II Geometric and physical characteristics of the equipment Equipment FORMA Reach-In Incubator KELVINATOR Refrigerator/freezer ASP Refrigerator Weight (N) H (cm) W (Wout ) (cm) D (Dout ) (cm) RW (cm) RD (cm) W D (rad) (rad) pW (rad/s) pD (rad/s) 3780 228.6 118.4 0.39 0.27 2.44 2.49 213.4 111.3 111.8 0.29 0.30 2.57 2.57 720 180.3 62.2 (78.7) 66.0 (82.6) 58.4 (67.3) 123.4 1565 92.7 (96.5) 63.5 (83.8) 76.2 (81.3) 97.8 94.7 0.40 0.31 2.74 2.79 The stockiness of a block is an indicator of its disposition to enter rocking motion The smaller the stockiness (more slender), the more likely for the equipment to uplift, enter rocking motion, and possibly overturn The frequency parameters pW and p D are measures of the size of the equipment and are given by [1–3, 13] pW = 3g 4RW and where g is the acceleration of gravity and RW = larger the block (larger R), the smaller the p √ pD = 3g 4R D H + W /2, R D = (2) √ H + D /2 (Figure 2) The 3.2 Friction tests The mechanical properties of the contact interface between the equipment and the laboratory floors were determined by conducting slow-pull tests on the equipment The floors throughout the UC Science Building are lined with vinyl tiles In order to simulate the actual conditions, a 2.4×3.6 m Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 832 D KONSTANTINIDIS AND N MAKRIS Figure Schematic diagram of the experimental setup for the quasi-static pull tests pressboard surface covered with identical vinyl tiles was constructed Atop it rested the equipment specimen Figure shows a schematic of the experimental setup of the quasi-static pull tests conducted on the equipment Figure plots load–displacement curves recorded during the quasi-static pull tests on the three pieces of equipment shown in Figure The pre-yielding elasticity in the load–displacement curves originates from the flexure of the legs of the equipment prior to sliding This pre-yielding elasticity of the legs and the friction force that develops along the vinyl surface combine to a yielding mechanism of the interface Simple idealizations of the yielding mechanism of each interface are the elastoplastic models shown with dashed lines in Figure The model parameters that define the elastoplastic idealization are the yield displacement, u y , and the normalized strength, k = Q/mg, where Q is the post-yield constant force Another idealization of the contact interface is that of classical Coulomb friction where a static friction coefficient, s , and a kinetic friction coefficient, k , are used Numerical simulation studies using the elastoplastic model with k values extracted from the slow-pull tests were conducted after the shake table tests in order to examine the validity of the mechanical model The analyses yielded results that were in fair agreement with the experimental data from the shake table tests (presented in the following section) The predicted response of all three pieces of equipment was appreciably improved when lower values of their respective friction coefficients were used The lower friction levels of the elastoplastic idealization are indicated in Figure with solid lines The heavy solid lines shown in Figure correspond to the Coulomb model with reduced friction coefficients The reduced values of s and k have been obtained by best-fitting results from numerical simulations using the commercially available software Working Model [14] to results obtained from shake table experiments Table III summarizes the values of friction coefficients and yield displacements that were obtained from the slow-pull tests and from the best-fitting procedure Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 833 LABORATORY EQUIPMENT TO EARTHQUAKE SHAKING Coefficient of Friction Tests Specimen: FORMA Incubator, Weight=3780N 1200 Elastoplastic (Slow Pull Tests) uy=0.35cm, μk=0.23 Lateral Load [N] 1000 800 FORMA Elastoplastic (Best Fit) uy=0.23cm, μk=0.15 600 400 Working Model μs=0.18, μk=0.13 200 0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 Coefficient of Friction Tests Specimen: KELVINATOR refrigerator, Weight=1570N Lateral Load [N] 800 KELVINATOR Elastoplastic (Slow Pull Tests) uy=0.07cm, μk=0.28 600 400 200 Working Model μs=0.23, μk=0.17 Elastoplastic (Best Fit) uy=0.05cm, μk=0.20 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 Coefficient of Friction Tests Specimen: ASP refrigerator, Weight=720N Lateral Load [N] 400 ASP Elastoplastic (Slow Pull Tests) uy=0.02cm, μk=0.31 300 200 100 Elastoplastic (Best Fit) uy=0.02cm, μk=0.24 0.2 0.4 0.6 Working Model μs=0.28, μk=0.20 0.8 1.2 1.4 1.6 1.8 Displacement [cm] Figure Recorded load–displacement plots obtained from the slow-pull tests (wavy lines); the elastoplastic idealization with k from the slow-pull tests (dashed line); the elastoplastic idealization with k from best-fitting (solid line); the Coulomb friction model (Working Model) with s and k by best-fitting (heavy line) Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 834 D KONSTANTINIDIS AND N MAKRIS Table III Coefficients of friction and yield displacements obtained from slow-pull tests and from best-fitting numerical simulation results to experimental results from the shake table tests Values from Slow-Pull Tests Equipment FORMA Incubator Kelvinator Refrigerator ASP Refrigerator s k u y (cm) Values used to fit Elasto-plastic Model simulation results to experimental results from the shake table tests k u y (cm) Values used to fit Coulomb (Rigid-Plastic) Model simulation results to experimental results from the shake table tests s k 0.30 0.23 0.35 0.15 0.23 0.18 0.13 0.37 0.28 0.07 0.20 0.05 0.23 0.17 0.43 0.31 0.02 0.24 0.02 0.28 0.20 The static friction coefficient, s , is known to exhibit time dependence; with time an object sinks into the surface of its base, as asperities deform, causing an increase in meniscus and Van der Waals forces, which are macroscopically manifested by an increase in s [15] Furthermore, s depends on the rate of application of the pull force [16–18] The kinetic friction coefficient, k , can also exhibit a pressure and velocity dependence [16–18] SHAKE TABLE TESTS OF FREESTANDING LABORATORY EQUIPMENT The three pieces of equipment shown in Figure were subjected to shake table tests at the UC Berkeley Earthquake Engineering Research Center’s (EERC) Earthquake Simulator Laboratory The same type of pressboard surface that was used as the base for the slow-pull tests was built on the shake table to support the equipment Figure shows a photograph of one of the freestanding equipment resting on the shake table The displacement of the shake table and the equipment were measured with wire transducers attached to a frame fixed on the laboratory floor Figure shows the locations of the wire transducers on the test specimen with heavy white lines Accelerometers were also installed on the positions shown with black arrows in Figure in order to capture horizontal and vertical accelerations The horizontal displacement capacity of the shake table at EERC is ±15 cm Given this constraint, experiments at full scale were run only for moderate ground motions (i.e with hazard level equal to 50% in 50 years and 10% in 50 years) In several occasions the equipment while shaken along the primary direction, exhibited rotations about its vertical axis In some cases these plane rotations were as small as 0.005 rad, while in others as large as 0.33 rad, indicating that even when the excitation is one-directional, the response is in fact in three dimensions Table I lists the earthquake records that were used to test the freestanding equipment Also listed are the recorded peak table acceleration (PTA) and the equipment peak sliding displacement, Umax The PTA level of the records used in this experimental program ranged from 0.43 to 1.70g and Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe LABORATORY EQUIPMENT TO EARTHQUAKE SHAKING 835 Figure The FORMA incubator resting atop the shake table at the UC Berkeley Earthquake Engineering Research Center The locations of the wire transducers are indicated with white lines, and the locations of the accelerometers are indicated with dark arrows the Umax from cm to as much as 60 cm The shake table tests indicated that although some uplift occurs, the primary mode of response of the three pieces of equipment is sliding Criteria provided by Shenton [19] allow the determination of the initiation of a response mode using the values of listed in Table II in conjunction with the values of s listed in Table III If the values of s obtained from the slow-pull tests are used, slide-rocking is anticipated to initiate for all three pieces of equipment used in this study However, sliding dominates the response In fact, the maximum recorded uplift rotation in all tests for the Kelvinator refrigerator was only 0.02 rad (0.069 ) and only 0.005 rad (0.015 ) for the ASP refrigerator The maximum recorded uplift rotation for the FORMA incubator was 0.13 rad (0.489 ) Although the FORMA incubator experienced larger uplift rotations than the other two specimens, these rotations were still well below the level for overturning (> ) Nonetheless, after about 20 tests one of the flimsy leg supports of the FORMA incubator failed The resulting instability caused overturning of the specimen Figure (bottom window) plots the OTE FP ground-acceleration history recorded during the 1995 Aigion, Greece, earthquake The graph on the window above the acceleration record plots the resulting shake table displacement, and the third window from the top plots with a heavy solid line the recorded sliding displacement of the FORMA incubator The recorded sliding displacement history shows that the equipment suddenly slides once a threshold table acceleration is exceeded (at about 3.8 s) The top two graphs in Figure that plot with heavy solid lines the equipment uplift and the plane rotation (rotation about the vertical axis) show that the specimen does uplift and twist just slightly even before the initiation of sliding (at about 3.65 s) In other cases, one mode of response does not seem to trigger the other, but rather both happen simultaneously The response mode coupling is less pronounced for the Kelvinator refrigerator which, although demonstrating a slightly larger coefficient of friction during the slow-pull tests than the FORMA incubator ( s = 0.37 versus s = 0.30), is stockier ( = 0.289 versus = 0.266) and therefore less susceptible to uplift The Kelvinator refrigerator exhibited relatively small rotations, never exceeding 7% its Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 836 D KONSTANTINIDIS AND N MAKRIS equipment: FORMA incubator uplift [rad] -0.1 10 20 15 10 -5 15 10 -5 10 10 10 10 0.05 -0.05 table acc [g] table disp [cm] sliding [cm] plane rot [rad] motion: Aigion, OTE FP 0.1 -1 time [sec] Figure Response of the FORMA incubator subjected to the OTE FP motion recorded during the 1995 Aigion, Greece, earthquake The heavy gray lines on the bottom two graphs plot a Type-B trigonometric pulse that approximates the main pulse of the motion record stockiness angle, , in any of the tests Moreover, the Kelvinator refrigerator exhibited very small rotation about its vertical axis The maximum recorded plane rotation in all tests performed on the Kelvinator refrigerator was 0.05 rad Figure plots the response of the Kelvinator refrigerator to the computed [9, 12] 6th-floor motion of the Gavilan College FN (10% in 50 years) record of the 1989 Loma Prieta, California, earthquake The ASP refrigerator is considerably stockier in one direction than in the other ( W = 0.400 versus D = 0.313; see Table II and Figure 2) When the table excitation is along the more stocky direction (W ), the configuration is designated Profile, whereas when the table excitation is along the more slender direction (D), the configuration is designated Face The recorded response for all the shake table tests performed on the ASP refrigerator in the Face configuration and the Profile configuration is almost identical The maximum recorded uplift rotation did not exceed 0.005 rad (1.5% of D ) Figure plots the recorded response of the ASP refrigerator (Profile configuration) subjected to the computed [9, 12] 6th-floor motion of the Gilroy Array #6 FN (50% in 50 years) record of the 1979 Coyote Lake, California, earthquake An interesting characteristic to note is the waviness of the heavy solid lines that plot the sliding displacement This wobbling is more pronounced for the FORMA incubator whose legs were very flexible, less pronounced for the Kelvinator refrigerator whose legs are fairly stiff, and almost non-existent for the ASP refrigerator whose legs are nearly rigid Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 837 LABORATORY EQUIPMENT TO EARTHQUAKE SHAKING equipment: KELVINATOR refrigerator motion: Loma Prieta, Gavilan College FN 6th Floor (10% in 50yrs) table acc [g] table disp [cm] sliding [cm] plane rot [rad] uplift [rad] 0.1 -0.1 0.05 10 10 10 10 10 -0.05 -5 -10 -15 15 10 -5 -10 -15 -1 time [sec] Figure Response of the Kelvinator refrigerator subjected to the UC Science Building 6th-floor motion of the Gavilan College (10% in 50 years) record of the 1989 Loma Prieta, California, earthquake The heavy gray lines on the bottom two graphs plot a Type-C1 trigonometric pulse that approximates the main pulse of the motion record REGRESSION ANALYSIS AND FRAGILITY CURVES 5.1 Governing parameters during sliding Many parameters influence the full behavior of a piece of equipment subjected to seismic motion However, since we are primarily concerned with sliding, the governing parameters become those that describe (a) the mechanical characteristics of the equipment–floor interface and (b) the kinematic characteristics of the base motion From the slow-pull tests performed on the equipment, the load–displacement curves show that there is a pre-yielding elasticity due to the flexibility of the legs A peak value of force is reached (associated with the static coefficient of friction s ) after which the equipment starts sliding with a relatively constant force (associated with the kinetic coefficient of friction k ) Among these three parameters, the one parameter that best describes the sliding resistance of the interface is the kinetic coefficient of friction k , designated hereafter simply as Figures 5–7 that plot the results of the shake table tests on the freestanding equipment also plot the results obtained by numerical simulations for two different models of the sliding interface: an elastoplastic model (MATLAB) and a rigid-plastic (Coulomb friction) model (WM2D) It was noted that when the values obtained from the slow-pull tests (see Figure and Table III) Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 838 D KONSTANTINIDIS AND N MAKRIS equipment: ASP refrigerator (profile) uplift [rad] -0.1 0.05 10 10 10 10 10 sliding [cm] -5 -10 -15 -20 -25 table disp [cm] -0.05 15 10 -5 -10 -15 table acc [g] plane rot [rad] motion: Coyote Lake, Gilroy Array #6 FN 6th Floor (50% in 50yrs) 0.1 -1 time [sec] Figure Response of the ASP refrigerator ( profile configuration) subjected to the UC Science Building 6th-floor motion of the Gilroy Array #6 (50% in 50 years) record of the 1979 Coyote Lake, California, earthquake The heavy gray lines on the bottom two graphs plot a Type-C1 trigonometric pulse that approximates the main pulse of the motion record were used, the numerical predictions of the elastoplastic model (plotted with dashed gray lines in Figures 5–7) are, in general, closer to the experimental results (heavy black lines) than the numerical prediction offered by the rigid-plastic (Coulomb friction) model (solid gray lines) Nevertheless, even the elastoplastic model predictions are only in fair agreement with the experimental results A considerably improved agreement between experimental and numerical results from either model is observed when the numerical value of is reduced in the numerical simulations The reader will note that the reduction proposed for the rigid-plastic (Coulomb) model is consistently larger in all three equipment than the reduction proposed for the elastoplastic model Part of the reason for the reduction of the friction coefficient in the elastoplastic model is that the friction coefficient between the steel equipment legs and the vinyl floors may exhibit some pressure dependence that was not captured during the quasi-static pull tests During the quasi-static tests, the equipment specimen was pulled from a height of approximately 15 cm from the sliding surface, which means that the pressure on all four legs was about equal However, during the shake table tests, the overturning moment on the equipment results in an uneven distribution of pressure on the equipment legs, which in turn may have resulted in lower overall frictional resistance This would be similar to the pressure-dependent behavior of a teflon–steel interface observed by other investigators [16, 17] The further reduction that is needed in the rigid-plastic (Coulomb) model is due to its zero yield displacement Recent studies by the senior author [20, 21] have shown that for the same value of the Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe LABORATORY EQUIPMENT TO EARTHQUAKE SHAKING 839 friction coefficient (strength) the sliding displacement increases with increasing yield displacement Consequently, given that in all three pieces of equipment the behavior at the sliding interface is elastoplastic (finite yield displacement), when a rigid-plastic (Coulomb) model is adopted to capture the behavior, a smaller value of the coefficient of friction is needed Interestingly, the amount of reduction in the value of the coefficient of friction is proportional to the yield displacement—a conclusion that is consistent with the aforementioned studies by the senior author 5.2 Kinematic characteristics of pulse-type motions Physically realizable pulses can adequately describe the impulsive character of near-fault ground motions both qualitatively and quantitatively The input parameters of the model have an unambiguous physical meaning The minimum number of parameters is two, which are either the acceleration amplitude, ap , and duration, Tp , or the velocity amplitude, vp , and duration, Tp [22–25] The current established methodologies for estimating the pulse characteristics of a wide class of records are of unique value since the product ap Tp2 is a characteristic length scale of the ground excitation and is a measure of the persistence of the most energetic pulse to generate inelastic deformations in structures [26] It is emphasized that the persistence of the pulse is a different characteristic than the strength of the pulse which is measured with the peak pulse acceleration The reader should recall that among two pulses with different acceleration amplitudes (say ap1 >ap2 ) and different pulse durations (say Tp1 1, and since generally s > , it is possible for sliding to occur even for negative values of the chosen IM Nevertheless, the numerical simulation studies that were done in parallel with the experimental studies showed that the coefficient of static friction s had little influence on the maximum sliding displacement that the equipment exhibited On the contrary, the maximum sliding displacement was considerably sensitive to the kinetic coefficient of friction For this reason, the expression for IM given by (3) features and not s The chosen EDP will be henceforth designated as It is given by = p Umax (4) PTA where Umax is the maximum sliding displacement recorded, p = /Tp is the circular frequency of the pulse that approximates the predominant pulse of the earthquake excitation and PTA is the peak table acceleration of the earthquake excitation The choice for the EDP emerges from dimensional analysis in conjunction with previously published results on the response of a sliding block [10, 29–31] The quantity of interest Umax can be expressed as a function of independent variables ap , p , and g Umax = f (ap , p, g) (5) The dependent variable Umax and independent variables ap , p and g, involve only two dimensions, those of length, L, and time, T The quantities of interest have dimensions [Umax ] = L , [ap ] = LT −2 , [ p] = T −1 , [ g] = LT −2 (6) Buckingham’s Pi Theorem states that a dimensionally homogeneous equation with a total of k variables and r reference dimensions can be reduced to a relationship among k −r independent dimensionless -products [30] Accordingly, in this case there are 4−2 = dimensionless products Two obvious choices for -products are 1= p Umax 2= and ap g ap (7) and the two are related by a function = ( 2) (8) For the rectangular acceleration pulse, this function is [29] =2 2 −1 (9) Note from Equations (3) and (9) that the IM chosen is exactly the quantity in parenthesis in (9) except that the IM uses for simplicity PTA instead of ap The two values are expected to be close For the case of trigonometric pulses such as Type-A, Type-B, and Type-Cn , the response of the rigid-plastic system is again described by Equation (8) [10, 26], and the form of the function is obtained numerically Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 842 D KONSTANTINIDIS AND N MAKRIS 5.4 The EDP as a lognormal random variable; regression of experimental data Figure (bottom left) plots the EDP, =Umax 2p /PTA, as a function of the IM = PTA/ g −1, for the shake table tests performed on the three pieces of equipment The value of the coefficient of friction, , is that obtained by the slow-pull tests It is obvious that the data exhibit considerable scatter, which suggests that has to be treated as a random variable When a random variable, , expresses a quantity that is only positive ( >0), it is common to assume that the variable is lognormally distributed In this study we hypothesize that the EDP, , is lognormally distributed, and we test this hypothesis against the experimental results The reason behind the name lognormal is that the lognormally distributed variable is related to a normally distributed variable X by X = ln Note that attains only positive values, >0, while the corresponding X variable is unrestricted, −∞c|IM) = 1− P( 2) = P(Umax >45 cm) ≈ 0.3 is read with engineering judgment Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 846 D KONSTANTINIDIS AND N MAKRIS Figure 10 Fragility curves for c = 0.5 (top left), c = 1.0 (bottom left), c = 2.0 (top right) and c = 3.0 (bottom right) Two curves are plotted in each graph The top curve corresponds to values of obtained by slow-pull tests, while the bottom curve corresponds to values of obtained by best-fitting numerical simulation results to shake table experiment results CONCLUSIONS In this study, a comprehensive experimental program investigating the seismic response of freestanding laboratory equipment located within several floor levels of a research facility was undertaken The study followed the PBEE approach Results from shake table tests experiments on the freestanding equipment subjected to ground and floor motions of 50 and 10% in 50 years hazard levels were presented For the equipment tested, although there was some rocking observed (particularly for the FORMA incubator), sliding was the predominant mode of response, with sliding displacements reaching up to 60 cm The experimental results were then used to calibrate and validate numerical simulation models Numerical simulation studies with MATLAB [34] on the sliding response of the equipment using the elastoplastic model with the values of the friction coefficient extracted from the slow-pull tests yielded results that were in fair agreement with the experimental results The predicted response of the equipment was appreciably improved when reduced values of their friction coefficients were used Numerical simulation studies with Working Model [14] using the Coulomb friction model also provided fair results that were considerably improved when the values of s and k were reduced The IM, IM = PTA/ g −1, and (random) EDP, =Umax 2p /PTA, were identified The results of the shake table tests were then used to test the hypothesized lognormal distribution of the random variable and to arrive at simple relationships for the mean m and standard deviation in Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe LABORATORY EQUIPMENT TO EARTHQUAKE SHAKING 847 terms of the IM Fragility curves, that give the probability that the EDP will exceed a specified threshold c as a function of the IM were generated, and an example was presented to illustrate how the fragility curves can be used ACKNOWLEDGEMENTS From the UC Science Laboratory Building, we thank Barbara Duncan and Todd Laverty for providing the equipment that was used for the shake table experiments; Eddy Kwan for providing us with invaluable information regarding the laboratory equipment and its in situ boundary conditions in the building laboratories From the Earthquake Simulator Laboratory at the Earthquake Engineering Research Center, UC Berkeley, where the shake table experiments were conducted, we greatly appreciate the technical assistance of Don Clyde, Wes Neighbour and David Maclam We also thank our PEER colleagues for their collaboration We are very grateful for their continual feedback throughout the course of this study This work was supported primarily by the Earthquake Engineering Research Centers Program of the National Science Foundation under award number EEC-9701568 through the Pacific Earthquake Engineering Research Center (PEER) REFERENCES Yim CS, Chopra AK, Penzien J Rocking response of rigid blocks to earthquakes Earthquake Engineering and Structural Dynamics 1980; 8(6):565–587 Makris N, Roussos Y Rocking response of rigid blocks under near-source ground motions G´eotechnique 2000; 50(3):243–262 Makris N, Konstantinidis D The rocking spectrum and the limitations of practical design methodologies Earthquake Engineering and Structural Dynamics 2003; 32(2):265–289 Shao Y, Tung CC Seismic response of unanchored bodies Earthquake Spectra 1999; 15(3):523–536 Lopez Garcia D, Soong TT Sliding fragility of block-type non-structural components Part 1: unrestrained components Earthquake Engineering and Structural Dynamics 2003; 32(1):111–129 Lopez Garcia D, Soong TT Sliding fragility of block-type non-structural components Part 2: restrained components Earthquake Engineering and Structural Dynamics 2003; 32(1):131–149 Hutchinson TC, Chaudhuri SR Bench-shelf system dynamic characteristics and their effects on equipment and contents Earthquake Engineering and Structural Dynamics 2006; 35(13):1631–1651 Porter KA An overview of PEER’s performance-based earthquake engineering methodology Proceedings of the Ninth International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP9), San Francisco, CA, 2003 Comerio MC (ed.) PEER testbed study on a laboratory building: exercising seismic performance assessment Report No PEER 2005-12, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, 2005 10 Konstantinidis D, Makris N Experimental analytical studies on the seismic response of freestanding and anchored laboratory equipment Report No PEER 2005-07, Pacific Earthquake Engineering Research Center, University of California, Berkeley, CA, 2005 11 Somerville PG Ground Motion Time Histories for the UC Lab Building URS Corporation: Pasadena, CA, 2001 12 Lee TH, Mosalam KM Seismic demand sensitivity of reinforced concrete shear-wall building using FOSM method Earthquake Engineering and Structural Dynamics 2005; 34(14):1719–1736 13 Housner GW The behavior of inverted pendulum structures during earthquakes Bulletin of the Seismological Society of America 1963; 53(2):404–417 14 Working Model User’s Manual MSC Software Corporation: San Mateo, CA, 2000 15 Gitis NV, Volpe L Nature of static friction time dependence Journal of Physics D: Applied Physics 1992; 25(4):605–612 16 Mokha A, Constantinou MC, Reinhorn AM Teflon bearings in aseismic base isolation: experimental studies and mathematical modeling Technical Report NCEER-88-0038, National Center for Earthquake Engineering Research, State University of New York at Buffalo, NY, 1988 Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe 848 D KONSTANTINIDIS AND N MAKRIS 17 Constantinou MC, Tsopelas P, Kim YS, Okamoto S NCEER-Taisei Corporation research program on sliding seismic isolation systems for bridges: experimental and analytical study of a friction pendulum system (FPS) Technical Report NCEER-93-0020, National Center for Earthquake Engineering Research, State University of New York at Buffalo, NY, 1993 18 Konstantinidis D, Kelly JM, Makris N Experimental investigation on the seismic response of bridge bearings Report No EERC 2008-02, Earthquake Engineering Research Center, University of California, Berkeley, CA, 2008 19 Shenton III HW Criteria for initiation of slide, rock, and slide-rock rigid-body modes Journal of Engineering Mechanics (ASCE) 1996; 122(7):690–693 20 Makris N, Black CJ Dimensional analysis of rigid-plastic and elastoplastic structures under pulse-type excitations Journal of Engineering Mechanics (ASCE) 2004; 130(9):1006–1018 21 Makris N, Psychogios T Dimensional response analysis of yielding structures with first-mode dominated response Earthquake Engineering and Structural Dynamics 2006; 35(10):1203–1224 22 Makris N, Chang SP Effects of viscous, viscoplastic and friction damping on the response of seismic isolation structures Earthquake Engineering and Structural Dynamics 2000; 29(1):85–107 23 Makris N, Chang SP Response of damped oscillators to cycloidal pulses Journal of Engineering Mechanics (ASCE) 2000; 126(2):123–131 24 Makris N Rigidity-plasticity-viscosity: can electrorheological dampers protect base-isolated structures from nearsource ground motions? Earthquake Engineering and Structural Dynamics 1997; 26(5):571–591 25 Mavroeidis GP, Papageorgiou AS A mathematical representation of near-fault ground motions Bulletin of the Seismological Society of America 2003; 93(3):1099–1131 26 Makris N, Black CJ Dimensional analysis of inelastic structures subjected to near-fault ground motions Report No EERC 2003-05, Earthquake Engineering Research Center, University of California, Berkeley, CA, 2003 27 Veletsos AS, Newmark NM, Chelepati CV Deformation spectra for elastic and elastoplastic systems subjected to ground shock and earthquake motions Proceedings of the 3rd World Conference on Earthquake Engineering, Wellington, New Zealand, 1965 28 Jacobsen LS, Ayre RS Engineering Vibrations, with Applications to Structures and Machinery McGraw-Hill: New York, NY, 1958 29 Newmark NM Effects of earthquakes on dams and embankments Fifth Rankine Lecture G´eotechnique 1965; 15:139–160 30 Barenblatt GI Scaling, Self-similarity, and Intermediate Asymptotics Cambridge University Press: Cambridge, U.K., 1996 31 Konstantinidis D, Makris N Seismic response analysis of multidrum classical columns Earthquake Engineering and Structural Dynamics 2005; 34(10):1243–1270 32 Crow EL, Shimizu K (eds) Lognormal Distributions Marcel Dekker: New York, NY, 1988 33 Scheaffer RL, McClave JT Probability and Statistics for Engineers (4 edn) Duxbury Press: Belmont, CA, 1995 34 MATLAB High-performance Language Software for Technical Computing The MathWorks, Inc.: Natick, MA, 2002 Copyright q 2008 John Wiley & Sons, Ltd Earthquake Engng Struct Dyn 2009; 38:827–848 DOI: 10.1002/eqe