Engineering Structures 59 (2014) 462–468
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Torsional response of seismically isolated structures revisited E.D. Wolff a,1, C. Ipek b,1, M.C. Constantinou c, L. Morillas d,⇑,1 a
Fyfe Co, 8380 Miralani Drive, San Diego, CA 92126, USA Department of Civil Engineering, Faculty of Civil Engineering, Istanbul Technical University, Maslak, Istanbul, Turkey c Department of Civil, Structural and Environmental Engineering, University at Buffalo, SUNY, 132 Ketter Hall, Buffalo, NY 14260, USA d Department of Mechanics of Structures, University of Granada, ETSICCP Avenida Fuentenueva, 18071 Granada, Spain b
a r t i c l e
i n f o
Article history: Received 21 June 2013 Revised 25 October 2013 Accepted 14 November 2013
Keywords: Seismic isolation Torsional response Torsional amplification Accidental eccentricity Scaling principles
a b s t r a c t The torsional response of seismically isolated structures is revisited. Experimental results on the torsional response of seismic isolated structures in the past have been conducted with unrealistically torsionally stiff seismic isolation systems. It is shown in this paper that the measured torsional amplification ratios correspond to accidental eccentricities of about half of the code-described value of 5-percent of largest plan dimension. Current practice to calculate the amplification factor for the displacement demand due to torsion is discussed and a rational and simple improvement for predicting these amplification factors is presented. The revised equations are useful in predicting the torsional response of torsionally stiff seismic isolation systems and in assessing the utility of shake table experimental results on the basis of scaling principles. Ó 2013 Elsevier Ltd. All rights reserved.
1. Introduction The calculation of the seismic response of seismically isolated structures requires consideration of torsion. Torsion results from eccentricities between the centers of mass and rigidity at various levels of the isolated structure (asymmetric buildings and/or asymmetric isolation systems), variations in the mechanical properties of structural elements and from the combined effects of torsional and rocking ground motions. The latter are not directly considered in response history analysis although it is known that they may be important [1–6]. Rather, current practice in seismically isolated structures is to: (a) Perform response history analysis with only translational components of the ground motion and indirectly account for uncertainty in the locations of the centers of mass and eccentricity and of torsional and rocking ground motion effects through the use of the accidental eccentricity. In this approach, the center of mass is artificially shifted by the code prescribed accidental eccentricity (typically a distance equal to 0.05 of the building plan dimension).
⇑ Corresponding author. Tel.: +34 (958)249960. E-mail addresses:
[email protected] (E.D. Wolff),
[email protected] (C. Ipek),
[email protected] (M.C. Constantinou),
[email protected] (L. Morillas). 1 Formerly: University at Buffalo, SUNY, Buffalo, NY, USA. 0141-0296/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2013.11.017
(b) Perform response history analysis with only translational components of the ground motion and without consideration of accidental eccentricity and then magnifying the results of the analysis to account for torsion on the basis of some rational amplification factors applied to forces, drifts and displacements. The accidental eccentricity approach is used to indirectly account for: (a) plan distributions of mass that differ from those assumed in design, (b) variations in the mechanical properties of structural components in the seismic force-resisting system, (c) non-uniform yielding of components in the seismic force-resisting system, and (d) torsional and rotational ground motions. It is meant to be used in static analysis calculations, as for example the Equivalent Lateral Force (ELF) method of the ASCE Standard 7, Minimum Design Loads for Buildings and Other Structures [7]. The shifting of the center of mass by the code prescribed accidental eccentricity in the response history analysis of seismic isolated structures is problematic for the following reasons: (a) the code prescribed shift of the position of the center of mass leads to unrealistic distribution of mass, (b) the shift in the center of mass does not result in the expected eccentricity in gravity based isolation systems (such as the Friction Pendulum) for which the centers of resistance and mass coincide, (c) the analysis considers the spatial distribution and the nonlinear hysteretic behavior of the isolators so that non-uniform yielding is directly accounted for, and (d) it has been shown [1] that this approach leads to incorrect results as increasing the eccentricities between the centers of
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mass and rigidity changes the dynamic characteristics of the analyzed structure so that one may calculate reduction of torsional response with increasing eccentricity. The authors believe that the approach of conducting response history analysis of seismically isolated structures without consideration of accidental eccentricity and then application of an amplification factor on the calculated response represents a rational procedure. The question then is what the amplification factor should be. An approach followed by some engineers is to utilize the amplification factor calculated by the ELF procedure. This is somehow conservative as the so-calculated factor includes some effects already accounted for in the response history analysis (i.e., non-uniform yielding). Nevertheless, it is a simple and rational procedure. Another approach is to perform nonlinear response history analysis with due consideration for realistic distribution of mass and stiffness and explicitly accounting for the effects of the rotational ground motion [1]. This is a highly complex procedure that requires numerous analyses and involves the generation of rotational ground motion histories [10,11]. Such a procedure would be most useful in calibrating the simpler approach of using an amplification factor. This paper concentrates on the ELF approach in ASCE 7 [7] for calculating the amplification factor for the isolator displacement demand due to torsion. Specifically, ASCE 7 has the following equation for calculating the displacement DT at the corner bearing by multiplying the displacement D at center of rigidity (subscripts of D and M for the design and maximum displacements, respectively, are omitted):
12e DT ¼ D 1 þ y 2 2 b þd
ð1Þ
In this equation, e is the distance between the center of rigidity of the isolation system and the center of mass of the structure above (actual plus an added accidental eccentricity of 5% of the plan dimension perpendicular to the direction of seismic action), b and d are the plan dimensions of a representative rectangular plan for the structure and y is the distance of the bearing to the center of rigidity. Note that the quantity (b2 + d2)/12 represents the square of the radius of gyration of the rectangular structure. ASCE 7 also includes an exception that allows the use of an amplification factor of not less than 1.1 provided that ‘‘the isolation system is shown by calculation to be configured to resist torsion’’. Note that the amplification factor predicted by Eq. (1) varies between 0.17 (for a square plan) and 0.33 (for a strip plan) when the e is taken as 5% of the largest plan dimension; arguably large values. Eq. (1) has been used in two ways: (a) In the ELF procedure calculations where often the minimum amplification factor of 1.1 is used on the basis of arguments that the system has coinciding centers of mass and rigidity or on the basis of experimental results. (b) In calculating an amplification factor for use in the response history analysis results where typically the 1.1 minimum value is utilized on the basis of the same arguments as in item (a) above. Some engineers disregard Eq. (1) in calculating the amplification factor for use with the response history analysis results where it is often claimed that the nonlinear response history analysis explicitly accounts for all contributors to torsional amplification. However, recent studies, although limited, have demonstrated that the rotational components of the ground motion may amplify the isolator displacement demands by as much as the amplification predicted by Eq. (1) [1]. Interestingly, the rocking components of
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the ground motion further affect the displacement demand, an observation also made earlier by Almazan and de la Llera [12]. The derivation of Eq. (1) is re-visited in this paper and a simple rational modification is proposed. The modification is useful in evaluating experimental results and in calculating amplification factors for torsionally stiff seismic isolation systems. Experimental results on the torsional response of seismically isolated structures are also presented in this paper and discussed in terms of their utility in establishing torsional amplification factors. This paper only addresses the effect of asymmetry in the isolated structure on the isolation system displacement demand when the structure is subjected to translational ground motions. The effects of rocking and torsional ground motion effects are not addressed. It is demonstrated that all test data are based on unrealistically torsionally stiff isolation systems so some analysis is required to extrapolate the results to full size structures. The modification to Eq. (1) provides the theoretical basis for the extrapolation of the results. It is argued that the modification of Eq. (1) be used for calculating the amplification factor for torsion in both the ELF and the response history analysis methods. Limitations of this approach are discussed. 2. ELF procedure for amplification of response due to torsion Eq. (1) originally appeared in a 1986 document of the Structural Engineers Association of Northern California [13]. It is derived by considering a rigid seismic isolated structure subject to a lateral force V (equal to the effective stiffness of the isolation system times the displacement D of the center of rigidity-CR) and a static moment equal to Ve at the CR of the system, where e is the eccentricity. Furthermore, the derivation is based on an ideal situation in which the effective isolation system translational period is equal to the effective isolation system torsional period. Some of the earliest studies on coupled lateral-torsional response of seismically isolated structures have made this assumption (e.g., Pan and Kelly [14]). An improvement to Eq. (1) is derived by relaxing the assumption of equal translational and torsional period. Let define the ratio of effective isolation system translational period T to the effective isolation system torsional period Th as Xh. That is,
Xh ¼
T Th
ð2Þ
Winters and Constantinou [15] have shown that the effective damping of the isolated structure in the torsional mode bh is related to the effective damping in the translational direction b:
bh ¼ Xh b
ð3Þ
Also, they showed that the ratio of the effective periods is given by the following simple expression provided that the effective translational stiffness of each isolator in each principal direction is the same:
1 Xh ¼ r
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PN 2 2 i¼1 ðxi þ yi Þ N
ð4Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 In Eq. (4), r is the radius of gyration (¼ ðb þ d =12 for a rectangular structure of plan dimensions b d and uniform distribution of mass), N is the number of isolators and xi and yi are the coordinates of the ith isolator with respect to the CM. Note that the derivation of Eqs. (3) and (4) involves an approximation that disappears as the eccentricity between the CM and CR diminishes. A properly designed seismic isolation system has very small or essentially zero actual eccentricity (this property is natural for gravity based systems such as the Friction Pendulum system). That is, Eqs. (3) and
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(4) are valid for the small accidental eccentricities considered in the design of seismically isolated structures. Use of Eq. (4) or the exact equation (see Pan and Kelly [14]) for a number of isolated structures gives essentially the same result. Table 1 presents values of parameter Xh for a number of structures calculated either by exact analysis or by the use of Eq. (4). Note that for the tested full scale structure in Japan [16], the value of Xh was calculated assuming uniform distribution of mass, which is approximate. The results of Table 1 show that seismically isolated structures have a translational period systematically larger than the torsional period and the two approach each other as the number of relatively uniformly distributed isolators increases. A value of Xh larger than unity denotes a torsionally stiff system. The authors were unable to identify an isolated structure with a value of parameter Xh less than 1.0; unlike the case of non-isolated buildings in which some studies reported values of this parameter less than unity [2–5]. Actually, many seismically isolated structures are intentionally designed to resist torsion by distributing isolators of higher effective stiffness on the perimeter of the isolation system. A notable example is the USC Hospital [17,18]. There are two important implications of the fact that parameter Xh is generally larger than unity: (a) the torsional component of displacement should be less than what is in Eq. (1) due to the torsional stiffness, and (b) the torsional component of displacement should be further reduced due to the increased effective damping by virtue of Eq. (3). Neglecting the increased damping in the torsional mode, it is a simple exercise to modify Eq. (4) to include the effect of Xh being different than unity:
" DT ¼ D 1 þ
y
12e
X2h b2 þ d2
#
" ¼ D 1 þ PN
eyN
2 i¼1 ðxi
# ð5Þ
þ y2i Þ
Note that the second version of Eq. (5) is derived by use of Eq. (4). Eq. (5) could also be written in the following more general form, where r is the radius of gyration:
"
# 1 ye DT ¼ D 1 þ 2 2 Xh r
ð6Þ
As an example of how parameter Xh affects the torsional amplification ratio DT/D, Table 2 presents values of this ratio for some of the isolated structures of Table 1 (the highly asymmetric examples of Ryan and Chopra [19] and the USC Hospital-also asymmetrichave been excluded) as calculated by Eqs. (1) and (5). Note that Eq. (6) gives the same results as Eq. (5) because, for the considered structures, the mass is approximately uniformly distributed. Eq. (6) represents a more accurate predictor of torsional response for structures with non-uniform distribution of mass. Evidently, the effect is important (especially when the structure is torsionally stiff, i.e: larger values of Xh) so that one can conclude that Eqs. (5) and (6) deserve consideration for inclusion in specifications for the design of isolated structures.
Ryan and Chopra [19] made an observation based on results of response history analysis of several asymmetric isolated structures that the torsional amplification ratio may be approximated by:
DT h yei0:88 ¼ 1þ 2 r D
ð7Þ
Given that quantity ye/r2 is much smaller than unity, Eq. (7) may be written, with very good approximation, as:
DT h yei 1 þ 0:88 2 r D
ð8Þ
Eq. (8) is in the form of Eqs. (5) and (6) with factor 0.88 representing the inverse of X2h . It is likely that Eq. (7) is valid but the exponent should have some relation to the translational to torsional period ratio of the isolation system. 3. Experimental results on torsional response of seismically isolated structures The earliest experimental study of torsional response of a seismically isolated structure is the one of Zayas et al. [20]. A one story model was supported by four single Friction Pendulum (FP) isolators and tested on a shake table. Parameter Xh was about 1.73 (based on rectangular shape of mass of the model and with the isolators placed at the corners of the mass). The mass of the model was shifted to create mass eccentricity. However, the inherent ability of the FP system to have its CR coincide with the CM, eliminated any eccentricities in the isolation system. Moreover, the model had one of its four columns rotated to bend about the weak axis so that a stiffness eccentricity in the superstructure was created. Torsional amplification ratios for the isolator displacements of 1.035 or less were recorded. This low value of the torsional amplification ratio should be assessed on the basis of the large value of parameter Xh = 1.73. Actually, the important parameter is ye=X2h r2 which will be discussed later in this paper. A more recent experimental study of Wolff and Constantinou [21] investigated the torsional response of a six-story seismically isolated model structure in which the isolation system was symmetrically placed, mass eccentricities were eliminated and severe stiffness eccentricities were created in the superstructure. A schematic and a picture of the tested structure are shown in Figs. 1 and 2. The eccentricity was created by adding x-bracing (angles L 1 ½ 1 ½ 1=4 ) on the east side of the frame only. Effectively, the CR and CM of the superstructure were separated by a distance of 21% of the plan dimension perpendicular to the direction of testing. Four isolators on a grid of 1.22 m by 2.44 m supported the isolated structure which had a mass lumped over an area larger that the isolator footprint. The parameters of the system were y = 0.61 m, e = 0.21 1.22 m, r = 1.2 m, and Xh = 1.13. Avoiding mass-related eccentricities and rather creating stiffness-related eccentricities in these tests resulted in a more realistic representation of isolated structures and prevented an unfair comparison of isolation systems. Note that isolated structures are
Table 1 Values of ratio of translational to torsional effective period of isolated structures.
a
Structure
Number of isolators
Xh
References
Erzurum Hospital, Turkey USC Hospital, Los Angeles Analysis example Analysis example Various asymmetric-plan examples Tested isolated full scale model in Japan Tested isolated models at Buffalo
386 149 45 18 16–20 9 4
1.04a 1.20 1.14a 1.22a 1.15–1.28 1.42a 1.13
Constantinou et al [22] Celebi [17], Nagarajaiah and Xiaohong [18] Winters and Constantinou [15] Basu [1] Ryan and Chopra [19] Dao et al [16] Wolff and Constantinou [21]
Calculation based on assuming uniform distribution mass over rectangular plan (Eq. (4)).
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E.D. Wolff et al. / Engineering Structures 59 (2014) 462–468 Table 2 Values of torsional amplification ratio of isolated structures. Structure
Xh
Erzurum Hospital, Turkey [22] Analysis example, Winters and Constantinou [15] Analysis example, Basu [1] Tested isolated full scale model in Japan, Dao et al [16] a
a
1.04 1.14a 1.22a 1.42a
r (m)
b (m)
d (m)
DT/D Eq. (1)
DT/D Eq. (5)
61.4 15.8 15.2 4.5
140.0 24.4 15.2 10.0
160.0 48.8 48.8 12.0
1.187 1.264 1.284 1.195
1.173 1.203 1.191 1.097
Calculation based on assuming uniform distribution mass over rectangular plan (Eq. (4)).
1220 typ
1220 typ
915 typ
∠ 37 x 37 x 6.25 typ
S 3x5.7 typ
8.9 kN typ
190
330
2440 Earthquake simulator platform South
W14x90 rigid base
Bearing
Load Cell
Testing Direction
North
Fig. 1. Schematic of tested six-story isolated model structure.
intentionally designed to substantial reduce or eliminate eccentricities between the CR of the isolation system and the CM. As an example, the Erzurum Hospital in Turkey [22] had a distance between the CR of the isolation system (calculated using the nominal effective stiffness of the isolators at 500 mm displacement) and the CM equal to about 0.5% of the largest plan dimension of 160 m. Lead-rubber, single FP, low damping elastomeric, combined elastomeric and sliding, and elastomeric or FP with linear and nonlinear viscous dampers isolation systems were tested in this program. The viscous dampers were connected to the geometric center of the isolation system so that they did not resist torsion. Figs. 3–5 present the torsional amplification ratio recorded in each of the conducted tests. The amplification ratio is presented as function of the peak shake table velocity (the latter is in the scale of the experiments for which the velocity scale factor is 2, that is, the prototype velocity is twice the one in the figures). The interested reader is referred to Wolff and Constantinou [21] for more details on the isolation system and structure properties. In general, the recorded torsional amplification ratio is about equal to or less than 1.1. Again, the tested model had four isolators but the mass distribution extended beyond the footprint of the isolation system resulting in a value of parameter Xh = 1.13. In these tests, the value of parameter ye=X2h r 2 was equal to 0.085
(y = 0.61 m, e = 0.21 1.22 m, r = 1.2 m, Xh = 1.13). Thus, the torsional amplification ratio per Eq. (6) is 1.085 in consistency with the experimental results (less than about 1.1). The experimental values on the torsional amplification ratio somehow differed from results obtained in analytical studies on the torsional response of isolated structures by Nagarajaiah et al. [23,24] as described below. (a) One of the analyzed cases had the following attributes: no isolation system eccentricity (as in the tested systems), a superstructure eccentricity es/d 0.2 (or es/r = 0.5) (the tested systems had es/d = 0.21), a value of the translational to torsional period of the isolation system Xhb equal to 1.0 (the tested systems had Xhb = 1.13) and a value of the translational to torsional period of the superstructure (if fixed at the base) Xhs equal to 1.7 (the tested superstructure had Xhs = 1.6). That is, the analyzed system had structural and eccentricity characteristics similar to the tested system. However, the isolation system properties differed, with the analyzed system having a period on the basis of the isolator post-elastic stiffness equal to 2.1 s and a characteristic strength to weight ratio of about 0.05, whereas the tested systems had a period (in prototype scale) in the range of
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Fig. 2. Photograph of tested six-story isolated model structure.
Fig. 4. Torsional amplification ratio for elastomeric isolation system.
Fig. 5. Torsional amplification ratio for lead-core and combined elastomeric-sliding isolation system.
Fig. 3. Torsional amplification ratio for FP isolation system.
1.6–3.5 s and strength to weight ratio of 0.08–0.16. The tested system with characteristics closest to the analyzed one was the combined elastomeric and nonlinear damper system for which the period was 2.0 s and the strength to weight ratio was about 0.13. (b) The analyzed system had torsional amplification ratio values of 1.04 and 1.32 in analysis with two different ground motions (El Centro and Taft, respectively). The substantial
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difference in the amplification factor for the two cases of ground motion has not been observed in the tested systems which included tests with these two ground motions. Rather, the test results show values of the amplification ratio between 1.0 and about 1.1 in tests with 18 different motions. (c) There was a tendency for increase in the torsional amplification ratio with reducing value of the translational to torsional period of the superstructure (if fixed at the base) Xhs, that is, for torsionally flexible superstructures. However, the increase was highly dependent on the ground motion used in the analysis. The experimental data on torsion in the works of Zayas et al. [20] and Wolff and Constantinou [21] have been utilized to justify use of the exception in ASCE 7 [7] and use of a torsional amplification ratio of 1.1 for calculating the isolator displacement demand, either in the ELF procedure or in the response history analysis procedure. However, the data are based on testing of systems with (a) very large eccentricities, and (b) large ratio of effective translational to torsional isolation system period. It is important to investigate how these test data can be used to extract information on the behavior of full size isolated structures with smaller eccentricities and smaller ratio of translational to torsional isolation system period. The results of the experimental torsional studies can be extrapolated to the prototype scale provided that a suitable theory exists. We propose that Eq. (6) provides the theoretical background to utilize the results of the scaled experiments. Quantity ye=X2h r 2 is dimensionless and should be preserved in the scaled and the prototype structure. Concentrating on the model of Wolf and Constantinou [21], which resulted in data for many isolation systems and without unfair distribution of mass, the value of this quantity is ye=X2h r 2 ¼ 0:085 (based on y = 0.61 m, e = 0.21 1.22 m, Xh = 1.13, r = 1.2 m). The test data (that is, torsional amplification ratio less than 1.1) are valid for any isolated structure for which this quantity is less than or equal to 0.085. For example, let investigate the applicability of the data for the case of the Erzurum Hospital for which the plan dimensions are 160 m by 140 m, r = 61.4 m, y = 80 m and Xh = 1.04. Using ye=X2h r 2 ¼ 0:085 a value of e = 4.3 m is calculated. That is, the data are valid for an accidental eccentricity of 4.3/160 = 0.027 (or less) of the largest plan dimension. In another example, considered the structure analyzed in Winters and Constantinou [15]. The plan dimensions are 48.8 m by 24.4 m (1 ft = 0.305 m), r = 15.8 m, y = 24.4 m and Xh = 1.14. Using ye=X2h r 2 ¼ 0:085 a value of e = 1.1 m is calculated. That is, the data are valid for an accidental eccentricity of 1.1/48.8 = 0.023 (or less) of the largest plan dimension. In a final example consider the structure analyzed by Basu [1] for which the plan dimensions are 48.8 m by 15.2 m, r = 15.2 m, y = 24.4 m and Xh = 1.22. Using ye=X2h r 2 ¼ 0:085, a value of e = 1.2 m is calculated. That is, the data are valid for an accidental eccentricity of 1.2/48.4 = 0.025 (or less) of the largest plan dimension. These three examples demonstrate that the experimental data are valid for an accidental eccentricity of about half of the codeprescribed 5% of the plan dimension value.
4. Conclusions and recommendations Eq. (5) or its equivalent Eq. (6) presented in this paper are simple and rational improvements of the equation in the Equivalent Lateral Force method of ASCE 7 [7] for predicting the amplification of the isolator displacements for the effects of accidental torsion. The improved equations are particularly useful in: (a) predicting
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the torsional response of torsionally stiff seismic isolation systems and (b) assessing the utility of shake table experimental results on the basis of principles of scaling. Experimental results on the torsional response of seismic isolated structures have been conducted with unrealistically torsionally stiff seismic isolation systems. The most contemporary and unbiased of these results have been presented in this paper in Figs. 3–5. Despite the fact that the experimental data presented herein have been obtained for systems with large eccentricities, the recorded torsional amplification factors for the isolator displacements have been substantially lessened by the torsionally stiff isolation system configurations used. It has been shown that the measured torsional amplification ratios correspond to accidental eccentricities of about half the code-described value of 5% of largest plan dimension. Experimentally measured torsional amplification ratios should be evaluated on the basis of rational scaling principles if they will be used to justify the exception of ASCE 7 for a torsional amplification value of 1.1. A rational scaling principle has been presented and its use demonstrated in examples in this paper. It is noted that the analysis and experimental results presented in this paper do not consider the effects of rocking and rotational ground motion. These effects have been shown to likely be important [1] and studies are needed to better quantify them and develop simple procedures for their consideration in analysis and design.
Acknowledgement Partial support for the work presented in this paper has been provided by the Multidisciplinary Center for Earthquake Engineering Research, University at Buffalo, Buffalo, NY.
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Multidisciplinary Center for Earthquake Engineering Research, Buffalo, NY; 2004. Available online at . [22] Constantinou MC, Whittaker AS, Kalpakidis Y, Fenz DM, Warn GP. Performance of seismic isolation hardware under service and seismic loading, Report No. MCEER-07-0012. Multidisciplinary Center for Earthquake Engineering Research, Buffalo, NY; 2007. [23] Nagarajaiah S, Reinhorn AM, Constantinou MC. Torsional coupling in sliding base-isolated structures. J Struct Eng ASCE 1993;119(1):130–49. [24] Nagarajaiah S, Reinhorn AM, Constantinou MC. Torsion in base-isolated structures with elastomeric isolation systems. J Struct Eng ASCE 1994;119(10):2932–51.
